The prisoner’s dilemma and repeated games: when future interaction changes incentives

Define the dilemma through payoff comparisons

Let each of two players choose C or D. These letters are labels, conventionally called cooperate and defect, rather than judgments about real people. The Yale lecture on payoffs and rationality explains why a strategic model needs preferences as well as outcomes. Here each player receives three when both choose C and one when both choose D. A player choosing D against C receives five, while the player choosing C against D receives zero. All values are hypothetical utilities, with no monetary interpretation.

Verify dominance in the one-shot model

If the other player chooses C, choosing D yields five instead of three. If the other chooses D, choosing D yields one instead of zero. D therefore strictly dominates C for each player in this particular table. The unique one-shot Nash equilibrium is (D,D), even though both prefer the payoff pair (3,3) obtained at (C,C). The conclusion follows from the assigned rankings. A different table, binding agreement or additional action can produce a different strategic problem despite using similar everyday language.

Inspect the full payoff table

The table contains every action pair, so the dominance and equilibrium checks are exhaustive for this one-shot model. It says nothing by itself about how frequently people cooperate in experiments or how an actual community should organise its relationships. Those are empirical and institutional questions.

Examine a known finite horizon

The Yale lecture on finite repeated interactions explains backward reasoning in a repeated dilemma. Under the usual complete-information and common-knowledge rationality assumptions, the final period has the one-shot incentive to choose D. Working backward leaves no future cooperative reward to sustain C in the preceding periods. This conclusion concerns the specified finite repetition with a unique stage equilibrium and the relevant equilibrium refinement. It should not be generalised to every finite repeated game: multiple stage equilibria or altered information can change the argument.

Derive a hypothetical continuation threshold

Now suppose the same game continues indefinitely, actions are perfectly observed and each future period’s utility is multiplied by a discount factor δ, with 0≤δ<1. Consider a specified rule: choose C while both have always chosen C, then choose D forever after any deviation. Staying on the cooperative path gives 3/(1−δ). Deviating once against C gives five immediately and then one per future period, or 5+δ/(1−δ). Cooperation meets this particular incentive constraint when 3≥5−4δ, equivalently δ≥1/2. The threshold belongs to these invented payoffs and assumptions, not to human cooperation generally.

At δ=1/2 the two displayed continuation values are equal; a larger δ makes the cooperative continuation strictly preferable to the specified one-time deviation. This boundary check helps verify the algebra. It does not establish that the proposed equilibrium is unique or that every possible strategy has been classified.

Check credibility after a deviation

The Yale treatment of continuation incentives stresses that future behaviour must also be consistent with incentives. Here the proposed continuation of choosing D is credible within the model because (D,D) is the stage equilibrium. A threatened response that its author would prefer not to carry out requires further analysis. The calculation is a teaching example of equilibrium reasoning, not interpersonal advice to punish someone indefinitely. Real situations can include mistakes, incomplete observation, changing participants and additional options absent from this deliberately simplified construction.

Distinguish discounting, uncertainty and observation

An uncertain chance that interaction continues can sometimes enter an effective continuation factor, but that requires a specified model rather than a loose analogy. Likewise, imperfect observation can make an apparent deviation ambiguous. A rule that reacts to every observed discrepancy may respond to measurement error as though it were a deliberate action. State whether histories are public, whether participants know the horizon and whether payoffs remain stable. Repeated-game conclusions are sensitive to those choices, so the model’s information structure deserves as much attention as its payoff table.

Repeated-game analysis checklist

• Verify the stage-game payoff rankings.
• Separate one-shot and repeated strategies.
• State the horizon and knowledge assumptions.
• Specify how future utilities are discounted.
• Write the continuation rule completely.
• Check incentives both before and after a deviation.
• Keep abstract equilibrium conditions separate from behavioural or ethical claims.

The same four stage payoffs can support different analyses when the interaction protocol changes. A useful report therefore gives the table, the timing and the information rules together. The mathematics becomes interpretable when readers can see exactly which future consequences enter the present comparison and which real-world complications remain outside it.

Sources and verification

• Game Theory, Lecture 2: formal ingredients, payoffs and common knowledge — Open Yale Courses — 2026-10-23
• Game Theory, Lecture 21: repeated interactions and finite horizons — Open Yale Courses — 2026-10-23
• Game Theory, Lecture 22: repeated games and continuation incentives — Open Yale Courses — 2026-10-23