Ultimatum game
The ultimatum game is a game that has become a popular instrument of economic experiments. An early description is by Nobel laureate John Harsanyi in 1961.[1] One player, the proposer, is endowed with a sum of money. The proposer is tasked with splitting it with another player, the responder (who knows what the total sum is). Once the proposer communicates their decision, the responder may accept it or reject it. If the responder accepts, the money is split per the proposal; if the responder rejects, both players receive nothing. Both players know in advance the consequences of the responder accepting or rejecting the offer.
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:
However, only the first set of Nash equilibria satisfies a more restrictive equilibrium concept, subgame perfection. The game can be viewed as having two subgames: the subgame where the proposer makes a fair offer, and the subgame where the proposer makes an unfair offer. A perfect-subgame equilibrium occurs when there are Nash Equilibria in every subgame, that players have no incentive to deviate from.[2] In both subgames, it benefits the responder to accept the offer. So, the second set of Nash equilibria above is not subgame perfect: the responder can choose a better strategy for one of the subgames.
Multi-valued or continuous strategies[edit]
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).[3]
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 - for some small number and the receiver's best response would still be to accept. The weak equilibrium is an artifact of the strategy space being continuous.
Sociological applications[edit]
The ultimatum game is important from a sociological perspective, because it illustrates the human unwillingness to accept injustice. The tendency to refuse small offers may also be seen as relevant to the concept of honour.
The extent to which people are willing to tolerate different distributions of the reward from "cooperative" ventures results in inequality that is, measurably, exponential across the strata of management within large corporations. See also: Inequity aversion within companies.
Variants[edit]
In the "competitive ultimatum game" there are many proposers and the responder can accept at most one of their offers: With more than three (naïve) proposers the responder is usually offered almost the entire endowment[52] (which would be the Nash Equilibrium assuming no collusion among proposers).
In the "ultimatum game with tipping", a tip is allowed from responder back to proposer, a feature of the trust game, and net splits tend to be more equitable.[53]
The "reverse ultimatum game" gives more power to the responder by giving the proposer the right to offer as many divisions of the endowment as they like. Now the game only ends when the responder accepts an offer or abandons the game, and therefore the proposer tends to receive slightly less than half of the initial endowment.[54]
Incomplete information ultimatum games: Some authors have studied variants of the ultimatum game in which either the proposer or the responder has private information about the size of the pie to be divided.[55][56] These experiments connect the ultimatum game to principal-agent problems studied in contract theory.
The pirate game illustrates a variant with more than two participants with voting power, as illustrated in Ian Stewart's "A Puzzle for Pirates".[57]