23.01.22 · economics / strategic-behavior

Game theory basics

draft3 tiersLean: none

Anchor (Master): Mas-Colell, Whinston & Green, Microeconomic Theory; relevant academic sources

Intuition [Beginner]

Game theory is the study of what happens when your best choice depends on what someone else chooses, and their best choice depends on what you choose. Most economic models assume people make decisions in isolation -- you decide how much to buy based on prices and your preferences. But many real decisions are strategic: businesses set prices knowing competitors will react, countries negotiate trade deals knowing the other side has its own interests, and employees decide whether to work hard knowing their colleagues might free-ride.

The most famous example is the Prisoner's Dilemma. Two suspects are arrested and held separately. Each can either cooperate (stay silent) or defect (betray the other). If both cooperate, each gets 1 year. If both defect, each gets 5 years. If one defects and the other cooperates, the defector goes free and the cooperator gets 10 years.

From each prisoner's perspective: "If my partner cooperates, I do better by defecting (0 vs. 1 year). If my partner defects, I also do better by defecting (5 vs. 10 years). Either way, I should defect." Both reason this way. Both defect. Both get 5 years. But if they had both cooperated, they would have gotten only 1 year each. Individual rationality leads to a collectively worse outcome.

This structure appears everywhere in economics. Two firms could both keep prices high and earn healthy profits, but each has an incentive to undercut the other. Countries could all reduce carbon emissions and avoid climate catastrophe, but each benefits if the others cut while it continues polluting. The tension between individual and collective rationality is the central problem of game theory.

Visual [Beginner]

Prisoner's Dilemma

                    Prisoner B
                    Cooperate    Defect
                +-------------+-------------+
  Cooperate     |  A: 1 year  |  A: 10 yrs  |
  Prisoner A    |  B: 1 year  |  B: goes    |
                |             |      free   |
                +-------------+-------------+
  Defect        |  A: goes    |  A: 5 years |
                |      free   |  B: 5 years |
                |  B: 10 yrs  |             |
                +-------------+-------------+

  For A: Defect is better regardless of what B does.
  For B: Defect is better regardless of what A does.
  Result: Both defect. Both get 5 years.
  Better outcome for both: Both cooperate, get 1 year.


  Dominant Strategy: A choice that is best regardless of
  what the other player does.

  Nash Equilibrium: A combination of strategies where no
  player can improve their outcome by changing alone.
  (Both defect is the Nash equilibrium here.)

Worked Example [Beginner]

Two gas stations sit across the street from each other. Each can set a high price ( $4/ g a l l o n) or a * * l o w p r i ce * * ($ 3/gallon).

	Station B: High	Station B: Low
Station A: High	A: $10, 000, B :$ 10,000	A: $2, 000, B :$ 14,000
Station A: Low	A: $14, 000, B :$ 2,000	A: $6, 000, B :$ 6,000

(Payoffs are daily profit.)

If Station B charges High: Station A earns $14, 000 b y c ha r g in g L o w (v s .$ 10,000 for High). Low is better. If Station B charges Low: Station A earns $6, 000 b y c ha r g in g L o w (v s .$ 2,000 for High). Low is better.

Low is a dominant strategy for Station A. By symmetry, Low is also a dominant strategy for Station B.

Nash Equilibrium: Both charge Low, each earning $6,000.

But if they both charged High, each would earn $10,000. The individually rational strategy (cut prices to steal customers) leads to a worse outcome for both. This is the Prisoner's Dilemma applied to price competition.

How might they resolve this? In practice: brand loyalty, location advantages, product differentiation, or -- if there are few enough competitors -- tacit collusion (each understands that a price war hurts both, so they match each other's high prices without explicitly agreeing to do so, which may run afoul of antitrust law).

Check your understanding [Beginner]

Formal Definition [Intermediate+]

Basic definitions

A game consists of:

A set of players $N = {1, 2, ..., n}$
A set of strategies $S_{i}$ for each player $i$
A payoff function $u_{i} : S_{1} \times S_{2} \times ... \times S_{n} \to R$ for each player $i$

Dominant strategies

A strategy $s_{i}^{*}$ is strictly dominant for player $i$ if:

$u_{i} (s_{i}^{*}, s_{- i}) > u_{i} (s_{i}, s_{- i}) \forall s_{i} \neq = s_{i}^{*}, \forall s_{- i}$

where $s_{- i}$ denotes the strategies chosen by all players other than $i$ .

If every player has a dominant strategy, the game has a dominant strategy equilibrium. This is the strongest solution concept.

Nash equilibrium

A strategy profile $(s_{1}^{*}, s_{2}^{*}, ..., s_{n}^{*})$ is a Nash equilibrium if:

$u_{i} (s_{i}^{*}, s_{- i}^{*}) \geq u_{i} (s_{i}, s_{- i}^{*}) \forall s_{i} \in S_{i}, \forall i \in N$

No player can improve their payoff by unilaterally deviating. John Nash proved (1950) that every finite game (finite players, finite strategies) has at least one Nash equilibrium (possibly in mixed strategies).

Mixed strategies

A mixed strategy is a probability distribution over pure strategies. In a mixed-strategy Nash equilibrium, each player randomizes so that the other players are indifferent between their strategies.

Example: In a penalty kick game (kicker chooses left or right, goalkeeper chooses left or right), the Nash equilibrium has both players randomizing. If the kicker always kicks left, the goalkeeper will always dive left. The kicker must randomize to keep the goalkeeper guessing.

Coordination games

Unlike the Prisoner's Dilemma, coordination games have multiple Nash equilibria and the challenge is selecting among them.

Battle of the Sexes: Two friends prefer different restaurants but strongly prefer going together over going alone. Two Nash equilibria: both go to restaurant A, or both go to restaurant B. The game has no dominant strategy; the challenge is coordination.

Stag Hunt (Rousseau): Two hunters can hunt a stag (large payoff, requires cooperation) or a hare (small payoff, can be done alone). Two Nash equilibria: both hunt stag, or both hunt hare. The stag equilibrium is better for both, but it requires trust -- if one hunter doubts the other will cooperate, the safe choice is to hunt hare.

Repeated games

When the Prisoner's Dilemma is played repeatedly, cooperation can emerge as an equilibrium. The Folk Theorem states that in infinitely repeated games (or games with unknown end date), any payoff between the minmax payoff and the cooperative payoff can be sustained as a Nash equilibrium, provided the discount rate is low enough (players value the future sufficiently).

The tit-for-tat strategy (cooperate first, then copy the other player's previous move) is a simple and effective strategy in repeated Prisoner's Dilemma tournaments, as demonstrated by Robert Axelrod (1984).

Sequential games and backward induction

In sequential games (players move in order), the solution concept is subgame perfect Nash equilibrium, found by backward induction: starting from the final decision and working backward.

Example: An incumbent firm faces a potential entrant. The entrant decides whether to enter. If the entrant enters, the incumbent decides whether to fight (price war) or accommodate. Backward induction: the incumbent accommodates (fighting is costly). Knowing this, the entrant enters. The threat to fight is not credible because it is not rational once the entrant has entered.

Key Concepts [Intermediate+]

Game theory: The mathematical study of strategic interaction among rational decision-makers.
Prisoner's Dilemma: A game where individual rationality leads to a collectively suboptimal outcome. The canonical model of cooperation problems.
Nash equilibrium: A strategy profile where no player gains by unilaterally deviating. The central solution concept in non-cooperative game theory.
Dominant strategy: A strategy that yields a higher payoff regardless of what others do.
Mixed strategy: Randomizing over pure strategies. Used when no pure strategy Nash equilibrium exists or to make oneself unpredictable.
Coordination game: A game with multiple Nash equilibria where players benefit from choosing the same strategy.
Repeated game: A game played multiple times. Repeated interaction can sustain cooperation that would not occur in a one-shot game.
Backward induction: Solving a sequential game by reasoning from the end to the beginning.
Subgame perfect equilibrium: A Nash equilibrium that is also an equilibrium in every subgame. Eliminates non-credible threats.
Folk theorem: In infinitely repeated games, many outcomes (including cooperation) can be sustained as equilibria.

Exercise (hard, short answer).

In a repeated Prisoner's Dilemma played an unknown number of times, explain why cooperation can be sustained even though it cannot be sustained in a one-shot game.

Answer

In a one-shot game, defection is always the best response because there is no future consequence. In a repeated game with no known end, each player's current action affects the other player's future behavior. If Player A defects today, Player B can retaliate by defecting tomorrow (and beyond). The future cost of retaliation can exceed the immediate gain from defection, making cooperation the best response. This is the logic behind tit-for-tat and other trigger strategies. The Folk Theorem formalizes this: with a sufficiently low discount rate (high patience), cooperation can be sustained as a Nash equilibrium.

Academic Perspectives [Master]

Neoclassical / mainstream view

Game theory is a foundational tool in modern neoclassical economics. It underpins the analysis of oligopoly (Cournot, Bertrand, Stackelberg models), mechanism design (auction theory, matching markets), and information economics (adverse selection, moral hazard, signaling). The Nash equilibrium is the standard solution concept, though refinements (subgame perfection, Bayesian Nash equilibrium, perfect Bayesian equilibrium) are used in specific contexts.

The mainstream approach assumes rational agents with well-defined preferences who can perform the strategic reasoning required by the model. This assumption is both the framework's strength (it generates precise, testable predictions) and its limitation (real people are boundedly rational).

Keynesian perspective

Keynesians use game theory to model coordination failures in macroeconomics. A recession can be understood as a coordination game: if all firms expect demand to be high, they hire and invest, producing high demand. If all expect low demand, they cut back, producing low demand. Both outcomes can be Nash equilibria. The economy can get stuck in a "bad" equilibrium (recession) even though a "good" equilibrium (full employment) exists. This provides a rationale for government intervention: fiscal or monetary policy can shift expectations and move the economy to the better equilibrium.

The concept of strategic complementarities (Cooper and John, 1988) formalizes this: when one firm's investment makes other firms' investments more profitable (through demand spillovers), multiple equilibria can arise, and the high-activity equilibrium may not be reached without coordination.

Marxian perspective

Marxian economists use game theory to analyze class conflict and bargaining power. The relationship between capital and labour can be modeled as a bargaining game: firms and workers negotiate over wages and profits, and the outcome depends on each party's outside option (what happens if negotiations fail). The decline of unions, the threat of offshoring, and the reserve army of unemployed workers all shift the bargaining game in favor of capital.

Marxian game theory also analyzes collective action problems among workers. The Prisoner's Dilemma structure explains why individual workers may not join strikes or unions (free-riding on others' efforts), even though all workers would benefit from collective action. Overcoming this requires institutions (union membership rules, picket lines, solidarity networks) that change the payoff structure.

Austrian view

Austrian economists are skeptical of formal game theory. Following Hayek, they argue that real economic decisions are made under radical uncertainty, not the well-defined payoff matrices of game theory. Players in real markets do not know the rules, the payoffs, or even who the other players are. The formal game assumes knowledge that real agents do not possess.

Austrians also criticize the equilibrium focus. In real markets, entrepreneurs are constantly discovering new strategies, new products, and new opponents. The game is never in equilibrium; it is a process of discovery and learning. Formal game theory, in the Austrian view, captures the logic of strategy but misses the dynamic, open-ended nature of real competition.

Institutional perspective

Institutional economists use game theory to study how institutions emerge and persist. The repeated Prisoner's Dilemma provides a model for how cooperation can arise without a central enforcer: repeated interaction creates incentives for mutual cooperation. Institutions (property rights, contract enforcement, social norms) can be understood as mechanisms that transform one-shot games into repeated games, making cooperation sustainable.

Elinor Ostrom's work on governing the commons (1990) is a landmark application. Common-pool resources (fisheries, grazing lands, irrigation systems) face a Prisoner's Dilemma: each user has an incentive to overuse, but overuse destroys the resource for everyone. Ostrom showed that communities can develop institutions (monitoring, sanctions, reputation systems) that sustain cooperation without privatization or government control, contradicting the prediction that common resources are inevitably overexploited ("the tragedy of the commons").

Behavioral perspectives

Behavioral game theory examines how real people play games, revealing systematic deviations from the predictions of standard game theory:

Ultimatum Game: One player proposes a split of $10; t h eo t h er a cce pt sor r e j ec t s (b o t h g e t n o t hin g i f r e j ec t e d) . S t an d a r d t h eor y p r e d i c t s t h e p r o p oser o f f er s$ 0.01 and the receiver accepts. In experiments, most proposers offer 40-50%, and offers below 20% are frequently rejected. People care about fairness, not just money.
Dictator Game: The proposer simply allocates the money; the receiver has no choice. Even here, most proposers give something (20-30%), contradicting pure self-interest.
Trust Game: The sender gives money to the receiver, which is multiplied (e.g., tripled). The receiver decides how much to return. Standard theory predicts zero sending. In experiments, most senders trust, and most receivers reciprocate.
Level-k thinking: People do not reason infinitely ("I think that you think that I think..."). Most people engage in 1-2 levels of strategic reasoning, leading to systematic deviations from Nash equilibrium predictions in novel games.

These findings have led to models of social preferences (Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000) that incorporate fairness, reciprocity, and inequality aversion into the utility function.

Historical Context [Master]

von Neumann and Morgenstern (1944): Theory of Games and Economic Behavior founded game theory as a mathematical discipline. The book introduced the concept of expected utility and zero-sum game solutions.
John Nash (1950): Proved the existence of Nash equilibrium in finite games. The concept became the cornerstone of non-cooperative game theory. Nash was awarded the Nobel Prize in Economics in 1994.
Prisoner's Dilemma (1950): Formulated by Merrill Flood and Melvin Dresher at RAND and named by Albert Tucker. Became the most widely used model in game theory and social science.
Folk Theorem (1950s-1970s): A series of results (credited to "folk" because the origins are murky) showing that cooperation can be sustained in repeated games. Formalized by Aumann and Shapley, and later by Friedman (1971).
Selten (1965, 1975): Introduced subgame perfect equilibrium and trembling hand perfection, refining Nash equilibrium for sequential games and games with small probabilities of error. Shared the 1994 Nobel Prize with Nash and Harsanyi.
Axelrod's tournaments (1981, 1984): Computer tournaments for the repeated Prisoner's Dilemma. Tit-for-tat (submitted by Anatol Rapoport) won both tournaments, demonstrating the power of simple reciprocal strategies. The results influenced thinking about the evolution of cooperation.
Mechanism design (1970s-present): The "reverse" of game theory: designing the rules of the game to produce desired outcomes. Hurwicz, Maskin, and Myerson received the 2007 Nobel Prize for this work. Applications include auction design (spectrum auctions, online advertising), matching markets (medical residency matching, school choice), and voting systems.
Behavioral game theory (1990s-present): Experimental and theoretical work incorporating psychological realism into game-theoretic models. The work of Colin Camerer, Matthew Rabin, Ernst Fehr, and others has transformed the field.

Bibliography [Master]

Axelrod, R. (1984). The Evolution of Cooperation. Basic Books.
Camerer, C. F. (2003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press.
Cooper, R., & John, A. (1988). Coordinating coordination failures in Keynesian models. Quarterly Journal of Economics, 103(3), 441-463.
Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. Quarterly Journal of Economics, 114(3), 817-868.
Myerson, R. B. (1991). Game Theory: Analysis of Conflict. Harvard University Press.
Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1), 48-49.
Ostrom, E. (1990). Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge University Press.
von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.