# Game Theory — Strategy when others strategize too

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Category: Mathematics
Slides: 14
Updated: 2026-05-17T20:55:53.161Z
Tags: catalog, math, game, theory

## Summary

A 13-slide tour of payoffs, equilibria, and the mathematics of mutual anticipation — from von Neumann's chessboard to the FCC spectrum auction. Key sections include: Game Theory /; Strategy when others strategize too.; The Setup; Zero-Sum Games; Non-Zero-Sum Games; Prisoner's Dilemma; Nash Equilibrium; Repeated Games & Tit-for-Tat; The Stag Hunt; Mixed Strategies.

## Slide Outline

1. Game Theory /
2. Strategy when others strategize too.
3. The Setup
4. Zero-Sum Games
5. Non-Zero-Sum Games
6. Prisoner's Dilemma
7. Nash Equilibrium
8. Repeated Games & Tit-for-Tat
9. The Stag Hunt
10. Mixed Strategies
11. Mechanism Design
12. Real Applications
13. Limits of Game Theory
14. Further Reading

## Slide Transcript

### Slide 1: Game Theory /

- Mathematics / Decision Sciences
- Strategy when others strategize too.
- A 13-slide tour of payoffs, equilibria, and the mathematics of mutual anticipation — from von Neumann's chessboard to the FCC spectrum auction.
- Deck 01 — Game Theory in 13 moves

### Slide 2: The Setup

- Slide 02 — Foundations
- Game theory is the study of strategic interaction: what should you do when the people you're playing against are doing the same calculation about you?
- Three ingredients
- Players — assumed rational, expected-utility maximizers.
- Strategies — the menu of choices each player can make.
- Payoffs — numerical reward for every combination of choices.
- The hidden ingredient
- Common knowledge. I know the rules. You know the rules. I know that you know. You know that I know that you know — recursively, all the way down.
- Strip any layer of that mutual recursion and the predictions wobble.

### Slide 3: Zero-Sum Games

- Slide 03 — Pure Conflict
- Every dollar I win is a dollar you lose. Chess, poker (heads-up), tennis. The total is fixed; we fight over slices.
- Minimax theorem
- von Neumann, 1928. In any finite two-player zero-sum game with mixed strategies, there is a value V such that:
- Player 1 can guarantee at least V. Player 2 can prevent more than V. The pessimistic upper bound and pessimistic lower bound meet.
- The first deep result of the field. Equilibrium exists, and it is computable.
- Chess — finite, zero-sum, perfect information.

### Slide 4: Non-Zero-Sum Games

- Slide 04 — Beyond Pure Conflict
- Most real life isn't a fight over a fixed pie. The pie can grow, shrink, or split unevenly. Both players might gain — or both might lose.
- The pie can grow
- Trade. Marriage. Joint ventures. The kingdom of cooperation.
- The pie can shrink
- Arms races. Price wars. Climate inaction. The kingdom of mutual harm.
- The interesting question: when does a non-zero-sum game offer mutual gain — and what stops the players from grabbing it?
- Cooperation is possible. It is not, however, always rational. That is the whole problem.

### Slide 5: Prisoner's Dilemma

- Slide 05 — The Canonical Trap
- Two suspects, separately questioned. Stay silent (cooperate) or rat out the other (defect)?
- B: Cooperate
- B: Defect
- A: Cooperate
- −1−1
- −100
- A: Defect
- 0−10
- −5−5
- Years in prison (negative). Shaded cell = Nash equilibrium.
- Whatever the other does, defecting is better for me. So both defect — landing on (−5, −5), worse than the (−1, −1) we could have shared.
- Individual rationality, collective tragedy. The single most-cited 2×2 in social science.

### Slide 6: Nash Equilibrium

- Slide 06 — The Big Idea
- John Nash, 1950, 28-page Princeton thesis. The definition that earned a Nobel Prize:
- A strategy profile is a Nash equilibrium if no player can improve their payoff by changing only their own strategy.
- Everyone's choice is a best response to everyone else's choice. The fixed point of mutual anticipation.
- What Nash proved
- Every finite game has at least one equilibrium (in pure or mixed strategies).
- Generalizes minimax beyond zero-sum.
- Used Kakutani's fixed-point theorem — pure topology applied to choice.
- What Nash didn't promise
- Equilibria can be bad (see: Prisoner's Dilemma).
- Multiple equilibria are common — which one are we in?
- Players may not actually find them.

### Slide 7: Repeated Games & Tit-for-Tat

- Slide 07 — Repetition Changes Everything
- If we play once, defection wins. If we play repeatedly, the future casts a shadow on the present — and cooperation becomes possible.
- Axelrod's tournament (1980)
- Robert Axelrod invited game theorists to submit programs to play repeated Prisoner's Dilemma. The shortest program won — twice.
- TIT-FOR-TAT by Anatol Rapoport: cooperate first, then copy whatever the opponent did last move.
- Why it wins
- Nice — never defects first.
- Retaliatory — punishes defection immediately.
- Forgiving — returns to cooperation when opponent does.
- Clear — opponent quickly learns the rule.
- The folk theorem: with patient enough players, almost any reasonable outcome can be sustained as equilibrium in a repeated game.

### Slide 8: The Stag Hunt

- Slide 08 — Coordination
- Rousseau's parable. Two hunters: cooperate to take down a stag (big payoff), or grab a hare alone (small but safe). The stag requires both — and faith in your partner.
- B: Stag
- B: Hare
- A: Stag
- A: Hare
- Two equilibria: payoff-dominant (Stag, Stag) and risk-dominant (Hare, Hare).
- Unlike the Prisoner's Dilemma, cooperation IS a Nash equilibrium here. The problem is trust: if you suspect your partner will hare, hare is safer for you too.
- Models: building institutions, signing treaties, joining a startup, going to a party only if your friends do.

### Slide 9: Mixed Strategies

- Slide 09 — Randomization as Rationality
- Some games have no equilibrium in pure strategies. The fix: randomize.
- Matching pennies
- I want our coins to match; you want them to differ. Any pure choice is exploitable. The unique equilibrium: each picks heads with probability 1/2.
- Soccer penalty kicks
- Empirically, professional kickers and goalkeepers randomize — and the observed frequencies match Nash predictions remarkably well (Chiappori, Levitt, Groseclose 2002).
- Decision tree — matching pennies extensive form.
- A mixed-strategy equilibrium is a probability distribution that makes the opponent indifferent. Indifference is the lock; randomness is the key.

### Slide 10: Mechanism Design

- Slide 10 — Reverse Game Theory
- Standard game theory: given the rules, predict behavior. Mechanism design flips the question:
- Given the behavior we want, design rules so that self-interested play produces it as the equilibrium.
- Key principles
- Incentive compatibility — telling the truth is in your interest.
- Individual rationality — players prefer participating to leaving.
- Revelation principle — anything achievable by any mechanism is achievable by a direct, truthful one.
- Heroes of the field
- Hurwicz, Maskin, Myerson — Nobel 2007.
- Vickrey — second-price auctions, Nobel 1996.
- Roth — kidney exchange & matching, Nobel 2012.

### Slide 11: Real Applications

- Slide 11 — Theory in the Wild
- $Spectrum auctions
- FCC, 1994 onward. Mechanism designers (Milgrom, Wilson) built simultaneous ascending auctions that have raised hundreds of billions while allocating spectrum efficiently.
- +Kidney exchange
- Alvin Roth's clearinghouse: incompatible donor-patient pairs are matched in chains and cycles. Thousands of transplants now happen that otherwise wouldn't.
- @Search ad auctions
- Google's AdWords / Generalized Second-Price auction. Every search query is a tiny auction. Game theory funds half the modern internet.
- !Matching markets
- Medical residents to hospitals (NRMP). Students to public schools. Deferred-acceptance algorithms produce stable, strategy-proof matches.
- A coordination/matching graph — pairs find each other through a market.

### Slide 12: Limits of Game Theory

- Slide 12 — Where the Theory Strains
- Bounded rationality
- Real players don't compute Nash equilibria. They use heuristics, satisfice, get tired. Behavioral game theory (Camerer, Thaler) folds psychology back into the model.
- Common knowledge is fragile
- Most equilibrium concepts assume infinite recursion of beliefs. Drop one level and predictions diverge. Robert Aumann's "agreement theorem" looks beautiful but rarely holds in the wild.
- Multiple equilibria
- The theory often predicts several stable points. Which one we end up in depends on history, focal points (Schelling), or culture — variables outside the model.
- Evolutionary games
- Maynard Smith's reframing: strategies as genes, equilibria as evolutionarily stable states. No rationality required — selection does the optimizing. Used in biology, networks, cultural transmission.
- The theory works best as a lens, less well as a crystal ball.

### Slide 13: Further Reading

- Slide 13 — End of the Match
- Books
- Theory of Games and Economic Behavior — von Neumann & Morgenstern (1944).
- The Evolution of Cooperation — Robert Axelrod.
- Thinking Strategically — Dixit & Nalebuff.
- The Strategy of Conflict — Thomas Schelling.
- Who Gets What — and Why — Alvin Roth.
- Watch
- YouTube: Prisoner's Dilemma — Game Theory
- YouTube: Nash Equilibrium — Explained
- People
- von Neumann, Nash, Schelling, Aumann, Roth, Axelrod, Maynard Smith, Milgrom.
- "In game theory, the assumption is that you know your opponent. In life, the discovery is that you barely know yourself."
- — end of deck —


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