For grown-ups

What this is

Winning Moves is a self-contained introduction to game theory, written for one specific young reader who is far ahead in mathematics and bored by anything that talks down to him. It has 21 chapters in four parts. Every chapter states one real idea using the words a mathematician would use, gives the reader one or more games or simulations to play against a robot opponent that plays perfectly, and ends with the actual mathematics underneath, exercises with hidden answers, one genuinely hard challenge, and a short true story about the people who discovered the idea.

Nothing here is simplified to the point of being false. Where an idea is normally taught with calculus or formal notation, it is taught here with counting, fractions, remainders, binary, and short proofs, which is how most of these ideas were first discovered anyway.

How to use it

• Read in order, or don't. Part I builds the vocabulary (players, strategies, payoffs, best response, equilibrium, mixed strategy). After that, the parts are independent. Part II is the most mathematical. Part III is about people. Part IV is chance and crowds.
• Robo never makes mistakes unless a "Robo goofs sometimes" box is checked. Losing to Robo is expected at first. Every board has a Hint button, and most have a "show the secret" toggle that reveals the structure. The learning is in turning the secret off again.
• Stars are earned by doing something clever: beating Robo fairly, solving a puzzle, predicting a simulation's outcome before running it. They cannot be earned by clicking around, and they are deliberately not earned when Robo is goofing. Stars are saved in the browser (nothing is sent anywhere), so they belong to one device and one browser. The home page has a reset button.
• Play with real things. Almost every game works on a kitchen table: 21 coins, a real chocolate bar (Chomp is much funnier with actual chocolate), a cookie to cut, pennies to match, a deck of cards for the auction chapter. The moment a child teaches the 21 Game to a grown-up and wins, the chapter has done its job.
• The "Try it on paper" and "Challenge" boxes are where the real thinking happens. The answers are hidden behind a click. Resist the urge to open them first.

What mathematics is in here

The site does not announce its mathematics, but it is all there, and a reader who works through it will have met, used, and mostly understood:

• Counting and combinatorics: products for grids, factorials for orderings, "n choose k" for shapes of a chocolate bar, symmetry to shrink a count (Chapters 1, 7, 8, 14).
• Probability: equally likely outcomes, probability as area, expected value as a weighted average, the law of large numbers as something you can watch, conditional probability by counting cases (Chapters 1, 4, 16, 17).
• Algebra: variables, solving a linear equation to find a mixing probability or an equilibrium split, inequalities for dominance and best responses (Chapters 2, 3, 4, 16, 19).
• Number theory: remainders and the mod notation, binary numbers, XOR as addition without carrying, periodic sequences (Chapters 5, 6, 10).
• Proof: backward induction, induction stated as "if it works for n it works for n + 1", proof by contradiction, strategy-stealing arguments, the difference between proving something exists and constructing it (Chapters 5, 8, 9, 18).
• Algorithms: minimax and game trees, pruning, memoized search, Monte Carlo sampling, best-response dynamics, replicator dynamics (Chapters 7, 9, 3, 16, 19).
• Real theorems, named and dated: Zermelo (1913), von Neumann's minimax theorem (1928), Nash's theorem (1950), the Sprague-Grundy theorem (1935 and 1939), Bouton's solution of Nim (1901), Arrow's theorem (1951), the Gibbard-Satterthwaite theorem (1973), the median voter theorem, Braess's paradox (1968), Vickrey's second-price result (1961).

What to read next

• Math Games with Bad Drawings by Ben Orlin (2022). Seventy-odd games with rules, strategy, and the mathematics behind them. The natural next book, playable at any age.
• The Art of Strategy by Avinash Dixit and Barry Nalebuff (2008). Game theory for grown-ups with no equations to speak of. A strong reader of this site could read it at 12.
• Winning Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy (1982). The bible of games like Nim and Chomp. Dense, funny, and full of things nobody else has written down. For later, but not much later.
• The Evolution of Cooperation by Robert Axelrod (1984). The full story of the tournaments in Chapter 11, readable by a teenager.
• A Beautiful Mind by Sylvia Nasar (1998), the biography of John Nash. Honest about the illness; a grown-up should read it first.
• Games of No Chance, a free collection of research papers on combinatorial games, for the day the reader wants to see what the frontier looks like.

About the site

The site is a handful of static files with no accounts, no tracking, and no server. It works offline once loaded, prints reasonably, and adapts to phones and to light or dark mode. The robot opponents compute their moves honestly (exact search, dynamic programming, or, where a game is too large to solve, Monte Carlo sampling, which the relevant chapter explains). If you find an error in the mathematics or the history, it is a real error and worth reporting.