Sequential Rationality¶

Definition¶

In an extensive-form game, let the other players follow $\sigma_{-i}$ . Player $i$ 's strategy $\sigma_i$ is sequentially rational if, at every information set $h$ at which $i$ moves, the continuation of $\sigma_i$ from $h$ onwards is a best response to $\sigma_{-i}$ , that is, it maximizes $i$ 's expected payoff from $h$ onwards. This must hold whether or not $h$ is reached when $\sigma$ is played.

If $h$ is a single node, the expected payoff from $h$ is well defined. If $h$ contains several nodes, it is computed with a belief $\mu(\cdot\mid h)$ $h)$ , a probability distribution over the nodes of $h$ , so sequential rationality is defined relative to these beliefs.

Interpretation¶

Sequential rationality says a player optimizes whenever she is called upon to move, not only when planning at the start of the game. It formalizes credibility: a threat or promise is credible only if carrying it out would be optimal at the moment it has to be carried out.

Entry game tree: the entrant chooses Out, ending at (0,2), or In; the incumbent then chooses Accommodate, giving (1,1) and highlighted with In, or the dashed Fight, giving (−1,−1).

The figure draws the entry game of the Examples, with payoffs listed as (entrant, incumbent). In the Nash equilibrium (Out, Fight) the incumbent's node is never reached, so the threat to fight costs her nothing. Sequential rationality asks what she would do if the node were reached: Accommodate gives $1>-1$ $- 1$ , so Fight is not a credible plan, and only the highlighted path (In, Accommodate) survives.

Nash equilibrium does not impose this. In a Nash equilibrium a player must optimize at information sets that are reached with positive probability, but at unreached information sets any action is consistent with ex-ante optimality, because it does not change her payoff. This is the gap through which non-credible threats enter.

Backward induction builds a profile that is sequentially rational at every node of a perfect-information game, and subgame-perfect Nash equilibrium imposes the same requirement subgame by subgame. Where information sets contain several nodes, beliefs are needed; this leads to perfect Bayesian equilibrium.

Key Results¶

In a Nash equilibrium, each player's strategy is optimal at every information set reached with positive probability; off the path it is unrestricted.
In a finite game of perfect information, a pure-strategy profile in which every player is sequentially rational at every node is a backward-induction profile, hence an SPNE, and conversely.
At a non-singleton information set, the optimal action can change with the belief $\mu(\cdot\mid h)$ $h)$ .
In multistage and repeated games, the one-stage deviation principle reduces the check to single-period deviations.

Examples¶

Entry game. Out gives $(0,2)$ to (entrant, incumbent); after In, Accommodate gives $(1,1)$ and Fight $(-1,-1)$ . The profile (Out, Fight) is a Nash equilibrium, but Fight is not sequentially rational: at the incumbent's node, Accommodate gives $1>-1$ $- 1$ . Only (In, Accommodate) is sequentially rational for both players.

Optimality depending on beliefs. An incumbent observes entry but not whether the entrant is ready ( $R$ ) or unready ( $U$ ); both nodes are in one information set. Her payoffs are: against $R$ , Fight $-1$ and Accommodate $1$ ; against $U$ , Fight $2$ and Accommodate $1$ . With belief $\mu$ that the entrant is ready,

\mathbb{E}[\text{Fight}]=-\mu+2(1-\mu)=2-3\mu,\qquad \mathbb{E}[\text{Accommodate}]=1.

Fight is sequentially rational if and only if $2-3\mu\ge1$ $1$ , that is $\mu\le\tfrac{1}{3}$ $\frac{1}{3}$ ; Accommodate is sequentially rational if and only if $\mu\ge\tfrac{1}{3}$ $\frac{1}{3}$ . At $\mu=\tfrac{1}{3}$ both are.

Prerequisites¶

Sources¶

Game Theory · Week 3

Aliases: sequentially rational, sequentially rational strategy Keywords: credibility, optimal at every information set, off-path optimality, non-credible threat, beliefs at an information set, entry game Symbols: mu, sigma

Last updated: 22 October 2026 19:04