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JGWeissman comments on A simple game that has no solution - Less Wrong
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Classical game theory says that player 1 should chose A for expected utility 3, as this is better than than the sub game of choosing between B and C where the best player 1 can do against a classically rational player 2 is to play B with probability 1/3 and C with probability 2/3 (and player 2 plays X with probability 2/3 and Y and with probability 1/3), for an expected value of 2.
But, there are pareto improvements available. Player 1's classically optimal strategy gives player 1 expected utility 3 and player 2 expected utility 0. But suppose instead Player 1 plays C, and player 2 plays X with probability 1/3 and Y with probability 2/3. Then the expected utility for player 1 is 4 and for player 2 it is 1/3. Of course, a classically rational player 2 would want to play X with greater probability, to increase its own expected utility at the expense of player 1. It would want to increase the probability beyond 1/2 which is the break even point for player 1, but then player 1 would rather just play A.
So, what would 2 TDT/UDT players do in this game? Would they manage to find a point on the pareto frontier, and if so, which point?
Two TDT players have 3 plausible outcomes to me, it seems. This comes from my admittedly inexperienced intuitions, and not much rigorous math. The 1st two plausible points that occurred to me are 1)both players choose C,Y, with certainty, or 2)they sit at exactly the equilibrium for p1, giving him an expected payout of 3, and p2 an expected payout of .5. Both of these improve on the global utility payout of 3 that's gotten if p1 just chooses A (giving 6 and 3.5, respectively), which is a positive thing, right?
The argument that supports these possibilities isn't unfamiliar to TDT. p2 does not expect to be given a choice, except in the cases where p1 is using TDT, therefore she has the choice of Y, with a payout of 0, or not having been given a chance to chose at all. Both of these possibilities have no payout, so p2 is neutral about what choice to make, therefore choosing Y makes some sense. Alternatively, Y has to choose between A for 3 or C for p(.5)*(6), which have the same payout. C, however, gives p2 .5 more utility than she'd otherwise get, so it makes some sense for p1 to pick C.
Alternatively, and what occurred to me last, both these agents have some way to equally share their "profit" over Classical Decision Theory. For however much more utility than 3 p1 gets, p2 gets the same amount. This payoff point (p1-3=p2) does exists, but I'm not sure where it is without doing more math. Is this a well formulated game theoretic concept? I don't know, but it makes some sense to my idea of "fairness", and the kind of point two well-formulated agents should converge on.