Grim Trigger Strategy¶

Definition¶

In an infinitely repeated game with discount factor $\delta\in(0,1)$ $(0, 1)$ , fix a target action profile $a^c$ and a stage-game Nash equilibrium $a^*$ . The grim trigger strategy of player $i$ is

play $a_i^c$ at $t=0$ and in every period $t$ whose history $h^t$ consists only of $a^c$ ;
play $a_i^*$ after every other history, that is, forever once anyone has deviated from $a^c$ .

It is the trigger strategy whose punishment phase never ends.

Interpretation¶

Among trigger strategies that punish with a given stage equilibrium $a^*$ that gives every player less than $a^c$ , grim trigger punishes longest, so it needs the least patience. Its incentive check is short. After any deviation, play is $a^*$ forever, which is a Nash equilibrium of every such subgame. On the cooperative path, a player compares the stream $v_i(a^c)$ forever with the best one-period gain followed by $v_i(a^*)$ forever.

Per-period payoffs in the Prisoner's Dilemma of the Examples: cooperation pays 3 every period, while a deviation pays 5 once and then 1 in every later period.

The figure draws this comparison for the Prisoner's Dilemma of the Examples. Cooperating gives $3$ in every period (blue); deviating gives $5$ , the payoff of defecting against a cooperator, once and then, because the opponent's grim trigger plays $D$ forever, the mutual-defection payoff $1$ in every later period (orange). The deviation gains $2$ now and loses $2$ in every later period, so it pays only when the future is discounted heavily: cooperation is optimal if and only if $\delta\ge\tfrac{1}{2}$ $\frac{1}{2}$ .

The filled parts of the bars are discounted payoffs. With the default payoffs $T=5$ $5$ , $R=3$ $3$ , $P=1$ and $S=0$ $0$ , deviation pays at $\delta=0.4$ and cooperation is optimal from $\delta=\tfrac12$ $\frac{1}{2}$ . Changing the payoffs changes this threshold; the punishment must also remain credible.

The profile in which everyone uses grim trigger is a subgame-perfect Nash equilibrium if and only if, for every player $i$ ,

v_i(a^c)\ge(1-\delta)\,d_i+\delta\,v_i(a^*),\qquad d_i=\max_{a_i\ne a^c_i}v_i(a_i,a^c_{-i}),

that is, $\delta\ge\big(d_i-v_i(a^c)\big)/\big(d_i-v_i(a^*)\big)$ when $d_i>v_i(a^*)$ $v_{i} (a^{*})$ .

Grim trigger is unforgiving: a single mistake destroys cooperation forever, even though both players would prefer to restart. This makes it fragile as a description of behaviour, but it is the standard device for showing which outcomes can be sustained at all.

Key Results¶

In a Prisoner's Dilemma, grim trigger sustains $(C,C)$ in every period as a subgame-perfect equilibrium if and only if $\delta$ is at least the one-period gain from defecting against a cooperator divided by the difference between that defection payoff and the mutual-defection payoff; in the Examples, $\delta\ge\frac{5-3}{5-1}=\frac{1}{2}$ $\frac{1}{2}$ .
The threshold lies in $(0,1)$ . It rises when the payoff from defecting against a cooperator or the mutual-defection payoff rises, and falls when the mutual-cooperation payoff rises.
Off the path, punishment is credible because $a^*$ is a stage-game Nash equilibrium.
Here $d_i$ is the best payoff from an action other than $a^c_i$ . If $a^c$ is itself a stage equilibrium, then $d_i\le v_i(a^c)$ $v_{i} (a^{c})$ , so deviating gains nothing today. The condition then holds for every $\delta$ if $v_i(a^c)\ge v_i(a^*)$ for all $i$ (in particular if $a^c=a^*$ $a^{*}$ ). If some player strictly prefers the punishment equilibrium, $v_i(a^*)>v_i(a^c)$ $v_{i} (a^{c})$ , it holds only for $\delta\le\big(v_i(a^c)-d_i\big)/\big(v_i(a^*)-d_i\big)$ $d_{i})$ : a strict stage equilibrium is sustained for small $\delta$ , and the condition fails for every $\delta>0$ when some deviation ties, $d_i=v_i(a^c)$ $v_{i} (a^{c})$ .

Examples¶

Prisoner's Dilemma. Take $v(C,C)=(3,3)$ $(3, 3)$ , $v(C,D)=(0,5)$ $(0, 5)$ , $v(D,C)=(5,0)$ $(5, 0)$ , $v(D,D)=(1,1)$ $(1, 1)$ . Cooperating forever gives $3/(1-\delta)$ in total; defecting now gives $5+\delta\cdot1/(1-\delta)$ $δ)$ . Cooperation is optimal if and only if

3\ge5(1-\delta)+\delta\iff4\delta\ge2 \iff\delta\ge\tfrac{1}{2}=\tfrac{5-3}{5-1}.

At $\delta=0.4$ a defector gets average $5(0.6)+0.4=3.4>3$ $3$ , so cooperation fails.

Collusion in a Cournot duopoly. In a Cournot duopoly with inverse demand $p=a-Q$ and unit cost $c<a$ $a$ , write $m=(a-c)^2$ $c)^{2}$ . Producing half the monopoly output, $(a-c)/4$ each, gives profit $m/8$ ; the best deviation against it is $3(a-c)/8$ with profit $9m/64$ ; Cournot play gives $m/9$ . Grim trigger sustains collusion if and only if

\delta\ge\frac{9/64-1/8}{9/64-1/9}=\frac{1/64}{17/576} =\frac{9}{17}\approx0.53.

Prerequisites¶

Sources¶

Game Theory · Week 4

Aliases: grim trigger, grim strategy, grim-trigger strategy Keywords: permanent punishment, delta >= (5-3)/(5-1), critical discount factor, cooperation in the repeated Prisoner's Dilemma, collusion 9/17 Symbols: delta, element of

Last updated: 27 October 2026 11:52