OT: Game Theory/Statistics

[geoschmo]

I came accross a description of a game some months ago and I'm trying to recall it now. It was a simple game that I believe was used in the discussion of game theory in a programming class, or it might have been during a statistics class. In the game you have two players with some number of tokens, I believe it was 100 each, and some number of boxes, I believe it was 10. Each player could distribute their tokens among the boxes in any proportion they wanted without knowing how many the other player had placed in each box, and you won a box if you put more tokens then the other player in it. The winner of the game was the person who won the most boxes.

Does this sound familier to anyone?

[MrToxin]

I think I may have heard of it, but that's a bit simplistic to be an interesting game.

One of two things would happen: you would put ten in each box and break even every game, or put 20 in five boxes each and break even in every game.

[geoschmo]

It's simple rules, but the strategy comes in trying to anticipate your opponents moves. If you think he's the type to put ten in each box, then you can put 11 in 9 boxes and 1 in 1 box and win 9 out of 10. But there is no perfect strategy because a style that will beat one type of player might lose to another.

The discussion I read was more from a theoretical game science point of view than an "This is fun, let's play" perspective anyway. The game had a particular name and was used in discussing classifying other games. As in "This game is a "blank-type" game."

[MrToxin]

Ah, k. I never worry too much about the names of the games, just the statistics and whatever.

But, yeah...the problem comes from the fact that anybody that knows probability and game theory is going to, eventually, play the guaranteed draw instead of the possible win. Either that or a jerk that nobody will want to play with any more.

I understand the theory behind the game but, without randomness or a sense of what your opponent is doing, "safe" is better than "uncertain."

Of course, I'm probably over analyzing things, as I am often caught doing.

[geoschmo]

I think you aren't thinking it completely through actually. You can't really play to a draw in the way that you suggest. Putting ten in each box would only be a draw if your opponent also put ten in each box. And as I suggested before if you think your opponent likely to do that you could easily defeat them.

[capnq]

MrToxin said: anybody that knows probability and game theory is going to, eventually, play the guaranteed draw instead of the possible win.

People who know probability are a small minority, and game theorists are even rarer.

I'm not convinced that there is a guaranteed draw, but then I managed to fail three different probstat courses in college.

[Glyn]

Geoschmo - interesting little game. I don't know the name. Also, I don't think there is a guaranteed draw. Two really good players might get to the point where they are winning half the game and losing half the games.

[douglas]

It's a simple enough game I'm sure there's a reasonably easy (for a game theorist) to prove optimal strategy, but it's probably one of those situations where the optimum is to randomly pick which of some number of strategies to follow each game. Any given fixed distribution of tokens can be beaten, but if you came up with a pair of distributions A and B such that any way to beat A loses to B and any way to beat B loses to A, flipping a coin to decide which of A and B to use each time would guarantee you a 50% win rate.

[AngleWyrm]

A mixed strategy example: A pair of bombers are attacking, and only one of them carries the nuke. Who does the SAM site target first?

If the SAM site is guaranteed to target the lead bomber, the nuke goes in the trailing bomber. If the nuke goes in the trailing bomber, the SAM site should target the trailing bomber...

For both players, the optimal strategy is 50% chance of the bomb in either plane, so flip a coin.

[Suicide Junkie]

capnq said:

People who know probability are a small minority, and game theorists are even rarer.

I'm not convinced that there is a guaranteed draw, but then I managed to fail three different probstat courses in college.

There is quite a simple way to show that there is no guaranteed draw.

Put all your tokens down, and pretend I get lucky.

If your biggest pile of tokens is less than 9, I could have put 10 in every box and win 10 to zero.
If your biggest pile of tokens is 9 or more, I could have used the same distribution of tokens as you, except for zero against your biggest box, and +1 against all the other boxes. Meaning I win 9 to 1.

Therefore, no matter how many tokens your largest pile has, there is a possibility of me winning.