The ‘individual rationality condition’ is about the payoffs in equilibrium, not about the strategies. It says that the equilibrium payoff profile must yield to each player at least their minmax payoff. Here, the minmax payoff for a given player is −99.3 (which comes from the player best responding with 30 forever to everyone else setting their dials to 100 forever). The equilibrium payoff is −99 (which comes from everyone setting their dials to 99 forever). Since −99 > −99.3, the individual rationality condition of the Folk Theorem is satisfied.
I think the “at least” is an important part of this. If it yields more than their minimax payoff, either because the opponents are making mistakes, or have different payoffs than you think, or are just cruelly trying to break your model, there’s no debt created because there’s no cost to recoup.
The minimax expectation is 99.3 (the player sets to 30 and everyone else to 100). One possible bargaining/long-term repeated equilibrium is 99, where everyone chooses 99, and punishes anyone who sets to 100 by setting themselves to 100 for some time. But it would be just as valid to expect the long-term equilibrium to be 30, and punish anyone who sets to 31 or higher. I couldn’t tell from the paper how much communication was allowed between players, but it seems to assume some mutual knowledge of each other’s utility and what a given level of “punishment” achieves.
In no case do you need to punish someone who’s unilaterally giving you BETTER than your long-term equilibrium expectation.
Oh yeah, the Folk Theorem is totally consistent with the Nash equilibrium of the repeated game here being ‘everyone plays 30 forever’, since the payoff profile ‘-30 for everyone’ is feasible and individually-rational. In fact, this is the unique NE of the stage game and also the unique subgame-perfect NE of any finitely repeated version of the game.
To sustain ‘-30 for everyone forever’, I don’t even need a punishment for off-equilibrium deviations. The strategy for everyone can just be ‘unconditionally play 30 forever’ and there is no profitable unilateral deviation for anyone here.
The relevant Folk Theorem here just says that any feasible and individually-rational payoff profile in the stage game (i.e. setting dials at a given time) is a Nash equilibrium payoff profile in the infinitely repeated game. Here, that’s everything in the interval [-99.3, −30] for a given player. The theorem itself doesn’t really help constrain our expectations about which of the possible Nash equilibria will in fact be played in the game.
The ‘individual rationality condition’ is about the payoffs in equilibrium, not about the strategies. It says that the equilibrium payoff profile must yield to each player at least their minmax payoff. Here, the minmax payoff for a given player is −99.3 (which comes from the player best responding with 30 forever to everyone else setting their dials to 100 forever). The equilibrium payoff is −99 (which comes from everyone setting their dials to 99 forever). Since −99 > −99.3, the individual rationality condition of the Folk Theorem is satisfied.
I think the “at least” is an important part of this. If it yields more than their minimax payoff, either because the opponents are making mistakes, or have different payoffs than you think, or are just cruelly trying to break your model, there’s no debt created because there’s no cost to recoup.
The minimax expectation is 99.3 (the player sets to 30 and everyone else to 100). One possible bargaining/long-term repeated equilibrium is 99, where everyone chooses 99, and punishes anyone who sets to 100 by setting themselves to 100 for some time. But it would be just as valid to expect the long-term equilibrium to be 30, and punish anyone who sets to 31 or higher. I couldn’t tell from the paper how much communication was allowed between players, but it seems to assume some mutual knowledge of each other’s utility and what a given level of “punishment” achieves.
In no case do you need to punish someone who’s unilaterally giving you BETTER than your long-term equilibrium expectation.
Oh yeah, the Folk Theorem is totally consistent with the Nash equilibrium of the repeated game here being ‘everyone plays 30 forever’, since the payoff profile ‘-30 for everyone’ is feasible and individually-rational. In fact, this is the unique NE of the stage game and also the unique subgame-perfect NE of any finitely repeated version of the game.
To sustain ‘-30 for everyone forever’, I don’t even need a punishment for off-equilibrium deviations. The strategy for everyone can just be ‘unconditionally play 30 forever’ and there is no profitable unilateral deviation for anyone here.
The relevant Folk Theorem here just says that any feasible and individually-rational payoff profile in the stage game (i.e. setting dials at a given time) is a Nash equilibrium payoff profile in the infinitely repeated game. Here, that’s everything in the interval [-99.3, −30] for a given player. The theorem itself doesn’t really help constrain our expectations about which of the possible Nash equilibria will in fact be played in the game.