31 December 2008
My Candyland post got a lot of attention, and I'm working on a post about Monopoly. Guess what? It's a Markov-chain too (nearly). There is a lot of material available on this (just Google Monopoly game math), but I think I can add something a little different.
Are there games people want to see analyzed? I'm not sure I can take requests like that, but the question itself might be interesting.
Someone asked what other games I've analyzed (maybe over-analyzed), and I'll have to think about that. Probably not all that many in any depth, but I've never made a list (maybe I should).
I've got a series of posts that require a fair bit of spreadsheet work to set up, and a laborious effort to describe so that someone else might be able to figure out what I've done. This is painful to write, and probably more painful to read. This is also my best way to "share the math" in a way which is accessible to others, and really essential to what I am trying to do. I need to find a way to make this type of post work.
The downside of writing a lot is I'm not doing much painting (or much laundry). I need to balance things a bit better, or become a faster/better writer. Of course, writing a lot is the key to becoming a better/faster writer.
There is a big topic I've been trying to work up to writing about: Game Theory. This is another one of those hard to write but important to do things I need to figure out how to do.
What else is in the works?
More miniatures pictures, an interview, a browser game with mecha in it, that long-promised Facebook Games followup, Elo scoring, some old-but-great games in my closet, some new games, the game I'm trying to write, some simple "demonstrate the math" games, not to mention following up on my blogroll and some of you nice people who are following this blog.
So that's a bunch of ideas even at one post each, and some of those need multiple posts. If I could hit on all those topics and do a decent job on them, that should easily keep me busy for a good part of 2009, and that's not counting the other games I find along the way.
So that's a quick rundown of some of the topics coming on GBR, at least the ones running through my head at the moment. Happy 2009, and happy gaming!
PS: 3-4 more ideas since I "finished", but this is enough
30 December 2008
The short answer is ...
... no. Don't let that stop you from reading the full post though.
Far more men play chess than women and based on that simple fact, you could actually predict the differences we see in chess ability at the highest level. It's a simple statistical fact that the best performers from a large group are probably going to be better than the best performers from a small one. Even if two groups have the same average skill and, importantly, the same range in skill, the most capable individuals will probably come from the larger group.
The article goes on to discuss Elo scoring as well, something I've mentioned before and will come back to again.
28 December 2008
More than that, I tend to obsess over them. I have had to set some games aside, never playing them again or refusing to buy them in the first place, otherwise I would spend all my time playing or thinking about them. Games in the Civilization, Warcraft/Starcraft series are maybe the most recent examples of this, but there are plenty of others (Doom!, Quake, etc), and the next one is calling to me from underneath the Christmas tree where I stashed it along with other gifts.
I wrote about Gran Turismo 2 a while back, and how I set up spreadsheets to calculate optimal gearing ratios for my cars in the game. This is something a serious gearhead or racer might do, but for a game? Objectively I'd have to say this is just way too much effort spent on a game. On the other hand, I think this is part of how I like to play a game - I really want to understand at a very basic level. In this case I actually learned a lot about the application of physics that I had never really thought about before. I had a lot of fun creating those spreadsheets tinkering with the real world physics as it applies to automobiles, both real and virtual. (This is also a real testament to the physics engine driving games such as Gran Turismo, it's so good you can study the real world by studying the game!)
I also spent a lot of time alone playing and working on Gran Turismo 2 (or Civilization, etc.), and contributing little to my household, my family, and my life in general. Those are some great games, enormous fun and I can play them for hours or days, but they tend to leave me feeling like I've wasted my time when I am done playing. I hate that feeling. Therefore I now budget my time to those games that give me back the most socially, creatively, and intellectually. For me, the games that give back the most are the tabletop games, and that is why I tend to focus on them in this blog.
So not to dis computer games, it's just that I love them too much.
26 December 2008
Gaming means a lot to me. It's a a social activity, an outlet for creativity, and a subject of study.
I meet regularly with friends for a weekly session of Battletech. It's a mixed groups of working professionals, craftsmen, and students. We are not there for drinkin' an smokin' (maybe a little cussin' tho); we are there to play, and to borrow a phrase; a bad evening gaming is better than a good day working. I am a bit unusual in the group because I don't play a lot of games other than Battletech (I do have limited time for gaming), and most of the guys also play other games on a regular basis. (Note aside: I also like computer games, but I find these do not have the same aspect of social interaction, so I don't play them nearly so much. Thoughts?)
A fair bit of my time (likewise many of the guys) goes into game related activity such as painting and assembling miniatures. Working with miniatures is a hobby within a hobby, and a topic for another day, but I love working with minis as much as I like to play. Maybe more. Check some great galleries of Battletech minis at LOTB, and consider the craftsmanship that goes into some of these.
The other creative aspect of the hobby (for me) is writing, and I don't just mean this blog. I've been writing related to gaming on and off for years. This might be writing up a scenario or adventure, describing something I'd like to represent in a game, writing my own game, or working out strategies. Most if this writing isn't interesting to anyone but myself, but I enjoyed writing it all the same. More recently I actually became somewhat organized about recording my game related ideas, and in the process I realized I had something I wanted to say to others (and so here we are).
Finally, I like to study games. I wrote my first computer simulation program when I was 13 or 14 on my Apple II+. Why? Because I wanted to settle an argument about whether it is better in the game Risk for the defender to roll one die or two (short answer: two whenever possible). Little did I know I was actually taking my first steps towards being a statistician, I just wanted to know the best way to play. More recently I decided to really look hard into the mathematics of the game Battletech. Rather than write a program I decided to try to get into the theory behind it. Part of me wonders why I never really tried to apply my skills this way before, but maybe I wasn't ready? What I quickly learned is that I wasn't just studying Battletech, but the math that applies to a whole class of games, sports, and even military studies. I've always loved games, but until recently I never knew they could be so deep.
So now I've rambled on too long and need to get to work. I invite you to post comments on why you like games, why you play them, and what intrigues you about them. How do you spend your play time?
25 December 2008
in a Giant Battle Mecha ...
O'er the tanks we go,
Smashing all the way!
Ok, so maybe I won't give up the day job to become a lyricist, and those missiles are delivering presents, honest!
There have been over 4000 new visitors to this blog since it was posted on Blogs of Note Tuesday, and I am gratified and overwhelmed by the attention. Thanks to everyone who stops by, and thanks to the folks at Blogspot (Santa Blog?, Saint Blogolas? Father Blogmas?) for noticing me.
Merry Christmas everyone, and I hope that one or two of those nicely wrapped boxes under your tree might have a games in them.
23 December 2008
[Image taken from lscheffer.com, who also has a nice analysis.]
In my own words, a Markov-chain is a series of random states. A trivial example might be flipping a coin; the coin is either in a state of "heads" or "tails" and has a 50% chance of changing state with each flip. It can get more complicated, with the probabilities of changing to a different state dependent on the current or even (correct me if I am wrong) previous states. A state which ends the series (if any) is called an absorbing state. In my recent Netrek post the number of planets a team controls would be the state, and the absorbing state would be one team conquering all the planets, thus ending the series and the game (oversimplified, but true).
Candyland is a Markov-chain if you reshuffle the deck of cards after every draw. Each square (rhomboid?) on the board represents a state, and players advance randomly towards the end of the game. Reshuffling is required so that the probability of changing states remains independent of previous card-draws (if you were wondering). There is a nice summary and history of the game itself here.
Speaking of fine work, Greg Costikyan wrote about this a few weeks back, and it has traveled far in the blogosphere. The really interesting part here is (for me) the discussion of whether or not it really is a game; all moves are random, so if the player doesn't actually do anything is it really a game? My answer: Yes. Not the most interesting game we may ever play by any means, but most definitely a game. Games where players make decisions and enact strategies are more difficult, and for most, more interesting.
I should not end without mentioning an older game that is a Markov-chain to start with - no reshuffling required - known to most as Snakes and Ladders.
[Image from DKimages]
21 December 2008
For each hex where a player might move a Battlemech, the player will typically consider how good that hex is from the standpoint of attacking the other player, and how bad it might be from the viewpoint of the other player counter-attacking. The player will evaluate many moves, and generally choose one that is most-beneficial and least-harmful. If we wanted to get analytical about it, we might even assign a number to each hex representing the value of that particular move (really we would need a number for each hex, facing, choice of targets, and maybe more if there are multiple ways to accomplish the same move, but the added complexity doesn't contribute anything here, so I'll leave it out).
If you can picture the number in each hex as a height, making the tallest hex on the map the best move, then we have a representation of the game (the current move) as a sort of fitness landscape. This "landscape" probably looks very different from the Battletech game map, and I think of it as an abstract representation (possibly in many dimensions) of the choices to be made in the game. Basically your best move is always to move towards the highest point in this landscape.
In a simple scenario where each player has one Battlemech it is actually not too hard to picture how such a landscape might look. However, when the scenario is more complex and each side has multiple 'Mechs to move this landscape can get very complicated indeed, and the choice of movement for one unit affects the landscape for all the others. If we want to get really deep, we might think about the entire game as a sort of landscape instead of just a single move.
This representation of a game as a landscape is the "converence" I was getting at in the post title, where a method (or at least the representation) from science has meaning for a game. I've never worked with genetic algorithms myself, but it seems like they might be very well suited to solving "best move" problems in games too. More importantly though, this gave me an opportunity to introduce the concept of a game landscape. Thinking about games in this way is probably an unusual concept for most gamers, but it is incredibly useful to be able to think about a game with the sort of mathematical representation that this allows.
20 December 2008
Jesper Juul has a book out, or rather a chapter he wrote which is collected in this book which is coming out soon.
Winning isn’t everythingIt is quite simple: When you play a game, you want to win. Winning makes you happy, losing makes you unhappy. If this seems self-evident, there is nonetheless a contradictory viewpoint, according to which games should be “neither too easy nor too hard”, implying that players also want not to win, at least part of the time. This is a contradiction I will try resolve in what follows.
I read the Fear of Failing article, and recognized his "Snake" game and the accompanying survey. I don't know if was one of his study participants or if I encountered it at some later time, but I recall play this game and completing the survey (some years ago). Now that I'm trying to write about games on my own, it is particularly interesting to have this particular research come back to me in this way.
But back to the article: Mr. Juul gives an excellent account of the balance between game difficulty and player skill/ability, and that the proper balance is what make a good game. He is writing about video games in particular, but this easily applies to the sort of tabletop games as well.
Before I forget, the image to the right (© 2004 Noah Falstein, reference: Falstein, Noah. 2005. "Understanding Fun—The Theory of Natural Funativity". In Introduction to Game Development, ed. Steve Rabin, 71-98. Boston:Charles River Media.),
Then I got to thinking, and although I still think this applies to tabletop games, is not so simple to describe what is going on you add in the dynamics of human players, either in tabletop games or multiplayer video/computer games. I touched on this briefly when I wrote about The Mathematics of Netrek, and I intend to spend some more time on this in future posts.
and Part 2:
Louis Zocchi is a great one ... but now I may have to replace all my dice.
[Update] Critical Gamers (whom I just discovered) have a post on this topic too, and a favorable review of Game Science dice in use.
19 December 2008
Playing Board Games Helps Prevent Degenerative Diseases
Many individual do not realize the benefits of playing board games especially in relation to health. There are certain ways in which board games can be implemented for the prevention of certain degenerative diseases like Alzheimer’s disease and Dementia. Alzheimer’s disease or AD is a fatal and progressive disease. This disease destroys brain cells and leads to memory loss, behavior and thinking, which can affect social work, hobbies and also work. Research was conducted that showed that when the brain gets challenged through leisure activity stimulation, it greatly reduces the chances of contracting this disease.
Click through for the full article
18 December 2008
17 December 2008
My favorite bit:
When I was young and naive, I came across a brilliantly excitingly named branch of mathematics called "Game Theory". Naturally I said to myself, "holy crap! Pretty darn smart of me to become a mathematician -- now I'll get to play games for a living!Now go read the post at Curmudgeon Gamer!
In case you aren't aware, "Game Theory" is a bait-and-switch ruse right up there with "Greenland".
I confess I had a similar first contact with Game Theory, and even would have signed up for a class has it been offered. Now I've gotten back to the point where it might be interesting again, but that is a tale for another day.
14 December 2008
[Click Through to Adventures in Gaming]
There is a lot of other good material on this blog as well, and I plan to spend some time exploring it.
10 December 2008
Player versus Player: One-on-one, ship-to-ship combat (dogfighting) is perhaps the most basic component of Netrek. Players combat each other with phasers and torpedoes, trying to inflict enough damage to destroy the other ship first. There is an aspect of Point Objective Games here, with the added complexity that shields regenerate and ships repair. This complexity aside, it comes down to a contest of skill between players, rather like a Chess match. Chess has a ratings system, Elo Scoring, which is used to describe the past performance of a player, and (ideally) has some usefulness in predicting the winner of a match between two players who have never met before. Elo scoring uses a scale where a 200 point difference between player ratings is interpreted as a 75% chance that the player with the higher rating will win (assuming the ratings accurately represent player skill). Chess is considered to be a very “deep” game, with a very great range or many levels of skill between the worst and best players, and this deepness can be measured (in part) by the range in scores. Beginning Chess players might have a rating of 800-1000 points, while the World Championship Chess players have ratings of 2500 or more (Go has even greater depth by this measure). Netrek also has considerable depth on player skill, and it would be interesting if this sort of ratings system could be implemented to measure it. Mathematics based measurements of player skill in games is one of those topics I intend to discuss at greater length in future posts.
Team versus Team: This is where the game of Netrek really shines. And if we could measure the skill of a “team” with a sort of Elo score in the same way we might measure individual players’ skill, I’m pretty sure the depth of Netrek would approach that of Chess. Tournament mode (T-mode) play in Netrek has an underlying mathematical structure that is actually fairly simple. I want to deconstruct the game and point out where the basic mathematics applies to the team games.
Suppose we have the following simple game: 20 Coins are laid out on a table with 10 heads and 10 tails. [edit: here the coins respresent planets, and each side wants to capture all the planets.] Two players, one represented by heads and the other by tails, each turn both players choose one coin and flip it, with the object of making all the coins either heads or tails (with 50/50 chance). Obviously the heads-player will select any coin showing tails and flip it, trying to get it to come up heads, and the tails player will do the opposite. This game is essential a discrete random walk, and with every turn (one flip for each player) there is a 25% chance a player will gain a coin, 25% chance they will lose a coin, and a 50% chance that things stay the same. This is in concept very close to a completely random T-mode Netrek game. (This might also be stated as a Markov chain.).
A purely random game like this could go on for quite some time (it might even be infinite), but suppose there were some element of skill involved so that one player (lets say heads) is a little bit more likely to get the desired result than the other. So now maybe each turn there is a 35% chance the heads-player will gain one coin, a 20% chance the tails-player will lose one coin, and a 45% chance that things stay the same [edit: both win, or both lose.] Out of every 20 turns we would expect that heads would gain 7 coins (35% of 20) while losing 4 (20% of 20) for a net gain of 3 coins. We should expect the heads player will win in 67 turns, on average.
Netrek takes this same mathematical structure and adds a lot to it (players of varying skill, a variety of ships and planets, etc.). I wonder if it is possible to measure the “team playing” skill of players in the same way as Elo scoring might be used to measure the individual dogfighting skills of players? This would be rather complicated, and likely impractical, but it is interesting to think about.
Resource Allocation: This is the final aspect I want to write about today, and the part that leads to the wonderful complexity of team play in Netrek (It will also be the least mathematical discussion). Resources in this game are not equal; The team with more planets has greater potential to gather armies and try to take planets. More planets than players mean that is it not possible to defend every planet equally. Players are not equally skilled, but have varying levels of ability at individual and team play. Finally, controlling a greater number of planets increases the distance the leading team must travel to front-line-combat, and players are out of position for a longer period of time (though Starbases help).
This last part has an effect not seen in most games; the closer one team is to winning (in an otherwise balanced game) the harder it is to successfully take more planets, because the losing side can respond to defend remaining planets much more quickly – and it tends to make the game self-balancing. I am not a very good Netrek player (I confess to being a twink), and I have never participated in a real “clue” game involving 16 really good players, but my understanding is that in these games an outright victory is rare; most games are given a set time limit where the team in control of the most planets at the end of time is declared the winner. This is entirely consistent with Netrek being a self-balancing game.
One common complaint I recall players making is about how hard it is to actually win a T-mode game. There are many reasons for this, certainly more than I have discussed, but I would propose the following: In a really balanced game with equally skilled players on each side, it may be nearly impossible to actually win, either by Genocide (all planets) or Timer (all but three planets for ~20 minutes). Therefore, most wins probably occur in unbalanced games.
[There is an alternate version of this post, co-authored by Zach Uram, Netrek player extraordinaire, that goes deeper in the game history and mechanics. Give that a read too, and check out Zach's blog at http://www.fidei.org/2011/01/mathematics-of-netrek.html.
08 December 2008
Ultrabots: Fairly simple to learn with wireframe graphics (state of the art) and very few glitches. I picked up my copy used in '95-95, played it through and beat it on normal difficulty the first try, then again on hard that same afternoon, at which taught me something about replay value. It was a good game though, if only for a single scenario where I pulled a glorious victory out of an "all is lost" situation. Turns out the AI was entirely too single-minded about its objective, and let my lone operating humanoid bot take out the attacking force in Sergeant York fashion.
06 December 2008
Try this simple experiment: First, take a small piece of metal and try to break it. A straightened paperclip will do nicely - literally try to pull it apart, use pliers to get a good grip if needed, and pull without bending or twisting. This ought to be difficult, and odds are you probably can't break the metal this way barehanded. For the second part of this experiment, bend the metal; work it back and forth several times bending and re-straightening it in the same place. Now pull again, and the metal should part fairly easily. If not, bend it some more and try again, OR just keep bending it until it breaks. Why does the metal, originally quite strong, now break easily?
The answer is metal fatigue (which I previously hinted at here). Metal is malleable and easily bent, and when subjected to stress it develops tiny cracks, The more times the metal is stressed, the longer these cracks become (growing in an essentially random manner). Eventually these cracks reach some critical length and the metal breaks.
Exactly how long a piece of metal can be stressed before it breaks is difficult to predict, but there is a nice mathematical results (Birnbaum and Saunders, 1969) that derives the theoretical probability distribution that describes how long a metal part under repeated stress will last. Furthermore, this is an application of the Central Limit Theorem, and will give a useful approximation to the large sample behavior of survival time in many similar circumstances, and not just for metal parts (exceptions are a topic for another day). So here is the formula that describes the Cumulative Distribution Function (CDF) of the Birnbaum-Saunders distribution for fatigue life:
(It is at this point I realize there may be some difficulty in expressing a formula within the format of a blog without some special software tools. Please forgive the crude scribbles done in Paint.)where Xi is the magnitude of the stresses or shocks, Xi > 0, which have mean m and variance V, t (where t>0) is the number of shocks, W is the critical length of a crack that will break the part when the sum of the shocks Xi is greater than W, and the Greek-looking-thingee is the Normal CDF function. I might do as well to present this is Excel-formula-like notation:
F(t) = NORMDIST(Z,0,1,1)
which is very close to something you can plug into an Excel spreadsheet and use**. Lets try this one more time in English: The formula F(t) gives the probability that the part will break on or before some number of shocks (t), or that the sum of t shocks (the Xi's, defined by m and V) will be greater than W. For a less scatterbrained definition of the Birnbaum-Saunders distribution, see this excellent source.
**I will provide a spreadsheet to demonstrate this, but that will get put off to another post so I can get to the point (yes, there is a point to this).
The mathematical theory doesn't care anything about metal, it is describing the probability that the sum of a series of random numbers X will be greater than some number W. This is a Central Limit Theorem result, which means that we can use almost any sort of random numbers, like those generated using dice or cards, as long as we know the mean and variance. Further, if we a playing a point scoring game with some fixed number of points that are needed to win, then we know W. Therefore, we have a mathematical tool that can describe the probability distribution of a Point Objective Game.
Dramatic pause and a deep breath. I have done a fair amount of work to get to this point, and it is going to pay off, because I now have a tool to help analyze a whole class of games. One of these games is Classic Battletech - which I do not deny is a very complex sort of Point Objective Game - but it can be done. I have a number of other topics to cover along the way too, and this is just the beginning.
05 December 2008
04 December 2008
By Tom Kaneshige
Everyone knows that techies love "Dungeons & Dragons," where they can prowl the bowels of a castle and cast spells on clueless managers, er, mages. After all, it's just a game.Click through to InfoWorld for an interesting read on the connection between Tech skills and gaming. This article considers computer gaming only, but I think it certainly extends to tabletop games as well.
Or is it?
02 December 2008
Les Chappell writes at The Escapist
I've always been behind the curve when it comes to videogames, going back to childhood when my parents refused to buy a Nintendo and my brother and I had to rely on older PC games for entertainment. While I never developed the manual dexterity necessary to win an Xbox deathmatch (my performance in Halo is a joke), I do have an interesting series of childhood recollections: top-down terrorist control in Syndicate, the quirky adventure game-style of Quest for Glory, spinning to avoid giant feline pilots in Wing Commander II.
With the old disks lost or rendered obsolete, I thought these games existed only in memory. That is, until a few months ago when my friend Pat mentioned that he'd been playing through System Shock 2, a game I'd heard many great things about but missed when it came out in 1999. Where'd he find it, I asked - garage sale? eBay?
Click through for the full article, and maybe revisit some old favorites? There is also a lively discussion of abandonwear in the comments.