Game theory becomes much more fun when we make it harder to adopt mixed strategies. I’d be interested to know what any game theorists reading this (or non-game-theorists) think the players should do in the following game.

A and B are playing for $1,000,000. Each of them has to choose a number: either 0 or 1. If they choose the same number, A gets the $1,000,000. If they choose different numbers, B gets the $1,000,000. Neither is capable of making a genuinely random choice, left to their own devices. They are able to make random choices by flipping coins, but any coin they flip while making the choice must be forfeited, and all coins have positive value. What should they do?

The idea for this game is borrowed from this paper by Roger Koppl and Barkley Rosser Jr that I’ve mentioned previously. One of the many things that make the game interesting is that there is (I think) no Nash equilibrium in the game.

By the way, in case you’re thinking “Why is he talking about this game? Is there a hidden agenda here?” there is a hidden agenda, but it isn’t very well worked out. At some level I think I think there shouldn’t be such a thing as game theory. What’s currently game theory should be viewed as just decision theory where one of the unknown variables is a decision by another agent. But any argument to that conclusion is a long way off. A long, long way off.