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Slide 01
GAME
THEORY
- The Mathematics of Strategy
- The mathematical study of strategic interaction — how rational agents make decisions when their outcomes depend on what others choose. From nuclear deterrence to auction design to evolutionary biology.
- Nash EquilibriumPrisoner's DilemmaDominant StrategyMechanism DesignZero-Sum
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Slide 02
What Is a Game?
- Foundations
- In game theory, a "game" is any situation where multiple agents make choices, and the outcome for each agent depends on the choices of all agents. Every game has three components: Players (who decides?), Strategies (what can each player choose?), and Payoffs (what does each player receive for each combination of strategies?).
- Players
- Decision-makers: can be individuals, firms, countries, species, or algorithms. Must have objectives and strategic capacity.
- Strategies
- The complete plan of action for each possible situation. A strategy specifies what to do at every decision point — not just the next move.
- Payoffs
- The outcome each player receives for every possible combination of strategies. Usually expressed as utilities — what each outcome is worth to each player.
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Slide 03
Origins of the Field
- History
- 1713James Waldegrave analyzes minimax strategy in card game le Her — first known game theory calculation.
- 1838Cournot's model of duopoly competition — firms choose quantities simultaneously. First economic game theory application.
- 1944Von Neumann & Morgenstern publish Theory of Games and Economic Behavior — founding text of modern game theory.
- 1950John Nash proves existence of equilibrium in any finite game. Nash's 27-page Princeton dissertation changes economics forever.
- 1994Nash, Harsanyi, and Selten share Nobel Prize. Game theory enters the economic mainstream institutionally as well as intellectually.
- 2007Hurwicz, Maskin, Myerson win Nobel for mechanism design — using game theory to design markets and institutions that achieve desired outcomes.
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Slide 04
Nash Equilibrium
- The Central Concept
- A Nash equilibrium is a set of strategies, one per player, such that no player can improve their payoff by unilaterally changing their strategy — given what all other players are doing. It is the mathematical definition of stability in strategic interaction.
- NE: for each player i, s*_i = argmax U_i(s*_i, s*_-i)
- What It Means
- At Nash equilibrium, each player is doing the best they can given what others are doing. No one has an incentive to deviate. The system is self-enforcing.
- What It Doesn't Mean
- Nash equilibrium does not imply efficiency. Players can be stuck in equilibria that are bad for everyone — like the arms race or traffic jams.
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Slide 05
The Prisoner's Dilemma
- The Iconic Example
- Two suspects are interrogated separately. Each can cooperate (stay silent) or defect (betray). The payoff matrix below shows prison years (lower is better):
- B: CooperateB: Defect
- A: CooperateA: -1, B: -1A: -10, B: 0
- A: DefectA: 0, B: -10A: -5, B: -5 ← NE
- Defect dominates Cooperate for both players — regardless of what the other does, you're better off defecting. Yet mutual defection (-5,-5) is far worse for both than mutual cooperation (-1,-1). The rational outcome is collectively irrational.
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Slide 06
When One Choice Always Wins
- Dominant Strategies
- A dominant strategy is one that yields a higher payoff than any alternative, regardless of what other players do. In the Prisoner's Dilemma, "Defect" is a dominant strategy for both players — making it easy to predict, even without communication.
- Many real-world games have dominant strategies. Understanding them reveals why certain behaviors persist even when they appear counterproductive at the group level.
- Advertising "arms race" — firms advertise because not advertising is dominated even if mutual silence would be better
- Nations arm because disarming unilaterally is dominated, even if mutual disarmament is preferred
- Professional athletes use performance-enhancing drugs because abstaining unilaterally is dominated when others use them
- Overfishing: each fisher harvests maximally because restraint is dominated when others don't restrain
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Slide 07
What One Gains, Another Loses
- Zero-Sum Games
- In zero-sum games, payoffs always sum to zero — one player's gain is exactly another's loss. Chess, poker, and military conflict are zero-sum in their outcomes. Von Neumann and Morgenstern proved the minimax theorem: each player should minimize the maximum loss the opponent can inflict.
- Minimax Strategy
- In two-player zero-sum games, the optimal strategy minimizes your maximum possible loss. This is provably optimal — an opponent cannot do better than the minimax outcome against a minimax player.
- Mixed Strategies
- When no pure strategy is optimal (rock-paper-scissors), Nash showed mixed strategies (probability distributions over pure strategies) always yield equilibrium. A soccer kicker should randomize — a predictable pattern is exploited.
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Slide 08
Games Where Both Can Win
- Non-Zero-Sum Games
- Most economic, social, and biological situations are non-zero-sum: there exist strategy combinations that make everyone better off. Trade, marriage, employment, and alliance all create surplus that didn't exist before agreement.
- The challenge in non-zero-sum games is not just finding joint gains but distributing them — and preventing defection. This is the realm of bargaining theory, contract design, and institutions.
- Stag Hunt
- Two hunters can cooperate to catch a stag (big reward) or each hunt rabbits alone (small reward). Cooperation is better for both — but requires mutual trust. If one defects, the cooperator gets nothing.
- Battle of the Sexes
- Two partners prefer to be together but disagree on the activity. Multiple equilibria exist; neither is uniquely "correct." Communication and coordination mechanisms solve this.
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Slide 09
The Shadow of the Future
- Repeated Games
- When the Prisoner's Dilemma is played once, defection dominates. When it is played repeatedly by the same players indefinitely, cooperation can emerge. The threat of future punishment changes incentives — a powerful insight with massive implications for real-world relationships.
- Folk Theorem
- In infinitely repeated games, any outcome that gives each player more than their minimum guaranteed payoff can be sustained as a Nash equilibrium — if players are patient enough. Cooperation, punishment, and revenge are all theoretically supportable.
- Robert Axelrod's Tournaments (1980)
- TIT FOR TAT
- The strategy that won both tournaments: cooperate on round 1; thereafter copy your opponent's last move. Nice, retaliatory, forgiving, clear.
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Slide 10
Games in Biology
- Evolutionary Game Theory
- John Maynard Smith and George Price (1973) applied game theory to evolutionary biology. Instead of rational players choosing strategies, they modeled populations of organisms with genetically fixed strategies, and asked which strategies would spread over generations.
- An evolutionarily stable strategy (ESS) is one that, if adopted by the population, cannot be invaded by a rare mutant with a different strategy. ESS is Nash equilibrium adapted for biology — it's self-reinforcing at the population level.
- Hawk-Dove game explains when animals fight vs. retreat over resources
- Parental investment game: why do fathers invest less in offspring in most species?
- Sex ratio evolution: 50:50 sex ratio is ESS even when 1:3 would maximize offspring
- Altruism in kin selection: Hamilton's rule explains self-sacrifice for relatives
- Reciprocal altruism: Axelrod-style cooperation evolves between non-kin
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Slide 11
Fight or Flee
- The Hawk-Dove Game
- Two animals contest a resource V. Hawks always fight; Doves always retreat. If a Hawk meets a Dove: Hawk wins V, Dove gets 0. If Doves meet: each gets V/2. If Hawks meet: each has 50% chance of winning V, 50% chance of injury cost C.
- Opponent: HawkOpponent: Dove
- I am: Hawk(V−C)/2V
- I am: Dove0V/2
- When V > C: Hawks dominate. When V < C: mixed ESS emerges — V/C proportion of Hawks in population. Real animal populations are rarely pure hawk or pure dove; mixed strategies explain animal conflict data.
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Slide 12
Costly Signals and Credibility
- Signaling Theory
- If words are cheap, they can be faked. Michael Spence's (Nobel 2001) signaling theory explains why costly, hard-to-fake signals credibly convey information. A peacock's tail is metabolically expensive and makes it vulnerable to predators — which is precisely why peahens believe it signals genetic quality.
- Spence applied this to education markets: if education is harder for low-ability workers to obtain (more years, more struggle), then employers can use education as a signal of ability even if the education itself doesn't increase productivity.
- Education degrees as employer signal (Spence)
- Advertising spending signals product quality — only firms confident in repeat purchase advertise heavily
- Military posture as signal of resolve to adversaries
- Warranties signal product reliability — only makers of durable goods can afford them
- Venture capital investments signal startup quality to other investors
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Slide 13
The Market for Lemons
- Asymmetric Information
- George Akerlof's 1970 paper showed how information asymmetry destroys markets. Used car sellers know the car's quality; buyers don't. Buyers must average over all possibilities — which drives down their willingness to pay — which drives out sellers of good cars — which further lowers average quality — leading to market collapse.
- Adverse Selection
- When the uninformed party can't distinguish types, the "wrong" types are disproportionately selected. Health insurance markets attract the sick; car insurance attracts bad drivers.
- Solutions
- Signaling (informed party sends costly signal), screening (uninformed party designs menus to separate types), and mandates (force participation to prevent selection).
- Moral Hazard
- After a contract, agents take hidden actions that benefit themselves at others' expense. Insured drivers drive more recklessly; insured homeowners maintain less carefully.
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Slide 14
Engineering Game Rules
- Mechanism Design
- Mechanism design — sometimes called "reverse game theory" — asks: given a desired social outcome, what rules of the game will make self-interested players produce that outcome? It is the engineering branch of game theory, applied to auctions, regulations, and institutions.
- Leonid Hurwicz, Eric Maskin, and Roger Myerson won the 2007 Nobel Prize for this field. The Vickrey auction (sealed-bid, pay second-highest price) is the landmark result: it is incentive-compatible — each bidder's dominant strategy is to bid their true valuation.
- FCC spectrum auctions designed by Milgrom & Wilson (Nobel 2020) — raised $45B+
- Medical residency matching designed by Roth (Nobel 2012) using stable matching algorithm
- Organ donation allocation markets — Roth's work on repugnant markets
- School choice assignment mechanisms in Boston, NYC, Chicago
- Carbon permit auction design
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Slide 15
The Art of Bidding
- Auction Theory
- English Auction
- Open ascending bid. Dominant strategy: bid up to your valuation. Winner pays their bid. Highest valuation wins. Revenue may be lower than sealed-bid formats.
- Dutch Auction
- Open descending bid. Clock starts high, falls until someone stops it and pays that price. Strategically equivalent to first-price sealed bid.
- Vickrey (Second-Price)
- Sealed bid, pay second-highest price. Incentive-compatible — dominant strategy is truthful bidding. Used in Google Ad auctions (generalized second price). Reduces strategic manipulation.
- Revenue Equivalence
- Under standard assumptions, all four basic auction formats yield the same expected revenue. Real auctions deviate due to information, risk aversion, and collusion.
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Slide 16
How Deals Get Done
- Bargaining Theory
- Nash's bargaining solution (1950) identifies the unique agreement that rational players reach when bargaining over a surplus, given outside options. The key insight: your bargaining power depends on how much you lose if talks collapse — not on how much you want the deal.
- Nash Bargaining Solution
- Maximize the product of gains above disagreement point: (u₁−d₁)(u₂−d₂). Implies equal split of surplus when outside options are symmetric. Asymmetric power shifts the split proportionally.
- Rubinstein Alternating Offers
- When players alternate offers, the first mover advantage is larger the less patient the opponent is. Impatience is weakness — the party with better alternatives and lower discount rate captures more surplus.
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Slide 17
Slide 17
- "You don't have to be nice to be good. You just have to be clear about your incentives."— Robert Axelrod, The Evolution of Cooperation
- 11Nobel Prizes related to Game Theory
- 1950Nash's 27-page dissertation
- ∞Applications found since
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Slide 18
Voting, Coalitions, Power
- Political Applications
- Political science absorbed game theory to analyze voting rules, legislative coalition formation, and war. The Median Voter Theorem (Downs 1957) explains why two-party systems converge to the political center: any candidate to the left of the median can be beaten by moving right.
- The Shapley Value assigns power to each coalition member proportional to their marginal contribution across all possible orderings — a mathematical definition of political power used in weighted voting systems and corporate governance.
- Arms race as Prisoner's Dilemma — Cold War deterrence modeled by Schelling
- Legislative vote trading (logrolling) modeled as cooperative game
- NATO deterrence: credible commitment problems and Schelling's "madman theory"
- Electoral system design: how voting rules shape outcomes
- Constitutional design: veto players and gridlock modeled formally
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Slide 19
When More Users Make More Value
- Network Effects and Coordination
- Many modern markets are coordination games where the value of a product increases with the number of users. This creates tipping points and winner-take-all dynamics that game theory explains precisely.
- Competing standards (VHS vs. Betamax, iOS vs. Android) are coordination games where multiple equilibria exist. Which equilibrium is selected depends on expectations, early movers, and network structure — not necessarily product quality.
- Social networks: worth joining only if friends are already there
- Currency: money is valuable because everyone accepts it, accepted because valuable
- QWERTY keyboard: inferior layout sustained by coordination equilibrium
- Programming languages: Python's dominance is partly coordination, not superiority
- Payment systems: credit cards vs. cash coordination between merchants and consumers
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Slide 20
When Individual Rationality Destroys the Commons
- The Tragedy of the Commons
- Garrett Hardin's 1968 essay described the logic of common-pool resource depletion: each individual rationally extracts as much as possible from a shared resource, collectively depleting it. Game theory formalizes this as an n-player Prisoner's Dilemma.
- The Problem
- Each additional unit extracted gives private benefit but distributes cost across all users. Dominant strategy: extract maximally. Result: depletion of the commons.
- Elinor Ostrom's Answer
- Nobel 2009. Real communities often manage commons without either privatization or state control — through local institutions, monitoring, and graduated sanctions. Hardin's "tragedy" is not inevitable.
- Real Examples
- Atlantic cod collapse, aquifer depletion, overfishing, traffic congestion, antibiotic resistance, carbon emissions — all Tragedy of the Commons scenarios at different scales.
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Slide 21
Coordination Without Communication
- Schelling Points
- Thomas Schelling (Nobel 2005) showed that when people must coordinate without communication, they converge on "focal points" — solutions that stand out by virtue of their prominence, symmetry, or convention. He called these Schelling points.
- Asked to meet a stranger in New York City without agreeing on a time or place, most people say noon at Grand Central Station's main clock. No communication needed — shared cultural knowledge creates a focal point that everyone expects everyone else to choose.
- Currency conventions — why dollars come in specific denominations
- Cease-fire lines along geographic features (rivers, mountains)
- Territorial disputes settled by historic lines, rivers, equidistance
- "50-50" splits as default in bargaining when other criteria are unclear
- Industry price points: $0.99, $9.99, $99, $999 — psychological landmarks
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Slide 22
Games with Hidden Types
- Incomplete Information
- John Harsanyi (Nobel 1994) showed how to analyze games where players have private information — about their own costs, values, or types. The key insight: treat uncertainty about types as uncertainty about which game is being played, resolved by nature before the game begins.
- Bayesian Nash Equilibrium
- Each player maximizes expected utility given their type and beliefs about other players' types and strategies. The extension of Nash equilibrium to games with private information.
- Perfect Bayesian Equilibrium
- Extends Bayesian Nash to sequential games: beliefs must be consistent with strategies on and off the equilibrium path. Eliminates implausible threats and promises.
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Slide 23
Machine Learning Plays Games
- AI and Game Theory
- Modern AI has achieved superhuman performance in complex games — Go (AlphaGo, 2016), Chess (Stockfish, AlphaZero), Poker (Libratus, 2017), StarCraft II (AlphaStar, 2019). Each represents a different game-theoretic challenge: perfect vs. imperfect information, discrete vs. continuous action spaces.
- Multi-agent reinforcement learning is an active research area combining game theory and machine learning — training AI agents that must interact strategically with other agents, not just a fixed environment.
- AlphaGo: MCTS + deep RL solved the largest perfect-information game
- Libratus: counterfactual regret minimization solved heads-up No-Limit Poker
- OpenAI Five: cooperative MARL trained at massive scale for Dota 2
- AlphaZero: self-play from scratch in chess, shogi, Go — no human knowledge
- Generative adversarial networks (GANs): two-player minimax game structure
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Slide 24
When Humans Don't Play Rationally
- Behavioral Game Theory
- Experimental economics revealed that real people in game theory experiments do not play Nash equilibria in one-shot games. Ultimatum game players reject unfair offers; public good game players contribute voluntarily; trust game players trust strangers.
- Colin Camerer's behavioral game theory incorporates social preferences (fairness, reciprocity), limited rationality (k-level thinking), and learning (how behavior adapts over repeated play) to predict actual human behavior more accurately.
- k-Level Thinking
- Real players have limited depth of strategic reasoning. Level-0: play randomly. Level-1: best response to level-0. Level-2: best response to level-1. Most people are level 1–2; Nash equilibrium requires infinite mutual recursion.
- Applications: beauty contest games, pricing models, advertising timing
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Slide 25
Game Theory in Competitive Markets
- Business Strategy
- Entry Deterrence
- Incumbents invest in excess capacity not to use it but to credibly threaten price wars against entrants. The threat must be credible — only costly commitments deter.
- Price Leadership
- In oligopolies, one firm signals price changes that others follow. This implicit coordination is a Nash equilibrium — and potentially an antitrust concern.
- Predatory Pricing
- Incumbents price below cost to drive out entrants, then raise prices. Game theory shows this is only credible with reputation for toughness or commitment to future price cuts.
- Patent Races
- R&D competition is a first-mover advantage game. Winner-take-all patent system creates races that may produce excessive investment or underinvestment depending on market structure.
- Platform Competition
- Two-sided markets are matching games between buyers and sellers. Platform pricing must balance both sides; network effects mean early decisions lock in long-term outcomes.
- Negotiation
- Every negotiation is a bargaining game. BATNA (Best Alternative To Negotiated Agreement) is the outside option; understanding your opponent's BATNA reveals their reservation price.
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Slide 26
Nuclear Deterrence as Game Theory
- Cold War Applications
- Thomas Schelling's The Strategy of Conflict (1960) applied game theory to nuclear deterrence. Deterrence works not by making attack impossible but by making it irrational — ensuring retaliation that makes attack a losing game for the aggressor.
- The problem: deterrence requires a threat to retaliate that must be credible. If the attack is already launched, retaliating only adds destruction. How do you commit in advance to an action that would be irrational after the fact?
- Massive Retaliation: commit to nuclear response to any Soviet aggression — credibility problem
- Mutually Assured Destruction: stable if both sides have secure second-strike capability
- Launch-on-warning: reduces decision time, increases risk of false positive
- Dead Hand: Soviet automated retaliation system — commitment device that removes human judgment
- Arms limitation treaties: cooperative equilibrium maintained through verification
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Slide 27
What Game Theory Cannot Do
- Limitations
- Game theory is a model — it abstracts from complexity to find tractable solutions. Several assumptions regularly fail in practice: rationality is bounded, preferences are uncertain, players may not know the rules, and many games have multiple equilibria that the theory cannot select between.
- The multiple equilibria problem is fundamental. When a game has many equilibria, game theory predicts all of them or none definitively. Real behavior is shaped by history, culture, framing, and communication — outside the formal model.
- Common knowledge of rationality — players may not know opponents are rational
- Equilibrium selection — multiple equilibria problem has no general solution
- Complexity — Nash equilibrium computation is PPAD-complete for general games
- Utilities unknown — real payoffs are not observable, must be inferred
- Behavioral deviations — real humans consistently deviate from game-theoretic predictions
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Slide 28
Game Theory in Daily Life
- Everyday Applications
- Traffic and Routing
- Braess's Paradox: adding a road to a network can increase average travel time when each driver optimizes individually. Adding highway capacity may not reduce congestion.
- Job Markets
- Signaling (degrees, portfolio), screening (assessment tests), matching (algorithms that power recruiting platforms) — all grounded in game theory.
- Online Platforms
- Google's PageRank is a fixed-point solution to a game between websites linking to each other. Ad auctions are generalized Vickrey mechanisms. Recommendation is a multi-armed bandit.
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Slide 29
How Societies Solve Dilemmas
- Cooperation and Institutions
- The deepest question in game theory is: how do human societies maintain cooperation despite individual incentives to defect? The answer involves repetition, reputation, monitoring, punishment, norms, and institutions that change the payoffs of the underlying game.
- Institutions as Commitment Devices
- Laws, contracts, and property rights are mechanisms that change the payoff matrix of social games — punishing defection sufficiently to make cooperation incentive-compatible without relying on altruism.
- Reputation Systems
- eBay seller ratings, Yelp reviews, credit scores — all are reputation mechanisms that extend the shadow of the future into one-shot interactions, transforming them into repeated games.
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Slide 30
STRATEGY IS NOT ABOUT
WHAT YOU WANT
IT IS ABOUT WHAT
OTHERS EXPECT YOU TO DO
- The Core Insight
- Game theory changed how we understand economics, biology, political science, computer science, and everyday life. Its central lesson: in a world of strategic interdependence, your best action depends on what others will do — and what they will do depends on what they expect you to do.
- NashSchellingAxelrodOstromMechanism Design
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