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James_Miller 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?
"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 "
No since this is not a subgame because of the uncertainty. From Wikipedia " In game theory, a subgame is any part (a subset) of a game that meets the following criteria...It has a single initial node that is the only member of that node's information set... "
I'm uncertain about what TDT/UDT would say.
Can you give an example a pair G1, G2 such that you consider G2 to be a "subgame" of G1?
To see that it is indeed a subgame:
Represent the whole game with a tree whose root node represents player 1 choosing whether to play A (leads to leaf node), or to enter the subgame at node S. Node S is the root of the subgame, representing player 1's choices to play B or C leading to nodes representing player 2 choice to play X or Y in those respective cases, each leading to leaf nodes.
Node S is the only node in its information set. The subgame contains all the descendants of S. The subgame contains all nodes in the same information set as any node in the subgame. It meets the criteria.
There is no uncertainty that screws up my argument. The whole point of talking about the subgame was to stop thinking about the possibility that player 1 chose A, because that had been observed not to happen. (Of course, I also argue that player 2 should be interested in logically causing player 1 not to have chosen A, but that gets beyond classical game theory.)
I'm sorry but "subgame" has a very specific definition in game theory which you are not being consistent with. Also, intuitively when you are in a subgame you can ignore everything outside of the subgame, playing as if it didn't exist. But when Player 2 moves he can't ignore A because the fact that Player 1 could have picked A but did not provides insight into whether Player 1 picked B or C. I am a game theorist.
I just explained in detail how the subgame I described meets the definition you linked to. If you are going to disagree, you should be pointing to some aspect of the definition I am not meeting.
If it is somehow the case that giving player 2 info about player 1 is advantageous for player 1, then player 2 should just ignore the info, and everything still plays out as in my analysis. If it is advantageous for player 2, then it just strengthens the case that player 1 should choose A.
I still think you are making a mistake, and should pay more attention to the object level discussion.
Let's try to find the source of our disagreement. Would you agree with the following:
"You can only have a subgame that excludes A if the fact that Player 1 has not picked A provides no useful information to Player 2 if Player 2 gets to move."
The definition you linked to doesn't say anything about entering subgame not giving the players information, so no, I would not agree with that.
I would agree that if it gave player 2 useful information, that should influence the analysis of the subgame.
(I also don't care very much whether we call this object within the game of how the strategies play out given that player 1 doesn't choose A a "subgame". I did not intend that technical definition when I used the term, but it did seem to match when I checked carefully when you objected, thinking that maybe there was a good motivation for the definition so it could indicated a problem with my argument if it didn't fit.)
I also disagree that player 1 not picking A provides useful information to player 2.
"I also disagree that player 1 not picking A provides useful information to player 2."
Player 1 gets 3 if he picks A and 2 if he picks B, so doesn't knowing that Player 1 did not pick A provide useful information as to whether he picked B?
The reason player 1 would choose B is not because it directly has a higher payout but because including B in a mixed strategy gives player 2 an incentive to include Y in its own mixed strategy, increasing the expected payoff of C for player 1. The fact that A dominates B is irrelevant. The fact that A has better expected utility than the subgame with B and C indicates that player 1 not choosing A is somehow irrational, but that doesn't give a useful way for player 2 to exploit this irrationality. (And in order for this to make sense for player 1, player 1 would need a way to counter exploit player 2's exploit, and for player 2 to try its exploit despite this possibility.)
"The reason player 1 would choose B is not because it directly has a higher payout but because including B in a mixed strategy gives player 2 an incentive to include Y in its own mixed strategy, "
No since Player 2 only observes Player 1's choice not what probabilities Player 1 used.