Psychological Games

Ennio Stacchetti, who demonstrate the concept through a series of simple games in which one of the players do not have any moves. Two of these games are instructive.

Consider first a man who has already asked a woman out on a date. He doesn’t know if the woman likes him or doesn’t like him. Whether or not she will accept is a function of whether she likes him and of whether she believes he is confident that she will accept. His own utility of her either accepting or rejecting the date depends upon whether he expects that she will accept or not. The more confident he is, the more he will enjoy the date. The more pessimistic he is, the less he will enjoy the date, but also the less disappointed he will be if she says no.

Suppose that nature determines whether the woman likes him or not, and that the probability of her liking him is 0.50. She wants to go out with a relatively confident man. If the woman likes him, her utility of accepting the date is given by uwomanacceptlike = 3q + s, where q is her belief regarding the man’s belief of the probability she would say yes if she likes him. The variable s represents her belief as to his belief of the probability she would say yes if she did not like him. Thus, she receives a higher utility from going on the date if she thinks he believes she will say yes whether she likes him or not. Her utility of saying no is given by uwomanrejectlike = 1. Alternatively, if she does not like him, her utility of accepting is uwomanaccept doesn’t like = 0 and the utility of rejecting is uwomanrejectdoesn’t like = 1.

His utility is given by uman accept = 1 + q + s, whether she likes him or not, where q is his belief regarding her probability of accepting if she likes him and s is his belief regarding her probability of accepting if she does not. Thus, his utility of her accepting is higher if it is a high-probability event—he doesn’t like the risk of being rejected even if he is not rejected. Alternatively, if she rejects, he will receive umanreject = − 4q + s whether or not she likes him. If he is rejected, he would rather rejection to have been a high-probability event—he wouldn’t have had high expectations.

The solution concept for a psychological game requires that all beliefs reflect reality, just as in the fairness equilibrium. In this case, this means s = s = r, where r is the actual probability of her accepting his offer if she doesn’t like him determined by her employed strategy, and q = q = p, where p is the actual probability of her accepting his offer if she likes him determined by her strategy. Clearly, if she does not like him, she will always reject in any equilibrium obtaining a utility of 1 rather than 0. Thus, s = s = r = 0. If we examine only pure strategies (where actions are taken with certainty) then her accepting with probability 1 given she likes him would imply q = q = p = 1, and thus she would receive uwomanacceptlike = 3 which is greater than uwomanrejectlike = 1. This would constitute one equilibrium. In this case, because she is so likely to accept, he is very confident and is thus a much more enjoyable date.

Alternatively, if she rejects with probability 1 given she likes him, then q = q = p = 0, and thus she would receive uwomanrejectlike = 1, which is greater than uwomanaccept like = 0. This also constitutes an equilibrium. In this case, because she is so discerning, he lacks confidence, which means he wouldn’t have been much of a date anyway.

If all players have realistic beliefs and the solution is a Nash equilibrium given those beliefs, we call it a psychological equilibrium. The key to finding the psychological equilibrium is noting that the payoffs to each strategy change depending on the beliefs of each player. Thus, in equilibrium, the beliefs must be consistent with reality, and the players must still have the incentive to play the strategies implied by their beliefs.

In general, we look for a psychological equilibrium as the solution to any game in which beliefs are incorporated into the payoff functions of the participants. It is possible to incorporate higher-order beliefs (e.g., Player 1’s belief about Player 2’s belief about Player 1 and so on) of any order into the psychological game. Such games have been key to interpreting the odd behavior often observed in practice when examining games in an experimental lab.

EXAMPLE 15.8 Guilt and Grit

The game that virtually all students of economics are familiar with is the prisoner’s dilemma, pictured in Figure 15.5. The story, as you recall, is that two men have committed a crime. The police have arrested the men but have insufficient evidence. They place the prisoners in separate rooms for interrogation. If both prisoners keep their mouths shut, both get short sentences. If one confesses, implicating the other, the one who confesses goes free and the betrayed prisoner gets a long sentence. If both confess, both get moderate sentences.

The Nash equilibrium for this game is given by (confess, confess). Given Prisoner 1 chooses to confess (and implicate Prisoner 2), Prisoner 2 does strictly worse by choosing not to confess (and implicate Prisoner 1). If Prisoner 1 chooses not to confess, Prisoner 2 does strictly better by confessing and vice versa.

But suppose that, in addition to the time in jail, we considered the prisoner’s feelings of guilt or vindication in their payoffs. For example, suppose that Prisoner 1 is the dominant thief and the mastermind of their heist. He feels Prisoner 2 is somewhat spineless and dimwitted. Thus, Prisoner 1 would take some satisfaction in demonstrating his grit by not confessing if he believes Prisoner 2 is going to waffle under pressure and confess. Thus, Prisoner 1’s perception of the probability that Prisoner 2 will confess will increase the payoff to not confessing and decrease the payoff to confessing whether Prisoner 2 actually confesses. Prisoner 1 would feel this satisfaction in addition to the disutility resulting from the substantial jail sentence he will serve. Prisoner 1 will feel like a wimp if he decides to confess and Prisoner 2 does not even though this would result in Prisoner 1 going free. Thus, Prisoner 1’s perception of the probability of Prisoner 2 confessing enhances the payoff to confessing and decreases the payoff to not confessing if Prisoner 2 doesn’t confess. Alternatively, Prisoner 2 is rather timid and also very worried about how Prisoner 1 perceives him. Thus, he would feel embarrassed and guilty if he is found to have confessed while Prisoner 1 did not. Thus, his utility of confessing if Player 1 does not confess is enhanced if it was highly probable Prisoner 1 was going to confess. Let p represent the beliefs of Prisoner 2 regarding Prisoner 1’s probability of confessing. Further, let q represent Prisoner 1’s beliefs regarding Prisoner 2’s probability of confessing. Then, we could represent the prisoners’ payoffs as in Figure 15.6.

In this case, the Nash equilibrium of the standard game could not be the Nash equilibrium of the psychological game. In that outcome, both choose to confess, so p = p = q = q = 1. In this case, Prisoner 1 obtains 0 and would clearly prefer not to confess. Alternatively, consider the outcome in which neither prisoner confesses. In this case, p = p = q = q = 0. Prisoner 1 is indifferent between confessing and not confessing, receiving a utility of 5 in either case. Alternatively, Prisoner 2 strongly prefers to not