Module 5 — Static, Dynamic, and Repeated Games

Core question

When each person's best action depends on others, which predictions are mutually consistent—and which threats or promises are credible?

Learning outcomes

You will be able to:

• specify players, actions, information, timing, and payoffs;
• find pure and mixed Nash equilibria;
• solve finite perfect-information games by backward induction;
• test whether repeated-game cooperation is incentive compatible;
• distinguish Nash, subgame-perfect, and Bayesian Nash equilibrium.

1. Write the game before solving it

A strategic-form game is:

A strategy profile a* is a Nash equilibrium if, for every player i,

No player wants a unilateral deviation. This is strategic stability, not cooperation, morality, or efficiency.

Best-response routine

1. Hold opponents' actions fixed.
2. Mark every payoff-maximising response.
3. Find cells where all players' marks intersect.
4. Check ties and weak dominance explicitly.

2. Dominance can produce inefficient equilibrium

Consider a prisoner's dilemma:

Defect strictly dominates cooperate for both, so (Defect, Defect) is the unique Nash equilibrium. Yet (Cooperate, Cooperate) Pareto dominates it.

The failure is not irrationality. Each player rationally responds to an institution that cannot enforce joint commitment.

3. Mixed strategies make opponents indifferent

In matching pennies, A wins when actions match; B wins when they differ:

There is no pure equilibrium. Let B choose L with probability q. A's expected payoffs are:

A mixes only if q=1/2. Symmetry gives A's probability p=1/2. The mixed equilibrium is (p,q)=(1/2,1/2).

Mixing is not indecision. Each probability makes the opponent exactly indifferent among actions used with positive probability.

Every finite game has at least one mixed-strategy Nash equilibrium. It may have several, creating a selection problem.

4. Dynamic games require complete strategies

An extensive-form game specifies decision nodes, actions, information sets, probabilities, and terminal payoffs. A strategy states what a player would do at every information set, including those not reached in equilibrium.

Entry-deterrence example

An entrant chooses Out or Enter. After entry, the incumbent chooses Fight or Accommodate.

• Out gives (0,2).
• Enter, Fight gives (−1,−1).
• Enter, Accommodate gives (1,1).

Backward induction:

1. after entry, the incumbent prefers 1 to −1, so accommodates;
2. anticipating this, the entrant prefers 1 to 0, so enters.

The subgame-perfect Nash equilibrium is Enter; Accommodate. “I will fight entry” is not credible because fighting is not optimal when the decision arrives.

5. Repetition can make future punishment valuable

Repeat the prisoner's dilemma indefinitely with discount factor δ. Under grim trigger, each player cooperates until any defection, then defects forever.

Cooperation yields:

A one-time deviation yields 5 now and 1 thereafter:

Cooperation is sustainable when:

Patience, monitoring, and credible punishment support cooperation. Noise can make grim trigger excessively harsh; forgiving strategies may perform better but require a different incentive calculation.

6. Incomplete information creates Bayesian games

Player i has private type t_i, belief \mu_i(t_{-i}\mid t_i), and strategy s_i(t_i). A Bayesian Nash equilibrium satisfies:

The distinction is crucial:

• random actions produce mixed strategies;
• random types represent private information;
• random states may be moves by nature observed by some players.

Auctions, signalling, screening, and contracts use this framework in Modules 7–8.

7. Equilibrium refinements answer different objections

A refinement is useful only if its behavioural and informational assumptions fit the institution.

8. Current extension: pricing algorithms as players

Johnson, Rhodes, and Wildenbeest model platforms where sellers use Q-learning pricing algorithms. They show theoretically and in simulations that demand-steering rules which reward lower prices with greater prominence can disrupt algorithms' ability to rotate demand and share profits; history-dependent, non-neutral rules can lower prices (2023).

This result links repeated interaction to institution design. Its boundary matters: simulated Q-learning agents in a specified marketplace are not proof that every deployed algorithm colludes or that one ranking rule is optimal under all consumer-search and platform incentives.

Practice

1. Find all equilibria in a 2×2 coordination game and explain the selection problem.
2. Solve a 2×2 zero-sum game with unequal payoffs by indifference.
3. Change the entry game's fight payoff to (−1,2) and resolve it.
4. Derive the grim-trigger threshold for payoffs (R,T,P,S).
5. Describe one platform interaction as a Bayesian game: identify types, beliefs, actions, and observables.

Quick check

• Nash equilibrium is mutual best response, not necessarily welfare maximisation.
• Mixed strategies equalise expected payoffs on their support.
• Dynamic strategies specify off-path actions.
• Repeated cooperation requires a profitable future to outweigh deviation.
• Bayesian equilibrium makes strategies contingent on private types.

Next: apply strategic models to oligopoly, entry, and algorithmic pricing.