Part I — Where Game Theory Came From
In 1928, a twenty-four-year-old Hungarian mathematician named John von Neumann proved a theorem about parlour games. It established that in any two-person, zero-sum game of perfect information, there exists a pair of strategies and a value such that neither player can do better by deviating. He called it the minimax theorem. It was, at the time, a curiosity — a piece of pure mathematics about poker and chess.
Sixteen years later, von Neumann and the Austrian economist Oskar Morgenstern published Theory of Games and Economic Behavior (1944). Their claim was audacious: that economics had been building on the wrong mathematics entirely. Physics had given economics the calculus of maximisation under constraint — one agent, one objective, a fixed environment. But an economy is not a fixed environment. It is a room full of other people, each of whom is also maximising, and each of whose choices changes the problem the others face.
“A Robinson Crusoe economy has one decision-maker and no strategy. Every real economy has many, and therefore has nothing else.”
Then came the Cold War. The RAND Corporation, established in 1948 and funded by the US Air Force, needed to think rigorously about nuclear deterrence — a situation with two players, catastrophic payoffs, and no possibility of experiment. Game theory was the only available language. It was at RAND, in 1950, that Merrill Flood and Melvin Dresher constructed a payoff matrix that Albert Tucker would soon dress in a story about two arrested criminals.
In the same year, a twenty-one-year-old graduate student at Princeton submitted a doctoral dissertation of twenty-seven pages. John Forbes Nash Jr. had generalised von Neumann’s result to games with any number of players and any payoff structure — not merely zero-sum ones. Von Neumann himself is said to have been unimpressed, dismissing it as a fixed-point theorem.
It was a fixed-point theorem. It also reorganised economics, biology, political science and computer science, and won Nash the 1994 Nobel Prize.
Part II — The Anatomy of a Game
Game — any situation of strategic interdependence, in which the outcome for each participant depends not only on their own choice but on the choices of others.
The word “game” is unfortunate. Nuclear brinkmanship, cartel discipline, and species evolution are games. Solitaire is not — nobody responds to you.
A game in strategic (normal) form has three elements:
- Players — the decision-makers, i = 1, …, n
- Strategies — a complete contingent plan for each player. Note the word complete: a strategy specifies what you would do at every decision point, including ones you never reach.
- Payoffs — a function ui(s1, …, sn) giving player i‘s utility from every possible strategy profile.
Two assumptions doing all the work
Players are rational — they maximise expected payoff. And rationality is common knowledge: I know you are rational; you know I know; I know you know I know; ad infinitum.
Common knowledge of rationality is what licenses iterated reasoning — “she knows that I know that she would never play X, therefore…”. It is also, as we shall see in Part VII, empirically false. Real people iterate one or two steps, not infinitely. Every strong exam answer eventually says so.
Part III — Types of Games
| Dimension | Type A | Type B | What Changes |
|---|---|---|---|
| Timing | Simultaneous — players move at once, or in ignorance of the other | Sequential — one moves, the other observes, then moves | Sequential games are solved by backward induction and admit first-mover advantage |
| Payoff structure | Zero-sum — one player’s gain is exactly the other’s loss | Non-zero-sum — mutual gains or mutual losses possible | Zero-sum games have no scope for cooperation. Von Neumann solved these; Nash solved the rest. |
| Binding agreements | Cooperative — players can sign enforceable contracts | Non-cooperative — no external enforcement exists | Cartels are cooperative in intention and non-cooperative in law. That is their entire tragedy. |
| Repetition | One-shot | Repeated | Repetition creates a “shadow of the future” that can sustain cooperation |
| Information | Complete — everyone knows everyone’s payoffs | Incomplete — types are private (Bayesian games) | Incomplete information generates signalling, screening, and reputation |
Cooperative game. Players may negotiate binding agreements and form coalitions. The analysis concerns how the joint surplus is divided (the Shapley value, the core, the Nash bargaining solution).
Non-cooperative game. No agreement is enforceable. Any cooperation must be self-enforcing — each player must find it in their own interest to comply. Almost all economic game theory is non-cooperative, because contracts to fix prices are illegal and contracts between nations have no court.
Part IV — Dominance and the Prisoner’s Dilemma
Strategy si strictly dominates s′i if it yields a strictly higher payoff against every strategy the opponents might play. A rational player never plays a strictly dominated strategy.
The canonical payoff matrix
Two suspects, held separately. Payoffs are years in prison, so lower is better; we write them as negative utilities. Format: (Row’s payoff, Column’s payoff).
| B: Stay Silent | B: Confess | |
|---|---|---|
| A: Stay Silent | (−1, −1) | (−10, 0) |
| A: Confess | (0, −10) | (−8, −8) |
Solving by dominance
Suppose B stays silent. A gets −1 by staying silent, 0 by confessing. Confess is better.
Suppose B confesses. A gets −10 by staying silent, −8 by confessing. Confess is better.
Confess strictly dominates. By symmetry, the same holds for B. Both confess. Both serve eight years — when mutual silence would have given them one each.
The tragedy is not that the prisoners are stupid. It is that they are clever, and that cleverness is precisely what destroys them.
Adam Smith’s invisible hand asserts that self-interested action produces collectively desirable outcomes. The Prisoner’s Dilemma is a rigorous, minimal counterexample. Individually rational choice can produce a Pareto-inferior outcome, and no amount of intelligence, information, or foresight on the players’ part can rescue them. Every subsequent economics of market failure — cartels, arms races, overfishing, climate change, antibiotic resistance — is a Prisoner’s Dilemma with more players.
Iterated elimination of strictly dominated strategies (IESDS)
Delete dominated strategies; this may make others dominated in the reduced game; repeat.
Eliminating strictly dominated strategies is safe: the result is independent of elimination order. Eliminating weakly dominated strategies (never worse, sometimes equal) is not — the surviving set can depend on the order in which you delete. Examiners test this. If a strategy ties, say so explicitly and do not delete it without comment.
Part V — Nash Equilibrium
A strategy profile (s*1, …, s*n) is a Nash equilibrium if no player can raise their own payoff by unilaterally deviating, holding all other strategies fixed.
- It is not the outcome that maximises joint payoffs. (Prisoner’s Dilemma.)
- It is not necessarily unique. (Coordination games have several.)
- It is not necessarily efficient, fair, or good.
- It is not a prediction that players will find it. It describes a state that, once reached, is stable.
Nash’s Existence Theorem
Every finite game — finitely many players, finitely many pure strategies — possesses at least one Nash equilibrium, possibly in mixed strategies.
The proof applies the Kakutani fixed-point theorem to the best-response correspondence. Consider the map that takes any strategy profile and returns the set of profiles in which everyone best-responds to it. An equilibrium is a fixed point of that map: a profile that maps to itself.
An equilibrium is a state of the world that, when everyone reacts optimally to it, reproduces itself. This is what a fixed point is — and it is why topology sits beneath both game theory and general equilibrium.
Finding pure-strategy equilibria: the underlining method
- Take each of Player 2’s strategies (each column) in turn. Find Player 1’s best response. Underline Player 1’s payoff there.
- Take each of Player 1’s strategies (each row). Find Player 2’s best response. Underline Player 2’s payoff.
- Any cell where both payoffs are underlined is a pure-strategy Nash equilibrium.
Two people would rather be together than apart, but disagree about where.
| She: Opera | She: Football | |
|---|---|---|
| He: Opera | (2, 3) | (0, 0) |
| He: Football | (0, 0) | (3, 2) |
Check (Opera, Opera): He gets 2; deviating to Football gives 0. No. She gets 3; deviating gives 0. No. Nash equilibrium.
Check (Football, Football): Symmetric. Nash equilibrium.
Two pure equilibria. Nash equilibrium cannot tell us which occurs. This is the equilibrium selection problem, and it is why Schelling introduced the idea of a focal point — a salient outcome that culture, not mathematics, makes obvious.
Mixed strategies — the indifference principle
Consider Matching Pennies. Player 1 wins if the coins match; Player 2 wins if they differ.
| P2: Heads | P2: Tails | |
|---|---|---|
| P1: Heads | (1, −1) | (−1, 1) |
| P1: Tails | (−1, 1) | (1, −1) |
No pure equilibrium exists. Whatever I do, if you knew it, you would beat me. The equilibrium requires randomisation.
In a mixed-strategy equilibrium, each player’s probabilities are chosen to make the opponent indifferent between their pure strategies.
Why? Because a player will only willingly randomise if they are indifferent — otherwise they would play the better strategy with probability 1. And what makes them indifferent is the opponent’s mix. So my probabilities are pinned down by your payoffs, not by mine.
Let Player 1 play Heads with probability p. For Player 2 to be willing to mix, Player 2’s expected payoff from Heads must equal that from Tails:
E[P2 plays Heads] = p(−1) + (1−p)(1) = 1 − 2p
E[P2 plays Tails] = p(1) + (1−p)(−1) = 2p − 1
Setting equal: 1 − 2p = 2p − 1 ⟹ 4p = 2 ⟹ p = ½
By symmetry, Player 2 plays Heads with probability q = ½. The unique Nash equilibrium is (½, ½) for both. Each player’s expected payoff is 0.
Part VI — Sequential Games, Credibility, and Commitment
Many strategic situations unfold over time. These are represented in extensive form — a tree of decision nodes — and solved by backward induction: start at the final nodes, determine the optimal action, replace the node with its payoff, and recurse toward the root.
Subgame Perfect Nash Equilibrium (SPNE) — a strategy profile that constitutes a Nash equilibrium in every subgame, including subgames that are never reached in equilibrium. Introduced by Reinhard Selten (1965), who shared the 1994 Nobel with Nash and Harsanyi.
The non-credible threat
An incumbent monopolist earns 10. A potential entrant may Enter or Stay Out.
- If the entrant stays out: Incumbent 10, Entrant 0.
- If the entrant enters and the incumbent accommodates: Incumbent 5, Entrant 5.
- If the entrant enters and the incumbent fights a price war: Incumbent −2, Entrant −3.
The threat: “Enter and I will destroy you.” If believed, the entrant stays out (0 > −3). And the incumbent has no incentive to deviate, since the threat is never tested. This is a Nash equilibrium.
Backward induction: Look at the subgame after entry has occurred. The incumbent chooses between accommodating (5) and fighting (−2). It accommodates. Knowing this, the entrant enters and earns 5.
The unique SPNE is (Enter, Accommodate). The threat was never credible. Nash equilibrium tolerated a threat no rational player would carry out; subgame perfection rules it out.
Commitment: how to make a threat credible
Thomas Schelling’s insight in The Strategy of Conflict (1960) — for which he shared the 2005 Nobel — is that a player can gain enormously by destroying their own options.
Burn the bridge behind your army, and retreat becomes impossible. The enemy now knows you will fight. Your loss of freedom is the source of your power.
In economics: build excess capacity, so that fighting a price war is genuinely cheap for you. Sign a contract with a penalty clause. Sink costs into a technology that only pays at high output. Each of these changes the payoffs in the subgame, converting a non-credible threat into a credible one. Delegating a negotiation to an agent with a reputation for stubbornness does the same thing.
Part VII — Repetition and the Emergence of Cooperation
Finite repetition: the unravelling
Suppose the Prisoner’s Dilemma is played exactly 100 times, and both know it.
In round 100 there is no future, so both defect. Knowing round 100’s outcome is fixed regardless of what happens in round 99, there is no future consequence in round 99 either — so both defect. Induct backward. Defection in every round.
This is Selten’s chain store paradox (1978), and it is deeply counterintuitive. A known endpoint destroys cooperation all the way back to the beginning.
Infinite (or indefinite) repetition: the Folk Theorem
Now suppose the game continues to the next round with probability δ (equivalently, players discount the future at rate δ). There is no last round to induct from.
Consider the grim trigger strategy: cooperate until the opponent defects, then defect forever.
Let T = temptation payoff (defect while other cooperates), R = reward (mutual cooperation), P = punishment (mutual defection), with T > R > P.
Payoff from cooperating forever: R + δR + δ²R + … = R / (1 − δ)
Payoff from defecting once, then being punished forever: T + δP + δ²P + … = T + δP / (1 − δ)
Cooperation is sustainable when R/(1−δ) ≥ T + δP/(1−δ). Rearranging:
δ ≥ (T − R) / (T − P)
Cooperation requires players to be sufficiently patient, or equivalently for the relationship to be sufficiently likely to continue. Trust, reputation, cartel discipline and social norms all reduce to a single inequality about δ. The shadow of the future is what enforces good behaviour in the present.
Fudenberg, D., & Maskin, E. (1986), Econometrica 54(3)
The result: With sufficiently patient players, any feasible payoff vector that gives each player at least their minmax value can be sustained as a subgame perfect equilibrium of the infinitely repeated game.
The good news: This explains, without appealing to altruism, how cooperation, trust, reputation and informal contract enforcement can exist in a world of self-interested agents facing no external enforcer.
The bad news: A theory that can rationalise almost any outcome as an equilibrium makes almost no prediction. The Folk Theorem is simultaneously game theory’s most powerful explanatory result and its most damning admission. Selecting among the infinity of equilibria requires criteria that game theory itself does not supply.
Axelrod’s tournament
The experiment: Political scientist Robert Axelrod invited game theorists worldwide to submit computer programs to play iterated Prisoner’s Dilemma round-robin against one another.
The winner: Tit-for-Tat, submitted by Anatol Rapoport — four lines of code. Cooperate on the first move; thereafter, copy whatever the opponent did last.
Axelrod ran the tournament again, publishing the results so entrants could design counter-strategies. Tit-for-Tat won again.
The four properties of successful strategies: be nice (never defect first); be retaliatory (punish defection immediately); be forgiving (return to cooperation once the opponent does); be clear (simple enough that the opponent can learn to cooperate with you). Notably, Tit-for-Tat never beats any individual opponent — it wins the tournament by never doing much worse than anyone, while eliciting cooperation.
Part VIII — Applications to Oligopoly
Oligopoly is defined by strategic interdependence. Game theory is therefore not an application to oligopoly; it is oligopoly’s native language.
Cournot competition (firms choose quantities)
Demand: P = a − b(q₁ + q₂). Constant marginal cost c for both firms.
Firm 1’s profit: π₁ = [a − b(q₁ + q₂) − c] q₁
Differentiate w.r.t. q₁ and set to zero: a − 2bq₁ − bq₂ − c = 0
Firm 1’s reaction function: q₁ = (a − c − bq₂) / 2b
By symmetry, q₂ = (a − c − bq₁) / 2b. Solving simultaneously (imposing q₁ = q₂ = q*):
q* = (a − c) / 3b Q* = 2(a − c) / 3b P* = (a + 2c) / 3
With n symmetric firms, q* = (a − c)/[b(n+1)]. As n → ∞, P → c: Cournot converges continuously to perfect competition.
Bertrand competition (firms choose prices)
With identical products, constant marginal cost c, and consumers buying from the cheapest seller, the unique Nash equilibrium is P₁ = P₂ = MC. Zero profit. With only two firms.
The logic: Any price above MC invites the rival to undercut by a penny and capture the entire market. The only price nobody wants to undercut is MC itself.
Resolutions: capacity constraints (Edgeworth), product differentiation (Hotelling), or repeated interaction permitting tacit collusion. That the paradox needs resolving at all is what makes it instructive.
Stackelberg (sequential quantity choice)
The leader chooses first, anticipating the follower’s reaction function. Substituting q₂ = (a − c − bq₁)/2b into the leader’s profit function and maximising gives q₁ = (a−c)/2b — more than the Cournot quantity.
In Stackelberg quantity competition, moving first is an advantage — but only because a capacity commitment is credible and irreversible. In Bertrand price competition, moving first is a disadvantage: you announce a price and are undercut. Whether first-mover advantage exists depends on whether strategies are strategic substitutes or complements. Never assert it without saying why.
Cartels: the Prisoner’s Dilemma with a logo
The structure: Each member agrees a production quota. Restricting output raises the world price and every member’s profit. But each member individually gains by producing above quota — selling more at the high price the others are maintaining.
Cheating is a dominant strategy in the one-shot game. The cartel’s stated agreement is not enforceable in any court. This is a non-cooperative game wearing cooperative clothing.
Why does OPEC persist? The Folk Theorem. Members expect to interact indefinitely (δ is high), production is observable via tanker traffic and export data (cheating is detected quickly), and Saudi Arabia holds the spare capacity to flood the market — a credible punishment.
The general condition for cartel stability: a high discount factor, rapid detection of cheating, and a punishment that is credible in the subgame following defection. Remove any one and the cartel dissolves. Nearly every cartel that has collapsed did so because detection failed or punishment lacked credibility.
Part IX — Where the Theory Fails
Güth, Schmittberger & Schwarze (1982), Journal of Economic Behavior and Organization 3(4)
The game: The Proposer offers a split of $10. The Responder accepts (both receive the split) or rejects (both receive nothing).
The prediction: Backward induction says the Responder should accept any positive amount, since something beats nothing. Therefore the Proposer offers one cent.
The result: Modal offers cluster near 50/50. Offers below roughly 20% are rejected about half the time. Responders pay real money to punish unfairness.
The cross-cultural extension: Henrich et al. (2001), American Economic Review, ran the game across fifteen small-scale societies on five continents. Offers varied systematically with market integration and local norms of cooperation. Behaviour was not universal — but nowhere did it match the subgame-perfect prediction.
Nagel, R. (1995), American Economic Review 85(5)
The game: Choose a number from 0 to 100. The winner is whoever is closest to two-thirds of the average.
The reasoning: If everyone chooses randomly, the average is 50 — so choose 33. But everyone reasons that way, so the average is 33 — choose 22. Iterate. The unique Nash equilibrium is 0.
The result: Subjects cluster around 33 and 22. One or two rounds of iterated reasoning. Almost nobody plays 0 — and anyone who does, loses.
The formalisation: Camerer, Ho & Chong (2004), QJE, proposed the cognitive hierarchy model: players are of different “levels,” with a level-k player best-responding to the belief that others are level-(k−1). Common knowledge of rationality is empirically false. Keynes anticipated exactly this in the General Theory with his newspaper beauty contest — investors do not pick the prettiest face, but the face they expect others to pick.
Nash equilibrium describes where a system settles after learning, imitation, and repetition. It does not describe what people do the first time. Evolutionary game theory formalises exactly this: equilibrium as the resting point of a dynamic process rather than the output of a calculation. Maynard Smith’s evolutionarily stable strategy requires no rationality at all — only differential reproduction.
Part X — Practice Questions
Consider the following game. Find all pure-strategy Nash equilibria.
| Left | Middle | Right | |
|---|---|---|---|
| Up | (3, 1) | (0, 0) | (5, 0) |
| Down | (2, 1) | (4, 2) | (3, 1) |
Show worked answer
Row’s best responses. If Column plays Left: Up gives 3, Down gives 2 → Up. If Middle: 0 vs 4 → Down. If Right: 5 vs 3 → Up.
Column’s best responses. If Row plays Up: Left 1, Middle 0, Right 0 → Left. If Row plays Down: Left 1, Middle 2, Right 1 → Middle.
Check each cell. (Up, Left): Row best-responds ✓, Column best-responds ✓ → Nash equilibrium. (Down, Middle): Row ✓, Column ✓ → Nash equilibrium.
Answer: two pure-strategy Nash equilibria — (Up, Left) and (Down, Middle). Note that Right is strictly dominated for Column by Left, so IESDS would have removed it first.
A striker chooses to shoot Left or Right; the goalkeeper dives Left or Right. Payoffs are the striker’s probability of scoring (goalkeeper’s payoff is the negative).
| Striker \ Keeper | Dive Left | Dive Right |
|---|---|---|
| Shoot Left | 0.3 | 0.9 |
| Shoot Right | 0.8 | 0.2 |
Find the mixed-strategy Nash equilibrium.
Show worked answer
Let the striker shoot Left with probability p. The striker’s mix must make the keeper indifferent. Keeper’s payoff is −(striker’s payoff):
Keeper dives Left: −[0.3p + 0.8(1−p)] = −(0.8 − 0.5p)
Keeper dives Right: −[0.9p + 0.2(1−p)] = −(0.2 + 0.7p)
Set equal: 0.8 − 0.5p = 0.2 + 0.7p ⟹ 0.6 = 1.2p ⟹ p = 0.5
Now let the keeper dive Left with probability q. This must make the striker indifferent:
Shoot Left: 0.3q + 0.9(1−q) = 0.9 − 0.6q
Shoot Right: 0.8q + 0.2(1−q) = 0.2 + 0.6q
Set equal: 0.9 − 0.6q = 0.2 + 0.6q ⟹ 0.7 = 1.2q ⟹ q ≈ 0.583
Equilibrium: striker shoots Left with probability 0.5; keeper dives Left with probability ≈0.583. Striker’s equilibrium scoring probability ≈ 0.55. Chiappori, Levitt and Groseclose (2002, AER) tested exactly this on real penalty kick data and could not reject the mixed-strategy prediction — one of game theory’s cleanest field successes.
Two firms play a repeated Prisoner’s Dilemma. Colluding yields each 10 per period. Undercutting while the rival colludes yields 15 (rival gets 2). Mutual undercutting yields 6 each. The game continues each period with probability δ. Using grim trigger, find the minimum δ that sustains collusion.
Show worked answer
Identify: T = 15, R = 10, P = 6.
Apply δ ≥ (T − R)/(T − P) = (15 − 10)/(15 − 6) = 5/9 ≈ 0.556
Interpretation: the firms must place at least ~56% weight on next period’s payoff. If the industry is expected to be disrupted, or if managers are rewarded on quarterly results, δ falls and the cartel collapses.
Evaluation point for an essay: notice that the condition also weakens if the punishment P rises (a weaker price war) or if the temptation T rises (a bigger short-run gain from cheating). Antitrust policy that increases the temptation — for example, by granting leniency to the first cartel member to confess — attacks the inequality from the numerator. This is precisely how modern cartel leniency programmes are designed.
Market demand is P = 120 − Q. Two identical firms have MC = 30. Find the Cournot equilibrium quantity, price and profit per firm. Compare with the monopoly and perfectly competitive outcomes.
Show worked answer
Here a = 120, b = 1, c = 30.
Cournot: q* = (a−c)/3b = 90/3 = 30 each. Q = 60. P = 120 − 60 = 60. Profit each = (60 − 30)(30) = 900.
Monopoly: MR = 120 − 2Q = 30 ⟹ Q = 45, P = 75, profit = (75−30)(45) = 2,025. Total industry profit is higher (2,025 > 1,800), which is exactly why they would like to collude.
Perfect competition: P = MC = 30, Q = 90, profit = 0.
The ranking: Qmonopoly (45) < QCournot (60) < Qcompetition (90). Cournot sits strictly between the two extremes — which is precisely the intuition oligopoly is meant to capture.
Part XI — Exam Technique
- Master the underlining method for a 2×2 matrix. It is fast and it does not fail.
- Identify dominant strategies, then state whether the resulting equilibrium is Pareto efficient. These are different questions.
- Know that the Prisoner’s Dilemma equilibrium is Nash but not Pareto optimal, and be able to explain in one sentence why the two concepts diverge.
“Discuss whether cartels are likely to be stable.”
- Define oligopoly by interdependence, and identify game theory as its formal language.
- Construct the cartel payoff matrix. Show that cheating is a dominant strategy in the one-shot game. Analysis marks.
- Conclude that the one-shot cartel is unstable — the Nash equilibrium is mutual cheating.
- Then complicate it. Introduce repetition and the Folk Theorem. Derive or state δ ≥ (T−R)/(T−P). This lifts an answer into the top band.
- Identify the three requirements for stability: high δ, rapid detection, credible punishment.
- Apply to OPEC: indefinite horizon, observable tanker traffic, Saudi spare capacity.
- Evaluate: leniency programmes attack T; demand volatility obscures detection; a finite horizon (an industry facing obsolescence) triggers Selten’s unravelling.
- Conclude conditionally on the structural conditions, not on a bare assertion.
- Confusing Nash equilibrium with the highest-payoff cell. They coincide only by accident.
- Deriving a player’s mixing probability from their own payoffs. It comes from the opponent’s.
- Calling a threat credible without checking the subgame. Always ask: at that node, would they actually do it?
- Claiming a first-mover advantage in a price-setting game. In Bertrand, moving first is a liability.
- Asserting the Folk Theorem “proves cooperation happens.” It proves cooperation can be an equilibrium — along with almost everything else.
- Deleting weakly dominated strategies without noting order-dependence.
Summary
Game theory is the mathematics of interdependent decision-making. Its central concept, Nash equilibrium, guarantees existence but not uniqueness, efficiency, or attainability. Its central lesson — that rational individual choice can produce collectively catastrophic outcomes — is the formal foundation for the entire economics of market failure, from cartels to arms races to climate change.
Its central weakness is that it assumes far more about human cognition than any experiment supports. Real people iterate one or two steps of reasoning, punish unfairness at their own expense, and cooperate in games where theory says they cannot. The most interesting frontier is therefore not new equilibrium refinements but the question of how — and whether — real people ever arrive at the equilibrium at all.
Exercise 1 — Who Is Irrational in the Finitely Repeated Prisoner’s Dilemma?
Backward induction predicts defection in every round of a finitely repeated Prisoner’s Dilemma. Experimental subjects cooperate for most rounds and defect only near the end. Selten called this the chain store paradox and conceded that the game-theoretic prediction is descriptively false.
Three possibilities: (a) backward induction is a bad model of rationality; (b) the subjects are irrational; (c) the game the subjects believe they are playing is not the game the experimenter thinks they are playing. Defend one position. Then design an experiment that would discriminate between them — noting that you must somehow observe beliefs, not merely actions.
📄 Read: Kreps, D. M., Milgrom, P., Roberts, J., & Wilson, R. (1982). “Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma.” Journal of Economic Theory, 27(2), 245–252. Note how a vanishingly small probability that your opponent is an irrational Tit-for-Tat automaton restores cooperation for most of the game — and consider what that implies about the fragility of backward induction.
Exercise 2 — Is the Folk Theorem a Triumph or a Confession?
The Folk Theorem establishes that with sufficiently patient players, almost any individually rational outcome can be sustained as an equilibrium of an infinitely repeated game. A theory that predicts almost anything predicts almost nothing.
Is this a triumph — it explains cooperation, trust and reputation without appealing to altruism — or an admission of emptiness, since it imposes no discipline on prediction? What criteria would you use to select among the infinity of equilibria: renegotiation-proofness, evolutionary stability, risk dominance? And crucially: are any of those criteria derived from game theory, or are they imported from outside it?
📄 Read: Fudenberg, D., & Maskin, E. (1986). “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information.” Econometrica, 54(3), 533–554. Then read Harsanyi & Selten’s A General Theory of Equilibrium Selection in Games (1988) and judge whether their solution is a discovery or a convention.
References
- Axelrod, R. (1984). The Evolution of Cooperation. New York: Basic Books.
- Camerer, C. F., Ho, T.-H., & Chong, J.-K. (2004). A Cognitive Hierarchy Model of Games. Quarterly Journal of Economics, 119(3), 861–898.
- Chiappori, P.-A., Levitt, S., & Groseclose, T. (2002). Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer. American Economic Review, 92(4), 1138–1151.
- Fudenberg, D., & Maskin, E. (1986). The Folk Theorem in Repeated Games with Discounting or with Incomplete Information. Econometrica, 54(3), 533–554.
- Güth, W., Schmittberger, R., & Schwarze, B. (1982). An Experimental Analysis of Ultimatum Bargaining. Journal of Economic Behavior and Organization, 3(4), 367–388.
- Harsanyi, J. C., & Selten, R. (1988). A General Theory of Equilibrium Selection in Games. MIT Press.
- Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., Gintis, H., & McElreath, R. (2001). In Search of Homo Economicus: Behavioral Experiments in 15 Small-Scale Societies. American Economic Review, 91(2), 73–78.
- Kreps, D. M., Milgrom, P., Roberts, J., & Wilson, R. (1982). Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma. Journal of Economic Theory, 27(2), 245–252.
- Nagel, R. (1995). Unraveling in Guessing Games: An Experimental Study. American Economic Review, 85(5), 1313–1326.
- Nash, J. F. (1950). Equilibrium Points in n-Person Games. Proceedings of the National Academy of Sciences, 36(1), 48–49.
- Nash, J. F. (1951). Non-Cooperative Games. Annals of Mathematics, 54(2), 286–295.
- Schelling, T. C. (1960). The Strategy of Conflict. Harvard University Press.
- Selten, R. (1978). The Chain Store Paradox. Theory and Decision, 9(2), 127–159.
- von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.