Introducing Fairness to Game Theory
In class, we made two major simplifying assumptions in our discussion of game theory: first, that everything a player cares about is summarized in his or her own payoffs; and second, that each individual chooses a strategy to maximize his or her own payoffs. Because these assumptions are so essential to the conclusions we draw, it is important that we take the time to examine when they hold and when they fail. In a seminal 1993 paper, Harvard economist Matthew Rabin challenges these two assumptions by incorporating notions of fairness into game theory, ultimately concluding that conventional game theory fails to accurately describe human behavior when it comes to games dealing with unequal payoffs between players.
In order to come to this conclusion, he makes three important assumptions:
- People are willing to sacrifice their own material well-being to help those who are being kind
- People are willing to sacrifice their own material well-being to punish those who are being unkind
- Both of these motivations have a greater effect on behavior as the material cost of sacrificing becomes smaller
He then proceeds to define a new type of equilibrium, which he calls a “fairness equilibrium,” that reflects the implications of these three assumptions. In particular, he uses the concept of a ‘mutual-max’ outcome (an outcome in which, given the other person’s behavior, each person maximizes the other’s material payoffs) and a ‘mutual-min’ outcome (an outcome in which, given the other person’s behavior, each person minimizes the other’s material payoffs) to define a fairness equilibrium as follows:
- Any Nash equilibrium that is either a mutual-max outcome or mutual-min outcome is also a fairness equilibrium
- If material payoffs are small, then, roughly, an outcome is a fairness equilibrium if and only if it is a mutual-max or a mutual-min outcome
- If material payoffs are large, then, roughly, an outcome is a fairness equilibrium if and only if it is a Nash equilibrium
This new definition of a fairness equilibrium highlights an important principle: when material payoffs are small, people care more about rewarding kindness and punishing unkindness than about material payoffs.
This principle is best illustrated by a game called the ultimatum game. The rules of the ultimatum game are simple: a proposer offers some division of a certain amount of money to a decider, and if the decider agrees, the two split the money as per the proposal, but if the decider says no, they both get no money. In theory, proposers should never offer more than one penny, and deciders should accept any offer of at least one penny.
Yet the empirical evidence in Rabin’s paper shows that when it comes to low material payoffs deciders are willing to punish unfair offers by rejecting them, and proposers tend to make fair offers. If the ultimatum game were conducted with $1, then most deciders would reject a proposed split of ($0.90, $0.10). However, once the material payoffs are large enough, people start to care less about fairness. Hence, data shows that if the ultimatum game were conducted with $10 million, the vast majority of deciders would accept a proposed split of ($9 million, $1 million).
Often, introducing fairness to a game creates a set of fairness equilibria that are totally different from the Nash equilibria. Let’s consider the game of chicken and the Nash equilibrium where player one swerves and player two does not. If the material payoffs are small enough, then this equilibrium is not a Nash equilibrium because player two might switch to swerving out of a desire to punish player one for acting unkindly by swerving—even if doing so means that player one will incur larger losses than if he or she were to continue not swerving. Thus, in this case, (swerve, swerve) is a fairness equilibrium, while (swerve, don’t swerve) is not.
Source: http://www.jstor.org/stable/2117561