Repeated Games and Cooperation

Watch the Prisoner's Dilemma repeated forever. The strategy punishes any betrayal forever.

Betray today and you gain a lot right now. But from tomorrow your partner betrays too, and the losses keep coming. How much you care about tomorrow decides the choice. Care enough about the future, and cooperation pays and holds. The boundary value comes from the formula below.

For example, say you and a 🧑 friend take turns lending a ⚽ ball every day. Hog the ball today and today is fun. But from tomorrow your friend never lends it to you. If you want to keep playing together, keeping today's promise wins.

R1−δ≥T+δP1−δ  ⟺  δ≥δ∗=T−RT−P\frac{R}{1-\delta} \ge T + \frac{\delta P}{1-\delta} \iff \delta \ge \delta^* = \frac{T-R}{T-P}

Part of these learning paths

Examples & Exercises

In the infinitely repeated Prisoner's Dilemma with T=5T = 5, R=3R = 3, P=1P = 1, find the threshold δ∗\delta^* for grim-trigger strategies to sustain cooperation, and verify cooperation holds at δ=0.6\delta = 0.6.

R1−δ≥T+δP1−δ  ⟺  δ≥T−RT−P\frac{R}{1-\delta} \ge T + \frac{\delta P}{1-\delta} \iff \delta \ge \frac{T-R}{T-P}
δ=0.6≥δ∗=0.5\delta = 0.6 \ge \delta^* = 0.5 — cooperation sustained.
Show proof

  1. Value of perpetual cooperation: VC=31−δV_C = \frac{3}{1-\delta}.

    Value of a one-shot deviation: 55 now plus 11 forever after, VD=5+δ1−δV_D = 5 + \frac{\delta}{1-\delta}.

    Solving VC≥VDV_C \ge V_D gives δ≥5−35−1=12\delta \ge \frac{5-3}{5-1} = \frac12.

  2. At δ=0.6\delta = 0.6: VC=30.4=7.5V_C = \frac{3}{0.4} = 7.5, VD=5+0.60.4=6.5V_D = 5 + \frac{0.6}{0.4} = 6.5.

    Since 7.5>6.57.5 > 6.5 there is no incentive to deviate — cooperation holds.

    As δ\delta shrinks, future punishment weighs less and deviation pays — which is why the curves cross at δ∗\delta^*.

Explain via the δ∗\delta^* formula why a larger temptation payoff TT makes cooperation harder to sustain, and compute δ∗\delta^* for T=7T = 7, R=3R = 3, P=1P = 1.

With T=7T = 7, δ∗=2/3>0.6\delta^* = 2/3 > 0.6 — cooperation collapses.
Show proof

  1. In δ∗=T−RT−P\delta^* = \frac{T-R}{T-P}, a larger TT raises the one-shot gain from deviation (T−R)(T-R). For T=7T = 7: δ∗=7−37−1=46=23≈0.667\delta^* = \frac{7-3}{7-1} = \frac{4}{6} = \frac23 \approx 0.667.

  2. With δ=0.6<2/3\delta = 0.6 < 2/3, deviation pays: VC=7.5V_C = 7.5, VD=7+0.60.4=8.5V_D = 7 + \frac{0.6}{0.4} = 8.5. Bigger temptation demands more patience (higher δ\delta) — the formal version of "patience buys cooperation."

8. Game Theory