• Content map: SMU H3 Game Theory Map

Chapter 10: Repeated Games

Repeated Games

Concept Introduction

• Repetition changes incentives because current actions affect future responses.
• Strategies now depend on the history of play, not only on the current stage.

Formal Definition

Definition:

A repeated game is obtained by playing a base game across multiple stages. A strategy specifies an action after every possible history of previous play.

Method / Reasoning

• Identify the stage game.
• List the feasible histories after each stage.
• Define strategies as history-contingent action plans.
• Evaluate payoffs over the full horizon.

Implications

• Cooperative behaviour may become sustainable when future punishment is possible.
• Repetition enlarges the strategy space substantially.

Result:

Repeated games are solved by analysing continuation payoffs, not just one-shot incentives.

Insight:

The same base game can generate different outcomes once players remember the past and care about the future.

Finitely Repeated Games

Concept Introduction

• In a finite repeated game, the last round behaves like the one-shot game.
• This logic rolls back to every earlier stage.

Formal Definition

Definition:

A subgame perfect Nash equilibrium in a repeated game is a strategy profile that is optimal after every possible history.

Method / Reasoning

• Solve the final stage first.
• Replace the continuation value by the equilibrium payoff of the last stage.
• Move backward one stage at a time.

Implications

• In finitely repeated Prisoner’s Dilemma, defection occurs in every round under backward induction.
• Non-credible promises to cooperate collapse once the endpoint is known.

Result:

For any finite horizon, the unique subgame perfect outcome in Prisoner’s Dilemma is defection at every stage.

Insight:

The last round destroys cooperation first, and then the logic destroys every earlier round.

Infinitely Repeated Games

Concept Introduction

• Infinite repetition is meaningful only if future payoffs are discounted.
• Discounting converts an infinite stream into a finite value (convergence).

Formal Definition

Definition:

If $0 < \delta < 1$ is the discount factor and $\pi_t$ is the stage payoff at time $t$, then the present discounted payoff is

Method / Reasoning

• Write the stream of stage payoffs generated by a strategy profile.
• Weight future payoffs by powers of $\delta$.
• Sum the resulting geometric series.

Implications

• A higher $\delta$ means players are more patient.
• Patience makes future punishments more effective.

Result:

Cooperation in repeated games is easier to sustain when $\delta$ is high.

Interest Rates and Discount Factors

Time Value of Money

• A payoff received today can be invested and earn interest.
• If the interest rate is $r$, then $1$ unit today becomes $1+r$ units next period.
• Therefore, $1+r$ units next period have present value $1$ today.

Deriving the Discount Factor

• Let $\delta$ be the present value today of $1$ unit received next period.
• Since $\delta$ units today grow to $\delta(1+r)$ next period, equivalence with $1$ future unit requires

• Solving,

Definition:

The discount factor associated with interest rate $r$ is

It converts next-period payoffs into present-value units.

Intuition

• Future payoffs are discounted because receiving money later sacrifices the interest that could have been earned by receiving it today.
• A higher $r$ raises the opportunity cost of waiting.
• Since $\delta=\frac{1}{1+r}$, a higher $r$ implies a lower $\delta$.
• A high $\delta$ means patience: future payoffs remain valuable today.
• A low $\delta$ means impatience: future payoffs receive little present weight.

Repeated-Game Link

• In repeated games, $\delta$ measures how much players value future cooperation and future punishment.
• Cooperation is sustainable only when the discounted continuation value from obeying is at least as large as the current gain from deviating.
• For a constant payoff $x$ each period, the continuation value is

• Since this is an infinite geometric series with common ratio $\delta$,

Insight:

Low interest rates raise $\delta$, making players more patient and making future punishment more powerful. High interest rates lower $\delta$, making immediate deviation more attractive.

One-Shot Deviation Principle

Concept Introduction

• Dynamic strategies prescribe actions after every possible history.
• To prove subgame perfection, it is enough to check whether any player wants to deviate at one decision point only.

Formal Definition

Definition:

The one-shot deviation principle states that a strategy profile is subgame perfect if and only if no player can gain by deviating at a single history and then returning to the prescribed strategy thereafter.

Method / Reasoning

• List the relevant histories or states where a player must choose an action.
• Compute the continuation value from following the prescribed strategy.
• Compute the value from one immediate deviation, holding future play fixed according to the original strategy profile.
• Require the prescribed action to give weakly higher continuation value at every relevant history.

Implications

• The test reduces a large dynamic problem to local incentive constraints.
• It checks credibility both on the equilibrium path and after off-path histories.
• If any one-shot deviation is profitable, the strategy profile is not subgame perfect.

Result:

A dynamic strategy profile is a subgame perfect Nash equilibrium exactly when every one-shot deviation inequality is satisfied.

Insight:

The principle tests sequential credibility: a strategy must be optimal after every history, not only along the intended path.

Cooperation, Externalities, and Public Goods

Concept Introduction

• Externalities and public goods often have the same logic as Prisoner’s Dilemma.
• Individual incentives underprovide socially valuable actions.

Formal Definition

Definition:

A public good is non-rival and non-excludable. In contribution games, each player receives only part of the return from own contribution, while everyone shares the benefit.

Method / Reasoning

• Write individual payoff as private return plus shared benefit.
• Compute each player’s best response.
• Compare equilibrium contributions with the welfare-maximising level.

Implications

• Private incentives produce free-riding.
• Repetition and credible punishment can support higher contributions.
• Policy design must specify punishments that are themselves incentive compatible.

Result:

Public-good problems create a wedge between Nash equilibrium and the social optimum.

Insight:

The key issue is not whether cooperation is valuable. It is whether the incentive system makes cooperation individually optimal.