You know the trick for two people to fairly divide a cake. ie. one person halves it, and the other chooses which half. I was thinking about this in the context of choosing rules for a boardgame, which might be asymmetrical, like Thud or Tafl. It may also involve random elements during the game.
Q. Suppose two evenly matched players Al and Ben would like some new rules, but aren't sure they can agree them fairly.
A. Let Al choose the rules, and Ben choose whether to play white or black. Al has an incentive to make the sides even.
Q. What if the sides are intended to be uneven? Say, it's traditional that white has a 2/3 chance of winning?
A. Let Al choose the rules. Ben, instead of choosing to play as white or play as black, chooses between "playing as black", or "Toss a coin. Only if you win the toss and the game as white, you win."
Then if Al defines the rules as desired, both Ben's choices give a 1/3 chance of winning, but if Al makes either side have a greater than designated chance, Ben can choose that for teh win.
Q. What if the advantage to white isn't known? Suppose there are a family of existing games where white has an advantage, but exactly how much isn't known? And we want to introduce an en passant rule or other change intended not to affect that advantage?
A. Hmmm. I'm not quite sure how to measure that. Can I try an intermediate step?
Q. OK. The players have a board and rules giving white an unknown probability P of winning. Al must estimate P at Pal, and then, what? Why is Al incentivised to make Pal accurate?
A. Let Ben choose either to play the game with Al as white or let Al win on a die throw with probability Pal. Al will not make Pal too low, or he'll reduce his chance of winning.
Q. That works to stop Pal being too low. What stops it being too high?
A. Um. You can't let Ben play white because white has an advantage already, and you can't adjust for it because you can't know what it is. You could let Ben let Al win with probability P2/Pal.
Q. No you can't. That's not guaranteed to be between 0 and 1.
A. Doh! OK, win with probability P2 (eg. winning if he wins two consecutive games with the same set up), but count that as a payoff of winning 1/Pal games!
Q. I don't like that. I can just about buy that I might equate wining this with winning two of those, but to feel like I won 1/Pal times? It won't feel like it counts, which blows the incentive away.
Can you do it, where the only payoffs are 1, winning, or 0, losing?
That's as far as my thoughts took me. Can anyone solve that, or show it's impossible?
Some notes:
* A binary payoff of 1 or 0 is the same as 1 or -1. I don't think it makes any difference if it's zero-sum.
* I started thinking this about Magic:TG, but *tried* to simplify the final question I was considering.
* We assume that the players have enough experience that they fairly can estimate the probability of winning.
* Ideally you'd never have to play multiple games, but I think we must allow that. Also approximations. It's cheating to say "Play 100 games and measure the proportion" though.
* In real life you might just agree the rules and cut the cake as fairly as you can anyway; but I'm still interested in the theoretical question.
Q. Suppose two evenly matched players Al and Ben would like some new rules, but aren't sure they can agree them fairly.
A. Let Al choose the rules, and Ben choose whether to play white or black. Al has an incentive to make the sides even.
Q. What if the sides are intended to be uneven? Say, it's traditional that white has a 2/3 chance of winning?
A. Let Al choose the rules. Ben, instead of choosing to play as white or play as black, chooses between "playing as black", or "Toss a coin. Only if you win the toss and the game as white, you win."
Then if Al defines the rules as desired, both Ben's choices give a 1/3 chance of winning, but if Al makes either side have a greater than designated chance, Ben can choose that for teh win.
Q. What if the advantage to white isn't known? Suppose there are a family of existing games where white has an advantage, but exactly how much isn't known? And we want to introduce an en passant rule or other change intended not to affect that advantage?
A. Hmmm. I'm not quite sure how to measure that. Can I try an intermediate step?
Q. OK. The players have a board and rules giving white an unknown probability P of winning. Al must estimate P at Pal, and then, what? Why is Al incentivised to make Pal accurate?
A. Let Ben choose either to play the game with Al as white or let Al win on a die throw with probability Pal. Al will not make Pal too low, or he'll reduce his chance of winning.
Q. That works to stop Pal being too low. What stops it being too high?
A. Um. You can't let Ben play white because white has an advantage already, and you can't adjust for it because you can't know what it is. You could let Ben let Al win with probability P2/Pal.
Q. No you can't. That's not guaranteed to be between 0 and 1.
A. Doh! OK, win with probability P2 (eg. winning if he wins two consecutive games with the same set up), but count that as a payoff of winning 1/Pal games!
Q. I don't like that. I can just about buy that I might equate wining this with winning two of those, but to feel like I won 1/Pal times? It won't feel like it counts, which blows the incentive away.
Can you do it, where the only payoffs are 1, winning, or 0, losing?
That's as far as my thoughts took me. Can anyone solve that, or show it's impossible?
Some notes:
* A binary payoff of 1 or 0 is the same as 1 or -1. I don't think it makes any difference if it's zero-sum.
* I started thinking this about Magic:TG, but *tried* to simplify the final question I was considering.
* We assume that the players have enough experience that they fairly can estimate the probability of winning.
* Ideally you'd never have to play multiple games, but I think we must allow that. Also approximations. It's cheating to say "Play 100 games and measure the proportion" though.
* In real life you might just agree the rules and cut the cake as fairly as you can anyway; but I'm still interested in the theoretical question.