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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?

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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 $1 - \epsilon$.

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