One-Stage Deviation Principle¶

Proposition¶

Consider a multistage game with observed actions that is either finite, or infinite with payoffs $\displaystyle \sum_{t\ge0}\delta^{t}v_i^t(a^t)$ , $\delta\in(0,1)$ $(0, 1)$ , and bounded stage payoffs. A strategy profile $s$ is a subgame-perfect Nash equilibrium if and only if there is no player $i$ , stage $t$ and history $h^t$ such that player $i$ strictly gains, in the subgame starting at $h^t$ , from a strategy $\hat s_i$ that

differs from $s_i$ only in the action chosen at $h^t$ , and
agrees with $s_i$ at every later history,

while the other players follow $s_{-i}$ .

Interpretation¶

A strategy can be changed in infinitely many ways, at many histories at once. The principle says it is enough to test the simplest deviations: change today's action once, then return to the strategy. The deviator still evaluates the full continuation, so tomorrow's punishment enters the comparison; "one-stage" refers to where the action changes, not to ignoring the future.

Two rows of stage outcomes under grim trigger: conforming gives (C,C) with payoff 3 every stage; deviating once gives (D,C) with 5 at stage t, then (D,D) with 1 ever after.

The figure draws the one-stage test from the grim-trigger example at a history with no deviation yet. The deviator changes only the action at $h^t$ (orange box) and then follows her strategy again, which after a deviation prescribes $D$ forever; the opponent does the same, so the continuation is $(D,D)$ with payoff $1$ . The comparison is between the whole rows, $3$ against $(1-\delta)5+\delta$ $δ$ , not between the stage- $t$ payoffs $3$ and $5$ alone; the deviation does not pay if and only if $\delta\ge1/2$ $1/2$ .

The proof idea: if some deviation is profitable, a finite one is (in a finite game trivially; in an infinite game because discounting with bounded payoffs makes far-future changes negligible). Take a profitable finite deviation with the fewest changes and look at the last history where it changes behaviour: if that change alone were not profitable, it could be undone without loss, giving a profitable deviation with fewer changes. So the last change is a profitable one-stage deviation. This is the same logic as the principle of optimality in dynamic programming.

The hypotheses matter. Observed actions make each history a subgame. Discounting (more generally, continuity at infinity) rules out deviations whose gain only appears "at infinity"; without it, a strategy can pass every one-stage test yet fail to be optimal.

Key Results¶

The check must be made at every history, including histories off the equilibrium path.
In a finite game the principle is equivalent to checking optimality by backward induction stage by stage.
For grim trigger strategies punishing with a stage-game Nash equilibrium, only the cooperative-phase test can fail.
Necessity is immediate (a one-stage deviation is a deviation); the content of the result is sufficiency.

Examples¶

Grim trigger in the Prisoner's Dilemma. Stage payoffs $v(C,C)=(3,3)$ $(3, 3)$ , $v(C,D)=(0,5)$ $(0, 5)$ , $v(D,C)=(5,0)$ $(5, 0)$ , $v(D,D)=(1,1)$ $(1, 1)$ ; both players use the grim trigger strategy. Two kinds of history:

No deviation yet. Conforming gives average payoff 3. Playing $D$ once and then following the strategy (which now prescribes $D$ forever) gives $(1-\delta)5+\delta\cdot1$ $1$ . No gain if and only if $\delta\ge1/2$ $1/2$ .
A deviation has occurred. Conforming gives 1. Playing $C$ once gives $(1-\delta)\cdot0+\delta\cdot1<1$ $1$ . Never profitable.

So the profile is subgame perfect if and only if $\delta\ge1/2$ $1/2$ .

A profile that fails. "Always play $C$ , whatever happened" fails the test at every history: at any history, playing $D$ once gives $(1-\delta)5+3\delta>3$ for every $\delta<1$ $1$ . It is not subgame perfect, and not even a Nash equilibrium.

Prerequisites¶

Sources¶

Game Theory · Week 4

Aliases: one-shot deviation principle, one-step deviation principle, OSDP, one-deviation property, single-deviation principle Keywords: single deviation, unimprovability, continuity at infinity, checking subgame perfection, profitable deviation, Prisoner's Dilemma Symbols: delta, element of, sum

Last updated: 17 October 2026 21:53