User:Editeur24/ultimatumgame

Nash equilibrium or subgame perfect equilibrium

It is very common for games to have only weak Nash equilibria. One reason is that a player may choose his strategy purposely to drive his rival to indifference between two actions.

For example, in the Ultimatum game, the proposer chooses a share s of a pie to  offer the receiver. 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 Nash 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 a Nash equilibrium, because the proposer would deviate to s' = s - \epsilon for some small number \epsilon and the receiver's best response would still be to accept. The weak equilibrium is an artifact of the strategy space being continuous. If instead of a pie the prize were a dollar and the proposer could only make offers in one-cent units, there would be two equilibria: the weak (s=0, Accept) and the strong (s=1, Accept s>=1, Reject s=0). The These equilibria are very similar in their outcome, so modelling the strategies as continuous makes little difference.