Business · A course in fourteen chapters

Game Theory: The Science of Strategy

Game theory is the study of choices that depend on other people’s choices: prices and bids, votes and threats, promises and penalty kicks. This course starts with a single table of payoffs and climbs, one chapter at a time, to the tools of a first-year PhD course: Nash equilibrium, backward induction, repeated games, Bayesian games, auctions, signaling and mechanism design. Every chapter opens with a game you play against the computer.

Start with Chapter 1 →The big ideas at a glance

1.85

What game theory is

Game theory is the mathematics of strategic situations, where what is best for you depends on what others do, and what is best for them depends on you. A game is described by its players, the actions each one can take, and what each player gets from every combination of actions; the theory asks what sensible players will do, and when they will cooperate, compete, threaten or bluff. It began with parlor games and war planning and now shapes how economists think about markets, auctions, contracts, elections and even evolution.

The table on the right is the most famous game of all, the prisoner’s dilemma, written the way this course writes every game. The row player’s payoff comes first, in gold; the column player’s comes second, in blue. Each player does better by finking whatever the other does, yet both finking leaves them worse off than both staying quiet. Chapters 2 and 7 take that puzzle apart.

The field at the top of this page is that game played over and over on a grid, after Martin Nowak and Robert May (Nature, 1992). Each square scores 1 for every cooperating neighbor if it cooperates, and b for every cooperating neighbor if it defects, then copies the best scorer around it. Defectors win locally, but clusters of cooperators protect each other, and neither side takes over. Raise b and defection spreads; lower it and cooperation wins.

Suspect 2
Suspect 1QuietFink
Quiet2, 20, 3
Fink3, 01, 1
The prisoner’s dilemma in Osborne’s numbers. Higher is better. The green cell is the only Nash equilibrium.

The chapters

Each chapter starts with an interactive and a list of missions, then teaches the ideas behind it with worked examples, practice problems with answers, more small games and a short quiz. The colors mark three levels. The first four chapters assume only high-school algebra; by the last ones you are working at the level of a first-year graduate course.

  • Level 1, chapters 1–4: foundations
  • Level 2, chapters 5–7: dynamics and competition
  • Level 3, chapters 8–14: information, design and evolution

  1. Games, Players and Payoffs Guess two thirds of the average against a crowd of level-k thinkers, then build games and play bots with personalities. ✓ Done
  2. Dominance and the Prisoner’s Dilemma Cross out dominated actions until a game solves itself, fall into the prisoner’s dilemma, and watch a public-goods game unravel. ✓ Done
  3. Nash Equilibrium Underline best responses to find Nash equilibria in coordination, chicken and the stag hunt, and test your feel for focal points. ✓ Done
  4. Mixed Strategies and Zero-Sum Games Play rock-paper-scissors against a bot that reads you, solve any 2×2 game by indifference, and take penalties at minimax. ✓ Done
  5. Competition: Cournot, Bertrand and Hotelling Watch Cournot firms settle into equilibrium, start a Bertrand price war, and walk ice-cream sellers to the median. ✓ Done
  6. Sequential Games and Backward Induction Build and solve game trees, unmask empty threats, deter entry with commitment, and lead a Stackelberg market. ✓ Done
  7. Repeated Games and Cooperation Run an Axelrod tournament with your own strategy, add noise and evolution, and find the δ that makes cooperation pay. ✓ Done
  8. Incomplete Information and Bayesian Games Run a used-car market where only sellers know the lemons, watch it unravel, and fix it with warranties and inspections. ✓ Done
  9. Auctions Bid against computer rivals in four auction formats, test revenue equivalence, and fall for the winner’s curse. ✓ Done
  10. Signaling and Screening Set what the good job requires and watch job seekers signal, then measure cheap talk and design a menu that sorts customers. ✓ Done
  11. Bargaining Haggle over a $100 pie with bots that differ only in patience, then find Nash’s solution as the biggest rectangle. ✓ Done
  12. Mechanism Design and Matching Run the matching algorithm behind the medical residency match, try to cheat it, and see a tax that makes honesty pay. ✓ Done
  13. Coalitions, the Core and the Shapley Value Pay each player what they add, find splits no group would reject, and discover who really holds power in a vote. ✓ Done
  14. Evolutionary Game Theory Watch hawks and doves find a balance, cooperators build kaleidoscopes on a grid, and lizards cycle like rock–paper–scissors. ✓ Done

The big ideas at a glance

Ten ideas carry most of the subject. Here is each in one line, in symbols, and where the course teaches it. In the symbols, ui is player i’s payoff, ai an action of player i, and a−i the actions of everyone else.

ui(a′i, a−i) > ui(ai, a−i) for all a−i
Dominance

If one action beats another whatever the others do, never play the worse one, and expect others not to either.

Chapter 2
a*i ∈ Bi(a*−i) for every i
Nash equilibrium

A combination of choices where nobody can do better by changing only their own choice: no one has regrets.

Chapter 3
maxp minq = minq maxp
Mixed strategies

When being predictable costs you, randomize, in exactly the proportions that leave your opponent with nothing to exploit.

Chapter 4
solve the last move first
Backward induction

In a game played in turns, look ahead to the end and reason back. Threats you would not carry out are not believed.

Chapter 6
δ ≥ (T − R) ÷ (T − P)
The shadow of the future

When people expect to meet again, the fear of losing future cooperation can hold up cooperation today.

Chapter 7
ai(θi) best against beliefs about θ−i
Bayesian games

When you don’t know what kind of player you face, you best respond to your beliefs about their type.

Chapter 8
signal cost depends on type
Signaling

An action that is cheaper for some types than for others can reveal what words alone cannot: a costly signal is a credible one.

Chapter 10
truth-telling is a best response
Mechanism design

Turn game theory around: choose the rules so that self-interested players produce the outcome you want, such as an auction where bidding your true value is best.

Chapter 12
φi = average marginal contribution
The Shapley value

A fair way to split what a team creates: each member gets what they add, averaged over every order in which the team could have formed.

Chapter 13
u(s*, s*) > u(s, s*), or = and …
Evolutionary stability

A strategy that, once common, can’t be invaded by a rare mutant. No one needs to be rational: success simply spreads.

Chapter 14

A short history

  1. 1838

    Antoine Augustin Cournot analyzed two firms choosing how much mineral water to sell, each reacting to the other. His solution is what we now call a Nash equilibrium, more than a century before the name (Chapter 5).

  2. 1913

    Ernst Zermelo proved that in chess either White can force a win, or Black can, or each side can force at least a draw. It is often called the first theorem of game theory, and it holds the seed of backward induction.

  3. 1928

    John von Neumann’s paper “Zur Theorie der Gesellschaftsspiele” (On the theory of games of strategy, Mathematische Annalen) proved the minimax theorem: in every two-person zero-sum game, each side has a randomized strategy that guarantees it the game’s value. Émile Borel had raised the question in the early 1920s (Chapter 4).

  4. 1944

    Von Neumann and Oskar Morgenstern published Theory of Games and Economic Behavior (Princeton University Press), which founded the field and gave economics its theory of expected utility.

  5. 1950

    At the RAND Corporation, Merrill Flood and Melvin Dresher had colleagues play a game in which each player’s own interest pulls against their joint interest. Albert W. Tucker of Princeton told it as a story about two prisoners to make it accessible to an audience of Stanford psychologists, and the name “prisoner’s dilemma” stuck (Chapter 2).

  6. 1950–1951

    John Nash, a Princeton graduate student, defined the equilibrium that carries his name and proved that every finite game has one, possibly in mixed strategies, in “Equilibrium points in n-person games” (Proceedings of the National Academy of Sciences, 1950) and “Non-cooperative games” (Annals of Mathematics, 1951). His “The bargaining problem” (Econometrica, 1950) started the theory of bargaining (Chapters 3 and 11).

  7. 1953

    Lloyd Shapley defined the Shapley value, a fair division of what a coalition earns (Chapter 13). In 1962, with David Gale, he published the deferred-acceptance algorithm for stable matching (Chapter 12).

  8. 1960

    Thomas Schelling’s The Strategy of Conflict (Harvard University Press) brought focal points, commitment and credible threats into the study of bargaining, business and nuclear deterrence.

  9. 1965–1975

    Reinhard Selten introduced subgame perfection (1965) and trembling-hand perfection (1975), which rule out equilibria that rest on threats no one would carry out (Chapter 6).

  10. 1967–1968

    John Harsanyi’s three-part paper “Games with incomplete information played by ‘Bayesian’ players” (Management Science) showed how to analyze games where players don’t know each other’s payoffs (Chapter 8).

  11. 1973

    John Maynard Smith and George Price, in “The logic of animal conflict” (Nature), defined the evolutionarily stable strategy and carried game theory into biology (Chapter 14).

  12. 1980–1984

    Robert Axelrod ran two computer tournaments of the repeated prisoner’s dilemma. The first had 14 entries plus a random strategy; the second had 62 entries from six countries. Both were won by the simplest program entered, Tit for Tat, submitted by Anatol Rapoport. Axelrod told the story in The Evolution of Cooperation (1984) (Chapter 7).

  13. 1994 to 2020

    Five Nobel prizes in economics (the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) have gone mainly for game theory:

    YearLaureatesThe prize committee’s citation
    1994John C. Harsanyi, John F. Nash Jr., Reinhard Selten“for their pioneering analysis of equilibria in the theory of non-cooperative games”
    2005Robert J. Aumann, Thomas C. Schelling“for having enhanced our understanding of conflict and cooperation through game-theory analysis”
    2007Leonid Hurwicz, Eric S. Maskin, Roger B. Myerson“for having laid the foundations of mechanism design theory”
    2012Alvin E. Roth, Lloyd S. Shapley“for the theory of stable allocations and the practice of market design”
    2020Paul R. Milgrom, Robert B. Wilson“for improvements to auction theory and inventions of new auction formats”

    Others rest heavily on game theory too: 1996 (James Mirrlees and William Vickrey, incentives under asymmetric information, including Vickrey’s second-price auction), 2001 (George Akerlof, Michael Spence and Joseph Stiglitz, markets with asymmetric information, including Spence’s signaling) and 2014 (Jean Tirole, market power and regulation).

Where game theory shows up

Business

Pricing against a rival, deciding how much capacity to build, whether to enter a market, how to read a competitor’s announcement, when a promise to customers or workers is believable. The site’s business cases have several: Intel leaving the memory-chip price war in 1985, Tata’s public promise of a 100,000-rupee Nano as a commitment, and Southwest keeping its people after 9/11 as a commitment that built trust.

Auctions and markets

The U.S. Federal Communications Commission has sold licenses for radio spectrum by auction since 1994, in formats designed with the help of game theorists. Doctors in the United States are matched to residency programs by a deferred-acceptance algorithm, and kidney exchanges use matching theory to pair donors and patients (Chapters 9 and 12).

Sports

A penalty kick is a zero-sum game with mixed strategies. Studies of professional kickers and goalkeepers found choices close to what minimax theory predicts (Chapter 4). Tennis serves, pitch selection in baseball and run-or-pass calls in football have been studied the same way.

Biology

Animals don’t calculate, but natural selection rewards strategies that do well against the population. Hawks and doves, the sex ratio, and cooperation among relatives and strangers are all studied as evolutionary games (Chapter 14).

Politics and conflict

Why two candidates drift toward the middle (Chapter 5), how deterrence and arms races work (Chapter 2), how committees vote and how much power each member really has (Chapter 13). Schelling’s work on commitment grew out of thinking about nuclear deterrence.

Everyday life

Merging into traffic, splitting the chores, negotiating a salary or a used car, deciding whether to stand at a concert when everyone in front of you stands, choosing a meeting spot with a friend whose phone is dead. Once you see the payoffs, you see games everywhere.

Free courses to go further

Textbooks

  • Martin J. Osborne, An Introduction to Game Theory (Oxford University Press, 2004). This course follows its notation.
  • Avinash Dixit and Susan Skeath, with coauthors, Games of Strategy (W. W. Norton, several editions). Gentle, full of examples.
  • Robert Gibbons, Game Theory for Applied Economists (Princeton University Press, 1992). Short and clear, the classic bridge to graduate work.
  • Drew Fudenberg and Jean Tirole, Game Theory (MIT Press, 1991), and Roger B. Myerson, Game Theory: Analysis of Conflict (Harvard University Press, 1991): the graduate references.
  • Martin J. Osborne and Ariel Rubinstein, A Course in Game Theory (MIT Press, 1994), and Michael Maschler, Eilon Solan and Shmuel Zamir, Game Theory (Cambridge University Press, 2013).
  • Colin F. Camerer, Behavioral Game Theory (Princeton University Press, 2003): what real people do in the lab.
  • For reading in a chair: Thomas Schelling, The Strategy of Conflict (1960); Robert Axelrod, The Evolution of Cooperation (1984); Avinash Dixit and Barry Nalebuff, Thinking Strategically (1991) and The Art of Strategy (2008).

Sources

@teddyrookbook GitHub