Multistage Game¶

Definition¶

A multistage game (with observed actions) is an extensive-form game played in stages $t=0,1,\dots,T-1$ $1$ , where the number of stages $T$ may be finite or infinite (then $t=0,1,2,\dots$ $0, 1, 2, \dots$ ). In each stage $t$ the players move simultaneously in a stage game $G_t$ , choosing actions $a_i^t\in A_i^t$ $A_{i}^{t}$ , and before stage $t+1$ every player observes the realized action profile $a^t=(a_1^t,\dots,a_n^t)$ $(a_{1}^{t}, \dots, a_{n}^{t})$ . Player $i$ 's total payoff is the discounted sum of stage payoffs

\sum_{t=0}^{T-1}\delta^{t}v_i^t(a^t),\qquad \delta\in(0,1],

with $\delta<1$ required when $T=\infty$ $\infty$ . Because play is observed, everything that has happened before stage $t$ is summarized by the history $h^t=(a^0,\dots,a^{t-1})$ $(a^{0}, \dots, a^{t - 1})$ , and a pure strategy for player $i$ assigns an action in stage $t$ to every possible $h^t$ . A repeated game is the special case $G_t=G$ for all $t$ .

Interpretation¶

The defining assumption is observed actions: at the start of each stage all players know the full history. In the game tree, the node at which stage $t$ begins after a given $h^t$ forms a singleton information set, so every history starts a subgame. This is what makes subgame-perfect Nash equilibrium tractable here: one checks that the continuation play after every history is an equilibrium.

Two-stage diagram: stage t = 0 plays (C,C) with payoffs 4,4; after the observed history (C,C) stage t = 1 plays (P,P) with 3,3, after any other history (D,D) with 1,1.

The figure draws the strategy from the example. The first stage ( $t =$ $t=0$ $0$ ) prescribes $(C,C)$ , which is not a stage equilibrium; the players then observe $a^0$ , and the second-stage action depends on that history: the good equilibrium $(P,P)$ after $h^1=(C,C)$ $(C, C)$ , the bad equilibrium $(D,D)$ after anything else. Both second-stage boxes are stage equilibria, so every second-stage subgame is fine. For player 1, conforming totals $4+3=7$ $7$ , while the best deviation in the first stage totals $5+1=6$ $6$ , so the threat of moving to the bad box sustains $(C,C)$ .

Conditioning on history is what creates new equilibria. A player can be rewarded with a good stage-game equilibrium tomorrow for playing a non-equilibrium action today, and punished with a bad one if she does not. With finitely many stages this works only if some later stage game has more than one equilibrium; when every stage game has a unique equilibrium there is nothing to condition on. With infinitely many stages, as in an infinitely repeated game, the threat of future punishment can work even with a unique stage equilibrium.

Do not read "multistage" as "players move one after another": within a stage the moves are simultaneous, and the dynamic structure comes from the sequence of stages.

Key Results¶

Each history $h^t$ with $t\le T-1$ begins a subgame; for $t=0$ the empty history gives the whole game, which is a subgame but not a proper subgame, and every $t\ge1$ gives a proper subgame.
Playing a stage-game Nash equilibrium of $G_t$ in each stage, independently of history, is a subgame-perfect equilibrium.
If $T$ is finite and every $G_t$ has a unique Nash equilibrium, the game has a unique subgame-perfect equilibrium, obtained by backward induction: play that equilibrium in every stage after every history.
If some later stage game has two equilibria with different payoffs, a non-equilibrium profile can be sustained in earlier stages, provided each player's payoff difference between the two equilibria outweighs her one-stage gain from deviating.
Subgame perfection can be checked one stage at a time by the one-stage deviation principle.

Examples¶

Reward and punishment with two stage equilibria. The stage game below is played twice with $\delta=1$ $1$ ; rows are player 1's actions and the first entry is player 1's payoff.

	$C$	$D$	$P$
$C$	$4,4$	$0,5$	$0,0$
$D$	$5,0$	$1,1$	$0,0$
$P$	$0,0$	$0,0$	$3,3$

Its pure Nash equilibria are $(D,D)$ and $(P,P)$ ; $(C,C)$ is not one, since $D$ earns $5>4$ against $C$ . Strategy for each player: play $C$ at $t=0$ $0$ ; at $t=1$ play $P$ if $h^1=(C,C)$ and $D$ otherwise. Every second-stage subgame prescribes a stage equilibrium. At $t=0$ $0$ , conforming gives $4+3=7$ $7$ ; the best deviation, $D$ , gives $5+1=6$ $6$ . Hence $(C,C)$ is played in the first stage of a subgame-perfect equilibrium.

Prerequisites¶

Sources¶

Game Theory · Week 4

Aliases: multistage game with observed actions, multi-stage game Keywords: sequence of stage games, observed actions, perfect monitoring, two-stage game, conditional strategies Symbols: delta, element of, infinity, sum

Last updated: 14 October 2026 08:21