Hmm. The results appear quite different if you allow communication and repeated plays. And they also introduce something which seems slightly different than Trembling Hand (Perhaps Trembling Memory?)
With communication and repeated plays:
Assume all potential Player 1′s credibly precommit to flip a fair coin and based on the toss, pick B half the time, and C half the time.
All potential Player 2′s would know this, and assuming they expect Player 1 to almost always follow the precommitment, would pick Y, because they would maximize their expected payout. (50% chance of 2 means expected payout of 1, compared to picking X, where 50% chance of 1 means expected payout of 1⁄2.)
All potential Player 1′s, based on following that precommitment universally, and having Player 2′s always pick Y, will get 2 50% of the time and 6 50% of the time, which would get an expected per game payout of 4.
This seems better for everyone (by about one point per game) then Player 1 only choosing A.
With Trembling Memory:
Assume further that some of the time, Player 2′s memory trembles and he forgets about the precommitments and the fact that this game is played repeatedly.
So If Player 2 suspects for instance, that they MIGHT be in the case above, but have forgotten important facts (they are incorrect about this being a one-off game, they are incorrectly assessing the state of common knowledge, but they are correctly assessing the current payoff structure of this particular game) then following those suspicions, it would still make sense for them to choose Y, and it would also explain why Player 1 chose something that wasn’t A.
However, nothing would seem to prevent Player 2 from suspecting other possibilities. (For instance, under the assumption that Player 1 hits player 2 with Amnesia dust before every game, knowing that Player 2 will be forced into a memory trembled and will believe the above, Player 1 could play C every time, with player 2 drawing predictably incorrect conclusions and playing Y at no benefit.)
I’m not sure how to model a situation with trembling memory, though, so I would not be surprised if I was missing something.
Hmm. The results appear quite different if you allow communication and repeated plays. And they also introduce something which seems slightly different than Trembling Hand (Perhaps Trembling Memory?)
With communication and repeated plays:
Assume all potential Player 1′s credibly precommit to flip a fair coin and based on the toss, pick B half the time, and C half the time.
All potential Player 2′s would know this, and assuming they expect Player 1 to almost always follow the precommitment, would pick Y, because they would maximize their expected payout. (50% chance of 2 means expected payout of 1, compared to picking X, where 50% chance of 1 means expected payout of 1⁄2.)
All potential Player 1′s, based on following that precommitment universally, and having Player 2′s always pick Y, will get 2 50% of the time and 6 50% of the time, which would get an expected per game payout of 4.
This seems better for everyone (by about one point per game) then Player 1 only choosing A.
With Trembling Memory:
Assume further that some of the time, Player 2′s memory trembles and he forgets about the precommitments and the fact that this game is played repeatedly.
So If Player 2 suspects for instance, that they MIGHT be in the case above, but have forgotten important facts (they are incorrect about this being a one-off game, they are incorrectly assessing the state of common knowledge, but they are correctly assessing the current payoff structure of this particular game) then following those suspicions, it would still make sense for them to choose Y, and it would also explain why Player 1 chose something that wasn’t A.
However, nothing would seem to prevent Player 2 from suspecting other possibilities. (For instance, under the assumption that Player 1 hits player 2 with Amnesia dust before every game, knowing that Player 2 will be forced into a memory trembled and will believe the above, Player 1 could play C every time, with player 2 drawing predictably incorrect conclusions and playing Y at no benefit.)
I’m not sure how to model a situation with trembling memory, though, so I would not be surprised if I was missing something.