# Subgame Perfect Equilibria in a one-stage game

Consider the following one stage game, with two players A and B.

There is a pie which is to be divided between the two players. A can offer B any fraction of the cake, which B can accept or reject. If B accepts the offer (say $$1-x$$), then A gets $$x$$ and B gets $$1-x$$. If B rejects the offer, both players get $$0$$.

As far as the Subgame Perfect Equilibria are concerned, ($$1$$,accept always) is definitely one such equilibrium. What about ($$1-\epsilon$$,reject only when $$x=1$$), where $$\epsilon$$ is a very small number?

This is not even a Nash equilibrium, because the proposer has profitable deviations: e.g., A can propose $$1 - (\epsilon/2)$$ and B would accept it. This gives A a higher payoff: $$1 - (\epsilon/2)$$ instead of $$1 - \epsilon$$.