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Slide 01
Game
Theory
- Economics • Game Theory
- The mathematics of strategic interaction — how rational agents make decisions when outcomes depend on the choices of others.
- From cold war arms races to auctions, evolution to climate negotiation
- Nash Equilibrium
- Prisoner's Dilemma
- Auction Theory
- Evolutionary Games
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Slide 02
What Is Game Theory?
- Foundations
- Game theory is the formal study of strategic decision-making among rational agents whose outcomes are mutually interdependent.
- Players
- The decision-makers — individuals, firms, nations, or biological organisms.
- Strategies
- Complete plans of action specifying what a player will do in every possible situation.
- Payoffs
- Outcomes or utilities each player receives for every combination of strategies chosen.
- The central question: What will rational players do, and what outcome will result? Unlike classical optimization, the "best" choice depends on what you expect others to choose.
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Slide 03
Origins of the Field
- History
- 1713
- James Waldegrave describes a minimax solution to le Her — an early card game — in a letter to Pierre-Rémond de Montmort.
- 1838
- Antoine Augustin Cournot models duopoly competition, anticipating equilibrium concepts by over a century.
- 1944
- John von Neumann and Oskar Morgenstern publish Theory of Games and Economic Behavior, founding modern game theory.
- 1950
- John Nash proves the existence of equilibrium in any finite game — the Nobel Prize-winning insight that defines the field.
- 1994
- Nash, Harsanyi, and Selten share the Nobel Memorial Prize in Economic Sciences.
- 2020s
- Game-theoretic methods power mechanism design, algorithmic auctions, and AI alignment research.
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Slide 04
Von Neumann & Morgenstern
- Founders
- John von Neumann, a Hungarian-American polymath, proved the minimax theorem in 1928: in any zero-sum two-player game, there exists a strategy pair such that one player minimizes their maximum loss while the other maximizes their minimum gain.
- Partnering with economist Oskar Morgenstern, he extended this framework into a comprehensive theory of economic behavior, introducing expected utility theory and the formal apparatus of cooperative game theory.
- "Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do."
- — John von Neumann
- Key Contributions
- Minimax Theorem — optimal strategy in zero-sum conflict
- Expected Utility — rational preferences under uncertainty
- Cooperative Games — characteristic function form
- Strategic Form — normal-form game representation
- The Book
- Theory of Games and Economic Behavior (1944) — 641 pages, created an entirely new scientific discipline.
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Slide 05
The Nash Equilibrium
- Nash Equilibrium
- A profile of strategies — one for each player — from which no individual player can profitably deviate, given that all others hold their strategies fixed.
- Formal Definition
- Strategy profile s* is a Nash Equilibrium if for every player i and every alternative strategy s'ᵢ:
- uᵢ(s*ᵢ, s*₋ᵢ) ≥ uᵢ(s'ᵢ, s*₋ᵢ)
- Key Properties
- Every finite game has at least one Nash Equilibrium (possibly in mixed strategies)
- Self-enforcing: no player wants to deviate unilaterally
- Not necessarily unique — many games have multiple equilibria
- Not necessarily efficient — the equilibrium can be collectively suboptimal
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Slide 06
Finding Nash Equilibria
- Nash Equilibrium
- Consider the Coordination Game: two drivers choosing which side of the road to drive on. The payoff matrix (row = Driver 1, column = Driver 2):
- Driver 1 \ Driver 2
- Drive Left
- Drive Right
- Drive Left
- 1, 1
- −1, −1
- Drive Right
- −1, −1
- 1, 1
- Highlighted cells are Nash Equilibria: (Left, Left) and (Right, Right). Neither player benefits from switching when the other holds firm. This explains why countries adopt uniform traffic conventions — any convention beats no convention.
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Slide 07
John Forbes Nash Jr.
- The Mathematician
- Born in 1928 in West Virginia, Nash submitted his 27-page doctoral dissertation at Princeton in 1950. The central result — that every finite non-cooperative game possesses at least one equilibrium in mixed strategies — transformed economics, political science, and biology.
- Nash's proof used Kakutani's fixed-point theorem, a topological result, applied to best-response correspondences. The elegance lay in showing existence without constructing the solution explicitly.
- After decades battling paranoid schizophrenia, Nash was awarded the Nobel Prize in 1994. His story was dramatized in A Beautiful Mind (2001). He died in a taxi accident in 2015.
- 1950 Dissertation
- "Non-Cooperative Games" — 27 pages that changed three disciplines.
- Nash's Insight
- Prior work (von Neumann) addressed only two-player zero-sum games. Nash tackled any number of players with any payoff structure.
- Nobel Citation
- "For pioneering analysis of equilibria in the theory of non-cooperative games."
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Slide 08
The Prisoner's Dilemma
- Classic Games
- The most studied game in all of social science — a parable of why individual rationality can lead to collective catastrophe.
- The Scenario
- Two suspects are arrested. Each can Cooperate (stay silent) or Defect (betray the other). They cannot communicate. The district attorney offers each: if you testify against your partner who stays silent, you go free and they get 10 years. If both testify, each gets 5 years. If both stay silent, each gets 1 year.
- The dilemma: each prisoner is better off defecting regardless of what the other does — yet mutual defection is worse for both than mutual cooperation.
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Slide 09
Prisoner's Dilemma: The Payoff Matrix
- Classic Games
- Player A \ Player B
- Cooperate (Silent)
- Defect (Betray)
- Cooperate (Silent)
- −1, −1
- −10, 0
- Defect (Betray)
- 0, −10
- −5, −5
- The unique Nash Equilibrium (highlighted) is Defect/Defect — yielding −5 each. Yet Cooperate/Cooperate would yield only −1 each. This gap between Nash equilibrium and social optimum is the "price of anarchy."
- Dominant Strategy
- Defect is dominant: it yields a better payoff for a player regardless of what the opponent chooses.
- Social Dilemma
- When every player follows their dominant strategy, the outcome is collectively inferior — a market failure logic.
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Slide 10
The Dilemma Everywhere
- Applications
- The prisoner's dilemma structure appears across economics, politics, and biology wherever individual incentives diverge from collective welfare.
- Arms Races
- Two nations each prefer to disarm if the other disarms, but prefer to arm regardless — leading to costly mutual militarization even when both would benefit from peace.
- OPEC Cartel
- Each oil producer benefits from others restricting supply (keeping prices high) but gains by cheating and producing more — a classic multi-player dilemma.
- Climate Change
- Nations bear the full cost of reducing their emissions but share the benefit globally — creating powerful incentives to free-ride on others' reductions.
- Corporate Advertising
- Rivals each spend heavily on advertising that largely cancels out — both would prefer low spending, but each defects from the cooperative outcome.
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Slide 11
Cooperation Emerges:
The Iterated Dilemma
- Repeated Games
- When the same players interact repeatedly — with no fixed end — cooperation can become rational.
- Folk Theorem
- In infinitely repeated games (or games with unknown end-date), any outcome yielding each player more than their minimax payoff can be sustained as an equilibrium — including full cooperation.
- Tit-for-Tat
- Robert Axelrod's 1980 computer tournament found that Tit-for-Tat — cooperate first, then mirror opponent's last move — beat all rivals. It is nice, retaliatory, forgiving, and clear.
- "The evolution of cooperation requires that individuals have a sufficiently large chance to meet again so that they have a stake in their future interaction."
- — Robert Axelrod, The Evolution of Cooperation (1984)
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Slide 12
Zero-Sum Games
- Game Types
- In a zero-sum game, one player's gain is exactly another's loss — the total payoff is constant at every outcome.
- Von Neumann's minimax theorem was proved specifically for zero-sum games: there exists a saddle-point strategy pair where the maximizer's minimum gain equals the minimizer's maximum loss. This value of the game is unique.
- The optimal strategy often involves mixed strategies — randomizing among pure strategies according to precise probabilities so the opponent cannot exploit any predictable pattern.
- Minimax Principle
- Choose the strategy that maximizes your minimum possible payoff (maximin = minimax in zero-sum games).
- Real Examples
- Poker — one player's winnings are others' losses
- Chess / Go — one player wins, one loses
- Currency Speculation — gains against a counterparty
- Military Conflict — territory captured = territory lost
- Non-Zero-Sum Reality
- Most economic interactions are positive-sum: trade, investment, and cooperation create value rather than merely redistributing it.
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Slide 13
Pure vs. Mixed Strategies
- Strategy Theory
- A pure strategy is a deterministic choice. A mixed strategy is a probability distribution over pure strategies — the player randomizes.
- Mixed strategies are not irrational indecision. They prevent opponents from exploiting predictability. A penalty kicker who always shoots left is easily countered; one who mixes optimally keeps the goalkeeper genuinely uncertain.
- Kicker \ Keeper
- Dive Left
- Dive Right
- Shoot Left
- 0.6, 0.4
- 0.9, 0.1
- Shoot Right
- 0.9, 0.1
- 0.7, 0.3
- Empirical studies of penalty kicks find that professional players' mixed strategies match theoretical Nash predictions remarkably well.
- Nash's Theorem
- Every finite strategic-form game has at least one Nash Equilibrium in mixed strategies.
- Indifference Condition
- In a mixed-strategy equilibrium, each player is indifferent between the pure strategies they are mixing over.
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Slide 14
Extensive Form & Backward Induction
- Game Representations
- Sequential games — where players move in turns, observing prior moves — are represented as game trees.
- Extensive Form
- A tree structure depicting the order of moves, information available to each player at each decision node, and terminal payoffs. Captures timing in a way the normal (matrix) form cannot.
- Backward Induction
- Solve sequential games by starting at the final decision node and working backwards. Rational players anticipate optimal future play. This yields subgame perfect equilibrium — Nash equilibria that remain optimal in every subgame.
- Centipede Game
- Backward induction predicts players defect immediately — but experiments show players cooperate for many rounds, challenging strict rationality assumptions.
- Stackelberg Duopoly
- A leader firm commits to output first; the follower then optimizes. Backward induction shows the leader benefits from moving first — the "first-mover advantage."
- Hold-Up Problem
- A party making relationship-specific investments is vulnerable to renegotiation — explaining vertical integration, long-term contracts, and hostage exchanges.
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Slide 15
Cooperative Game Theory
- Cooperative Theory
- When binding agreements are possible, the relevant question shifts from "what will each player do?" to "which coalitions will form and how will they divide the gains?"
- The Core
- Allocations that no coalition can improve upon by acting alone. A stable distribution that no group can profitably deviate from collectively.
- Shapley Value
- Lloyd Shapley's 1953 solution assigns each player their average marginal contribution across all possible orderings of coalition formation. Uniquely fair by axiomatic definition.
- Bargaining Theory
- Nash's 1950 bargaining solution maximizes the product of utility gains over the disagreement point — used in wage negotiations, treaty design, and divorce settlements.
- "The Shapley value is the only efficient, symmetric, additive allocation rule satisfying the null player property."
- — Lloyd Shapley, 1953
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Slide 16
Evolutionary Game Theory
- Evolutionary Games
- Developed by John Maynard Smith and George Price in the 1970s, evolutionary game theory applies strategic logic to biology — without assuming rationality.
- Instead of rational optimization, selection pressure does the work: strategies that earn higher payoffs reproduce more. The population gradually shifts toward successful strategies.
- This reinterpretation transforms game theory from a theory of rational choice into a theory of natural selection, explaining the evolution of animal behavior, cooperation, altruism, and conflict without invoking conscious reasoning.
- Evolutionarily Stable Strategy
- A strategy that, when adopted by a population, cannot be invaded by any small group of mutants playing a different strategy.
- Replicator Dynamics
- Strategies that perform above average grow in population share; below-average strategies shrink. The dynamical counterpart of Nash equilibrium.
- Key Insight
- ESS ⊆ Nash Equilibrium — every evolutionarily stable strategy is a Nash Equilibrium, but not vice versa. ESS are stable equilibria under replicator dynamics.
- Applications
- Animal contest behavior, sex ratios, altruism and kin selection, signaling, immune system dynamics, cultural evolution.
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Slide 17
The Hawk-Dove Game
- Evolutionary Games
- The paradigmatic model of animal conflict over a resource of value V, where escalating conflict costs C.
- Opponent →
- Hawk
- Dove
- Hawk
- (V−C)/2, (V−C)/2
- V, 0
- Dove
- 0, V
- V/2, V/2
- When V > C (cheap conflict)
- Hawk is dominant — pure Hawk ESS. Every player escalates. Explains territorial aggression when costs are low relative to the prize.
- When V < C (costly conflict)
- Mixed ESS with proportion V/C of Hawks. The population stabilizes at a mix — explaining displays, rituals, and why escalation is relatively rare in nature despite competition.
- Maynard Smith's insight: we do not need to assume animals think strategically — selection pressure produces the same equilibrium that rational calculation would.
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Slide 18
Incomplete Information & Harsanyi
- Information Economics
- Real-world players often don't know opponents' payoffs, costs, or types. John Harsanyi (Nobel 1994) showed how to model such uncertainty.
- Bayesian Nash Equilibrium
- Each player has a type drawn from a probability distribution. Strategies map types to actions. Equilibrium requires each type's strategy to maximize expected payoff given beliefs about opponents' type distributions.
- Harsanyi's Insight
- Transform incomplete information into imperfect information by introducing "Nature" as a player who randomly assigns types. The resulting game can be analyzed with standard techniques.
- Signaling Games
- Informed players can communicate types through costly signals. Key examples: education as a signal of ability (Spence), advertising as a signal of product quality, peacock tails as fitness signals.
- Screening
- Uninformed parties design menus of contracts that induce informed agents to self-select, revealing their type. Used in insurance, labor contracts, and second-degree price discrimination.
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Slide 19
Mechanism Design:
"Reverse Game Theory"
- Mechanism Design
- Instead of analyzing a given game, mechanism design asks: what game should we construct to achieve a desired social outcome?
- Leonid Hurwicz, Eric Maskin, and Roger Myerson won the 2007 Nobel Prize for developing mechanism design theory — the engineering branch of game theory.
- A mechanism is a set of rules specifying: what messages players send, and what outcome results from each message profile. The designer chooses the mechanism; players respond strategically.
- "Mechanism design is to game theory what engineering is to physics — it's the constructive, prescriptive side."
- — Eric Maskin, Nobel Lecture, 2007
- Revelation Principle
- Any outcome achievable by some mechanism is achievable by a "direct" mechanism where players truthfully report their types — massively simplifying design problems.
- Applications
- Spectrum auctions
- School choice algorithms
- Organ donor matching
- Carbon permit trading
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Slide 20
Auction Theory
- Auctions are mechanisms for allocating goods under uncertainty about buyers' valuations. Paul Milgrom and Robert Wilson received the 2020 Nobel Prize for auction theory and practical auction design.
- English Auction
- Open ascending-bid. Dominant strategy: bid until you reach your true value. Efficient allocation.
- Dutch Auction
- Open descending-bid. Strategically equivalent to first-price sealed-bid. Winner bids below true value.
- First-Price Sealed-Bid
- Highest bidder pays their bid. Optimal strategy: shade bid below true value (bid shading). Equilibrium involves solving differential equations.
- Vickrey (Second-Price)
- Highest bidder pays second-highest bid. Dominant strategy: bid your true value. Strategy-proof and efficient.
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Slide 21
Revenue Equivalence & Spectrum Auctions
- Auction Theory
- Revenue Equivalence Theorem
- Under symmetric independent private values, all standard auction formats yield the same expected revenue to the seller. This fundamental result, due to Myerson (1981) and Riley & Samuelson (1981), guides auction choice.
- Winner's Curse
- In common-value auctions (everyone has the same true value, but different estimates), the winner is the bidder who most overestimated value. Rational bidders discount their bids accordingly — but many do not.
- FCC Spectrum Auctions
- Milgrom and Wilson designed the simultaneous multi-round auction for radio spectrum licenses, used by the US since 1994. Raised over $100 billion for the US government. The design accounted for complementarities between licenses.
- Combinatorial Auctions
- Bidders can bid on packages of items when complements matter. The assignment problem becomes computationally hard — mechanism design meets computer science.
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Slide 22
Matching Theory
- Market Design
- Some markets don't use prices. Matching theory designs rules for who gets who in markets where money cannot (or should not) clear the market.
- Gale-Shapley Algorithm
- David Gale and Lloyd Shapley's 1962 deferred acceptance algorithm produces a stable matching — no two agents would prefer each other to their assigned partners. Shapley shared the 2012 Nobel with Alvin Roth.
- Medical Residency (NRMP)
- The National Resident Matching Program assigns medical graduates to residency programs. Roth discovered the 1952 algorithm was equivalent to Gale-Shapley and redesigned it to eliminate strategic manipulation.
- School Choice
- Boston, New York, and many cities replaced ad-hoc assignment with strategy-proof mechanisms. Roth's work showed the old "Boston mechanism" created perverse incentives for gaming.
- Kidney Exchange
- When a donor is incompatible with their loved one's recipient, chains of exchanges allow compatible matches. Roth's algorithmic matching expanded the donor pool dramatically.
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Slide 23
Signaling Theory
- Information Games
- How can informed agents credibly communicate private information to skeptical audiences? Through costly signals that low types cannot afford to mimic.
- Michael Spence's 1973 job market signaling model showed that education can serve as a signal of ability — even if it imparts no productive skills. If high-ability workers find education less costly, they will acquire it to distinguish themselves.
- The key requirement for a separating equilibrium: single-crossing — the marginal cost of the signal must be lower for high types than for low types.
- Spence's Insight
- Education can be valuable even if it teaches nothing if it credibly separates worker types — a troubling possibility for education policy and social investment.
- Examples
- Peacock tails — fitness signal (Zahavian handicap)
- Warranties — product quality signal
- Advertising — brand investment as quality signal
- Dividends — firm profitability signal
- Burning money — costly but informative
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Slide 24
Behavioral Game Theory
- Behavioral Extensions
- Real humans systematically deviate from Nash equilibrium predictions. Behavioral game theory documents these departures and builds richer models.
- Ultimatum Game
- A proposer splits a sum; the responder accepts or rejects (both get nothing if rejected). Standard theory: offer the minimum. Observed: most offers are 40–50%, and low offers are rejected — fairness matters.
- Public Goods Games
- Players contribute to a public good that benefits everyone. Standard theory predicts zero contribution. Observed: substantial contribution in early rounds, declining with experience — conditional cooperation and punishment.
- Level-k Thinking
- Players reason about how many steps of strategic thinking their opponents perform. Level-0: random. Level-k: best responds to level-(k−1). Explains beauty contests and initial auction behavior.
- Colin Camerer, Ernst Fehr, and Matthew Rabin are leading figures connecting psychology and game theory — finding that fairness, reciprocity, and bounded rationality shape strategic interaction profoundly.
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Slide 25
Game Theory & Oligopoly
- Market Structure
- Modern industrial organization applies game theory to analyze strategic interaction among firms — pricing, entry, investment, and collusion.
- Cournot Competition
- Firms simultaneously choose quantities. Nash equilibrium yields prices above marginal cost but below monopoly. More firms → more competitive outcome (approaches perfect competition).
- Bertrand Competition
- Firms simultaneously set prices. With identical goods, the only Nash equilibrium has both firms pricing at marginal cost — the Bertrand paradox: two firms can suffice for competitive pricing.
- Entry Deterrence
- Incumbent firms invest in excess capacity as a commitment device — a credible threat to expand output and punish entrants. Without commitment, threats are incredible (subgame imperfection).
- Collusion & Stability
- Repeated interaction enables tacit collusion. The incentive to defect from a cartel is balanced by future punishment. Collusion is more stable with fewer firms, higher discount rates, and observable prices.
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Slide 26
Voting, Politics & Game Theory
- Political Economy
- Electoral competition, legislative bargaining, and international relations all involve strategic actors whose choices are interdependent.
- Median Voter Theorem
- Under single-peaked preferences and majority rule, the median voter's preferred outcome wins. Competing parties converge to the center — explaining the "crowding" of political positions in two-party systems.
- Arrow's Impossibility
- No voting rule satisfies all "fairness" axioms simultaneously (unanimity, independence of irrelevant alternatives, non-dictatorship) when there are three or more candidates.
- Legislative Bargaining
- Rubinstein alternating-offers bargaining applied to legislative coalition formation. Proposal power gives agenda-setters substantial advantage — explaining committee power in Congress.
- Nuclear Deterrence
- Schelling's analysis of credible commitments, brinkmanship, and mutual assured destruction — game theory applied to the existential risks of the Cold War.
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Slide 27
Climate Change as a Global Game
- Global Challenges
- International climate negotiations are among the highest-stakes multi-player games in history — a global commons problem with sovereign players and no enforcement.
- The atmosphere is a global public good: each nation bears the full cost of its own emissions reductions but shares the benefit of all reductions globally. This creates classic free-rider incentives.
- Game-theoretic analysis of the Paris Agreement shows it relies on reciprocal pledge-and-review rather than enforceable commitments — a repeated game where reputation matters and ratcheting mechanisms build over time.
- Key Mechanisms
- Carbon clubs, border adjustments, technology transfer, and side payments can shift equilibria — making emission reduction individually rational by changing the payoff structure.
- Coalition Formation
- IEA (International Environmental Agreement) theory shows stable coalitions are typically small — but even small committed coalitions can be significant if they include key emitters.
- Linking Agreements
- Linking climate agreements to trade, aid, or security agreements can change outside options and make cooperation credible — an application of richer mechanism design.
- Tipping Points
- Coordination games with tipping thresholds can generate multiple equilibria — clean and dirty — with history and expectations determining which the world ends up in.
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Slide 28
Algorithmic Game Theory
- Modern Frontiers
- As markets and interactions move online, computer science and game theory have merged into a powerful new discipline.
- Price of Anarchy
- Measures how much worse Nash equilibrium outcomes are versus the social optimum. In routing games, selfish behavior can degrade network efficiency by up to 33%.
- Sponsored Search
- Google's ad auction is a generalized second-price mechanism. Advertisers bid for keyword positions; mechanism design ensures (approximate) truthful bidding dominates.
- AI and Game Theory
- DeepMind's AlphaGo, AlphaStar, and OpenAI Five use game-theoretic training (self-play, fictitious play). Game theory also guides multi-agent AI safety and alignment research.
- Computational Complexity
- Finding Nash equilibria is PPAD-complete — computationally hard in general. This challenges the behavioral foundation: can real agents compute equilibria they supposedly play?
- Fairness in Algorithms
- Mechanism design techniques from game theory inform fair division algorithms — envy-free allocations, proportional apportionment, and equitable resource distribution in computational settings.
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Slide 29
The Nobel Impact of Game Theory
- Legacy
- More Nobel Memorial Prizes in Economics have been awarded for game theory and related fields than any other single area.
- 1994
- Nash, Harsanyi, Selten — non-cooperative game theory
- 1996
- Mirrlees & Vickrey — incentives under asymmetric information
- 2001
- Akerlof, Spence, Stiglitz — markets with asymmetric information
- 2005
- Aumann & Schelling — conflict and cooperation through game theory
- 2007
- Hurwicz, Maskin, Myerson — mechanism design
- 2012
- Shapley & Roth — stable allocations and market design
- 2020
- Milgrom & Wilson — auction theory and design
- Fields Transformed
- Economics — industrial organization, labor, trade, finance
- Political Science — voting, war, negotiation
- Biology — evolutionary dynamics, animal behavior
- Computer Science — algorithm design, AI
- Law — contract theory, antitrust, regulation
- Nobel Prizes with game theory at core
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Slide 30
The Enduring Power of Strategic Thinking
- Economics • Game Theory
- Game theory began as pure mathematics and became one of the most widely applied frameworks in human knowledge — because strategic interdependence is everywhere.
- The Core Lesson
- Optimal decisions depend on what others do. Ignoring this interdependence leads to systematic errors — in policy, business, and life.
- Open Questions
- Equilibrium selection, bounded rationality, dynamic mechanism design, and the game theory of artificial agents remain vibrant research frontiers.
- The Bigger Picture
- From bacteria to nations, from card games to climate treaties — strategic interaction shapes our world. Game theory gives us the language to understand and design it.
- "Life is a game. Money is how we keep score."
- — Ted Turner (adapted) — but game theory reminds us the game is far richer than any single payoff.
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