Giant Battling Robots

My Wikipedia Problem

2012-01-29T22:07:00.001-06:00

Some months back I started getting a lot of traffic from the Wikipedia page on Lanchester's Laws. It seems that someone had noticed my efforts on the subject and linked to me as a reference. It wasn't much really, just a reference in support of a single sentence - Here it is, and [5] links to my first post on Lanchester's Laws:

In modern warfare, to take into account that to some extent both linear and the square apply often an exponent of 1.5 is used.^[4]^[5]^[6]

Citation: Lanchester's laws. (2011, December 12). In Wikipedia, The Free Encyclopedia. Retrieved 03:02, January 30, 2012, fromhttp://en.wikipedia.org/w/index.php?title=Lanchester%27s_laws&oldid=465382667

I've put a lot of study into Lanchester's Laws, so I was happy that someone thought I was worth a reference, but I have a problem with that sentence. It's wrong.

To be fair, perhaps I ought to say it is incomplete statement on a complex topic, and the complete explanation would be much, much longer. It is true that an exponent between 1 and 2 is often used to approximate situations where both Linear and Square laws are in effect, but this exponent represents (in a very abstract sort of way) that a portion of each force are subject to the Linear law (exponent of 1), while the remainder is subject to the Square law (exponent of 2). There is no mathematical rule that makes any other values for the exponent correct, it just sort of works to describe how battles actually play out in an average sort of way. The article makes no previous mention of exponents at all, so it's hard to see how anyone could come away with a proper understanding of the statement. To my mind that makes it wrong.

Fixing it though, is another matter. I've thought about fixing it myself, and even contacted a Wiki editor about it, but I haven't had time or energy to take on the task. I still hope to get back to writing regularly again, but I have a stack of other topics to address, and I am not sure I really want to spend my time fixing someone else's problem.

Tangent: If you are new to the subject of Lanchester's Laws, the Linear law (exponent 1) describes combat attrition in a one-on-one combat setting, such as might occur with archaic weapons or between aircraft in air-to-air dogfights. The Square law (exponent 2) applies when multiple combatants can attack the same target, and vice-versa, such as a naval gunnery battle. Those are ideas though, and in practice there is almost always some complex mixture of these situation.

The Grinder

2011-08-20T10:48:00.003-05:00

Assorted bits of internet, selected from a stratified sample, sorted in ascending sequence, collated, assembled in logical order, then just sort of thrown in a pot and given a good hard shake. Completely by accident, I seem to have a fine selection of game design article for this edition. How'd that happen?

--- The Grinder - August 2011 ---

PAXsims - The latest Game Design blog I am following. Some good stuff here I need to check out.

IO9: Robot Art Wallpapers

A Fafnir crashes the party

Wargamer's Notebook: The Moment -- "I wish this was something I experienced more often. The point during a wargame after several plays or turns when things magically click. The moment when the scaffolding of the the rules and latticework of the bits fall away and the narrative zooms into the foreground and you - as the player - are completely absorbed by the game. Events occur that could not have been imagined, but that are totally plausible. Victory hangs in the balance. And you are not moving counters or playing cards, but making choices that give you hope of victory."

More irreverent humor to be found at Perry Bible Fellowship.

Desert Scibes brings us a bit bit of gaming history at Supergalactic Dreadnaught:

The Making of Starfleet Wars

NOTE: Scott R. Spicer (credited as "S.R. Spicer, Lt., TFSF"), along with his father, Ron Spicer (R.E. Spicer, Lt. Commander, TFSF), developed the Starfleet Warsminiatures line for Superior Models. As you probably know if you've read this blog before, the SfW universe contains five starfaring factions, each with a distinctive design style. Curious about the origin of this game and its minis, I emailed Scott Spicer about his work on these spaceship models. He was gracious enough to reply to my inquiry, and his response follows in its entirety ...

Make your own custom dice (with a bit of work).

From SMBC: (If only it were this easy!)

Gamasutra: Game Design Essentials: 20 Real-World Games

Chess, Go, and Life. Which one is the Best. Game. Ever. A nice bit of gamey mathiness from Patterns in the Void.

Finally, this is just cool.

Taking a RISK - the distribution of armies lost

2011-07-25T22:40:00.000-05:00

I came across the following question at boardgames.stackexchange.com:

How can I estimate my chances to win a Risk battle?

Elliot Avedon Virtual Museum of Games, Courtesy Canadian Museum of Civilization.

Now there is already plenty of material on the web about probability in the popular boardgame RISK*, but maybe I can add just a little bit more.

Most people already know that, given the choice, the Attacker should always 3 dice and the Defender should always roll 2, since this always gives the best results (a Nash equilibrium). Since it's always the Attackers choice to roll an attack or not, the relevant question seem to be "How many armies [X] will the Attacker lose if they make [N] attack rolls?"

Elliot Avedon Virtual Museum of Games, Courtesy Canadian Museum of Civilization.

Using some probabilities from a RISK FAQ for the probabilities of losses, I calculated the average attacker losses (about 0.921 per roll) and standard deviation (~0.81). It's not possible to lose 0.9 armies in RISK! as the attacker losses vary between 0, 1, and 2 per roll. However, as the results of many attack rolls are added up, the losses will begin to resemble the normal distribution. We can plug the average and standard deviation into a normal approximation formula, and get back a probability for losses in a fairly simple calculation. For a given number of attack rolls "A" and number of attacking armies lost "K",

Z = (K - 0.921*A) / (0.811*sqrt(K))

where Z is a standard normal variable (mean 0, standard deviation 1), and the probability of K-or-fewer losses in A attack rolls can be evaluated with the standard normal probability CDF function, otherwise known as the NORMSDIST(Z) function in Excel. That's pretty much it, except maybe for some graphs to show off the results.

Cumulative probability of K attacking armies lost in 25 attack rolls

The blue line shows the cumulative probability of K losses in A attack rolls (vertical red line). The yellow triangles show an approximate 50% confidence interval, meaning that your actual losses should be within this range 50% of the time. The red diamonds show a 90% interval for the same. These intervals are actually slightly wider than the stated 50%/90%; because I rounded outwards to the nearest whole number of armies, and there is no other good way to do it.

Cumulative probability of K attacking armies lost in 35 attack rolls

Recall this is an approximation, and it depends on there being lots of independent random events (dice rolls!) for the approximation to work well. It should start to work very well somewhere between 20-30 attack rolls, and depending on how fussy you are may give usefully accurate results for as few as 10-15 rolls. For smaller battles, or deciding whether or not you should attack just one more time, you might consider more accurate methods (look here, for starters).

Cumulative probability of K attacking armies lost in 70 attack rolls

You can think of this in terms of Defender losses too. Two armies are lost with every attack (between the attacker and defender), so if there are A attack rolls and K attacking armies are lost, that means the defender will be losing 2*A - K armies. For instance, in 25 attack rolls a total of 50 armies are lost; the probability of the attacker losing 20 armies the same as the probability of the defender losing 30.

Cumulative probability of K attacking armies lost in 50 attack rolls

Nostalgic Tangent: Calculating the probabilities of losses for the attacker and defender was one of my first mathematical efforts to figure out a game. I didn't know how to do the calculation, but I wrote a program on my Apple II+ to roll lots of electronic dice for me, and calculated the probabilities that way. Later I learned that this technique is called Monte Carlo simulation, and statisticians do this regularly to examine the properties of new statistical methods.

*** There WILL BE a link to the spreadsheet for these calculations, but it's getting late, so I'll have to add that in tomorrow. ***

Having written this, I realized that I haven't quite answered the question. I've given the probability for a given number of attacks/losses, but the question is "How many armies will it cost me to win?"

There is another approximation to answer that, but it's much less well known. I guess I'll have to write a part 2. Stay tuned!

---------------------------------------------------------------------------------

* RISK is a registered trademark of HASBRO, Inc., of course.

Footnote: The RISK! game images used in this post are from Elliott Avedon's Virtual Game Museum, and are used with permission of the Canadian Museum of Civilization. The Virtual Game Museum has much more interesting game related information, and I may be posting about it again.

The Grinder - July 4th Edition

2011-07-04T10:15:00.000-05:00

A collection of red-glaring rockets and bombs bursting in air, without the rockets and bombs.

The Grinder for 7/4/2011:

Paint-It-Pink : Ashley has some Battletech math going on --> How I Learned to Stop Worrying and Love the Medium Laser. A good discussion of how to evaluate the relative strength of weapons in Battletech, or any game.

"Can I haz tactix?"

Found on Operation Odyssey Dawn: If the animation doesn't work, go see it here.

Linkback! --> 程阳：Probability versus Odds

MathOverflow: Which popular games are the most mathematical?

Proof: The 120 cell is a 4 dimensional figure that can be considered the 4 dimensional analog of the dodecahedron. It has 720 five sided faces, 1200 edges, and 600 vertices. This animation shows 3 dimensional cross sections of the 120 cell in a way that is similar to taking 2 dimensional cross sections of a 3 dimensional figure. Translation --> Very Cool animation!

The Number Warior: Q*Bert Teaches the Binomial Theorem (an award winner too). Sort of a long (2-part) video.

Proof-of-False: Do games offer a solution for US Tax Reform?

From doctormatt: A Collection of Dice Problems with solutions and useful appendices

The Math Less Traveled: The Wonderful World of Mike Reilly.

Mike Reilly is a Toy/Puzzle designer and screenwriter, see what he has done at Reilly4Puzzles.

Reinwood's CBT Workbench gives us an AAR for Fourth Succession War: Skondia The Final Battle Kublacon. AND it's got no math in it. Honest!

A small update to my Graph Paper Race post (added link to a relevant article). This continues to be more of my more popular posts.

I didn't plan this, but somehow this has ended up being the most math-heavy edition of The Grinder to date. Oh well, it's all sausage now.

Dice Distributions Revisited

2011-07-03T11:23:00.000-05:00

Recent thoughts about calculating the distribution of the maximum sum of several dice (ex: roll 3, sum the highest 2) made me realize I needed a better tool for calculating the distribution of sums of dice in the first place. I first wrote about this some time ago in Dice Distributions, so I knew how to do it better, I just hadn't gotten around to doing it.

Tangent: While researching this I can across a great set of mathematical Dice Problems from Doctormatt (Web Page, Blog).

And now back to our story -

I first set up a spreadsheet to give me Pascal's Triangle, which looks like this:

This gets used in lookup functions to calculate (in a second worksheet) what I'm calling the "Dice Triangle", The number of dice [N] rolled is indicated in the column headers, and the sum of N D-sided dice [X] rolled in indicated in the first column. The number of ways to roll that sum is indicated in the table. The number of sides on the die [D] can be changed by entering a different value into the green shaded cell.

The first D rows of the table come directly from Pascal's Triangle. Subsequent rows are calculated from previous rows of this table. A few more details of how this is done in my earlier post (Dice Distributions), otherwise you can ask me, or dig into the spreadsheet for yourself (sorry, in a hurry today).

Here is the spreadsheet: Dice Distribution Calculator
I have not made this public, so you cannot change it directly online. You can download a copy for yourself (under the File dropdown) and play with it to your hearts content.

Maximums and Minimums of Dice Rolls

2011-07-01T17:54:00.012-05:00

A ~~week~~ ~~month~~ a while back I received questions from Christian and a read post from Saxywolf on essentially the same question: What is the probability of rolling a given value on an D-sided die, if you roll N dice and take the highest (or the lowest).

Here's the trick
The probability of rolling a 1 on 1 d6 is 1/6. (regular 6-sided dice)

For 2d6 the probability of rolling a 1 as the maximum is 1/6 times 1/6, or 1/(6*6) = 1/36.

For Nd6 the probability of rolling a 1 as the maximum is 1/6 times itself N times, or (1/6)^N.

For D-sided dice just substitute D for 6 above, so that the probability of rolling a 1 as the maximum of N D-sided dice is 1/D times itself N times, or (1/D)^N.

Now consider the problem of rolling 2-or less as the maximum. The probability of rolling 2-or-less is 2/D, and the probability of rolling a 2-or-less as the maximum is 2/D times itself N times, or (2/D)^N.
That's 2-or-less, but we really just want the probability of rolling 2, not 1 or 2. BUT we already know the probability of rolling a 1 as the maximum on the same dice, so we can subtract that to get what we want:

The probability of rolling a 2 as the maximum is 2/D times itself N times, minus the probability of rolling 1 as the maximum, or (2/D)^N - (1/D)^N.

And that's it. Using the same math you can work the complete distribution of the maximum for any number of dice which any number of (equally likely) faces. For minimums, just turn the problem around and find the probability of X-or-less.

Still too much math? Fear not for there is a spreadsheet to do the calculations for you:
Link to Google Docs Spreadsheet

See the blue numbers in the green-shaded cells? Change those to the number of side on your dice and the number you want to roll, and it will calculate the distribution for you. ~~It might even work inside the blog?~~Nope, but it was worth a try. The spreadsheet is now public and can be edited at the link above (also downloaded). Changes made there WILL show up here when the page is reloaded, which means you are looking at whatever was most recently entered.

And a chart to display the results:

This is a bit of an experiment, both linking to a shared spreadsheet, and adding the HTML code to it directly inside my blog post. One upshot of this is that when one person changes the spreadsheet, it will change it for everyone. Play nice! Let me know if it works too. [Fixed!]

The Grinder

2011-06-04T10:17:00.001-05:00

[A dazzling display of delightful de ... um ... I need a D-word ... deviations ... detritus ... de-links?

The Grinder - 6/4/2011 edition

And speaking of dazzle, could Dazzle-camouflage make a comeback? This recent research supports says Dazzle Camouflage Affects Speed Perception. [Hat-Tip IO9]

Image: WWIaviation.blogspot.com

This just in! Check out some dazzling WWI aviation paint schemes.

Terrain table pictures ... Shiny!

This could be interesting ...

Invasion3042 is a massive multiplayer online game that is based off the game Battletech. It is a free game and is not for profit.

Has anybody tried it?

Discoblog brings us Tiny Toss-able Robots.

World Peace Games, with teacher John Hunter. Video from TED. It's a bit slow to get started, but gets interesting about 8 minutes in. Never cross a 9-year-old girl with tanks!

[Hat-Tip Greg Laden]

[The Endeavour] There are exactly five platonic solids*, and you can prove it!
* Perhaps more familiar to my readers as dice - the d4, d6, d8, d12, and d20.

Another video, this one with singing and dancing! Roll A D6.

[Hat-Tip: Operation Odyssey Dawn]

[IO9] MTG as an RPG?
The was an MTG computer game, long ago, with very nearly this premise. With a bit of creativity it could be good for multiplayer too. Before that was a great game called Master of Magic. Also, there could be a new Star-Trek animated series!

Play JAM! Can you beat the computer? Can you beat it every time?? Can you figure out the secret??? (without peaking!) Here is a hint - You have almost certainly played this game before, and many times. [Hat-Tip Terrace Tao]

Non-Transitive Dice

Starcraft Humor from Abstruse Goose. I didn't get it until I saw the caption.

Green Cube: The Physics Boardgame

Enough bedazzlement for two sittings, but that's happens when I don;t post for a whole month. Writers-block sucks. Want to preview the next Grinder, before I post it? Check out my Google Reader Shared Links page for GRB.

Doug Chaffe

2011-04-27T15:23:00.001-05:00

Update [7/14/11]: Catalyst Game Labs is hosting a sale of Doug Chaffe original artwork at Gencon 2001.

Randall Bills writes about the passing of Doug Chaffee. Better go read that first. Here is Randall's opening paragraph.

In 1994 I walked into a game store and beheld BattleTech’s CityTech Second Edition on a shelf. The cover was by an artist I’d not seen before, but it was action-packed, evocative, and wonderfully done…everything a fan could ask from a cover. Of course I left with it under my arm. For the next 15 plus years Doug Chaffee would be an indelible part of crafting the visuals for BattleTech.

I confess I didn't know anything about Doug Chaffee before this, but Battletech players should know his work on sight ...

Image: Chaffee Studios

... because it is nothing short of iconic. Much more to be found at Chaffee Studios.

Here are a few links that my be of interest:
Battletech Wiki: Doug Chaffee
Book Illustrations

The Grinder

2011-04-27T07:47:00.000-05:00

\ This edition of The Grinder is brought to you with the assistance of Darwin the cockatiel, who inserted the backslash at the beginning of this post, and bit at my fingers as I type this. I haven't been posting much lately, but I have been writing, and hopefully some of that will find its way back here soon.
Now on with the Grind.

Some fun with Game Theory on the British show QI (short for "Quite Interesting").

[Via Terrence Tao's Buzz]

Airships for the 21st Century
Which has nothing to do with games or math; I just think it's cool.
[Hat Tip David Brin on Twitter]

Some cool new art for Magic-The-Gathering at IO9.

Mo Rocca visits Gen-Con and plays boardgames: Video at CBS News.

Is it just me, or does everybody have friends that ask them questions about counting problems and combinatorics? (I'm looking at YOU, RR!)
Somehow, I think it's just me.

Ian Schreiber writes about education and games at Teaching Game Design: My Problem With Gamification.

The Last Cause is a movie in pre-pre-production, but it has mechs and clones, so Battletech players are likely to take notice.

Game Designer, Graphic Designer, Wargamer Extrodinaire, and a bit of gaming history I should probably already know, but somehow didn't: Redmond Simonsen. Go read about him. [Hat Tip 2 Grog News]

Fletcher Pratt's Naval Wargame

2011-04-04T22:26:00.000-05:00

John Curry, editor of the History of Wargaming Project, has a new book out:

I first wrote about Fletcher Pratt's Naval Wargame almost two years ago in The Origin of Battletech. At the time a new edition of the book was still in the works, so I waited to get it. This edition includes some previously unpublished material from some of Pratt's original players and umpires.

My copy is on order, and I'm looking forward to seeing it soon. Maybe I will organize a local play session?

Information link: http://www.wargaming.co/rules/fletcherpratt/homepage.htm

Purchase link: http://www.wargaming.co/purchaserules/homepage.htm

Sicherman's Dice

2011-04-02T14:00:00.010-05:00

There is something different about these dice - can you spot it? I'm guessing you'll get it right away ...

Image found at Chuck-A-Con *

You can't see the non-facing sides, but the d6 on the left is labeled with 1-2-2-3-3-4, and on the right labeled with 1-3-4-5-6-8 (like this). That's not our standard 1-2-3-4-5-6, and if someone rolled these on the gaming table the 8-pip is a dead giveaway that something is off.

Now here's the trick: The probability distribution for the sum of these Sicherman Dice is identical to the distribution of the standard 2d6, so if you only see the results (the sum) there is no difference at all.

The mathematics for this gets into Generating Functions and Combinatorics, but essentially is involves doing the algebra to show that:

(x + x² + x³ + x⁴ + x⁵ + x⁶)² = (x + 2x² + 2x³ + x⁴)(x + x³ + x⁴ + x⁵ + x⁶ + x⁸)

Where the left-hand-side is the generating function for the sum of two standard 6-sided dice, and the right-hand-side is the appropriately factored generating function for the sum of Sicherman's dice. (OK, maybe a little harder than that.) There is only one way of doing this with 6-sided dice, but such variations exist for other polyhedral dice. It seems to be possible in general to do this with N-sided dice, and there might be multiple ways of doing this for some. The Mathematics Magazine article "Renumbering of the Faces of Dice" by Duane Broline (1979) goes into some detail, but I cannot access the full article from home. If I can grab it at work maybe there will be an addendum.

The Hard Way

I tried this working out possible numberings for Sicherman-type 2d8 dice by scribbling with pencil and paper until I found a combination that worked. On my third-and-a-half attempt I came up with 1-2-2-3-3-4-4-5, and ~~1-3-4-6-6-8-9-11~~. CORRECTION: TPC checked more carefully that I did, and offers 1-3-5-5-7-7-9-11 in place. ~~It took me a while to work this out "the hard way", but~~ it was probably still faster than I could have factored a 16^th-order polynomial**.

[Hat-Tip to The Endeavor/John Cook. Again!]
[As seen on Eon.]
* Sicherman Dice are available from Amazon, or directly from Gamestation - Gamestation.net is likely the original source for the image I used above.
** "Dammit Jim, I'm a statistician, not a combinatrician!"

Fun with Physics Special Topics

2011-03-26T10:59:00.000-05:00

I recently discovered the online journal Physics Special Topics (current issue, archives). These are essentially collections of student articles on various topics. Here is how the journal describes itself:

FOCUS AND SCOPE
This is an undergraduate journal for year 4 (MPhys) students in the Department of Physics and Astronomy at the University of Leicester. The journal accepts brief papers on topics original to the authors. It does not accept reviews or summaries of other peoples work. It is managed by an editorial board which rotates round the student body overseen by a member of staff. The journal forms part of the assessed element of the MPhys degree. Assessment is by number and quality of accepted publications and referee reports.

The students are writing serious article on a wide variety of topics, some of which are inspired by by games and science fiction. As science goes, these is not exactly cutting edge, but it sure is fun. Some of the articles that caught my eye are:

How Heroic is a Hero? An exploration of the probability distribution behind character creation in the D20 system.

Niven Rings looks at some of the physics of Larry Niven's Ringworld.

Can superhuman muscles stop bullets? Ask the question, "Just how bulletproof is Superman, anyway?"

And much more.

This is perhaps slightly off the usual theme of this blog, but surely playing games and reading SciFi has raised a few of these questions in the minds of gamers - "Just how what that work, if we actually tried it?" - Like many things, the fun is in the finding out.

If one of these articles catches your interest, please post about it below. Have fun!

A few more minutes in the library

2011-03-07T12:31:00.038-06:00

I received the following email, apparently from a young person, so I won't reveal the name:

Hi im 14 and ive loved modeling since i was 10. I have 3 dioramas Railroads Cut, Gettysburg, Pennsylvania/ The Battle of the Hurtgen Forset, France/ and one that i made up and created when i was 11 Bloody Ridge, Kentucky. I am know working on the battle of waterloo and was wondering do you have any tips for me i have already painted 80 bristish and prussian and 15 French calvalry and am working on the french foot soldiers. I am hoping to get a large board than i have know but any terrain tips or details?

There are many sites on the internet where you can find such advice, but I have a different suggestion: Go to your local library and look for books on "military modeling", "model railroading", "Civil War history", and anything you can find by Sheperd Paine. Browsing those shelves of related hobbies can show you more in a few minutes than you will find in hours of Googling the internet, because library shelves aren't sorted by keywords.

That doesn't mean you won't find the same information with your favorite search engine, but it's hard to search for ideas you don't know exist. Look in your library - because libraries are absolutely full of good ideas. You will be pleased with the results.

This is good advice for anyone who enjoys games, miniatures and modeling. DO NOT limit yourself to learning only from others within a narrow segment of your hobby. Reading some of the bulletin boards, you might get the impression that painting miniatures is nearly exclusive to just one or two games, and nothing could be further from the truth. Make a serious effort to study what others are doing, and bring back what you learn to make your own hobby work even better.

The Grinder - Valentine's Day Edition

2011-02-08T00:01:00.017-06:00

Links for the one you love, ground up and arranged in a nice bouquet.

The Dot and the Line: A Romance in Lower Mathematics

From Norton Juster, and the Immortal Chuck Jones

[IO9] - Mathematicians figure out how to fend off gold-diggers – with game theory --- This should be probably be re-titled: Two male Mathematician's figure out yet another way not to get girls.

Spiked Math

Spiked Math has an experimental simplified viewing page, and a new Spiked Math app for the iPhone/iPad.

[The Fallacy Files] - The Puzzle of the Absent-Minded Professors

Because no one can ever have enough logic puzzles.

That's a Terrible Idea: Actually has some pretty good idea's, and I have added them to my blogroll under Game Design Blogs (somewhere down there, but hey, a link is a link). Here are two good samples:

Dancing Robots! The party really gets going about 3 minutes in.
[Found on Engadget, via Pharyngula]

Proof

More goodness from Proof: Snowhedron

[Dark Roasted Blend] If you like the artwork for Catalyst Games Labs new game Leviathans, check out The Great Eastern, a leviathan steamship on the Victorian era.

[Discoblog] Can video games make smarter spies? Oh come on guys, get Sirius!

That's all for this edition. Be sure to set aside the games you love for a little time with the one you love. Or better yet, get them to play a game with you!

Pink Panthers, and a few Dropships

2011-02-06T15:50:00.003-06:00

I have a couple of Galactic Knights Carnivorian Panther Class Cruisers that I'm try to finish up and get off my workbench. I've been meaning to post some pictures for a while, and a recent post at Super Galactic Dreadnought prompted me to action.
While I had my camera out, I also snapped some pictures of some AeroTech Union and Overlord class dropships that I have been wanting to show off. With a little help from Picassa, you can see these pictures in the embedded slideshow below, or you can see the full size images in my miniatures gallery.

For the Panthers I employed a non-standard method for painting these. It went something like this:

Start with a black primer coat.
Drybrush a dark grey, then gradually work up to lighter shades.
Accidentally splatter with Pink Paint. Really dump it on.
Wash off most of the pink paint, and spread the rest around a bit.
Accent with magenta ink.
Highlight with Metallic Inks, as shown here.
Apply matte varnish.
Apply a flat black glaze to "deep" areas to further knock down shininess and bring up the contrast.
Add lettering in white ink for the ship names (in process).
(next) Mount on Ninja Magic flight adapters?

Since these are "Panther" cruisers, I decided to name them after cats I have known. The first is "Telstar" after a big black cat we had when I was very young. The second is "Boris" after an even bigger black* Maine Coon who has assisted in the development of Squadron Strike (Boris allows Ken to type, sometimes).

* Boris informs me he is actually a mackeral tabby.

The AeroTech dropships were an experiment in how much I could do with metallic inks, and I am very pleased with the result. I started with a black primer, and applied a number of thin layers of silver ink until I got a solid tone. Some highlighting with copper ink, black ink on the doors and some black-lining the details finished them off nicely. These came out with a much better metallic look than I get from metallic paints.

Jane McGonigal on The Colbert Report

2011-02-06T08:43:00.001-06:00

Jane McGonigal on the Colbert Report

McGonigal is plugging her new book, Reality is Broken. I just added that book to my wish list too.

More McGonigal!

UPDATE: Gamasutra has a short excerpt from McGonigal's new book.

The Grinder

2011-01-26T18:37:00.003-06:00

Fresh ground links*, salt, chili pepper, onion powder, tomato powder, oats, soy lectithin, sugar, soybean oil, garlic powder, yeast extract, citric acid and cocoa powder.

* If the total link content is less than 40%, can I still call them links?

First a bit of administrivia: Though I have previously refused any sort of ads or commercial links, I recently received and accepted a polite offer to link to an online Games Software Development degree program from Westwood college (that's it over in the sidebar). This makes my ORIGINS trip this year a bit more affordable, especially when tuition costs are up.

Now back to our the grind ...

Spiked Math brings us a Magic the Gathering card for the Banach-Tarski paradox.

[IO9] Giant Teen Robot Angst ... no ... Teen Robot Giant Angst? ... still no ... Giant Robot Teen Angst. That's it! (Evangelon)

[more IO9] Chevy Chase to play D&D. Hilarity ensues.

A Gentleman's Duel (Bugs Bunny homage at 5:00)

[Found on SFSFW (Society of Fantasy and Science Fiction Wargamers)]

[22 Words, via John Cook's shared items] Racist Robot Joke

Found on Web 2.0 RC1

Speaking of sharing, any Reader or Buzz users out there?

[Pat's Blog] Q: Which type of polyhedral die most closely resembles sphere? Think about it carefully, then go see it you were right.

[Cinerati] This looks like fun: Adventure Gamebooks as RPGs Part 1 -- Fighting Fantasy's Warlock of Firetop Mountain

Image: Geeks are Sexy

[Geeks Are Sexy] Sexy D&D Pics.

Hint: It's not the D20.

[IO9] 55 Science Fiction/Fantasy Movies to Watch Out for in 2011, but I probably won't see most of them.

[Schlock Mercenary] Howard does a Happy Dance! Don't read Schlock? Try this.

Image: Ottawa Gamer

The Ottawa Gamer launchers his Avalon Cruiser.

All ground up. Keep rolling those dice!

After Action Report

2011-01-23T22:55:00.000-06:00

I hosted a Saturday Battletech session at my home. It was a very cold morning (-11 F) and there had been a light snow - a good day to stay inside to play a game (and absolutely perfect for chilling the beer and soda!). The scenario was a rather generic "Assault" on a defended area, but I set myself some very specific guidelines for unit selection and pilot/gunnery skills. One guideline was that all units should be selected from miniatures I actually have (which I almost accomplished), and that the battle should start off light mechs on both sides. Someone also requested I add some armor/tanks into the mix, which complicated my plans a bit. Another guideline was that the skill levels of mech pilots should be very mixed. I actually created a quite elaborate system for generating skills, and I might write that up some other day, but for now it's enough to know that about half of each force had an gunnery skill of about 4, with a range from 2 to 5. I created over 20 lances (groups of 4 units), which was MUCH more than could possibly be needed, but I wasn't sure how many people could make it or how late we might play. Pilots (and vehicle crews) were randomly generated, but within each lance I assigned pilots where I thought they were the best fit.

Each side started with 1 lance of vehicles and 2 of mechs. The mechs were all small and medium (because I stacked them that way), but vehicle selection was random, and by chance both sides started with some BIG tanks - 2 Demolisher's for the defenders, and 2 Alacorn's (heavy gauss version) for the attacker. The attackers started with an advantage of two extra medium mechs, and the only objective was to try to make the defenders "call for reinforcements" before 7 turns or so, with the idea the battle could continue indefinitely with each side alternately calling for reinforcements.

The defenders "home edge" was the lower/right corner of the photo above, and could set up on either of these two maps. Attackers entered on the far side. I was playing on the defenders side, but I stayed out of the initial setup decisions to make it more fair. Consequently, the defenders set up "forward" and took the fight immediately to the defenders on turn one (pictured above).

Here's another turn 1 photo, and it must be after combat because I see a "prone" marker in the mix. I'm not going to give all the gory detail, just a few that time and memory allow. With so many fast mechs, mediocre gunners, and big guns (on the tanks) there was a lot of wild maneuver going on. This is sort of what I was hoping would happen when I set up the scenario, and if anything it was even better than I expected.

Here's a turn 3 photo (I forgot to take any photos on turn 2). The attacker Alacorns are perched on a hill in heavy woods (lower left). They kept this position from turn 1 and made life very unpleasant (and much shorter) for several of the defenders. The defending Demolisher tanks are near the "frac tower" (upper mid) facing an uppity pair of UC/20 Saladin hovertanks.

Turn 4: The attackers have two units, a Cicada and the surviving Saladin, on the town map (lower) and are menacing a nearsighted LRM Carrier and an outmatched Flea-17. On the right an attacker Javelin is about to have a showdown with a Demolisher tank (guess who has a bigger gun?)

Another view of turn 4.

We called it quits after 5 turns. The defenders had lost 4 mechs and 2 vehicles, with 2 other mechs seriously damaged or crippled. The attackers lost 3? mechs and 1 Saladin hovertank. Had we played on the defenders would be forced to call for reinforcements immediately (well before 8 turns), so this is a clear win for the attackers. The defenders put up a good fight though, and the attackers were not too far from needing their own reinforcements.

I am quite pleased with how this turned out. The effort I put into the setup paid off with an interesting game, and the balance worked out just as had hoped. There are survivors of this battle and 14 lances of reinforcements we never got to use, so there could be a "part 2" to this battle in a month or two.

And the ice-cold beer was fantastic too. :-)

The Grinder

2011-01-08T20:48:00.125-06:00

Carefully selected links, ground into a Truely Gritty substance (watching Duke right now).

from Doghouse Diaries

Battlestar Galactica versus Lanchester's Squared Law ... from the The Dog House Diaries webcomic.

Blogs of War - a list of Wargaming blogs with ~~951~~ 993 entries so far. You might find something you like. You might have a hard time not finding something you like.

From Pulsipher Boardgame Design, some comments on the value of playtesting: "Most players are not like us"

IO9: You might think something like this could only exist in a game ...

The manufacturer will not stand behind it.

From Vi Hart: Sick Number Games

Generating random numbers is a deep subject. Check out this free book to see how deep.

Game in the Brain: Nikolas comments on the Purpose of Realism in Games.

XKCD: Explorers

I put in my two-bits on Dice and Information (1,2,3), here's a great article about the information in a coin toss, from Tom Moertel's Blog:

More on the evidence of a single coin toss

That's all for this session. (I'm almost out of movie too, just the final shootout left.)

More Dice, More Information, but not as much as you think

2011-01-02T00:05:00.003-06:00

I may have created some confusion in my previous posts on the information in dice (1,2). That's understandable, because the concept of information is complex and has multiple interpretations, most of which I would not claim to really understand either. Let's see if I can sort this out without making an even bigger mess.

Shannon Information measures the information content of a random distribution of discrete events. A coin flip (heads/tail), a To-Hit roll (hit/miss), and a Hit-Location roll (arm, leg, torso, etc.) are all examples of this. What really matter here is not the number of dice rolled or coins flipped, but the number of possible outcomes and the probability of each. So you might use 4d6 to determine the result of a to-hit roll, but there are still only two outcomes - hit or miss. In game terms, you can think this as the information you don't know yet, just before you roll the dice, or the variability of outcomes of that roll.
Shannon information is measured on a logarithmic scale (base 2), so each additional bit represents a doubling of information. In absolute terms, the difference between 10 and 11 bits of information is MUCH more than the difference between 2 and 3 bits. Be careful with this sort of comparison though, because it's a bit like comparing apples and oranges (or comparing to-hit and hit location rolls). Such comparisons may not be meaningful.

There is an extension of Shannon Information to continuous outcomes, but this also requires changing the definition somewhat. I won't go too far into this, but there is one key point I'd like to make. When dealing with the sum of multiple dice, the distribution of the sum tend to become more like a normal distribution as the number of dice increases. Calculating the entropy of the sum of 10D6 is a bit of work, but the entropy of a normal distribution is easy to calculate. Long story short, I can get a good guess at the entropy for the sum of a large number of dice by using the normal distribution as an approximation.

I had speculated about calculating the information in an entire game. This was rather silly of me, because this would mean framing the outcome of an entire game as a single probability distribution. I can't do that, but now I know how to make an educated guess. If I only considered the win/lose aspect of a game this would be easy, because that is just a complicated sort of discrete "coin-flip" outcome. The more interest way to look at this is to consider ALL the ways a game might play out, and to treat this as a sort of continuous outcome. The law of averages comes into play, some ways the game will play out are more likely than others. For instance, if at some point in the game you make multiple attacks to achieve an objective, perhaps to destroy an enemy tank, then in the final outcome of the battle it might not matter which attack was successful, so long as one of them was - they all lead to the same outcome. This might be stretching the concept too far, but I should be able to use the entropy of the normal distribution to approximate the amount of information of a very complicated random distribution - like that of an entire game.

Now I can make an educated guess about the information in a game. I'll use a Battletech example, but there is surprisingly little dependence on the game. The most common random event in Battletech is weapons fire, which includes the to-hit and hit-location rolls, which each have about about 3 bits each (as calculated here). In a two player game where each player has 4 battlemechs, and each mech makes about 5 weapons attacks per turn, there will be about 10 random events per turn for the first 5 turns or so, about 200 random events, then a decreasing number of attack for the next 5 turns, call it 150 random events. That's 350 random events in one game, but I left out anything else that might require a die roll, so I'll round it up to 400. The basic random event in Battletech is about 3 bits, and 400 repeat random events adds about log2(400) or 8.6 bits, for a total of 11.6 bits.

Sooooo ... now that I've gone through all that, it seems that the information in a game is just log2 of the number of random events, plus a few bits of overhead. Does this mean anything all? I need to think on that.

Here is a partial answer: A game doesn't need any randomness at all, it could be completely strategic, like Chess. Add just a little bit of randomness, the whole game may depend on just a few rolls of the dice. As randomness increases the law of averages will come into play - Games like Risk and Battletech have many dice rolls - so many that the average effect of many rolls is almost always more important than single roll. Too much randomness, and players lose the ability to affect the outcome. The trick is getting is the right balance of randomness, and I don't think there is a formula for that.

PS: My friend Ashely also recently noted that it might be a good idea to eliminate any rolling of the dice that doesn't significantly add to the game. Wisdom!

More:
Dice and Information
Dice and Information, So What?

New Years Blogolutions

2011-01-01T00:24:00.001-06:00

IT has been a pretty good year, all in all. I didn't accomplish all I wanted - I only wrote 61 posts (not counting this), which is considerably less than the 149 I wrote in 2009, but many of those went into depth or required considerable research and preparation. That is also 61 posts while contending with a super-busy year at work and home, so I shouldn't complain. I have also made some new friends, which is always good.

StatCounter tells me I've had 25,381 page loads, 17,168 unique visitors, 14,364 first time visitors, and 2,804 returning visitors in 2010. This is actually down somewhat from 2009, but I've been less active, and 2009 included the remains of massive traffic spike I got from the Blog-of-Note in December 2008. Google stats tell me I now have over 200 RSS subscribers - Thanks everybody!

(and the ball drops - 10 - 9 - 8 - 7 - 6 - 5* - 4 - 3 - 2 - 1 - Happy New Year!)
(* about this point I jump up to kiss my wife and nearly knock over my painting tray in the process. If what you do on New Years Eve is a preview of what you will do in the New Year, then apparently I'm going to be spending a lot of time kissing my wife and cleaning up disasters. Could be worse!)

What I hope to do in 2011:

I think I've got one or two more good Lanchester's Laws posts in me, which ought to really nail that topic down. I might even use some historical data and do some of my own modeling, which would be good for a whole new series. I have a nagging doubt about the existing Lanchester's models - I think there may be something fundamentally wrong about the existing historical analyses, and I want to take my own poke at the problem. I'm probably wrong, but doing my own research on this idea could be fun.

The Lanchester posts have helped me see how point system can be built, and I've already written bit of this, but I want to consolidate this into a few concise posts detailing what I know about how it should work, and what parts are still open questions. This has been one of my long term goals, but now that I am finally within reach it is opening up new questions, so the quest will continue.

I want to write games. I have that grand idea in my head of writing my master game that combined every idea I have ever had (and that's a lot) into a single game, and I just know that's a bad idea. I need to write some smaller games and practice expressing these ideas in a form that others can try.

I really really need to get back to my painting again. When time is short my painting tends to fall by the wayside. Having a wet palette helps though, and I've got space clear in the basement for what is supposed to be our painting studio.

I need ... to get to bed! Goodnight and Happy New Year. Keep those dice rolling!

Dan

Battletech: Sword and Dragon Sessions

2010-12-28T17:11:00.000-06:00

Image: ClassicBattletech.com

Our local Battletech group has been playing through the Sword and Dragon Starterbook scenarios. I already posted my Mission Matrix, which has been a big help in planning missions, but I wanted to spend a little time with some other aspects of these scenarios. We have one player acting as gamemaster, two regularly attending players one each of the Fox's Teeth/Sorenson's Sabres sides, and 2-4 irregular players who sit-in as needed for the players or the OpFor. We've been having fun, but we are starting to run into some difficulties. One of our group made an especially good comment the other night as we were packing up for the evening, and I will quote him here to the best of my recollection:

JZ: "For a scenario book that is supposed to be an introduction to Battletech for new players, there are a lot of problems that only an experienced player would know how to deal with."

A lot of problems indeed. This book is intended as a sort of bridge between the Boxed Set and Total War rules, but it assumes players already know the full rules. Many missions descriptions are also quite vague as to exactly how they are supposed to be set up. Our veteran crew and GM take these problems in stride, but it would be difficult for new players to figure these things out.
The book could have done with more playtesting too, and mea culpa, I was one of the playtesters! My recollection is hazy, but we only had 18 days - enough for proofreading, but nowhere near enough time to play many missions. This also came at a busy time for most of our group, and I don't recall that we played any of these missions. This may also be a problem of having veteran players review a product intended for new players; the bugs that will trip up new players are nearly invisible to players that have been playing the game for years.

Errata problems aside, there are issues with the scenario tracks too. An inexperienced player could easily run out of WarChest Points (WP) with just a few unwise decisions. Taking the wrong combination of force and mission options (plus a little bad luck) turned into a big setback for our Sabres players. Now it's OK if players run out of WP, but I don't see how inexperienced players could avoid running into trouble. These scenarios really need an experienced gamemaster to supervise and help keep the players on a steady course.

There is another issue with these missions that is quickly becoming apparent; the scenario balance is just awful. If you play with the original mechwarriors and mechs from the book, and if the opposing force gets a few good rolls of the dice during the setup, then some of these missions can be challenging. --BUT-- If you are playing (as we are) where mechs can "lightly" modified with prototype weapons and upgrades, or replaced with other mechs purchased or captured, then the scenarios become heavy unbalanced in favor of the Teeth/Sabres sides.
I must admit my bias: I like well balanced scenarios because they are the most fun. I have been playing Battletech long enough that simply winning has little attraction for me - I want a challenge, because challenges are fun! Winning is still cool, but I want to win in a fair fight, not in some goofball setup where one side has no real chance to win (Our group even have a special name for these sort of scenarios). Consider how the following aspects of the Sword and Dragon missions lead to unbalanced scenarios:

Random opposition selection is highly variable, and the players could be up against a tough fight. This encourages them to ALWAYS field the strongest lance of mechs available, even when a weaker lance might be able to pull it off with a little luck.
There are heavy penalties for not accomplishing mission objectives, so players are again encouraged to field a very powerful force, even if they might get by with less, simply because they cannot afford to lose.
Random force selection does not consider that players may have upgraded their force. Even if the scenario might have been balanced originally, any improvements players make with push this towards unbalanced missions.

There is a common theme here. Given any amount of control other the scenario parameters, players will tend to optimize things for themselves even if that does not lead to a fun game. This can be OK if the other side has an equal chance to optimize, but in this sort of one-sided random scenario generation it gets broken pretty quickly. A a basic principle of good game design is broken here: There is no flow, no Fear of Failing to challenge players to the limits of their abilities.

I know I am asking for a lot. Computer games might be be able to adapt to players in this way, but can a boardgame possibly do this? The answer is yes, after a fashion. In a series of Battletech scenarios, there is no reason you couldn't have measure of performance or margin of victory, and use this to adjust the difficulty of future scenarios. To do this you need to start with a good way to rate the strength of force and opposition, which is one of the reasons I keep going on and on about point balance systems. You would also need a way to measure margin of victory over a series of scenarios. Chess, basketball, and many other games have ratings of skill and ability as measured by their performance against other players or teams. You could do this for Battletech too, with a bit of work. Finally, instead of just rewarding players for winning, make the rewards contingent on the difficulty of scenario they choose. Given the choice of an easy win or a challenging game, I think players will go for the challenge every time.

Hint: This is a topic I hope to spend some time with in 2011 (now where did I leave my notes?).

I should add a comment about scenario objectives, since there were some recent comments about objectives and balance in my recent post on Point Values for Squadron Strike. Sword and Dragon does use scenario objectives, but most of them are useless for maintaining any sort of balance, and some actually make it worse. A way to rate the difficulty of scenario objectives is something else I'm going to have to consider (and that's a tough one!).

And here is a bit of copyright info, just to make sure everyone is happy.

© 2001-2010 The Topps Company, Inc. MechWarrior, BattleMech, ‘Mech and AeroTech are registered trademarks of The Topps Company, Inc. All Rights Reserved.

The Grinder - Christmas Edition

2010-12-24T00:01:00.005-06:00

A selection of items ground into mincemeat, just in time for the holidays!

Like spaceship art? Check out Concept Ships.

And as that isn't cool enough, there is also Concept Robots. I think I already ground this up a while back, but it's worth a second pass.

A while back I wrote about a Wired article How the Allies used math against German Tanks. John Cook has a follow on post about how America tried to do the same thing with Soviet bombers in 1958, but the Soviets were wise to the trick: Military Intelligence from Serial Numbers. [From The Endeavorer]

Mathematics in Movies - with short video clips for most.

[from IO9] A problem that every parent faces - you know what I mean - sooner or later you have to have talk to them about that difficult subject ... you know the one ...

... Star Wars

[From Web 2.0 RC1] Sparkley Dice!

A calculus joke from Proof: Calculus and Transformers

Also from Proof:

Q: what do you get when you cross a mosquito and a mountain climber?

A: nothing. you can’t cross a vector and a scalar.

{Drum-riff-cymbal-crash}

Merry Christmas!

Support Net Neutrality

2010-12-20T14:05:00.000-06:00

Not much time left. Make yourselves known.

In less than 24 hours, FCC Chairman Julius Genachowski will unveil the FCC’s plan for Net Neutrality. From what’s leaked to the public so far, the proposed rules are not Net Neutrality and in fact would make things worse for consumers. Tiered pricing and network management that causes degrading of video are just some of what we can expect if these rules pass. This is the FCC siding with the telecommunication industry.
Take 30 seconds to write to the FCC and tell them to give us the Net Neutrality we were promised.
This is the perfect example of corporate lobbyists and behind closed door dealings running our government. We’ve been organizing for years on this and tens of thousands of you have written the FCC over the years. But, this is it. Make sure your voice is added and you speak out to demand Net Neutrality.
Happy gaming,
Brett Schenker, Online Advocacy Manager
Entertainment Consumers Association (ECA)

Point Values for Squadron Strike

2010-12-18T23:42:00.000-06:00

I had a long discussion with my friend Ken yesterday about point value systems in games, and specifically how this can be applied to his game Squadron Strike (Ad Astra Games). We spent several hours hashing out the details of a Lanchester type scoring system, and how various aspects of the game ought to affect the point value. I came prepared with the best I could muster from the theoretical standpoint, and Ken kept the focus on taking what the math says ought-to-be and turning it into something that works. At times we argued - a good sort of argument - and the very best work people can achieve often comes from such discussion. I can't reproduce all the details, but I think I can recall the high points:

There is no theoretically correct way to make a point system where values and simply be added together, but ...
There is a way to balance point values, somewhere between Lanchester's Linear and Square laws, that is optimal. That is, a point system can be structured so that there is a small advantage to be gained by taking a moderate strategy - with moderate meaning the right combination of intermediate sized ships should be superior to both a swarm of small ships and a single Dreadnought.
We ~~argued~~ discussed at length how various ship capabilities should affect the balance point. The result is that player ship-building decisions are going to change the balance point, and building The Ultimate Fleet that beats all challengers is going to be quite difficult (perhaps nigh impossible).
Squadron Strike has a detailed ship construction system that already has it's own checks and balances built in. I encouraged Ken not to penalize point costs for aspects that should already be self-balancing. For instance, I suggested that movement didn't need to be part of this point cost, because ships with more hull space devoted to maneuverability automatically have less space left for weapons and defenses. Making movement increase costs too much could penalize players for building maneuverable ships, and Ken wants plays to build maneuverable ships. (I think that's when Ken accused me of being a closet Republican.)

I think we ended up hammering out a good basis for Squadron Strike point values for scenario balance. It isn't the the last word in the mathematics of game balance. Squadron Strike is far too complex to be able to write out as any sort of "equation for victory", but then most games are far too complex to solve in that way. That very complexity is why game are fun, and why we play them. A point system doesn't have to be perfect. It is after all just another rule by which a game can be played. A lot of hard work went into making this rule, and I think it's going to be a good one.

Lanchester's Game

2010-12-18T13:30:00.001-06:00

I've been kicking ideas around about using Lanchester's Laws as a way to create a point-value system (1) for balancing sides in a game, and one of these involves designing a game around this mathematical principle. Usually point system seem to evolve after the game is created; I want to create the point system first, and build a game to fit the math. I'm not sure this is possible, but it should be interesting to try. So I set out to write out the rules for a simple game to demonstrate the difference between Lanchester's Linear, Square, and Logarithmic (see footnote 2) Laws, which I will abbreviate with L1, L2, L0. This turned out to be a really good idea, because it gave me an insight about how these laws arise, and a simpler way to explain them.

[A note aside: I wrote most of this post two months ago, and it gave me many good ideas for other post in the process, but I never quite figured out how to finish this one. Now I need this basic discussion to go along with some new posts I'm working on, so I'm making a second effort to finish this one.]

The Game
This game is really very simple, actually more of a thought experiment that explains Lanchester's laws.

Map, Movement, and Range: There is no map, and so no movement, and no range. This is an abstract game, and attacking another unit depend on which Attack Rule is being played (see below). It would be a more interesting game with these elements, but they only complicate the discussion. Maybe I will try to add these back in for another post. (Also good discussion points.)

Forces: Two sides, Blue and Red, each side have a number of tokens (B or R tokens, respectively) or markers representing the strength of each force (These might represent soldiers, tanks, etc.). Each player should start with 20 to 30 tokens, but not necessarily the same number.

Lethality: an attack is resolved by rolling a die: success kills one enemy, remove that marker/token. Assume equal lethality for simplicity, or allow to be different for completeness. Each attack has a lethality, or probability of a kill equal to b for Blue and r for Red. Lethality does not have to be the same for each side, but it simplifies this discussion if it is. For a good demonstration this should be a fairly small probability, so that the game will last 10-20 turns. The following discussion will assume a lethality of b = r = 1/6, so a roll of 1 on 1d6 can be used to resolve this easily.

Attacks: Every turn each player makes one or more attacks. The number of attacks a player makes depends on the Attack Rule in play, and could depend on the current size of each force (B or R).

Sequence of play:
1) Set up the game, decide force sizes, lethality, and Attack Rule.
2) Begin turn: players make one or more attacks, as determined by the Attack Rule in play.
3) Resolve attacks for each player based on the size of their force at the beginning of the turn.
4) Remove destroyed forces.
5) If both played still have forces remaining, go back to step 2 and play another turn. Play continues until one side is eliminated.

Attack Rules:

L1: Each player makes one attack every turn. No matter what casualties occur over the course of the game, each player will have same same total number of attacks, and this number will be proportional to the small of the two forces. This is exactly what is expected under the Linear Law.

L2: Each player makes one attack for every 5 tokens they have remaining (round up or carry fractions o the next turn). Over the course of the game the total number of attacks will be larger for the player with the larger initial force. The ratio of total Blue attacks to Red attacks will be proportional to (B/R)^2 [the ratio B/R, quantity squared]. (It does not have to one attack for every 5 tokens, it only need be some small proportion of the current size of the force. 5 was just convenient).

L0: Each player makes one attack for every 5 (convenience again) of the other sides tokens. Here the number of attacks made against you is proportional to the size of your own force (see footnote 2 again). This seems like a strange rule, but war in unhealthy! Putting your army in the field makes if subject to direct and indirect threats. Starvation, disease, accidents, mules kicks, artillery and bombing, are all hazards that put the entire force at risk. Sometimes the more you bring, the more you lose.

Discussion:
Each of these "attack rules" will lead to distinctly different outcomes for the game. More importantly, a form of one or more of these rules is inherently present in all war games and combat simulations. Even if it is not written explicitly, but it will still arise from how the game plays.
Game combine these rules in interesting ways. For instance, terrain, stacking rules, and range limits will tend to restrict some units in a game to the L1 attack rule. Other units will have a clear field of fire to attack (and be attacked) will use the L2 rule. Some units might stay in relative safety and threaten the other force from afar (like artillery) and subject the other side to the L0 rule. A unit firing from a bunker might only be attacked under the L1 rule, may be able to attack other using the L2 rule. so it's not necessarily the same rule in effect for both sides.

I have read many papers trying to model data from historical battles as if there is a new rule that somehow combines two or more of these rules. From a certain standpoint that is the wrong approach. These rules might mix, affecting different parts of armies in different ways, but there is no rule that says you will always get the same sort of mixture every time. In fact, you will almost certainly get a little different mixture in every battle. Lanchester's laws are not something over which either side has total control - they are something that happen to you during the battle. In a close fight, the army that is better able to exploit the rules is more likely to win.

Footnotes:
(1) Such as BattleValue in Battletech, which is the one I know the best. Ogre/GEV has a very simple point system. Warhammer 20K+/-20K has a point system but I know nothing about it other than it exists. If you can suggest other games that use point systems please post or email me about it so I can look into this topic further.
(2) The "Logarithmic" law arose from attempts to fit actual data of battle casualties to either the Square or Linear law, and finding that sometimes neither one is a very good fit. I have avoided mentioning it thus far in order to simplify discussion. The interpretation of the Logarithmic Law implies that the casualties suffered by one force are proportional to the size of one own force (not the opposing force). This seems unusual, but is sometimes observed in historical data describing large scale battles. See Fricken (1997) for the an excellent discussion and justification for the Logarithmic law.

Granular Skill Checks, and Interpolation

2010-12-12T12:06:00.026-06:00

I just read a discussion in the CBT forums (Thanks PiP!) about the "granularity" of skill rolls in Battletech. The basic problem is that skill level changes in the boardgame make a BIG difference in play, but if you are running A Time Of War (Battletech roleplaying) then you want may small changes so that characters can improve gradually in many small steps - as opposed to 3-steps to a Superman.

I'll keep this discussion about the Battletech RPG, but these comments should apply to any game where characters have skill levels that seem too granular. I originally posted this whole thing in the CBT forums, then UN-posted it because what I had was broken. Now it is fixed, but much longer, so I hope nobody minds that I'm linking back to myself.

One way of doing this would be to use a different random distribution for skill checks. The second editions of the Battletech roleplaying game (Mechwarrior) uses 2d6 rolls just like the boardgame, and all skill were very granular. The 3rd edition of the game made a switch to 2d10 "exploding" dice, which greatly reduced the granularity problem, but suffered because it was difficult to make meaningful improvement in character skills.

There are also various "house rules" for doing the same sort of thing, but these generally require changing other aspects of the game to balance out the change in probability distribution. For instance you might switch to a 2d10 or 4d6 to-hit roll. There is less granularity now, which is good for your RPG, but the game has changed! On this new scale a +1 or -2 modifier will have relatively less effect or results, potentially "breaking" the usual balance of the game. You might fix this by adjusting all these modifier, but you won't ever get the original balance back this way.

I have an alternate suggestion: Add a decimal point to the skill levels, and an extra 1d10 roll when rolling for a skill check. Differences between Battletech skill level are BIG changes, so the idea is to add steps in-between. For example, instead of Gunnery 5 and 4, some possible skill levels are 5.0, then 4,9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3, 4.2, 4.1 and finally 4.0. Likewise any skill level, just adding a fractional skill level to it. There is a word for this - "Interpolation". We can interpolate between whole number skill levels, filling in with smaller changes in probability.

To use this, calculate the target number (TN) normally adding the skill level and any modifiers, and round the final number down. Make the usual 2d6 roll;
if this is less than the TN, you fail;
if this is more than the TN, you succeed;
if you roll exactly the target number, then you must also roll the 1d10, read it as 0-9, and this must be equal or greater than the decimal in your skill to succeed (if the decimal is "0" then this always succeeds, no need to roll).

Example: Suppose the base gunnery skill is 3.6, and after various modifiers the target number to-hit is 8.6, which rounds to 8. You roll 2d6 and ...
On a 9 or better you hit,
On a 7 or less you miss,
On exactly 8, you roll 1d10 (0-9), and if this is a 6,7,8, or 9 then you hit, otherwise you miss.

With TN=9 probability of success would be 0.278, and TN=8 it would be 0.417. The effect of the decimal in the skill level and the 1d10 die roll is to interpolate, or smooth out, between those two probabilities. The final probability of success for a TN of 8.6 is 0.333.

This gives 10 steps of skill improvement to every 1 in the regular rules, which ought to be fine-grained enough to satisfy the pickiest Game Master. In fact it may be too fine, and you might want to restrict it to just 5 steps (.0, .2, .4, .6, .8) or even 2 (.0, .5). Further, you will need to adjust the experience needed for fractional skill improvements accordingly. If it cost 100 experience points to improve Gunnery skill from 5 to 4, then it should cost about 10 to improve from 5.0 to 4.9. Most GM's love to tinker with this sort of thing anyway, so I'll leave the application in your capable hands (or fangs, tentacles, whatever).

Now the really good news - interpolating skills does not "break" any other parts of the game by changing the probability distribution the game is designed on, it just smooths it out, so a +1 or -2 modifier still has the same effect it always did. There is nothing special about using 2d6 with this either, so you might easily apply this to any RPG with granular skills.

Here is a chart with the probabilities of success on a 2d6 skill check. The way I have set this up makes it look a bit like a wavy staircase:

Probability of success on 2d6 with standard skills

You could walk up those steps! And the granularity is obvious. This chart has target number up to 13 because I need to show the probability go all the way down to zero for the next chart. Now a chart showing the same for interpolated "Skills with a decimal":

Probability of success with interpolated skills

Nice and smooth. If you look really hard you might notice this is actually 11 straight line segments joined together. This method of interpolation really just connects the dots between probabilities for whole number skill levels.

Now we have wiped out the granularity problem, but at the cost of some extra dice rolls. If you don't want to roll so many extra dice, you could make a single 1d10 interpolation roll (after fire declaration) and apply it to all skill checks results for that turn. This will be weirdly granular, because it is like changing your skill level randomly from turn to turn, but it will average out to the same effect over the course of many rolls.

Number Needed to Hit

2010-12-10T22:37:00.000-06:00

In the comments on my previous post on Dice and Information, I mentioned a statistic called Number Needed to Treat (NNT). This is used in epidemiology and medical science as a way to compare the benefit and cost of two treatment. For example, this might be two different drugs that might be used to treat an illness. Here "benefit" means any beneficial effect, maybe something major like preventing fatal heart disease, or something minor like prevention of allergy symptoms. For this discussion I will assume it is a BIG benefit, like saving a life.

Benefit: Suppose that in a clinical trial running one year, 80% of patients on drug A benefit from the drug, and 82% of patients on drug B benefit equally. Now drug B is better by 2%, but what does that mean? If we treat 100 patient on each drug for one year, we expect 80 "benefactors" from Drug A and 82 benefactor from drug B, a difference of two patients saved every year for every 100 treated, or one saved for every 50 patients treated (for one year).

A shorter way to calculate this is NNT = 1/(0.82-0.80) 1/(0.02) = 50 patients "needed to treat" for one to benefit. Make sure to subtract the smaller probability from the larger, or you up with a negative number.

Cost: Now suppose that drug B costs $10,000 per year, compared to $1000 per year for drug A; Is drug B worth ten times the expense? NNT helps answer that question. Consider that treating 50 patients for one year (the NNT) would cost $250,000 on drug B and and $50,000 on drug A - a difference of $200,000 for one life saved each year. Now is drug B worth the cost? If the benefit is strong, like not dying, then it may be worthwhile. It certainly is if you are the 1-in-50 patient, but you can't know that is advance. Also, resources are limited. It is not good practice to make patients pay large expenses, or tie-up hospital resources, for treatments with very little benefit.

If anyone is wondering, this is not Obama-Care Death-Panel stuff, this is a serious sort of decision that determines what doctors call "Best Practice".

Well so far this has been pretty boring, but now we can take the idea behind NNT and turn it into a statistic for games.

If we replace the probability a treatment will be effective with the probability of a successful "to-hit" roll, we get a measure of Number or rolls Needed to Hit (NNH). This is now the number of to-hit attempts you will need to make, with some higher probability of success, in order to hit one more target on average. In the sniper rifle example from the previous comments, the difference between 80% and 82% means one more hit out of every 50 shots fired. This 2% gain could be useful from the perspective of a first-person-shooter game, or even critical to ensure taking out an important target, but on a larger scale other things could matter more. For instance, if the more accurate rifle weighs more, or costs more, and this is reflected in game usage and costs, then 2% gain in accuracy might not be worthwhile.

Dice and Information ... So What?

2010-12-10T19:24:00.131-06:00

After I finished my previous post on Dice and Information I had the thought, "These are some pretty neat numbers, but So What? What's it good for?"

One way to look at information is as a measure of uncertainty in the game, or at least the uncertainty in the outcome of a single move that is part of a larger game (The information in an entire game, start to finish, is a matter for another day.). Consider the game of Chess, which has nothing random about it. There is no information at all in a chess move, your just make your move, perhaps taking another piece, and it always works. Suppose now we change Chess so that it requires an attack roll if you want to take another piece, with a 50% chance of success (put the piece back where it started if the attack fails). Now there is uncertainty with each move (1 bit of entropy) and the outcome of any move is far from certain. We might change this to 90% success, which works out to about 0.08 bits* (oops. Thanks Bradley!) about 0.47 bits* of entropy per attack. This means that any attack is ~~nearly certain~~ quite likely to succeed. If the chance of success is 10%, then the entropy is again ~~0.08~~ 0.47 bits*, and the attack is ~~nearly certain~~ likely to fail.

So entropy is measuring uncertainty in the sense of the predictability of results, but NOT the predictably of a preferred result, such as a successful attack roll.

* It might help to think of 10% or 90% success as the flip of an unbalanced coin. Information is maximized, and the result is most uncertain, when the coin is fair (50% success).

This post hasn't gone the direction I thought it would - I was thinking (incorrectly) I could describe Entropy as a measure of the uncertainty in the outcome, but this is rather different. Consider a player making three to-hit rolls in a game, at 90%, 50%, and 10% success. The first (90%) has just a little entropy (0.47 bits) and the outcome is quite likely. The player has a high degree of control, because the decision to attack is very likely to succeed. At 50% the entropy of this roll is maximized at 1 bit, and the player will be most uncertain of the result either way. At 10% entropy is again 0.47 bits, but the player is very likely NOT to succeed. Now the player has very little control, or very little influence on the outcome of the game (with this single roll).

Back to the drawing board? The paragraph above hits a few rough spots, because Entropy and player control out outcomes in a game are maybe two different things. This gives me something new to think about.

...

And before I can finish hammering the bugs out of this post, Ashley has her response up over at Paint-it-Pink. At risk of quoting Ashley out-of-context ...

Ashley: Now when one applies modifiers to a 2D6 roll, if one knows that it only encodes 3.27 bits, then a modifier of one is equivalent to one bit. If I'm correct then a modifier of three is equal to three bits of information, which has the effect of reducing surprise in the diced for result? Such modification of the base 2D6 roll is therefore highly significant, which seems to me to support my proposition about the modifiers for targeting computers and pulse lasers in the Battletech game being too coarse.

Not quite right, but Ashley's intuition is basically correct. We need to sort this out a bit. First, the modifiers she mentions are for a Battletech "To-Hit" roll with Hit-or-Miss outcomes, and this is the same situation as flipping an unbalance coin. As in my "battle chess" example above, modifying this roll doesn't necessarily decrease the information. Second, the 3.27 bits a for a 2d6 hit-location roll** is separate from the To-Hit roll. However, we might combine the To-Hit and Hit-Location rolls by considering a "miss" to be a no-location result and grouping it with actual location results.

** Irrelevant quibble: this is actually about 3.0, because there are multiple ways to roll "Arm" hits.

Here is a new table, similar to my earlier table where I calculated the Entropy of a 2d6 result, but now the possibility of a miss of a 2d6 roll of 7 or less.

Now suppose the To-Hit roll is more difficult. Ashley's intuition says the entropy ought to decrease. Here's another table with a "12" needed to-hit.

Sure enough, the entropy has decreased. Unlike the battle Chess example, here the Entropy of hit-location (including no-location) will usually decrease with more difficult to-hit rolls (it hits maximum entropy with a to-hit roll of 3+).

Lesson learned: When thinking about entropy, it is important to include all possible outcomes of the random result.

Somehow I think this topic is not done yet, but that is all I have time for today.

A small update (12/12/2010, 4 PM): I just made the following comment on Ashley's blog, and I'm copying it here so I might remember to come back to the idea later.

Now here is a brain bender - Suppose you could roll one set of dice to resolve a whole turn of Battletech play, or a whole game - How much information would be in that? I don't know myself, but I'll think on it. My intuition is a simpler game should have less total information than a complex one, but I'm not sure yet what it would mean to compare them.

More:
Dice and Information
More Dice, More Information ...

Dice and Information

2010-12-07T00:01:00.416-06:00

There is a concept is statistics and the information sciences of information. Several concepts actually, as there are different types of information, but I want to focus specifically on Shannon information or Entropy. Entropy is a way of measuring the amount of variability or uncertainty in a probability distribution, and a simple way to illustrate this is with the example of a coin flip.

But first, a comment of notation, since the Blogger editor is not too equation friendly. Calculating entropy requires a logarithm function, usually denoted ln(x) or loge(x) for base-e or natural-log, and Shannon Information specifically uses a base-2 logarithm, which I denote here as log2(x). If my equations are not clear, any mention of the log function (outside this paragraph) always means the base-2 logarithm. If you are following along with a calculator, you probably have a natrual-log button ln(x), but can calculate the base-2 log as log2(x) = ln(x)/ln(2).

Image source, and quite interesting in itself.

Assuming a fair coin with a 0.50 probability of heads or tails, then the first step is to calculate a quantity called the Self-information or "surprisal" of all events. This is a measure of how surprising a given event is relative to the other possible events in the distribution. This less likely the event, the higher the value of its surprisal.

Surprisal is equal to -log2(p), where p is the probability of a given outcome. Calculating ...

log2(.5) = -1,

-(-1) = 1

... and the NOT so surprising result here is that heads and tails are equally surprising, with a value of 1 each.
Now that we have the surprisal, we can calculate the Entropy as the average or expected value of the surprisal over the entire distribution. For this example the calculation is trivial; the of 1 (for heads) and 1 (for tails) is just 1. Shannon Information is measured in "bits", the basic unit of information used in calculation by computers. To relate this to games it might help to think of one bit of information being equal to the amount of variability in the flip of a coin.

Here I have a table representing the information in discrete uniform distributions from 1 to N. In gaming terms this is the information in single N-sided dice, with each face of the die being equally likely as all others. I included all the values representing true polyhedral dice, and some additional values for comparison (most of these are powers of 2 or 10).
The second column p(x) gives the probability of each "face", the third the surprisal, and the forth the entropy.
Here we can see that a 2-sided die (a coin!) again has 1 bit of entropy, a 4-sided die (d4) has 2 bits, a d8 has 3 bits, and a hypothetical d16 has 4 bits, following powers of 2 as you might expect. I put in some extreme values just for fun - the final row, a one million-sided die, would have nearly 20 bits (or 20 coin-flips) of entropy.

As in the example of the fair coin, when all outcomes are equally likely, the surprisal and entropy are equal. This also maximizes the value of the entropy - meaning that if any result was more or less likely than another, the result cam only become more predictable, and the value of the entropy must be less, as will be seen in the next example.

For the second example I'm calculating the entropy of the sum of two six-sided dice. This table show the possible results from 2 to 12, the probability of each result (twice) as the odds-in-36 and a probability. Next (4th column) is the surprisal of each result, and unlike the uniform distributions this values varies with the probability of the outcome. A roll of 7 has a surprisal of 2.58 bits, and a roll of 12 (or 2) 5.17 bits; a 12 is the more surprising result, relatively speaking.
The final column is the surprisal multiplied by the probability, and these are summed to determine the Entropy at the bottom, which is 3.27.

In terms of information, a 2d6 roll is in-between the d9 and d10 rolls from the first table. This doesn't mean they are the same, but that they have a similar amount of variability.

For the third table I have calculated the entropy for some commonly used dice-rolls in games and listed them in order of increasing entropy. The 2d10- designates the difference of two ten-sided dice, as used for penetration damage in Squadron Strike.

Note the entropy of 4d6 is less than twice than of 2d6, and likewise 2d6 is not twice that 1d6. As numbers from single dice are summed, the distribution becomes less uniform, more like the bell curve of the normal distribution, is more predictable, and therefore has less entropy. If we were using two separate d6 rolls to generate a uniform random number between 1 and 36, we should expect the d36 entropy to be twice than of a d6, and it is; log2(1/36) = 5.17. We also see this with the entropy of d100 being twice that of d10.

What strikes me from this is rolls of 1d6, 3d6, and everything in-between, vary by only about 1 coin-flip of entropy, so maybe the many variations of dice used in games really don't make so much difference in terms of the variability of play.

A final note: Just because there might be more information in some combinations of dice does not mean the game takes full advantage of that variability. For instance if you are making a to-hit roll with some probability of success (hit or miss), then there is at most 1 bit of information in that result no matter what kind of dice you roll it with. There are only a full 6.64 bit of information in a d100 roll if there are 100 unique outcomes.

More:
Dice and Information, So What?
More Dice, More Information ...

Blue versus Gray, or maybe Federation versus Rebels?

2010-12-03T22:55:00.000-06:00

"Mr. Business" left me a good question on my last post about Gratuitous Space Battles, and it deserves a good answer.

N.B. writes: As an alternative, what about special anti-fighter weapons? Perhaps a frigate type ship that can act as picket against fighter waves. Specialized weapons can overcome numbers in many cases.

And

Like American Civil War. 100 Grays with rifles vs 3 Blues with a Gatling gun. On the open fields of a typical A.C.W. battlefield, your 3 Blues are going to win.

To the first part about specialized weapons, GSB lets you do just that. There are anti-fighter missiles and tractor beams to help shoot down those pesky little space-piranhas. Of course, big ships can carry heavy armor that is nearly impenetrable to fighter weapons, so it helps to have a few big ships with big guns to soften up those hard targets. Real battles are complicated, with many factors influencing the outcome. Lanchester's Laws are a simplification down to the bare basics of the number of men and how effective the are at killing the other guys. This demonstrates a principle of warfare; the advantage of superior numbers and why it exists, but it doesn't begin to cover all the possibilities that real situations allow.

The second part though - the Civil War example - is something that can be described nicely by Lanchester's Law (squared law). I'll change the number just a bit for my convenience though.

Suppose we start off with equal sides: 100 Blue and 100 Gray soldiers, each side equivalently armed with Rifle Muskets, and each able to shoot freely at any soldier on the other side. These sides should be approximately equal, with 100 soldiers on one side going to receive about as many casualties as they deal out to the other.
A game-like example will be useful: If we set this up as a game played in rounds, and give each soldier a 2% change of killing one of the other soldiers each round (completely arbitrary, just bear with me), then each side will lose about 2 soldiers each round (at the start) and the winner is probably just a matter of chance.

Now lets change it up - lets give the Blues Gattling guns (or maybe Uzi's?), and lets say that these BIG guns inflict casualties on the other sider 25 times faster that the old muskets of the Gray's. Now the Blue team is 25 time more effective, and therefore 100 Blues should now be equal to 25 times 100 equals 2500 Gray soldiers. Right?

Nope, 'fraid not, at least not according to Lanchester. The Squared law tells us that the advantage of greater number is proportional to the square of the ratio of the number of soldiers on each side. The Gray's can balance the Blue advantage with just 500 soldiers, not 2500, because a 500-to-100 is a ratio of 5, and 5 squared is 25. The reason behind this is that ever time Blue suffers a single casualty, they lose a greater portion of their total firepower.

Back to my game-like example: If each Blue soldier now has a 50% chance of killing a Gray soldier each round, and the Grays are still at 2% but there are now 500 of them, then there should be about 50 Gray and 10 Blue casualties in the first round. Now the ratio is 450 to 90, still 5-to-1, and the sides are still balanced. Under the differential equations of Lanchester's Laws, these to forces will grind each other down always at this 5-to-1 ratio until both sides are dead.

Now if this game has any bit of randomness, or as in reality things are not always equal, that ratio will not stay at 5-to-1 for very long. Before many rounds of fire are exchanged one side will gain a slight advantage, and that advantage will expand in to a victory without killing all the soldiers on both sides. If Gray bring 600 soldiers instead of 500 they should win this game every time, and this still isn't half way to 2500.

We could play this game differently too. Instead of giving Gray more soliers we can take away some of the Blues. With their BIG guns, just 4 Blues should be equal to 100 Grays. Now the difference in the casualty ratio really shows, because after just one round Blue will have 2 soldiers left, and Gray will still have 98. 20 Blue to 100 Gray should be about even (but don't hold me to that, because I make mistake when I'm tired and do math in my head).

That said, I haven't really done justice to N.B.'s Civil war example. There probably were Civil War examples of 100 Confederates charging a Union Gattling gun and getting torn to peices. As I said before, it's complicated! I'm no student of Civil war history, but I know it was a time of changing technology and tactics. A few decades earllier and 200 musket-wielding Grays charging straight at 100 similarly armed Blues was probably a winning tactic, but 200 Grays charging 100 Blues - who by-the-way have a few of Mr. Gattling's latest inventions - I think that might not work out so well for the Grays.

Gratuitous Space Battles

2010-11-29T00:01:00.001-06:00

This is actually part 3 of my series on Lanchester's Laws, sort of. Since writing parts 1 and 2 have been reconsidering the whole topic, and I seem to have a number of ideas that branch off of this topic but one doesn't necessarily follow another. Therefore I've decided to worry a little less about how this all fits together, and just write something and get it posted. The figuring out can come later.
That said, it time for something I've been putting off for far too long ...

... which is a game called Gratuitous Space Battles. GSB is my first computer game purchase in several years, and I consider it money well spent. Let's start with a video of the action!

This operates a little differently from most space battle games; this is a Tower Defense game. If a game is a series of interesting decisions, then all the decisions in this game come at the beginning. First you design ships, choose your fleet, give then orders, and click on START. Then you sit back and WATCH the battle unfold. If you lose the battle, you go back and try a different set-up. if you WIN the battle, you can go back and try to win with few resources. The replay value of this game is very high.

Gratuitous Space Battles has been called "massively single player", because you can submit and download scenario challenges from other players in the online GSB community. I haven't done much of this yet, but my experience so far has been very positive, with other players happily responding to my challenges, politely pounding me to newbie-snot, and offering tips on how I might do better. This offers the sort of difficulty level you can only get from human players, even though you aren't playing online, or even simultaneously.

GSB has a downloadable Demo that I would like you to try. This isn't just because I think it is a cool game; which it is, but I have something specific in mind. GSB is a perfect way to demonstrate Lanchester's Square Law in action. If you want to give this a try, download the demo and play a few rounds to get the idea how it works. Once you have that under you belt, try the following setup:

Create a fleet with as many fighters as you can manage, and tune it until it wins consistently. Next, start removing fighters a few at time until your fleet no longer wins easily. Lanchester's Square Law predicts that the casualties you will inflict are proportional to the square of the ratio of friendly/enemy units. With a bit of experimentation you should be able to find the balance point - the point where your fleet doesn't always win, but the addition or subtraction of just a few fighters changes the balance drastically. Try to find the point where adding 2-3 fighter practically guarantees a win, but subtracting 2-3 fighters is a sure loss. In GSB, fighters are as close to single soldiers as you can get, and it is interesting to find that balance point where "just a few men" can completely turn the tide of the battle.

Some hints if you want them, in white text in case you prefer to figure things out for yourself. Select the text below to view:
[hidden text]
Choose a few cruisers with 2-3 Plasma Launchers, and a few shorter range weapons.
One cruiser should have a Fighter Support Bay, and order your fighters to be "Cautious" so they will will return to the bay for repairs.
Faster, lightly-armed fighters attacking at minimum range (inside cruiser shields) are much more effective than slow, heavily armed fighters.
Give your fighters orders for "Cooperative", "Vulture", and "Stick Together". You should also order them to "Escort" one of your cruisers (at range 600) so they do not wander too far from the rest of your fleet.
[/hidden text]

OK, now for just a tiny bit of mathematical consideration. Suppose we take the source code for GSB and simplify it - stripping out everything that makes it look like a game, removing all options except for how many ship there are and the firepower of each ship, removing movement (assuming every ship can fire at any other), leaving just the input of forces at the beginning and a report of the battle outcome. What we have left is a basic attrition model; a simple program that conducts a simulated "battle", even though we don't get to watch the progress of that battle any more. With all the simplifications it is barely resemble a game, but it become a very effective tool for studying the effect small changes in force selection (or force orders) can have on the outcome of a battle. Consider putting this tool inside of a computer program and run through hundred, thousands, or even millions of scenarios, each with slightly different starting conditions, then look at which starting conditions lead to success or failure.

Just like my suggested example above, tinkering with the number of fighters to find a balance point, you might tinker with the behavior of forces. You can examine things like the effect of the priority in which enemy units are attacked, maybe add ammunition considerations, command and control, whatever you like. The point is, there can be value in studying very basic considerations in a model that has been stripped of all but the most basic structure. It's true that the effect of something complex (like command and control) probably depends on any number of other factors, but if you don't understand how something works in a simple model, then it is unlikely you can understand how it works in a more complex model.

I'm a big fan of simplifying thing down to base elements, and trying to understand them from the bottom up. Gratuitous Space Battles, even in demo form, is a good tools for studying Lanchester's Laws. It's not an ideal tool for such - that would require your own program - but it is very accessible to anyone with an interest in the topics, and it's a heap of fun too. I give GSB a four-robot rating :-)

Over at the developer's blog, you can read about The Fighter Spam Issue he is having with the GSB campaign game. Essentially, by taking advantage of Lanchester's Squared Law, it becomes fairly simple to construct a fighter-heavy fleet that is nearly unbeatable. Balancing the game requires in this situation require putting some limits on the usefulness or availability of fighters.

Credits: Some of the images here are taken from the GSB developer's site, or the GSB Facebook page, and the rest are screenshot I created.

A question in the comments below inspired this response and example, and this related post.

Wet Palette

2010-11-28T11:40:00.002-06:00

I've been working on finishing up an Arctic Wolf mini. I took this picture of it (on the right) for a post back in September, and I've been using it to "brush up" my technique, since I'm a bit out of practice. This is not amazing.

I wasn't happy with the browns, so I broke out those colors again and got back to work on it. This also, is not amazing.

Here's a close-up. I started reworking this back in September, but took a long break because I've been busy. The Thanksgiving holiday has given me more free time, and since last week I've been painting again in what time I could manage in the evenings. Some progress has been made, but still not happy with it. And still not amazing.

A picture from few days later; I've got a better blend and a start at highlighting. I'm still working at the browns, and this work has been spread out in short spurts over 3-4 days.

But not amazing.

Since this work has been spread out over time, I've been storing my paints in my new wet palette I bought back in September. The paints in the round well-palette are the same batch that I started this work with back in September. That was about two-and-a-half months ago.

AND THIS IS AMAZING

I don't get many long stretches of painting time these days, and being able to store my paints open and wet is incredibly useful. I also like to blend my colors, and even the blending spots I make outside of the wells are staying wet.

AMAZING!

To be fair, I haven't exactly left these paints unattended for over two months. Every few weeks I've opened them up and stirred-in a little airbrush medium to keep them wet. I also dosed the palette sponge with rubbing alcohol to help keep the paint from curing. The pigments do settle, and the airbrush medium I added made them very thin. I had to stir them throughly, and I added a little regular acrylic blending medium to get them back to a good consistency. I always work with thin paints anyway, so this didn't me a bit, but if you are used to working with paint straight out of the bottle this might not be to your liking.

This wet palette far exceeds my expectations for what it could do, and will change the way I paint. Now I should be able to pick up the previous colors I've been working without having to spend time remixing and getting the consistency to how I like it.

Bonus Pic: If you recall my little adventure with the pink paint, my might be amused to learn there is still a spot of pink on my living room furniture.

Happy Painting!

Sword and Dragon Mission Matrix

2010-11-18T00:10:00.002-06:00

Our local Battletech group has been playing through Starterbook: Sword and Dragon, and we are having a lot of fun doing it. Peter, my cohort in planning for team Fox's Teeth came up with this state diagram as a way to follow the possible mission tracks, to help us plan in advance:

Nifty, but very difficult to follow as many of the arrows are on top of one another. This put the idea in my head to redo it in matrix form. This entailed repeating most of the work Peter had already done, but in the process I gained a much better understanding of how the Warchest Point system works. I'd never paid much attention to this before, so it was worth the effort:

The left-hand side lists the possible current missions, along with prerequisites and Warchest Points (WP), number of 'Mechs in the scenario, and WP rewards for a successful mission. If you follow the row for the current mission over to the right-hand side you can see what missions are available following the current mission. There is a "1" in the cell if that mission can be played next, and blank otherwise. I also added color codes for missions that are only available to either Fox's Teeth or Sorenson's Sabres. This has been corrected for the published errata too, so if you are looking for the mysterious Probe or Holding Action missions, they have gone away. You can download this in spreadsheet format:

Download as Google Docs spreadsheet.
Download as Excel 2003 spreadsheet.
Download as Excel 2007 spreadsheet.

The Creative Commons license I apply to this blog obviously can not apply material derived from this Catalyst Game Labs product. However, I would appreciate being credited as author if you pass this on. The Mission Matrix form is an idea I will develop for my own purposes. Send any suggestions or comments along too, and I'll see about working them in.

I should probably blog about the play sessions too, because there is a lot going on there. Not tonight tho, it's late!

The Grinder

2010-10-29T19:20:00.000-05:00

Why did I ever think it was a good idea to put the date in the title of these things?

The ~~Daily~~ ~~Weekly~~ ~~Occasional~~ Whenever I Get Around To It Grinder

Just in time for Halloween: The Battlemech costume!

Found in this Gizmodo article

The LOL in MADCAT

MadCat

It's hard enough to face one of these when playing Battletech, but now my allergies can also suffer!

DeviantART MURO a very cool little drawing application (Java?)
Hat-tip --> John Schutte

I started a Tumblr, but I'm not sure quite what I'll do with it. I don't seem to have any time for more bloggish stuff, but I could divert the cartoons I post on Google Buzz here.

Top 10: Los mejores robots gigantes
This appears to be a childs' playset.
These kids have a very cool Dad.

That is all. Keep those dice rolling.

Actually, no. That's not all. I've got a bunch of good stuff half written and no time to fix/finish to publish here. It's very frustrating.

The Grinder (25/10/2010)

2010-10-24T10:41:00.004-05:00

Grinding ...

Found on Utopia Circus

Free Wargames from White Dog Games, brought to you via Grog News.

Found on Letters of Note

The birth of Wonder Woman

Nerd Anthem, by Marion Call

from Wondermark

Roshambo!

The meme of honourable death, a nice summary of a little bit of Anglo Boer War history from Greg Laden's Blog.

Whodunnit?

"Evil Incorporated, how may I harm you?"

A daily comic strip about a corporation run for super-villains by super-villains.

Math versus Tanks

2010-10-22T18:09:00.098-05:00

This story at Wired tells how statistical methods were used to estimate German tank production during World War II:

How Allies Used Math Against German Tanks

The Germans had given each tank a production number, and each captured tank provided information about how many more tanks were in the German army. If you capture 10 tanks, and the highest production number among these is #800, then you know there are at least 790 other tanks out there. If you can assume these tanks were captured more or less at random, then it is unlikely that tank #800 just happened be be among these 10, and the 10 numbers observed ought to be spread out unevenly between 1 and 800. The total number of tank should be something just a bit more than the highest number you happen to know.

Allied intelligence noticed each captured tank had a unique serial number. With careful observation, the Allies were able to determine the serial numbers had a pattern denoting the order of tank production. Using this data, the Allies created a mathematical model to determine the rate of German tank production. They used it to estimate that the Germans produced 255 tanks per month between the summer of 1940 and the fall of 1942.

Turns out the serial-number methodology was spot on. After the war, internal German data put der Führer’s production at 256 tanks per month — one more than the estimate.

That's a very good estimate. Are the statistics here really that good, or did they just get lucky? Using the formulas in the article and my example above with 800 the highest number observed out of 10 captured, the estimate of total number is equal to 800+800/10 - 1 = 879, with a margin of error (~1 standard deviations) of plus-or-minus 88, or about 10% of the estimate That's not so bad given that I only "sampled" 10 tanks, but there is a lot of room for error. It gets better as the sample gets larger though. It turns out the variability of this estimate is inversely proportional to the sample size, 10 tanks give a standard deviation of about 1/10, or 10%. Bump that up to 100 tanks and the standard deviation is 1/100, or 1%, and now the estimate is precise enough that your military planners won't care about the error.

This isn't the only example of statistics in war; many basic quality control methods were originally devised as part of the effort for World War II.

10 Games To Be Forgotten

2010-10-15T12:00:00.002-05:00

image source

As a follow-up on my contribution to the 15 Games That Will Always Stick With Me meme, my better half suggested "10 Games I Would Rather Forget" as a follow up. The rules are a simple modification of the previous ones:

The rules: Don't take too long to think about it. List 10 games you've played that somehow didn't click with you. These might good games, but something about them wasn't fun, a game that just didn't reach its best potential, or even a bad experience with a game you really like. List the first 10 you can recall in no more than ten minutes. Pass the meme along if you can, and link back so other can follow the responses (or just post them here).

To be clear, I intend to include games I like and play,

Striker (GDW) - Miniatures rules for Traveller, with a detailed construction system that let you create you own soldiers, weapons, armored vehicles and aircraft in the full range of Traveller tech levels. VERY COOL. The down side is that it came with no very little in the way of pre-designed soldiers, weapons, armored vehicles, or aircraft, so if you wanted to play you had to create all of these things for yourself. This meant hours of paperwork and hand-calculations (this was 1981) to create the gun you wanted to mount in a fast scout tank, only to find out that it was too heavy and you have to redesign the gun or redesign the tank. Very frustrating. File this under "Too far ahead of it's time."
D&D - Yes, this was on my "Stick With Me" list too, but this time I am specifically referring to persons or player groups that were just unbearable. My example is a Dungeon Master who only wanted to feed his own ego, and letting him be in charge of running the game was just a bad idea. File this under "Bad Experience."
Mechwarrior 3 - The Lifepaths character creation system from this RPG is practically a game in itself, and is perhaps it's best aspect. Character creation can also be tortuously complex, so it's also a bad aspect. After all this work, you either end up with a character that is mediocre at everything, OR, if you work extra hard to twist things to turn out a certain way, you could actually have a really good skill level in one area. None of this matters much, because the 2D10 exploding dice skills system tends to minimize the importance of skills anyway. File this under "Why did I bother?"
Life - Sort of a personal thing, but I got "fired" in the game shortly after losing a real job, and it stung. File this under "Well, that sucked."
Trivial Pursuit - Fun once in a while, but the amusement quickly fades. There's little worse than being stuck in a group of people who are keen to play "just one more round" after you have already played 2 or 3 games. File this under "Enough Already."
Battletech Total War - Battletech is another game from my "Stick With" list, and the single game I have played more than any other. - BUT - The were some rules changes in the new Total War edition of Battletech that never should have been. The Battlemech will always own this game, but in older editions it was possible to play armored vehicles (tanks!) and have fun challenging the new kings of battle. Under the Total War rules, already weak vehicles were totally emasculated by critical hits and motive hits. In a typical game, your vehicle is quickly immobilized, and then it explodes. If you are lucky enough not to explode, then you can enjoy playing a pillbox for as long as you can (but usually not long). The critical and motive hits rules are a good addition to the game - they add interest - but they are unbalanced. This has sort of ruined vehicles as being interesting and fun, where they used to be a good feature of the game. (Don't ask me what I think about WiGE's.) Mea Cupla: You will find my name among the playtesters for this book; I complained then, but now I wish I had complained louder. File this under "I'm going to catch some heat for this."
Diablo - I enjoyed playing Diablo, until I tried playing it online. I thought I knew what "bad behavior" in online gaming was, until I tried playing it online. If the transcendence of WoW is any indication, some of those issues with early online communities have been worked out. I sure hope so. File this under "Does not play well with others."
Zynga games - 'Nuff said (but I'm lookin' at you, Facebook.). File this under "Exploitation of privacy."
The Lottery (most any lottery will do) - Otherwise known as a tax on people that can't do math. File this under "Tax the poor."
Dawn Patrol (TSR) - I like air combat games, or at least I think I do, because I own a small fleet of them. I don't own Dawn Patrol, but it illustrates a serious flaw. I expect an air combat game to be about crafty maneuvers and outwitting your opponent in an aerial duel. In Dawn Patrol the planes are generally mobile enough to get a good shot at whomever may have moved before them, and moving first is a death sentence. There is no strategy, no outwitting your opponent, just luck in who happens to move first or last. Interestingly, this same thing often occurs in Batteltech, with 'Mechs forming a "conga line of death". The outcome in these situations is much more uncertain in Battletech, and the player generally has a chance to either escape or go down fighting. File this under "Barely Interesting."

That's my 10, pass it on and link back (please). Or not.

15 Games That Will Always Stick With Me, Too

2010-10-06T19:57:00.014-05:00

I picked up this meme from D20 Sapphire's post at 20 Sided Woman:

The rules: Don't take too long to think about it. 15 games you've played that will always stick with you. List the first 15 you can recall in no more than fifteen minutes. Tag fifteen friends, including me, because I'm interested in seeing what games my friends choose.

You can probably tell this originated from Facebook, but like my friend Sapphire I'm going to skip the tagging part and link back to her instead. When I started my list I thought I would have trouble coming up with 15 games, but within 5 minute I was already trying to decide which ones to leave out. SO many choices! At risk of over-thinking the question, I have split this into two parts: board games/RPG's and computer/console game (of any sort), and Honorable Mentions are included along the way. Each will be roughly in chronological order, starting with the board games:

Risk (Parker Brothers) - I would go to slumber-parties with my friends and play Risk and Monopoly for 12 hours straight, fall asleep, then wake up and play for another 12 hours. We never got tired of playing (but we did eventually get tired). Honorable mention: Monopoly.
Dungeons & Dragons (TSR) - Starting from the blue box, I played all I could when I was a young teenager.
Ogre/GEV - Originally published by Metagaming, and Steve Jackson's first published game. This was my introduction to the "map, chits, and dice" wargame. Honorable mention: Warpwar, also by Metagaming.
Air War (SPI) - I'm skipping a bunch of other games to list this one. I played many combat/battle board games, but Air War was sort of the apex of these games for me. Air War is quite complex, and as much a simulation as it is a game. Honorable Mention: Armor (Avalon Hill?), a great tank combat game.
Traveller (then GDW) - Science Fiction role-playing. I played quite a bit, but it wasn't until I hooked up with a college gaming group that it really started to shine for me.
Champions (Hero Games) - Another game I found in college, and with the same game master, my friend Pat (who also GM'd Traveller). Honorable mention: The Fantasy Trip (Metagaming) - Another game by Steve Jackson. This was an a very simple RPG based on the Microgames Melee and Wizard, and the basis of the two best fantasy RPG campaigns I ever played (One GM'ed by Pat, again!).
Battetech (then FASA, now Catalyst) - Fastforward about 10 years - graduation, job, grad school, job-hunting, marriage - where I didn't play board games at all. I did play some computer game version of Battletech, Solaris 3025 (online) and Mechwarrior 2, and I bought a rulebook so I could understand the computer versions better. I wan't until I took a job in Milwaukee, started going to GENCON (when it was still here), and fell-in with a group of die-hard Battletech players that I really learned to love the board game. Maybe more importantly, Battletech re-awakened my love of painting miniatures, something I still enjoy just as much as playing.
Squadron Strike (Ad Astra Games) - 3D science fiction space combat, and I get to help the author with some of the mathematical aspect of the game too - something I ought to be writing about!

And now the computer games, which I hate:

Asteroids (arcade) - I plunked a lot of quarters into arcade video games, and this is the game that started it. Gamers much younger than me won't recall a time when video and computer games didn't exist, but I do. I'd like to think that all of today's gamers would still be gamers even if there were not any electronic involved at all, but for many kids today, computer and console games are all there is (which is sad). Honorable mention: Donkey Kong.
Hack/Rogue - This amazing little dungeon crawl game was being passed around among the computer science students when I was in college, and I found it incredibly addicting - and still do. I have a version on my Palm Pilot now, and still play it from time to time. Honorable mention: Empire.
Harpoon - A computer version of Larry Bond's miniatures game, and a great naval warfare simulation. Honorable mention: 688 Attack Sub and Sea Wolf - Two submarine simulations that I spent a lot of time with. These games typically have long intervals of increasing tension as you hunt your target (or vice-versa), punctuated by a few minute of heart-and-keyboard-pounding excitement when the torpedoes start to run.
Master of Magic - Sort of a cross between Civilization and Magic, the Gathering. This game offered many different choices during setup that allowed for variety and interesting play.
MUME (Multi Users in Middle Earth (MUD) - A text-based "dungeon" adventure devoted to re-creating Middle Earth. This was my first experience in online gaming, and a terrible this to discover during graduate school, because it threatened to suck-up my every waking moment. I was never able to get a fast connection though, and this generally resulted in BAD things happening to my characters when this started getting tough, so I finally wised-up and focused on studying instead. I still log in every few years to see if there are still any of the old players still playing there (and there are). Honorable mention: Doom + Quake - Two games I refused of purchase because they definitely would have made me flunk out of grad school.
Starcraft - I've got to have an RTS in here somewhere, don't I?
Gran Turismo 2 (Playstation) - The only console game I'm listing here, because it's the only console game I've spent much time with. Even then, I had to get up early on Saturday mornings to run the longer races (100 laps) before my son want the PS2 for himself. This isn't so much a game as it is a driving simulator, and it's as close as I'll ever get to driving a real performance car too. I have the newer editions of the game too, but they don't have the same memories. Honorable mention: Netrek - Another online multi-player game, one if the first of its kind, and a whole lot of fun. This got demoted to "mention" status because I haven't actually played it all that much.

That's my 15 games I'll always remember, but I already know what number 16 will be:

Honorable mention: Gratuitous Space Battles - My first computer game purchase in a long time, and I am really loving it. Look for a proper review on these pages sometime soon.

The hazards of dog ownership

2010-10-05T07:00:00.000-05:00

If you happened to be paying very close attention to my last yesterday, you might have spotted two things which didn't belong in the photos below. (Pencils and shoes don't count)

The first is easy to spot above, but the second is harder. This other item is more obvious below:

Got it figured out yet? Scroll down when your are ready for the reveal ...

...

...

...

Yes, I own a dog, and some days there might be dog kibble scattered about the carpet. Such are the hazards of dog ownership. :-)

Battletech 4th Edition

2010-10-04T19:16:00.001-05:00

I made a good find yesterday at Half-Price Books: The Battletech 4th Edition boxed set.

The box itself was a bit beaten up, and I already own a beaten-up 4th edition box minus the original contents.

One quick peek inside the box was enough to show me at the original rulebook was still there. That, and a very distinctive lump metal, shown just to the right. I still wasn't sure the $25 price was worth it, even for an out-of-print Atlas miniature, but what the heck.

Later, out in the parking lot, when I broke the box open again to show her the Atlas, and found the Zeus mini, THAT'S when I was sure I'd made a good buy.

The rest is icing on the cake. The box may have had some wear and tear, but all the components appear to be in perfect condition.

There is also a set of stickers (above, on the left) with the Inner Sphere House insignia that I didn't expect. (Also a nice Power Rangers pencil.)

This is my second stroke of luck with used FASA games. Needless to say, I am quite pleased.

Banner Day

2010-09-19T20:57:00.073-05:00

I spent a few hours today pushing pixels around for my new banner - enough time that I figured it was worth a post in itself. Here it is ...

My new banner in all it's glory.

It's not quite right yet - as you can see up top, some of the best bits are covered up by my title text. I'll see about fixing that later.

Yellow Jacket working image

All the parts of this image are on separate layers in Paint.Net so that sort of change won't be too hard to do.

Here are the working images, and the original images (pretty sure I found the right ones).

A not-quite-final ...

Yellow Jacket original image

... and the original.

Guillotine original image

All the mini images were put through the "Ink Drawing" filter in PDN to give them a more stylized look.

A final layer below.

Two minis on one layer

Spider original image
(photo by Sam "Wasp" Snell)

Spider working image

The original "Jumping Spider" photo was given a perspective rotation using GIMP. The original was too small without resizing, and this me stretch the top and shrink the bottom, getting the whole effect in the picture. The jump-jet splash looks a bit flattened those, maybe I can tweak that too.

Vindicator final image.

The Vindy is the only one where I couldn't find the original image. I obviously took the photo, and this mini does feature here, but I guess I never posted any other photos of it. I might have over-done the Ink Drawing tool here, and I need to clean up the edges, which look messy in the final banner.

Hex grid original image

I tried rotating some images of a hex grid, but I found a Hex Grid renderer for GIMP that worked better. I made a REALLY big grid, gave it a perspective rotation, then cropped out the over-stretched parts. GIMP is a good tool, and I'm glad this gave me a chance to start using it.

Random Walks

The random walks image is derived from the same spreadsheet I used for my Monopoly post way back when. This image is a just a screencap from Excel. The random walks were in my original banner too, but with the addition of the hex grid I cut down to just four lines, and made them heavier. Here is the old banner for comparison (below).

Dice Original image (photo by Maximaximax)

These dice are the only part of the banner I didn't create myself - Thank You WikiMedia! I might take a stab at photographing some of my own dice if I get ambitious, but these will do nicely for now.

Finally, thanks to Steve and PiP for prodding me in the comments to get this done. It good when friend bug you to do stuff. :-)

How awful is it?

2010-09-02T21:52:00.000-05:00

I'm trying out the new Blogger template design tool. Unfortunately that means scrapping the entire scheme from the old blog and starting over. It's a bit of a mess yet, but I think it might be a bit easier to read now, at least to my middle-aged eyes. There is work yet to be done; I need to redo the banner, and the sidebar gadgets are a complete mess. If it doesn't work out, I can still got back to the old template. All comments appreciated.

--- Dan

The Grinder (02/09/2010)

2010-09-02T20:02:00.001-05:00

--- A collection of links in need of Grinding, September 2nd, 2010 ---

Tree Lobsters are poking fun at vampires and {spoiler deleted}.

From IO9: 10 Awesome Science Fiction Stories for Gamers

Also IO9: 25 Classic Science Fiction Movies ...
I don't agree with all of these choices. Some of these movies were very good entertainment, but not particularly good SciFi.

RISD Fleet Library: A gallery of Dazzle camouflage patterns.
Though I'm sure I've linked this before, it's worth seeing again. "Dazzle" is one of my favorite paint scheme for my Battletech miniatures. It's hard to camouflage a 100 ton vehicle, but you can make it harder to focus on. On the tabletop this scheme stands out attractively, yet still makes the other player look twice to be sure what they are looking at. (The mini on the right is still on the workbench, and will look much more like the other when it is done.)

Spiked Math! has a tribute to Martin Gardner. Check out the links to various fun mathematical topics below the cartoon.

That's all for this week - keep those dice rolling!

Avalon Hill founder Charles Roberts passes away

2010-08-29T21:04:00.007-05:00

Charles S. Roberts, Train line expert and founder of Avalon Hill games, died on August 20th. He is more well known for his writing about railroads than for his games, but the games are what I know.

In 1958, he founded Avalon Hill Co., a Baltimore game publishing company that specialized in war and other mental combat games such as "D-Day," "Stalingrad," "Battle of the Bulge," "Gettysburg" and "Victory in the Pacific."

See the full article at the Baltimore Sun:

Charles S. Roberts, train line expert, dies at 80

[Big Hat-Tip to Brant at Grog News]

Update: Christian has a much better article on Charles Roberts at Cinerati.

The Grinder (20/08/2010)

2010-08-20T22:12:00.002-05:00

The Grinder has been a regular event at the some of the gaming conventions for quite a while now. At our local our Battletech group, we sometimes play this as a "pickup" game, using whatever sheets and resources are available, sort of a "Battletech Stew" we can cook up on short notice from leftovers of other games - and an easy alternative when our planned scenarios doesn't work out.

Photo Courtesy of Scrapyard Armory

My plan for another part in my Lanchester/Attrition Modelling series (1,2) is taking me a long time to prepare, but I've got a bunch of cool stuff I was saving for a "miscellaneous" post, which is a good alternative plan. I've been thinking about making such a post an occasional anyway. All I really need is to think up a good name it.

Think up, or maybe ~~steal~~ borrow. See where I'm headed with this yet?

So with no further ado I bring you the first edition of an occasional feature:

THE GRINDER

(insert clever subtitle here)

From Clifski's blog - The developer of Gratuitous Space Battles writes about the issue of game length, and concludes that Size Doesn't Matter.

Completely by chance I came across the source of the original artwork for the game OGRE. This was one of the first wargames I ever bought, and still a favorite. The artist turns out to be Winchell Chung, who occasionally posts here, and has posted links to this blog at various other forums, bringing me a lot of web traffic. Thanks Winchell!
You can find info about the game OGRE at Steve Jackson Games.

Another post about why gigantic robotic war machines simply are not practical, or maybe even possible. Heathen!

It's actually a pretty good read, and he hits a few new reasons that hadn't occurred to me already. See Too many words about why mechas are stupid at Chizumatic.

The Ultimate D&D Gaming Room
I think I've posted this one before, so no piccy.

From The Pasqualian:

On Lanchester’s Differential Equations and their Transform into a Markov Transition Matrix (or, the Markovization of the Lanchester Equations)

If that looks sort of mathy, it is. ("Mathy" is a technical term, you know ;->)
I found a mathematician writing about expressing the Lanchester differential equations in the form of a Markov chain. Carlos Pasquali, whom I might describe as one smart cookie, was kind enough to answer some questions for me. We had a good and most helpful discussion (see comments), which I'll be writing more about this soon, I hope.

That's it for the first edition of The Grinder. I'll cook this up again soon.
This post is also my first use of the new "jump break" tool in the Blogger editor. Like?

Fair Dice, Unfair Dice, and Dice Rolling Machines

2010-08-05T23:14:00.000-05:00

Today's rambling blog post started off with my searching for a material in reply to a post at My Little Soldier. Looking for one thing I stumbled across much more, and ... well ... here it comes ...

Fair Dice and more from the Mathematical Association of America. Be sure to read about Lou Zocchi's father's Believe-it-or-Not accomplishment.

Unfair Dice, also from the MAA. Notable because someone calculated the theoretical unfairness of dice that are unbalanced due to hollowed out pips. This would include most of the dice in my bag, since I prefer pips to numbers.

The Physics of Dice, this page has links to pictures of a dice-rolling machine custom built to test dice, and a bit more that might be interesting. This is what I was looking for in the first place!

And Finally ...

A bit more about this machine at Gizmodo
and Boing Boing is better still.

God Bless Nerds

2010-07-24T18:39:00.000-05:00

If you follow any web-comics or comics-blogs, you might have heard something about ComicCon happening this week in San Diego. You might also have learned that the despicable Fred Phelps and his Westboro Baptist ~~Church~~ Hate Group* were going to be there picketing the event. Con-goer heard this too, and they have responded in fine fashion. Blog posts with pictures, commentary, and video are available from:

Comics Alliance [photo to right]

TV or not TV [photo below]

Phyrangula [That wretched hive of scum and villainy where I acquired the other links]

WAY TO GO NERDS!

* You DO NOT want to know any more than what is on the Wiki page. Trust me on this.

Transformers 3

2010-07-12T05:24:00.001-05:00

Some local news: filming is set to begin in Milwaukee this week on a scene from Transformers 3.

The locations will be the Milwaukee Art Museum (1,2,3), ...

... and the former Tower Automotive site.

The MAM does a bit of transforming itself, and seems particularly appropriate.

image source?

See the Transformers 3 trailers here .

Lanchester's Laws and Attrition Modeling, Part II

2010-07-09T18:27:00.003-05:00

In Part One I introduced the basic concept behind Lanchester's Laws, and now I want to write more about how this applies to games. This time I'll start off with Ernest Adams' excellent 2004 Gamasutra article, and follow up with my take on actually applying this to a game.

Kicking Butt By the Numbers: Lanchester's Laws

[Alternate Link to Ernest Adams article, in case the Gamasutra link flakes out again.]

Adams makes a particular good point about Lanchester's Laws and "victory".

Now, Lanchester's Laws are far from perfect. For one thing, they only apply to battles of attrition, in which the object is to wipe out the other side. If "winning" a battle is defined in some other way -- and modern Western militaries don't normally consider slaughtering every last opponent to be a legitimate objective -- then Lanchester's Law has nothing to say about who wins. Napoleon "won" every battle he fought in his march into Russia, but he still lost 98 percent of his men and was forced to retreat without achieving his objective.

On the other hand, some wargames are often fought to the last man standing, so if you are trying to understand game balance it is well worth considering. Some early comments tell me that rest of this post may be a confusing slog through a lot of math - I'm afraid that is the nature of the beast. All I can do is ask the reader to bear with me for now, and stay tuned for Part III where there will be some more accessible examples. I will start of again by stating my assumptions, fixing some errors I made last time.

This is a very simplified model of combat; Each side has identical soldiers; Each side has a fixed strength which governs the proportion of enemy soldiers killed. Range, terrain, movement, and all other factors that might influence the fight are either abstracted to the proportion killed or ignored entirely. In the derivation from differential equations casualties are inflicted continuously over time, but it also works to think of casualties inflicted in rounds or turns, which will be a more familiar setting to most gamers.

My first error last time was stating this in terms of the probability of killing an enemy soldier instead of a proportion or kill-rate. The differential equation derivation of Lanchester's Laws is a deterministic relationship that does not involve probabilities at all. I'm a statistician and tend to think in terms of probabilities rather than fixed proportions, and I'm also working my way up to demonstrating a probabilistic derivation of the same thing, so forgive me if I got a bit ahead of myself. A definition involving probabilities should be even more useful because it will demonstrate the degree of uncertainty involved in a balanced game.

My second error was in describing the total casualties for the winning side in my example. Side 2 loses all 2000 soldiers, and side 1 suffers casualties equal to 4/9ths (44%) of 2000 soldiers, for a total of 889. I had incorrectly stated that side 1 would lose 4/9ths of its total force (which would be 1400 soldiers), and the previous post has been corrected.

Notes on some abbreviations I may use (some changes since part one):

Pk = "Proportion killed" or combat effectiveness, with subscript k1 or k2 if it matters. This is usually a small proportion.
R = Pk1/Pk2 , the ratio of combat effectiveness. (1/R = Pk2/Pk1)
N1, N2 = The initial force sizes (before combat) of each side.
n1, n2 = The remaining force sizes at the end of combat.
C1, C2 = The total casualties suffered by each side at the end of combat (C1 = N1 - n1).
a = Exponent on the force size. For the Linear Law a=1, and for the Squared Law a=2.
"~=" is used to represent "approximately equal to".
Be aware that the !@#@!#%&*#!!! Blogger editor is subverting every effort to use smaller fonts as subscripts in the text. Hopefully the subscripts will be clear from context.

Here is a an expanded version of the formula from part one. That formula only worked in a special case; this is more general:

This is a more general form that accounts for both the Linear and Square Laws, adds the relative combat effectiveness Pk's, and allows for situation in between with 1<a<2. This assumes the condition of combat and acquiring targets are the same for both sides, and so the value of a is the same for both. (Yes it can be different, but that really complicates things). I should also clarify that in a battle to the last man standing, where either C1 = N1 or C2 = N2, or equivalently n1 = 0 or n2 = 0 (one side is completely destroyed).

If a game is perfectly balanced then both sides should be destroyed at the same time, which results in both n1 = 0 and n2 = 0. Plugging these in and a bit of algebra gives:

Some important points:

THE important point of Lanchester's Laws, is that the exponent representing the advantage of a higher rate of target acquisition applies to the size of the force (N), but not to the fighting effectiveness of the force (Pk). Under the Squared Law, any advantage that army 1 might have in combat effectiveness is usually quickly overcome if army 2 has the larger force.
Lanchester's Laws are not a governing rule of combat, they are a description of an ideal principle. In actual combat/play this ideal is unlikely to hold. If one player has a range advantage (tending towards a=2), other other will make use of terrain to counter this (tending towards a=1). Battles tend to occur in sets of smaller skirmishes, each with it's own circumstances, and the final result may represent some average of these skirmishes.
There is a third variation on Lanchester's Laws, known as the logarithmic law, which has relevance to interpreting historical data, but that will have to be a topic for another day.

So how is this used to balance a game? Let's suppose we have a hypothetical game where combat can potentially operate under the Squared law, but due to range and movement limitations effectively operates somewhere between Linear and Squared. I'll use a=1.5 to represent the actual state of this game, and work out examples under three cases:

Example 1 (Balance relative effectiveness):

For the Linear Law a=1.0, N1 = 3000, N2 = 2000, Pk1 = 0.01, and solving for Pk2 gives Pk2 = 0.015 and R = 0.67 (or 1/R = 1.5).
For the Squared Law a=2.0, N1 = 3000, N2 = 2000, Pk1 = 0.01, and solving for Pk2 gives Pk2 = 0.0225 and R = 0.444 (or 1/R = 4/9 = 2.25).
For my in-between value of a=1.5, and again N1 = 3000, N2 = 2000, Pk1 = 0.01, and solving for Pk2 gives Pk2 = 0.018371 and R ~= 0.554 (or 1/R ~= 1.84).

If I want to balance a scenario with these forces (or any 3:2 force ratio) side 2's forces will need to be between about 1.84 times as effective as side 1's in order for this to be a fair fight. What actually happens in the game will still depend on chance and the ability of players to take advantage of opportunities during play, but that's the hypothetical balance point.

Example 2 (Balance force sizes):

For the Linear Law a=0.01, N1 = 3000, Pk1 = 0.01, Pk2 = 0.015 and solving for N2 gives N2 = 2000.
For the Square Law a=2.0, N1 = 3000, Pk1 = 0.01, Pk2 = 0.015 and solving for N2 gives N2 ~= 2450.
For the in-between a=1.5, N1 = 3000, Pk1 = 0.01, Pk2 = 0.015 and solving N2 for gives N2 ~= 2289.

If I want a balanced scenario where the sides have this relative effectiveness (2:3 ratio, R = 0.67) side 2 will need an initial force size of about ~~2450~~ 2289 to balance with side 1 in a fair fight. The outcome still depends on what happens in the game, but that's the hypothetical balance point.

Note that I arbitrarily chose a=1.5 for this example, pretending that I have a game where this is true. In practice I don't think it is possible to design a game to a particular value of a. What you can do is assume a value for a and design a game around it, assigning values and balancing scenarios with different types of units in this manner. This is difficult though, because in a game with many different types of units there is potentially a different balance point for every pair of units that might face each other, and there isn't any simple formula that can solve it for for us. This process gets VERY complicated.

At this point I have probably created more questions than I have answered. That is a problem, but I need to stop somewhere. The next part will hopefully include some more accessible examples, and some toy spreadsheets you can download to play with.
In my research I found two recent papers by Perry that were very helpful (especially 6 below) and greatly improved my own understanding of the topic. The MacKay paper (4) is maybe a simpler article to read first. You don't necessarily need to understand differential equations to read these, but it helps.

Part III of this series is here.

Bonus Link!

Lanchester Systems and the Lanchester Laws of Combat
by "The Custodian" at everything2.com

This is another basic introduction to Lanchester's Laws. The author (semi-anonymous) spends more time on the differential equations and some practical aspects that I glossed over completely, so it might be worth your time.

References and Reading (updated since part one)

Ernest Adams, "Kicking Butt By the Numbers: Lanchester's Laws", a Designer's Notebook, Gamasutra webzine, August 4, 2004.
Bruce Fowler, De Physica Belli: An Introduction to Lanchestrial Attrition Mechanics Part One, Defense Modeling Simulation and Tactical Technology Information Analysis Center, Huntsville, AL, 1995. [Early versions of this series can be found online at DTIC: 1,2,3.]
Michael J. Artelli and Richard F. Deckro, Modeling the Lanchester Laws with System Dynamics, The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 2008 5: 1-20
Niall MacKay, Lanchester combat models, arXiv:math/0606300v1 [math.HO] (2006)
Wikipedia contributors. "Lanchester's laws." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 May. 2010. Web. 23 Jun. 2010.
Perry, Nigel. Defence Science and Technology Organisation (Australia). Joint Operations Division 2009 Fractal effects in Lanchester Models of Combat [electronic resource] / Nigel Perry Defence Science and Technology Organisation, Canberra.
Perry, Nigel. Defence Science and Technology Organisation (Australia). Defence Science and Technology Organisation (Australia). Defence Systems Analysis Division. 2006 Verification and validation of the fractal attrition equation [electronic resource] / Nigel Perry DSTO, Edinburgh, S. Aust.

Range Testing for Little Wars

2010-07-02T21:49:00.000-05:00

Over at Toy Soldiers Forever, Mannie is preparing to play one of the original miniatures rules wargames.

Last winter I acquired a reprint of H.G. Wells' Little Wars, that most original and elegant of wargaming tretises. First published in 1913, this very slim volume set an early standard for wargaming in which the obvious agenda is fun and the not-so-hidden agenda is the futility of war and the advantages of peace.

The clever prose and the droll attention to simplicity and pragmatism make this a delight to read.

Mannie clearly takes delight in testing his new toy cannon too. He's even done a very scientific study of projectile performance. I like it!

I'm looking forward to the report on the full scale battle too, coming later this month.
[Image From Mannie Gentile: Toy Soldiers Forever]

Lanchester's Laws and Attrition Modeling, Part I

2010-06-27T13:15:00.012-05:00

Consider of problem of two armies facing each other of a field of battle. One army is larger, but the other army is better armed. Once battle is joined, the two sides wear each other down until one is completely destroyed, or more likely, until one has suffered so many casualties it can no longer hold the field, and so retreats in defeat. There is a mathematical way to describe this sort of battle, a battle of attrition, and it is a topic I've been wanting to introduce here since I first started. I hesitated though, because it's a difficult topic with few really satisfying answers. Rather that reinvent the wheel, I'm going to link to a source articles and add comments of my own. This should easily make a series, maybe a long series if I don't get tired of it first.

So let's get things started - this first link is good introduction to the topic. After you get back, I'll repeat the definition in my own words, and explain where the Wiki article, and most articles on this topic, get it wrong.

Wikipedia - Lanchester's Laws
[edit April 2010: Of course Wikipedia changes sometimes! The article linked above is still good, but I have quoted the relevant proportion of the classical interpretations here.]

Lanchester's Linear LawIn ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons.[...]Lanchester's Square LawWith firearms engaging each other directly with aimed fire from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons firing. Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. (reference 6 below).

My turn. This is a very simplified model of combat; Each side has identical soldiers; Each soldier has an identical probability of killing a soldier on the other side, if they can (the probability does not have to be the same for both sides). Range, terrain, movement, and all other factors that might influence the fight are either abstracted to the probability of a kill or ignored entirely. In the derivation from differential equations casualties are inflicted continuously over time, but it also works to think of casualties inflicted in rounds or turns, which will be a more familiar setting to most gamers.

Note on some abbreviations I may use:
Pk = "~~Probability~~ Rate of a kill", with subscript k1 or k2 if it matters. This should be a small value relative to the size of the force and time interval. [Correction: There is no probability involved here. This should be the proportion or rate of casualties that results from combat over a short length of time.]
N1, N2 = The initial force sizes (before combat) of each side.
C1, C2 = The total casualties suffered by each side at the end of combat.

The Linear Law applies when one soldier can only fight one other soldier at a time. If one side has more soldiers, some of them won't be fighting all the time as the wait for an opportunity to attack. In this setting, the casualties suffered by both sides are proportional to the number actually fighting (and the relative probability of a kill). If the Pk is the same for both sides, then both sides will suffer casualties equaly to the size of the smaller force. This was originally called Lancherster's Law of Ancient Warfare, because it tries to model what happens if neither side has ranged weapons, and so are fighting with swords or spears (but it works equally well with ba'tleth or light-sabers).

The Squared Law, sometimes known as Lanchester's Law of Modern Warfare, is intended to apply to ranged combat, and it quantifies the value of the relative advantage of having a larger army. With the Linear Law, this advantage is proportional to the size of the forces, but when the entire force of both sides can engage the other simultaneously, the relative advantage is a function of the square of the force size. Again assuming equal Pk, the casualties of the larger army will be proportional to the ratio of the squared forces sizes.

So for example, if N1 = 3000 and N2 = 2000, then this ratio C2/C1 is equal to (dropping zeros, 3^2 / 2^2 = ) 9/4 or 2.25. By the end of the battle side-2 will have suffered 2.25 casualties to every 1.0 on side-1. Conversely, side-1 will lose 4/9 or 44.4% of side-2's loses, for total casualties of (4/9)*2000 = 889 soldiers.

Now for the bug. There is nothing wrong with the mathematical derivation, but there is considerable confusion about the interpretation. That the Squared Law describes the advantage of superior numbers in ranged combat is the practical interpretation, but range is not even considered in the derivation. The Squared Law really has nothing to do with range - what really matters is the rate of acquiring new targets. Having ranged weapons generally let's your soldiers acquire new targets as fast as they can shoot, whereas with a spear or sword (Linear Law) you have to locate a target and then move to engage them. In real life this may be a trivial distinction, because the "advantage of range" interpretation makes sense in most situations. However, games offer some alternatives where the Squared Law applies, but it clearly has nothing to do with range. Some examples of the Linear and Squared Laws in action, both on the gaming table and in real life:

For a platoon of Battle-axe wielding Dwarven warriors the Linear Law would generally apply, but make that a platoon of motorcycle mounted Battle-axe wielding Dwarven warriors that can move to engage any target on the gaming table, and suddenly the rate of target acquisition is as high as that of unit with ranged capability, and so the Squared Law applies.
(For the Battletech players) Consider a Battlemech like the Dasher H that carries powerful but short range weapons. This would usually imply the Linear Law. The limitation of short range weapons is irrelevant here because it moves so fast it can effectively engage most targets immediately.
In a game where some units may be effective invisible, either through stealth technology, "cloaking", or magical invisibility, then the advantage of range for acquiring targets may be effectively nullified, and the Linear Law will prevail in the battle.
The US military is increasingly making use of battlefield information systems to give field commanders knowledge of where the enemy is - to allow them to acquire targets first, and in the most advantageous way possible. This gives the advantage of the Squared Law to the US military, where the opposition with limited information is effectively fighting under the Linear Law.
Guerrilla warfare, in setting such as Iraq and Afghanistan, it is much easier for insurgent to find US targets than is it for the US to find insurgent targets, and the Square Law applies.

And here is the take-home lesson: Lanchester's Laws are NOT about range. Range doesn't matter, "ancient" or "modern" doesn't matter - It's all about the rate of target acquisition. It's OK to think about range being the key concept in most settings, because that is the mechanism which allows new targets to be attacked immediately. However, if you want to apply Lanchester's Laws when designing a game, or in understanding how game balance works, this distinction may be important.

There is more, much more, which is why I'm spliting this up into a series of posts. I have some references below, some of which didn't even get mentioned here, so if you don't want to wait for me you could peek ahead at some upcoming topics. Finally, here are some closing notes that didn't make it into the text above:

Though Lanchester generally gets the credit, a Russian mathematician named Osipov also wrote on the same topic at about the same time.
In application of Lanchester's Laws to historical data, it is generally found that some mix of the Linear and Squared Laws is the rule, not one or the other exclusively. I don't have any references on this below yet, but a Google search on Lanchester and Helmbold ought to turn up something relevant.
This has implications on point systems for balancing game scenarios, such as Battle Value in Battletech. Such point systems tend to have serious flaws, and Lancherster's Laws illustrate why: no single point system can be correct in every setting.
I implied, but did not state, that it is possible for one side/army to be operating under the Linear Law and the other under the Squared Law. This may be a topic in coming posts.
The assumptions for Lanchester's Laws are rarely true in a game setting, much less in reality. However, they do demonstrate the superiority of numbers principle in combat, which is a very important lesson, even if somewhat obvious.

References and Reading
(Please pardon the hodgepodge of styles. Organizing my math & gaming references is an ongoing project.)

Ernest Adams, "Kicking Butt By the Numbers: Lanchester's Laws", a Designer's Notebook, Gamasutra webzine, August 4, 2004.
Bruce Fowler, De Physica Belli: An Introduction to Lanchestrial Attrition Mechanics Part One, DEFENSE MODELING SIMULATION AND TACTICAL TECHNOLOGY INFORMATION ANALYSIS CENTE R HUNTSVILLE AL, 1995. [Early versions of this series can be found online at DTIC: 1,2,3.]
Michael J. Artelli and Richard F. Deckro, Modeling the Lanchester Laws with System Dynamics, The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 2008 5: 1-20
Niall MacKay, Lanchester combat models, arXiv:math/0606300v1 [math.HO] (2006)
Wikipedia contributors. "Lanchester's laws." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 May. 2010. Web. 23 Jun. 2010.
Wikipedia contributors. Lanchester's laws. (2011, March 30). In Wikipedia, The Free Encyclopedia. Retrieved 00:19, April 3, 2011.

Update: Part II is up.

Epilogue: I'm getting a lot of hits via Facebook, maybe thru Networked Blogs, or possibly posted on a group somewhere. If you came to this page via Facebook, please leave me a comment about how you got here. --- Thanks --- Dan

"Mister Eastwood, how about THIS Trebuchet?"

2010-06-22T17:44:00.001-05:00

Steven Satak sent me a "taste test" of some of the new art for his TRO:3063 project. Steve figured out early on that I have a sweet-spot for this particular Mech.

The work of kSeth.

And, if I'm not going insane from lack of sleep, Steve has recently given his blog layout a thorough redesign: Nice work! I'll have to ask him where he got that beautiful banner art.
Comments on the recent post indicate I'm not going insane, but I was worried for a second.

It's All In The Cards

2010-06-17T18:38:00.001-05:00

A two articles related to Collectible Card Games, with some math and game design aspects.

Gotta Catch 'em All? Just how many of those collectible card pack do you have to buy in order to get a complete collection, or maybe just that one card you are looking for. Not About Apples has a good discussion of the problem:

Getting a Complete Collection.

[Image kotaku.com]

Once you figure out how many cards you might have to buy, you could start to wonder if getting into such a game is a good idea in the first place. Ethan Ham article at GameStudies.org discusses problems and answers to creating balance in a CCG.

Rarity and Power: Balance in Collectible Object Games

For collectible card games (CCGs), game designers often limit the availability of cards that have a particularly powerful gameplay effect. The conventional wisdom is that the more powerful a card is, the more rare it should be. The long-term implications of such an approach can have negative consequences on a game’s suitability for casual play. Digital Addiction (a company that produced online, collectible card games in the 1990s) developed a different game design philosophy for balancing collectible card games. The approach called for the most obviously and generally useful cards to be the most common and to equate rarity to specialization rather than raw power.

If you need more math about math and cards, then John Cook has it for you at The Endeavour:

Chances a card doesn’t move in a shuffle

Finally, this doesn't have anything directly to do with cards, but it's still a good read on understanding probability:

Game Developer Column 9: Playing the Odds

The Pods Are Back!

2010-06-02T07:30:00.000-05:00

The PODS are coming back to Origins and GenCon.

Nickolas 'PropWash' Smith writes:

MechJock.Com (not to be confused with our friends at MechCorps.com) will once again be bringing 12 of its Virtual World Entertainment, Tesla II BattleTech: Firestorm cockpits to both Origins and GenCon.

This will be our 4th (or 5th?) appearance at GenCon, and our 2nd at Origins.

Here is the latest release for Origins at BattleTech.Com

http://battletech.catalystgamelabs.com/2010/05/27/mech-forces-gathering-origins/

Hope to see you there.

Want to rule the pods? Brush up or get a head start on your skills; Check this post from last year for a link to the Online Training Manual.

Auctions and More

2010-05-26T18:02:00.002-05:00

A collection of links I've read recently, most having something to do with game design.

From Play This Thing!

Auctions as a Game Balancing Tool:
Auctions are a widely used in the board games as an in-game mechanic or as a meta game. Read this if you want to cross train or at least avoid being a vidiot

Battletech players should be well aware of auctions in the Battletech universe - the Clans call it the Batchall. As the article describes, actions and board gaming have a long history together.

From Pulsipher Boardgame Design

Game design is no place for “perfectionism”

No game can be perfect–it depends so much on the audience, the individual player, the mood, the group (if played by a group), even the timing of creation and publication.

From Jon Radoff's Internet Wonderland, where you can view this at full size.

History of Social Games

[Tip-O-Hat 2 The Ludologist]

From Once Upon A Geek:

From TED: Will Wright makes toys that make worlds
SPORE has dropped off my horizon, but his comments about games and toys are interesting.

UPDATE: Because this is too cool not to share:

From Ethical Technology

Martin Gardner, 1914 – 2010

2010-05-24T12:36:00.003-05:00

Martin Gardner passed away Saturday. If you like math and games then you probably know who he is already. If you don't, it will be well worth your time to find some of his books. Steven Novella has some nice words about him.
[photo by Konrad Jacobs, Erlangen, via Wikimedia Commons]

More: I found this remembrance at Improbable.Com
Still More: At Greg Laden's Blog (am I behind the ball or what?)

Giant Battling Robots

My Wikipedia Problem

The Grinder

Taking a RISK - the distribution of armies lost

The Grinder - July 4th Edition

Dice Distributions Revisited

Maximums and Minimums of Dice Rolls

The Grinder

Doug Chaffe

The Grinder

Fletcher Pratt's Naval Wargame

Sicherman's Dice

Fun with Physics Special Topics

A few more minutes in the library

BlogRolling

The Grinder - Valentine's Day Edition

Pink Panthers, and a few Dropships

Jane McGonigal on The Colbert Report

The Grinder

After Action Report

The Grinder

More Dice, More Information, but not as much as you think

New Years Blogolutions

Battletech: Sword and Dragon Sessions

The Grinder - Christmas Edition

Support Net Neutrality

Point Values for Squadron Strike

Lanchester's Game

Granular Skill Checks, and Interpolation

Number Needed to Hit

Dice and Information ... So What?

Dice and Information

Blue versus Gray, or maybe Federation versus Rebels?

Gratuitous Space Battles

Wet Palette

Sword and Dragon Mission Matrix

The Grinder

The Grinder (25/10/2010)

Math versus Tanks

10 Games To Be Forgotten

15 Games That Will Always Stick With Me, Too

The hazards of dog ownership

Battletech 4th Edition

Banner Day

How awful is it?

The Grinder (02/09/2010)

Avalon Hill founder Charles Roberts passes away

The Grinder (20/08/2010)

Fair Dice, Unfair Dice, and Dice Rolling Machines

God Bless Nerds

Transformers 3

Lanchester's Laws and Attrition Modeling, Part II

Range Testing for Little Wars

Lanchester's Laws and Attrition Modeling, Part I

"Mister Eastwood, how about THIS Trebuchet?"

It's All In The Cards

The Pods Are Back!

Auctions and More

Martin Gardner, 1914 – 2010