A 13-slide tour of payoffs, equilibria, and the mathematics of mutual anticipation — from von Neumann's chessboard to the FCC spectrum auction.
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?
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.
Every dollar I win is a dollar you lose. Chess, poker (heads-up), tennis. The total is fixed; we fight over slices.
von Neumann, 1928. In any finite two-player zero-sum game with mixed strategies, there is a value V such that:
The first deep result of the field. Equilibrium exists, and it is computable.
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.
Trade. Marriage. Joint ventures. The kingdom of cooperation.
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?
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 |
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.
John Nash, 1950, 28-page Princeton thesis. The definition that earned a Nobel Prize:
Everyone's choice is a best response to everyone else's choice. The fixed point of mutual anticipation.
If we play once, defection wins. If we play repeatedly, the future casts a shadow on the present — and cooperation becomes possible.
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.
The folk theorem: with patient enough players, almost any reasonable outcome can be sustained as equilibrium in a repeated game.
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 | 44 | 03 |
| A: Hare | 30 | 33 |
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.
Some games have no equilibrium in pure strategies. The fix: randomize.
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.
Empirically, professional kickers and goalkeepers randomize — and the observed frequencies match Nash predictions remarkably well (Chiappori, Levitt, Groseclose 2002).
A mixed-strategy equilibrium is a probability distribution that makes the opponent indifferent. Indifference is the lock; randomness is the key.
Standard game theory: given the rules, predict behavior. Mechanism design flips the question:
FCC, 1994 onward. Mechanism designers (Milgrom, Wilson) built simultaneous ascending auctions that have raised hundreds of billions while allocating spectrum efficiently.
Alvin Roth's clearinghouse: incompatible donor-patient pairs are matched in chains and cycles. Thousands of transplants now happen that otherwise wouldn't.
Google's AdWords / Generalized Second-Price auction. Every search query is a tiny auction. Game theory funds half the modern internet.
Medical residents to hospitals (NRMP). Students to public schools. Deferred-acceptance algorithms produce stable, strategy-proof matches.
Real players don't compute Nash equilibria. They use heuristics, satisfice, get tired. Behavioral game theory (Camerer, Thaler) folds psychology back into the model.
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.
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.
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.