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Game Theory — 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.

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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.

Key sections

01Game Theory /
02Strategy when others strategize too.
03The Setup
04Zero-Sum Games
05Non-Zero-Sum Games
06Prisoner's Dilemma
07Nash Equilibrium
08Repeated Games & Tit-for-Tat
09The Stag Hunt
10Mixed Strategies
11Mechanism Design
12Real Applications
13Limits of Game Theory
14Further Reading

Topics covered

catalog math game theory

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Slide outline

01Game Theory /
02Strategy when others strategize too.
03The Setup
04Zero-Sum Games
05Non-Zero-Sum Games
06Prisoner's Dilemma
07Nash Equilibrium
08Repeated Games & Tit-for-Tat
09The Stag Hunt
10Mixed Strategies
11Mechanism Design
12Real Applications
13Limits of Game Theory
14Further Reading

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Category: Mathematics
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Updated: 2026-05-17
LLM text: https://shipslides.com/d/catalog-math-game-theory/llms.txt

Presentation Transcript

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Slide 01

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 02

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 03

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 04

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 05

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 06

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 07

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 08

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 09

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