to deviate from. The theory relies on the assumption that players are rational and utility maximising. In both subgames, it benefits the responder to accept the offer. So, the first two Nash equilibria above are not subgame perfect: the responder can choose a better strategy for one of the subgames. Multi-valued or continuous strategies 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 his 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 his 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 he accepts or rejects. 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. Experimental results The first experimental analysis of the ultimatum game was by Werner Güth, Rolf Schmittberger, and Bernd Schwarze: Their experiments were widely imitated in a variety of settings. When carried out between members of a shared social group (e.g., a village, a tribe, a nation, humanity) people offer "fair" (i.e., 50:50) splits, and offers of less than 30% are often rejected. One limited study of monozygotic and dizygotic twins claims that genetic variation can have an effect on reactions to unfair offers, though the study failed to employ actual controls for environmental differences. It has also been found that delaying the responder's decision leads to people accepting "unfair" offers more often.