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. 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 his 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. Equilibrium analysis 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. There are two strategies available to the proposer: propose a fair split, or propose an unfair split. For each of these two splits, the responder can choose to accept or reject, which means that there are four strategies available to the responder: always accept, always reject, accept only a fair split, or accept only an unfair split. A Nash equilibrium is a pair of strategies (one for the proposer and one for the responder), where neither party can improve their reward by changing strategy. It always benefits the responder to accept the offer, as receiving something is better than receiving nothing. Meanwhile, it benefits the proposer to make an offer that the responder will accept; furthermore, if the responder would accept any offer, then it benefits the proposer to switch from a fair to an unfair offer. So, there are three Nash equilibria for this game: The proposer makes a fair offer; the responder would only accept a fair offer. The proposer makes an unfair offer; the responder would only accept a fair offer. The proposer makes an unfair offer; the responder would accept any offer. However, only the last Nash equilibrium satisfies a more restrictive equilibrium concept, subgame perfection. The above 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