Subgame-Perfect Equilibrium

Backward induction rules out non-credible threats in sequential games.

The idea

Subgame-perfect equilibrium is the solution concept for games in which players move in sequence and observe the moves before theirs; it refines Nash equilibrium by discarding equilibria that rest on threats no one would carry out.

In such a game, a strategy must name an action at every decision point the player could face, including points they never expect to reach. A subgame is the entire game that remains from one decision point onward. A strategy profile is a subgame-perfect equilibrium when it is a Nash equilibrium in every subgame, not merely in the game as a whole.

The condition has force off the equilibrium path. Nash equilibrium asks only that no player gain by deviating given the play that actually occurs, so an announced punishment that play never reaches goes untested, even when carrying it out would hurt the punisher. Such a plan is a non-credible threat, and subgame perfection rejects it, because the threatened action is not optimal in its own subgame.

In a finite game with perfect information, backward induction finds the subgame-perfect equilibria: solve the last subgames, where the mover takes the best available payoff; replace each solved subgame by the payoffs it delivers; and work backward until the first mover chooses among known outcomes.

The figure shows one such fold on a schematic tree: at the last decision point player $2$ compares the leaves $(b_{1}, b_{2})$ and $(c_{1}, c_{2})$, keeps the branch with the larger own payoff ($b_{2} > c_{2}$), and the whole subgame is replaced by its value $(b_{1}, b_{2})$, beside the root's other branch worth $(a_{1}, a_{2})$.

Ways to work on it

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