ECON90034 Chap.7 Repeated Games and Bank Runs
Repeated Games and Bank Runs
Repetition changes what players can achieve. If a prisoners' dilemma is played a known, finite number of times, backward induction unravels any cooperation and the stage-game equilibrium is played in every round. If it is repeated without end, a grim trigger strategy, cooperate until anyone defects and then defect forever, can be a Nash equilibrium when the discount factor is high enough.
You check the condition by comparing the present value of cooperating with a one-period gain from cheating followed by permanent punishment. Bank runs, the last game-theory application, show a different shape: a coordination game with an efficient equilibrium and a self-fulfilling run.
The chapter tests a grim-trigger agreement between two banks at a given discount factor, finds the threshold, and then solves a two-date bank-run game with numbers you can recompute.
What this chapter covers
- 01
Finitely repeated prisoners' dilemma and unravelling
- 02
Infinitely repeated games and the discount factor
- 03
Grim trigger strategies
- 04
Checking whether grim trigger is a Nash equilibrium
- 05
Tacit collusion and why it is hard to prove
- 06
Bank runs: two dates, liquidation and the game tree
- 07
Two equilibria: an efficient outcome and a run
- 08
How a bank run differs from a prisoners' dilemma
Worked example · free
Does grim trigger hold at a discount factor of 0.6?
- 1Cooperating forever is worth 7 ÷ (1 − 0.6) = 17.5.
- 1Deviating is worth 12 now plus 0.6 × 4 ÷ 0.4 = 6 later, a total of 18.
- 118 exceeds 17.5, so a player gains by deviating: grim trigger is not a Nash equilibrium at 0.6.
- 1The threshold is (12 − 7) ÷ (12 − 4) = 0.625.
Key terms
- Discount factor
- The value today of one dollar received next period, a number between zero and one; the closer it is to one, the more patient the player.
- Grim trigger
- A strategy that cooperates until any player deviates and then plays the stage-game equilibrium action in every later period.
- Bank run
- A situation in which depositors withdraw early because they expect others to, forcing a costly liquidation that makes the fear self-fulfilling.
- Tacit collusion
- Cooperation among competitors sustained by the threat of future punishment rather than by any explicit agreement.
- Stage game
- The single-period game that is played again in every round of a repeated game; its Nash equilibrium is the punishment phase of grim trigger.
- Liquidation
- Selling a long-term investment early, at a loss, to meet withdrawals; its cost is what makes an early run on a bank damaging.
Repeated Games and Bank Runs FAQ
Why does cooperation fail in a finitely repeated prisoners' dilemma?
In the last round there is no future to protect, so both defect. Knowing that, the second-last round is also effectively the last, and the argument repeats back to the first round, so defection is played throughout.
How do you check whether grim trigger is a Nash equilibrium?
Compute the present value of cooperating forever and of the best deviation followed by punishment forever. Grim trigger is an equilibrium when cooperating is worth at least as much, which requires a high enough discount factor.
Is a bank run a prisoners' dilemma?
No. In a prisoners' dilemma defection is dominant and the good outcome is not an equilibrium. In the bank-run game neither action is dominant and both the run and the efficient outcome are equilibria, so beliefs decide.
What does the discount factor measure?
It is the value today of a dollar received one period later, between zero and one. A player with a discount factor close to one is patient and weighs future punishment heavily, which is what makes cooperation sustainable.
Why is tacit collusion hard to prosecute?
There is no agreement to point to. Each firm simply follows a rule that punishes any rival who undercuts, so matching high prices can look like ordinary competition even when it is sustained by the threat of a price war.
Exam move
Learn the grim-trigger comparison as a three-line template with symbols, then practise it with numbers until you can find the threshold discount factor directly. For the finite case, practise the backward-induction argument in words. For bank runs, solve date 2 first, write the date 1 matrix, and practise a short paragraph comparing it with the prisoners' dilemma.
Add one more drill: change a single payoff in a grim-trigger question and recompute the threshold, so you see how a bigger temptation or a harsher punishment moves it.
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