subgame perfect equilibrium

Definition

A strategy vector $\sigma^*$ (in mixed strategies or behavior strategies) in an extensive form game $\Gamma$ is called a subgame perfect equilibrium if for every subgame, the restriction of the strategy vector $\sigma^*$ to the subgame is a Nash equilibrium of that subgame: for every player $i \in N$, every strategy $\sigma_i$, and every subgame $\Gamma(x)$,

Remark

Since every game is a subgame of itself, by definition, every subgame perfect equilibrium is a Nash equilibrium.

Definition (subgame)

SPNE for strategy profile in terms of horizon restriction

A strategy profile $\{\mu_i^*\}^N_i$ is a subgame perfect Nash equilibrium (SPNE) if for any $k$, and its associated $h_k \in H_k$, the restriction $\{\mu_i\}^N_{i=1}$ to the horizon $t \in \{k,k+1,...,T\}$ is also a Nash equilibrium.

$J_i^{(k)}(\mu_i^*, \mu_{-i}^* | h_k) \geq J_i^{(k)}(\mu_i, \mu_{-i}^* | h_k)$ for every feasible $\mu_i \in \Gamma_i$, $h_

Notes:

• consider finite horizon problem

SPNE for infinite horizon game

#incomplete

Theorem

In an extensive form game without non-trivial subgames, every Nash equilibrium (in mixed strategies or behavior strategies) is a subgame perfect equilibrium.

Theorem

Let $\sigma^*$ be a Nash equilibrium (in mixed strategies or behavior strategies) of an extensive-form game $\Gamma$, and let $\Gamma(x)$ be a subgame of $\Gamma$. If $\mathbf{P}_{\sigma^*}(x) > 0$, then the strategy vector $\sigma^*$ restricted the subgame $\Gamma(x)$ is a Nash equilibrium (in mixed strategies or behavior strategies) of $\Gamma(x)$.

See also

• perfect equilibrium

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 254-255.