OT: Game Theory/Statistics

[geoschmo]

Suicide Junkie said:
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.

While I agree with your conclusion there are some problems with how you got there. First of all, each player has an 100 tokens and there are ten boxes. Simple division should show you the lowest possible number you can have for your biggest pile is 10, not 9. And that's if yo udivide them evenly. If any pile has less then ten then at least one other pile must have more than ten.

Secondly, each player has an equal numebr of tokens, so it's also impossible to outscore an opponent 10 to zip. The highest possible winning score would be 9 boxes to 1.

Geoschmo

[Suicide Junkie]

It is true that you couldn't have a maximum stack smaller than 10 unless you did not use all your tokens.

However, that is irrelevant to the proof.
The point being that if your maximum stack is any natural number at all, a random opponent will sometimes beat you no matter what you do.

In fact, if you allow a negative number of tokens in a box, it still works.

[Spoo]

What happens when both players put the same number of tokens into the same box?

[Suicide Junkie]

Then it would be a tie, naturally. Whether both score 1, half or zero points for the box, dosen't affect who wins the game.

PS:
Sorry, hit edit instead of reply

[narf poit chez BOOM]

Basically, it's a guessing game.

To make it a strategy game, there has to be some sort of in-game interaction between the playing pieces. Perhaps change it so you can select three boxes and see what your opponent put in them.

[Suicide Junkie]

Or, play it repeatedly, observing the outcomes after each trial, then try to predict the other person's reactions.

[narf poit chez BOOM]

That's still a guessing game. You've simply added a meta strategy game over a series of games.

(You know you've hung out over at The Forge for too long when...)

[AngleWyrm]

True, and a great summary of a lot of Game Theory games. they tend to have both players secretly select a move and then reveal the results. Not necessarily a bad thing -- the opening moves in most RTS and 4X games are done without the other player knowing what you're doing.

But it gets so much attention -- is it because it's easy to do on a chalk board? How would you quantify building a scout to shorten the time until you know something about your opponant's build order?

Secrecy does have merit in war. I once saw an interesting definition of a secret: Any info that, if others found out, would damage your score. Something like that.

Another problem is framing games of perfect knowledge like chess, where you see all and take turns. They are usually drawn as a tree of possibilities, that quickly grows enormous. Too much, really. Maybe that's the problem that needs to be worked on: How to represent a game where the players take turns, and see the pieces, but doesn't draw out to an exploding tree.

Like maybe some sort of greedy current situation heuristic, that only remembers a short list of successful things tried when in the current/similar position. And maybe also a short list of tragic things to definitely not try when in that situation. Isn't that sort of how people do it?

[capnq]

AngleWyrm said: Maybe that's the problem that needs to be worked on: How to represent a game where the players take turns, and see the pieces, but doesn't draw out to an exploding tree.

I dunno, a game that doesn't develop into an exploding tree sounds kind of dull to me.

Lots of things are like that: the interesting stuff is rarely easy to analyze.

[AngleWyrm]

Oh no, I didn't mean that the game would be different. Take chess for example. You can still model the game as an exploding tree if you like, but it's not possible to draw the whole thing out. It's too big.

Now think of another way to model the game of chess (no changes to the game of chess allowed).