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.
Player 2 observes “not A” as a choice. Doesn’t player 2 still need to estimate the relative probabilities that B was chosen vs. that C was chosen?
Of course Player 2 doesn’t have access to Player 1′s source code, but that’s not an excuse to set those probabilities in a completely arbitrary manner. Player 2 has to decide the probability of B in a rational way, given the available (albeit scarce) evidence, which is the payoff matrix and the fact that A was not chosen.
It seems reasonable to imagine a space of strategies which would lead player 1 to not choose A, and assign probabilities to which strategy player 1 is using. Player 1 is probably making a shot for 6 points, meaning they are trying to tempt player 2 into choosing Y. Player 2 has to decide the probability that (Player 1 is using a strategy which results in [probability of B > 0]), in order to make that choice.
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.
Player 2 observes “not A” as a choice. Doesn’t player 2 still need to estimate the relative probabilities that B was chosen vs. that C was chosen?
Of course Player 2 doesn’t have access to Player 1′s source code, but that’s not an excuse to set those probabilities in a completely arbitrary manner. Player 2 has to decide the probability of B in a rational way, given the available (albeit scarce) evidence, which is the payoff matrix and the fact that A was not chosen.
It seems reasonable to imagine a space of strategies which would lead player 1 to not choose A, and assign probabilities to which strategy player 1 is using. Player 1 is probably making a shot for 6 points, meaning they are trying to tempt player 2 into choosing Y. Player 2 has to decide the probability that (Player 1 is using a strategy which results in [probability of B > 0]), in order to make that choice.