Infinitely Repeated Games: Discounting, SPNE, Grim Trigger

There is no natural last period, so the “last‑period effect” that forces defection in finite games disappears. Without a fixed endpoint, the set of possible equilibria becomes very rich, allowing cooperation to be sustained even in games that are hostile in a one‑shot setting.

Framework Setup

The stage game (G) defines the action set (A) and the one‑period payoff function (u_i). The repeated interaction runs over an infinite horizon (t = 0,1,2,\dots). Payoffs are discounted by a factor (\delta \in (0,1)), so a player’s total payoff is

[ \sum_{t=0}^{\infty}\delta^{t}u_i(a_t). ]

The discount factor can be interpreted as (1/(1+r)) for monetary payoffs, as the probability that the game continues to the next period, or simply as a measure of impatience. The notation (G(\delta)) denotes the infinitely repeated game with discount factor (\delta).

Strategy Definition

A strategy for player (i) maps every possible history (h\in H) to an action in (A_i). Histories record the sequence of past action profiles, and superscripts indicate the period while subscripts identify players. This mapping determines the action path ((a_0,a_1,a_2,\dots)).

Analyzing Subgame Perfect Nash Equilibrium

The one‑shot deviation principle states that a strategy profile is a subgame perfect Nash equilibrium (SPNE) if and only if no player can profit from deviating in a single period, given any history. Because past payoffs become constants once a history is reached, the subgame payoff reduces to the discounted sum of future flow payoffs starting from the current period.

Prisoner’s Dilemma in an Infinite Horizon

In a finite Prisoner’s Dilemma the unique SPNE is always defect. When the game is repeated infinitely, alternative strategies can enforce cooperation. Two classic strategies are examined:

Grim Trigger Strategy

The grim trigger strategy prescribes cooperation as long as no defection has occurred; a single defection triggers permanent defection thereafter. Cooperation is sustained as an SPNE when

[ \delta \ge \frac{1}{3}. ]

A high discount factor makes the future punishment large enough to outweigh the immediate gain from defecting. The intuition is that the “future looms very large,” so the loss from the permanent switch to the stage‑game Nash equilibrium outweighs the one‑shot benefit.

Tit‑for‑Tat Strategy

Tit‑for‑tat starts with cooperation in period 0 and then copies the opponent’s previous action. Unlike grim trigger, tit‑for‑tat is generally not an SPNE; it only works under knife‑edge conditions such as (\delta = 1/3). Small deviations from this precise discount factor break the equilibrium because the threat of future retaliation is insufficiently strong.

Mechanisms & Explanations

• One‑Shot Deviation Principle – Guarantees that checking only single‑period deviations at every history is enough to verify SPNE.
• Subgame Payoff Calculation – After any history, only future discounted payoffs matter; past payoffs are fixed constants.
• Grim Trigger Logic – Defection yields an immediate payoff boost but switches the game forever to the Nash equilibrium of mutual defection, producing a lower long‑run payoff.

The geometric series (\sum_{t=0}^{\infty}\delta^{t}=1/(1-\delta)) underlies all discounted calculations, and the relationship (\delta = 1/(1+r)) links discounting to interest rates.