Discounted Payoffs
In an infinitely repeated game, player $i$’s total payoff is the discounted sum of stage payoffs:
\[U_i = (1 - \delta) \sum_{t=0}^{\infty} \delta^t u_i(a^t)\]where $\delta \in (0,1)$ is the common discount factor. The $(1-\delta)$ normalisation keeps $U_i$ in the same units as stage payoffs. Higher $\delta$ reflects greater patience and makes future punishments more credible.
Grim Trigger Strategy
The simplest trigger strategy sustains cooperation in the prisoner’s dilemma. Define:
- Cooperate in period 0.
- Cooperate in period $t$ if no player has ever defected; otherwise Defect forever.
Cooperation is a subgame perfect equilibrium if the gain from one-shot defection does not exceed the discounted loss from permanent punishment:
\[T - R \leq \frac{\delta}{1-\delta}(R - P)\] \[\Longleftrightarrow \quad \delta \geq \frac{T - R}{T - P}\]For the standard parameterisation $T=5, R=3, P=1$, cooperation requires $\delta \geq 1/2$.
Folk Theorem
Let $V^* = {v \in \mathbb{R}^n : v = \sum_a \alpha(a) u(a),\, \alpha \in \Delta(A)}$ be the set of feasible payoffs, and let $\underline{v}i = \min{\sigma_{-i}} \max_{\sigma_i} u_i(\sigma)$ be the minimax payoff. The folk theorem (Fudenberg-Maskin 1986) states that for any $v \in V^*$ with $v_i > \underline{v}_i$ for all $i$, there exists $\bar{\delta} < 1$ such that for all $\delta \geq \bar{\delta}$, $v$ is a subgame perfect equilibrium payoff vector. Cooperation is therefore generically sustainable with sufficiently patient players.