Survival Distributions

[OOPS. It seems I meant to publish this post almost a month ago, and now I find it is still stilling here in draft status. That is easy enough to fix {click}. It's kind of a boring post, but I thought it necessary to offer some basics of what I'm working towards.]

It is not my intent to turn this blog into a statistics textbook, but there are some topics I feel like I ought to introduce in a simple form before I launch into more complex discussions. One of these is the concept of a survival distribution.

A probability distribution is a type of function that describes the behavior certain types of random events. Most gamers should intuitively understand the probabilities that result from a single roll of a die, which is formally known as the Discrete Uniform distribution, and those with a basic knowledge of statistics will be familiar with the Normal distribution.

Survival distributions are a subset of probability distributions that can describe the probability that some length of time T will pass before an event X occurs. In the discrete case, this becomes the number of attempt or trials T until event X is observed. An example of the latter is the Geometric distribution. Here there is some probability p that the event will occur on each trial. A simple example would be: “How many times will you roll a six-sided die before getting “6” as the result. The probability of getting a “6” is p=1/6 = 0.1667, and the average or expected value is 1/p or six trials (rolls of the die) before getting a six.

The other reason for this post (the truth comes out) is that I had a nice plot left over from my previous example using the game Hammer’s Slammers, but really needed to describe the Geometric distribution first. Here is a chart illustrating the distribution of the number of attacks on a unit before it is destroyed. For example, a unit subject to 6 1:1 attacks has about 35% chance of survival (or 65% chance of destruction). As you can see, the lower attack-defense ratios (1-3 to 1-2) might require a large number of attempts before destroying the unit, but the higher ratios have very good change of destroying the target in just a few (or one) attempts.

So that, in a nutshell, is my introduction to survival distributions. In the ongoing discussion of the value of units within a game, this is going to be a key concept; the longer a unit survives within a game, the more opportunity there will be for it to be useful - to destroy the other players units - and the longer it can survive the greater its value will be. Next up (well, soon) will be an introduction to a type of survival distribution that can apply to Point Objective Games.

UPDATE: See the next post in this series.

Anatomy of a Black Hawk H - Part III

I've added a few more images. Not a lot of progress to show, but some better pictures. You can find the entire gallery here. These should be images 41-46 in the gallery.

From GBR: Anatomy of a Blackhawk H

From GBR: Anatomy of a Blackhawk H

I've been working in a very thin white-green tone to even out the shading and smooth out spots I missed before. Likewise with the blue stripes, I'm lightening them with a very thin white-blue mix. I don't know if this is efficient, but it is my usual approach - multiple layers to blend the colors together.

Posting policy

Posting policy for Giant Battling Robots and Dread Tomato Addiction -  It's nice to have visitors, but SPAM, not so much. I want to allow as much freedom for comments as I can, but there have to be some sort of limits too. Therefore, some rules:

1. Any posts or links which require any significant effort or my part to determine if they may or may not be legitimate comments, or otherwise wastes my time, are subject to immediate deletion.
2. If you want to promote your own blog*, do so by posting a relevant comments, or risk wasting my time (see rule #1). Promote yourself by participating.
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* Good for you! I'd like to help. Now go back and finish reading the rule.

Please note that comments on posts more than 14 days old are subject to moderation, and will not appear immediately.
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General(s) George from Le Petit Soldier Shop

On a recent visit to New Orleans, the wife and I spend a pleasant morning exploring the French quarter near Jackson Square. After coffee and beignets at Cafe Du Monde, and a stroll through the touristy shops on Decatur Street, we started exploring some of the nearby streets. I'm glad we did.

At 528 Royal Street, we made a nice find: The Sword And Pen / Le Petit Soldier Shop. They specialize in historical miniatures and toy soldiers, both modern and antique. Historical miniatures are not my main interest, but I really didn't want to leave this shop empty handed.
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From GBR: Public

Fortunately these two small figures caught my interest among a box of similar figures: George Washington and George Custer.

From GBR: Public

As you can see, someone started to paint the Custer figure, but didn't get very far.
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From GBR: Public

The shop owner told me these were probably produced by the Marx company sometime during the 1960's. There are some similar figures in the Marx Toy Museum (another place to visit one of these days!).
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I wish I'd had my camera to take some pictures of the displays, but now at least I have an excuse to go back.

Netrek: The Greatest Star Trek Game

Last year on a dark November evening I stumbled across an old computer game, one that I had never heard of before. Little did I know that I was going to be spending a fair portion of my free time dueling with other players for control of the galaxy! NETREK is a multiplayer online game loosely based on the Star Trek universe. Up to 16 Players battle each other in 2-D ship-to-ship dogfights, and team play starts when there are enough players for 4-8 players per team (depending on the server). NETREK also has a unique distinction, not only is it a progenitor of modern online games (versions have been around since the mid 1970's), but it was the first game to really take advantage of the online multiplayer setting, making teams and team play the best part of the game.

The game itself is amazingly simple; Right-Click to change course, left click to fire phasers, "t" for torpedos, "0-9" to set speed, etc. You can learn the basics of play and rudimentary "dogfighting" skills very quickly, and there are practice servers populated with "bots" to aid you in this. Of course, basic skills won't be enough to save you from others who may have been playing for years, but they will let you begin to understand the greater part of the game - the part that transcends the Eighties-tech simplicity of the game. This is because dogfighting is only how you play against other players, it is not the goal of the game.

The goal of NETREK is to transfer armies about between planets and so take control of all the other team's planets, thus winning the game. In a game full of players who all have the basic skill and more, this becomes amazingly difficult. Teams must struggle to accumulate and defend the armies needed to "take" another planet. They must cooperate to mount a successful attack and "take", and then fiercly defend the prize from counter-attack, all the while defending their other planets for similar attacks. To do so successfully requires a level of understanding and cooperation beyond far beyond the "twitchy" skills of dogfighting. Team play arises out of the simple mechanics of the games, transforming it from a simple shooter into a strategic masterpiece as complex as any sport.

Unfortunately, life started making other demands on my time, and my time for NETREK became very limited. Then there was the lightning strike in the spring that fried various bits of electronics about the house, including the network card on the computer I use for games, which ended my NETREK play entirely. Winter is coming around again though, and with it I'll be spending more time indoors looking for something to do. I now have a brand new computer, and I hear there is a new and improved NETREK client available - I think I'll give it a try.

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I also grabbed this image from the NETREK site, which didn't quite fit with the rest of the post. It is a full screen shot from an older version of the Netrek client, and if you click this link, it will take you to an animated GIF of some NETREK action.

Hammer's Slammers Math (Take 2)

I tried this post before, but I decided I really wanted to do a better job of it. The game Hammer's Slammers has point value for units to aid in creating scenarios, not unlike Battletech's Battle Value. Many games have such point-value systems, but this Hammer's Slammers game is particularly suitable for the sort of simple examples I need to get started. The entire games rules are only about 5-6 pages, and the portion that deal with resolution of combat is maybe half of than. My main goal is to demonstrate a statistical way of looking at the basic mathematics of the game. I'll start off with a bit more about how the game actually plays, or at least the detail of how combat is resolved.

Here is the description of the games counters, and if you have any familiarity with typical wargames then this is pretty standard. This particular counter represents a Supertank, and is the most powerful unit in the game. It has attack and defense factors of 8 each, a range of 4 hexes (8 hexes at 1/2 attack factor) and can move 6 hexes per turn.

To the right is the Combat Results Table (CRT) from Hammer's Slammers as I have recreated it. The attacker declares which units (within range) will attack the defending unit(s) in a single hex (stacking limit is 3). They sum up the attack factors of all attacking and defending units, then find the column that is closest to the ratio of attack to defense factors (rounding down), then roll 2d6 and look up the result. I am mostly interested in the probability of "Dis" and "DE" results, and so will ignore the additional results related to turning hexes to rubble.

An example may be helpful: An attacker with one Supertank (attack factor 8) is attacking a Combat Car (defense factor 4). The ratio is 8 to 4, or 2 to 1. The attacker rolls 2d6 and gets an 8. Looking this up on the "2-1" column of the CRT gives the result "Dis". The Combat Car is disrupted and the counter is flipped over to indicate this. If the Combat Car had already been disrupted, this result would destroy it. If the attacker had rolled a 9 instead of an 8 it would have been destroyed outright.
In this game most Infantry units are never disrupted, and are only affected by "DE" results. The exceptions are not relevant for this discussion.

There are two probabilities to be considered. The first if the probability that an attack either Disrupts or Destroys the defending unit. The second is the probability the defending units is Destroyed. These probabilities are represented in the chart at the right with the Attack-Defense ratio on the X-axis and the probability on the Y-axis. (Actually, we might also consider the probability of a unit being Disrupted and not destroyed, but I'll save that for another day.) These probabilities are basic to understanding the game, but this is not the only way they can be represented.

This is the same graph except the Y-axis now represents Odds instead of probability. If the probability of an event is equal to P, then the Odds of the event are P/(1-P). I have fudged a little bit where the probability that a unit is either Disrupted or Destroyed is equal to 1.0; here the odds are undefined (divide by zero) and I have arbitrary substituted of an odds of 90 so I could have complete graphs here. While probabilities are between zero and one, the odds range from zero to infinity, and large values are not uncommon. It is often useful to consider the odds on a different scale of measurement.

Now here is a different scale. The Y-axis now represent the natural log of the odd (log-odds), and the X-axis is log-base-2 of the attack-defense ratio. This is an odd decision of my part to use two different logarithmic scales. I chose log-base-2 for the X-axis simply because the graph looks nicer: "0.0" is at zero on this scale, and each doubling (or halving) of the A/D ratio is a shift of 1.0 on the log-2 axis. Otherwise, there is no reason to used log-base-2 instead of the natural log.
Note that the relationships between the log-odds and log A-D ratio can be approximately represented by a straight line, or by the equation of a line, such as
Log(Odds) = A*Log(A/D) + B,
where A and B are coefficients that define the line.

Representing this relationship by the equation of a line is easily accomplished by use of linear regression - a basic statistical technique with a multitude of applications. The implication I am (finally!) getting at, is that there is a statistical way to describe the relationship between attack factor, defense factor, and the probability of a unit being destroyed (or disrupted).

There is more to consider: In a typical game units may be attacked more than once before being destroyed (and in Battletech this is a certainty). To look at what happens when a unit is subjected to a sequence of attacks I will need to discuss survival distributions. Stay tuned.

Crisis = Danger + Opportunity

There is a common saying that the Chinese symbol for "Crisis" is a combination of the symbols for "Danger" and "Opportunity". While this is not really true, the underlying truth about the nature of crisis remains. We currently have a global financial crisis that represents a danger and opportunity for many businesses, which brings me to today's announcement from Catalyst Game Labs:

Catalyst Game Labs has tendered an offer with The Topps Company, Inc. to acquire various WizKids’ properties, including such dynamic game lines as HeroClix, the Pirates Pocket Model Game, BattleTech/MechWarrior, and Shadowrun. After carefully reviewing options, Catalyst firmly believes that to protect the properties and ensure the best possible continuity for all communities, acquiring all brands will ensure the on going success of these game lines.

I agree. Catalyst is the obvious choice to pick up and carry these game lines, and this represents a great opportunity for them. It is also a danger, and I am concerned that these are difficult times for Catalyst to be taking on greater financial responsibility. Economic hardship hits everyone, and especially the player base of these games, many of whom are young adults without much expendable income to start with. When money is hard to come by, games are going to be pretty low down on everyone's priority list.

So Catalyst faces opportunity and danger, but is the business to be gained worth the business risk at this time? I'm going to hazard my opinion: YES.

This is only my opinion, and I don't have any sort of special information to back it up, but here is my reasoning: The people who own and run Catalyst are highly motivated and enthusiastic about what they do. Creating games isn't a day job for most (any?) of these folks. They work very hard at their jobs, and then they work harder at the games they love. These folks are also busy. Really busy. There are always pressing deadlines and jobs to do. Normally adding yet more to manage might be a bad thing, but managing these new properties seems to be such a natural fit here that I think it can only help them. The remaining question in my mind is "how much is it going to cost them?", and here Catalyst seems to be in a great spot, because if the licensing rights cost too much they can simply walk away no worse than they were before.

There is perhaps one more obvious point to mention; people at Catalyst obviously thought enough of this opportunity to publicly announce they are negotiating with Topps, which may indicate they are already close to a deal. I wish them the best of luck. More: I'll be cheering for them too!

[UPDATE - 12/16/2006]
Catalyst and Pinata Games are teaming up. Maybe. And who is Pinata Games anyway?

Anatomy of a Blackhawk H - Part II

Picking up the brushes again after a break. This is where I left off last time. You can also pick up the entire photo album here (picture below is about number 22).

I applied a white primer and rough green base coat (Vallejo Game Color Dark Green) . I actually did that last session, but had already put the camera away.
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I evened out the base coat with more Vallejo Dark Green with some black (Vallejo Model Color Glossy Black) mixed in where I want a darker base tone. I thin out my paints with blending medium and extender to get the consistency and working time I like.
I haven't put the paint on so thick where I intend to put the "stripes" because that will get painted over anyway. Not that you can tell from this picture.
It's about at this point that a little voice in my head started telling me:

GAWD that is ugly! You should drop it in the Pine Sol and start over!

Ignoring this little voice is one of the hardest things I have to do, but some of my best work has come out of these jobs that started out looking so very wrong, and that little voice yelling in my ear. I haven't done much blending in green-tones though, a lot of which will be needed to make this work, so I'm a bit uncertain of myself at this point.
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Some time later --> The blue stripes are mostly on (but not finished), and the green tones are working out well. I'm beginning to think this might work (The little voice seems to have gone home in disappointment, but it will be back with the next mini). I haven't begun to obsess over the details yet though, and there is a lot more to do.
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Camospecs Diorama, GENCON 2007

Some pictures of the CAMOSPECS diorama I took at GENCON 2007. This set makes a nice post.
I only wish I had made a few notes about this, and factions were taking part in the battle. Maybe if I zoom in ...
[UPDATE: Randall Bills writes - The diorama depicts the battle on Acturus where Word of Blake forces attempted to destroy a coalition force that “Uncle Chandy” had assembled to attack the Ruins of Gabriel (the 2008 diorama). - Except I think he means 2007.]
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There is also a video of this from Catalyst Game Labs:

Wizkids Closes Shop

Yesterday was not a good day at WIZKIDS.

Announcement [11/10/008]

The Topps Company announced today that WizKids will immediately cease operations and discontinue its product lines.

Scott Silverstein, CEO of Topps, said “This was an extremely difficult decision. While the company will still actively pursue gaming initiatives, we feel it is necessary to align our efforts more closely with Topps current sports and entertainment offerings which are being developed within our New York office.”

[date added]

Good news from Topps and Catalyst: CBT is not affected.

VASSAL

Here is something I keep meaning to look into, but never seem to have the time to actually figure it out:

The VASSAL Game Engine

VASSAL is a game engine for building and playing online adaptations of board games and card games. It allows users to play in real time over a live Internet connection (in addition to playing by email). It runs on all platforms, and is free for personal use.

This seems like just the thing for running long distance boardgames.

Dice Wars

Here is a nice little Risk-like strategy game: Dice Wars


Battles are resolved by highest total dice roll (ties to the defender), with a winner take all result. Replacement dice are given based on the number of adjacent territories you control, and are assigned randomly. A sound strategy is to leave paths of weakly defended territories open to your biggest piles of dice, inviting other (computer controlled) players to attack you so that you can counter-attack with greater force. You also want to set up paths for your opponents to fight each other, so they do not spend all their efforts on you.

[Updated 11/10]

The Dungeon Librarian

Ed Greenwood recalls an amusing tale of ...

The Dungeon Librarian

at

Kobold Quarterly

Lots of other fun stuff in their dungeon too.

Point Objective Games

I previously discussed how Classic Battletech (CBT) shares basic characteristics with other point scoring games. I think I need to be a little more careful with this definition. In CBT the objective is to accumulate points of damage on the opposing player up to the maximum (damage) that player can take (destruction), at which point a "win" is achieved. There is a fixed goal of reaching some point total, and reaching that objective before your opponent does. There are many games with this sort of fixed goal structure: Hearts, Cribbage, and Star Fleet Battles, for starters.

The distinction I want to make is between scoring points towards some fixed goal, and trying to score more points than the other player with the win going to the player with the highest score at the end of play. Some examples of this might be Scrabble which is limited by the number of tiles, or any game where the score is kept until a time limit is reached. Many competitive sports are of this type, though we don't typically thing of those in the same class as the tabletop games. Some classic two player video games probably also fit into this category: Pong, Tank!, or anything else where you compete for points/scores simultaneously with another player and some amount of skill is involved.

Now I can make some definitions. All the games mentioned above involve some sort of point scoring, but the first paragraph describes games with a fixed objective to win, and in the second paragraph there is no predefined point limit. I will refer to the first type as "Point Objective" games, and the second as "Point Scoring" games. I'm sure there are some other details to be considered, so these definitions should improve with time.

There are still other games where players compete to remove ALL the other players' pieces from the board. (I'm going to exclude games like Chess and Checkers for the moment: Chess doesn't consider the any piece other than the King for a win, and neither Chess nor Checkers have any element of randomization in how the game plays out.) Many games - not just typical wargames - fit into this category: Ogre, Hammer's Slammers, Warhammer (including 40K), and most typical wargames. Collectible trading games games such as Pokemon and Magic the Gathering could also be consider part of the group. However, these games should also be considered in the same class as Point Objective games: Here the points are represented by the individual pieces on the board or map (or starting strength of the player in the collectible card games). Each player starts with a limited supply, and loses when they run out of pieces. This is "Point Attrition" instead of "Point Objective", but it makes no difference if we count points from zero up to a fixed goal, or start at some number and count down to zero.

So now I have definitions. Maybe I need my own Wiki?

[update]
Thinking about the dynamics of Collectible Card Games (CCGs), these games don't exactly have a fixed objective, or at least the path to the objective depends on the cards in play, and so may need their own category. I haven't actually played any of these games since my son out-grew Pokemon and so haven't given them much thought. There are aspects of playing for both offense and defense I have not considered.

www.ihatelorenjaffrey.com

An inside joke, regarding the death of Loren Jaffery:

It's ABOUT TIME! NOW STAY DEAD!!!

This message has been brought to you by ihatelorenjaffrey.com. When you care to hate an annoying CBT character, well, there's a lot more worse characters and novels out there, but I just plain don't like the guy.

Now at least the search engines will find something. :-)

[Update] He's not really dead, not so long as we remember him, and the Mech was equipped with CASE.

Wednesday Battletech: Battle of Stalingrad

Last week was a small group at AiG, and none of us had a key to the filing cabinet or any maps. Therefore we had to improvise a bit to put a game together. I had some sheets left over from past games, so we set up a the player battle on the Stalingrad table. Captions below the pictures:

Steve's Hunchback HBK-4SP advances down the bombed-out street. Unfortunately for Steve, those are Jeff's Plainsman hovertanks (proxy) behind him.

From the opposite angle --> Insert Cloud-of-Smoke where the Hunchback used to be because Jeff "snaked" a critical to the SRM6 ammo. In the background Dan's Vindicator VND-5L swats at Steve's Bandit hovertanks (This is the debut of this mini as the Mech the mini represents).

Another view, revealing Dan's Saladin and (Clan) Bandit hovertanks safely tucked away in the alley.

Still another view of Steve's hovertanks menacing Dan's Vindicator.

Having dispatched one of Steve's Bandits and damaged the other, Dan moves to engage Jeff. Off picture to the right Dan's Saladin explodes under a hail of fire from one of Jeff's Plainsman. Jeff's Bushwhacker is partially visible at the top of the picture.

The first clear view of Jeff's Bushwhacker, and one Plainsman peeking through a window to attack Dan's Bandit, which is quickly immobilized (and basically out of the fight). Jeff's other tank meets its end somewhere off picture.

Two turns later, overhead view: Jeff's Bushwhacker is wisely giving ground to the combined advance of the Vinicator and Steve's Bandit.

The Bandit is in position for a rear shot on the Bushwahcker, but is immobilized by the Plainsman. At this point the Bushwhacker has taken very little damage.

Another angle, and the blurriest. The Vindicator immobilized the Plainsman, but has been taking significant fire from the Bushwhacker.

Next turn - All out of pictures now: The Vindy jump behind the now immobile Plainsman and finishes it off, but is knocked down by fire from the BW. Steve's Bandit is still taking pot-shots at the BW, but it about to be out of the fight anyway.

At this point the vindicator has finally closed to where it might make physical attacks. The Bushwhacker turns and runs to increase the distance while the Vindy regains its feet. A pursuit of several turns ensues. The Vindy catches up as the Bushwhacker runs out of running room and it gets in one good TSM enhanced kick, but has lost the left (Sword) arm AND it gets knocked down again.

The final turn - the Vindy stands again but cannot reach the BW for physical attacks, so it let loose with a full alpha strike; Jeff's Bushwhacker loses an arm and takes some internal damage but remains standing and in good fighting form. The Vindicator shuts down at 26 heat, and we call it an evening.

Gary Gygax must be rolling (polyhedral dice) in his grave

Oh those EVIL D&D players!

CORRECTION: It has been brought to my attention this story is in error. See this link for the details. (Oops)

As has been reported by a few other gaming blogs and news sites, the Charity Auction at this year's GenCon Indianapolis was held to benefit Gary Gygax's favorite charity, which I will not name here for reasons that will soon become obvious. The fine folks at GenCon raised over $17,000 for this charity, which helps starving children in impovershed areas of the world--only to have that money actually turned down by the charity. The charity refused due to the fact that the money was raised partly by the sales of Dungeons and Dragons materials, which as we all know, puts an irrevocable taint of evil on the filthy lucre that us demon-worshipping gamers might want to use to, say, donate to starving children. Not only is this a slap in the face to every gamer, but it is especially insulting to Mr. Gygax himself, who I understand donated to their cause many times over the years. Plus, I'm sure the children who would have gotten food or clean drinking water with that money would be sort of upset, too. (Emphasis added)

from Giant in the Playground (please do click through).