Game Theory: Strategic Decision-Making in Economics and Beyond

Foundations of Game Theory

Game theory: mathematical study of strategic interaction between rational agents. Components: players, strategies, payoffs. Normal form (matrix) games: simultaneous decisions. Extensive form (tree) games: sequential decisions with information sets. Dominant strategy: optimal regardless of opponents' choices. Nash equilibrium (Nash, 1950, Nobel 1994): no player can improve by unilaterally changing strategy. Mixed strategy: randomizing over pure strategies. Zero-sum games: one player's gain = opponent's loss (minimax theorem, von Neumann 1928). Non-zero-sum: cooperation possible, e.g., Prisoner's Dilemma. Pareto optimality: no player can improve without making another worse off.

Classic Games

Prisoner's Dilemma: individually rational defection leads to collectively suboptimal outcome. Iterated PD: cooperation emerges through reciprocity (Tit-for-Tat, Axelrod 1984). Chicken Game (Hawk-Dove): two Nash equilibria, coordination problem — arms race, brinkmanship. Battle of the Sexes: coordination with conflicting preferences — two pure NE, one mixed. Stag Hunt: cooperation risk-dominates defection — trust and assurance. Ultimatum Game: proposer offers split, responder accepts/rejects — fairness concerns violate rational prediction. Public goods game: free-riding problem, tragedy of the commons. Repeated games (Folk Theorem): cooperation sustainable through punishment threats in infinitely repeated games.

Mechanism Design and Auctions

Mechanism design: "reverse game theory" — designing rules to achieve desired outcomes (Hurwicz, Maskin, Myerson, Nobel 2007). Revelation principle: any mechanism can be replaced by one where players truthfully report preferences. VCG mechanism (Vickrey-Clarke-Groves): dominant strategy incentive compatible, efficient allocation. Auction theory (Vickrey, Nobel 1996; Milgrom & Wilson, Nobel 2020): First-price sealed bid: shade bids below true value. Second-price (Vickrey): dominant strategy to bid true value. English auction: ascending, equivalent to second-price. Revenue equivalence theorem: under certain conditions, all standard auctions yield same expected revenue. Spectrum auctions: simultaneous multiple-round auctions designed by economists for FCC.

Applications

Economics: oligopoly pricing (Cournot, Bertrand models), bargaining (Nash bargaining solution), market design (matching algorithms — Nobel 2012, Roth & Shapley). Political science: voting theory (Arrow's impossibility theorem, Gibbard-Satterthwaite), coalition formation, international relations (deterrence, arms control). Biology: evolutionary game theory (Maynard Smith), evolutionarily stable strategies (ESS), replicator dynamics. Computer science: algorithmic game theory (price of anarchy), incentive-compatible protocols. AI: multi-agent reinforcement learning, adversarial training (GANs). Real-world: kidney exchange programs (Roth), school choice (deferred acceptance), spectrum allocation, ad auctions (Google, Facebook).

Behavioral Game Theory

Classical game theory assumes perfect rationality — humans systematically deviate. Behavioral observations: people cooperate more than predicted (ultimatum game rejections), exhibit fairness concerns, reciprocity, and altruistic punishment. Bounded rationality (Simon): limited cognitive resources, satisficing instead of optimizing. Level-k thinking: level-0 randomizes, level-1 best-responds to level-0, etc. Quantal response equilibrium: players make errors, choose better responses more often. Social preferences: inequality aversion (Fehr-Schmidt), reciprocity, warm-glow giving. Neuroeconomics: neural basis of strategic decision-making (fMRI, game theory experiments). Prospect theory (Kahneman & Tversky, Nobel 2002): loss aversion, reference dependence, probability weighting in strategic contexts.

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