Ultimatum game

The ultimatum game is a popular experimental economics game in which two players interact to decide how to divide a sum of money, first described by Nobel laureate John Harsanyi in 1961. The first player, the proposer, proposes a division of the sum with the second player, the responder. The responder can either accept the proposed division or reject it. If the responder accepts, the money is split according to the proposal; if the responder rejects, neither player receives anything. Both players know in advance the rules of the game.

The game is typically designed as a one-shot interaction to isolate immediate reactions to fairness, thereby minimizing the influence of potential future interactions. However, even within this one-shot context, participants' decision-making processes may implicitly involve considering the potential consequences of repeated interactions, due to the fact that humans have evolved within societies that interact repeatedly. This design is crucial for observing pure, unadulterated responses to the proposed division.

For ease of exposition, the simple example illustrated above can be considered, where the proposer has two options: a fair split, or an unfair split. The argument given in this section can be extended to the more general case where the proposer can choose from many different splits.

A Nash equilibrium is a set of strategies (one for the proposer and one for the responder in this case), where no individual party can improve their reward by changing strategy. If the proposer always makes an unfair offer, the responder will do best by always accepting the offer, and the proposer will maximize their reward. Although it always benefits the responder to accept even unfair offers, the responder can adopt a strategy that rejects unfair splits often enough to induce the proposer to always make a fair offer. Any change in strategy by the proposer will lower their reward. Any change in strategy by the responder will result in the same reward or less. Thus, there are two sets of Nash equilibria for this game:

In a non-repeated or finite-horizon ultimatum game, the first Nash equilibria (unfair offer, always accept) are the only that satisfy a stricter condition called subgame perfection equilibrium (SPE). The game can be viewed as having two subgames that repeat themselves: the subgame where the proposer makes a fair offer, and the subgame where the proposer makes an unfair offer. An SPE occurs when there are Nash Equilibria in every subgame, that players have no incentive to deviate from. Using backward induction, we see that in the final stage, the responder will always accept any offer. Therefore, in previous stages, the proposer will always offer the minimum amount. Thus, the responder's threat to reject unfair offers in the second Nash equilibrium is not credible in a finite setting.

However, in an infinite-horizon ultimatum game, the analysis changes significantly. Repeated interactions allow for strategies based on reputation and reciprocity. Discount factors become crucial, and the Folk Theorem suggests that many payoff distributions, including "fair" outcomes, can be supported as Nash equilibria, and potentially as subgame perfect equilibria. The one-shot deviation principle is used to verify SPE in these cases. Therefore, the conclusion that only the "unfair offer, always accept" equilibrium is SPE is specific to finite horizon games. Infinite horizon games can have many SPE.

The simplest version of the ultimatum game has two possible strategies for the proposer, Fair and Unfair. A more realistic version would allow for many possible offers. For example, the item being shared might be a dollar bill, worth 100 cents, in which case the proposer's strategy set would be all integers between 0 and 100, inclusive for their choice of offer, S. This would have two subgame perfect equilibria: (Proposer: S=0, Accepter: Accept), which is a weak equilibrium because the acceptor would be indifferent between their two possible strategies; and the strong (Proposer: S=1, Accepter: Accept if S>=1 and Reject if S=0).

The ultimatum game is also often modelled using a continuous strategy set. Suppose the proposer chooses a share S of a pie to offer the receiver, where S can be any real number between 0 and 1, inclusive. If the receiver accepts the offer, the proposer's payoff is (1-S) and the receiver's is S. If the receiver rejects the offer, both players get zero. The unique subgame perfect equilibrium is (S=0, Accept). It is weak because the receiver's payoff is 0 whether they accept or reject. No share with S > 0 is subgame perfect, because the proposer would deviate to S' = S - ${\displaystyle \epsilon }$ for some small number ${\displaystyle \epsilon }$ and the receiver's best response would still be to accept. The weak equilibrium is an artifact of the strategy space being continuous.