Winning Moves
Made for Garrett, who is going to be better at this than the person who built it.
Game theory is the mathematics of choices that depend on other people's choices. Chess is a game. So is splitting a cookie, bidding at an auction, choosing which road to drive on, and deciding whether to trust someone who might not trust you back.
This site has 21 chapters in four parts. Every chapter teaches one real idea, the way a mathematician would say it, and gives you a game to play against Robo, a robot that has read every game theory book ever written and never makes a mistake unless you ask it to. Do something genuinely clever in a chapter and you earn a star. Nobody can earn a star by clicking around.
Start at Chapter 1 if you are new. Jump anywhere if you are not. The Big words page collects every term, and the Arcade collects every game.
Stars are saved in this browser.
The rules of the game
What a game is, how to write one down as a grid of numbers, and the two ideas everything else is built on: best responses and equilibrium.
- Chapter 1
What is a game?
How to write a game down as a grid, and a surprising answer to "what is the best way to play Rock, Paper, Scissors?"
- Chapter 2
The Cookie Game
Payoffs become numbers, a choice that is best no matter what gets a name, and the most famous puzzle in game theory shows up with cookies.
- Chapter 3
Nobody wants to switch
John Nash's question about every game, five puzzles with one, two, or zero answers, and arrows that find the settled place.
- Chapter 4
Being unpredictable
Matching Pennies has no settled place until you pick at random with exactly the right odds. One line of algebra finds them, and it explains penalty kicks.
Games with turns
Games you can solve completely by thinking backwards from the end. Secrets hidden in remainders, in binary, and in proofs that tell you a winner exists without telling you how to win.
- Chapter 5
Thinking backwards
The 21 Game, and the trick of solving any game with turns by starting at the end.
- Chapter 6
Nim and binary
Several piles, one secret: write the sizes in binary, add the columns without carrying, and the losing positions appear.
- Chapter 7
Seeing the whole tree
How a computer plays tic-tac-toe perfectly by drawing every possible future, how big that drawing is, and the trick for skipping most of it.
- Chapter 8
A proof that you can win
A chocolate-bar game where mathematicians have proved the first player can always win, without knowing how.
- Chapter 9
Hex
Connect your two edges before Robo connects its own. A game with no draws, a proof that the first player wins, and a Robo that plays by imagining thousands of random games.
- Chapter 10
Every game is secretly Nim
One number per position, the mex rule that finds it, and why any pile of games played at once is just Nim in disguise.
Playing with people
When the other player is a person: repeating a game changes everything, sharing fairly is harder than it looks, and auctions and elections are games too.
- Chapter 11
Playing again and again
Play the Cookie Game twenty times against the same opponent and grabbing stops being smart. Meet Tit for Tat, the four-line strategy that won the famous tournaments.
- Chapter 12
I cut, you choose
How to split a cake so nobody can complain, and the place where 'fair' turns into two different ideas.
- Chapter 13
Going once, going twice
Four ways to sell one rare card, the bidding tradeoff, and the auction where telling the truth cannot be beaten.
- Chapter 14
How to pick a winner when nobody agrees
Thirty students, three pets, and four fair-sounding ways to count the same ballots that pick three different winners.
- Chapter 15
Take it or leave it
One offer, one answer. Backward induction says offer a single coin. Real people say no to that, and five pirates show why the captain keeps 98.
- Chapter 16
Hawks and doves
Animals that never calculate still land on an equilibrium, and one line of algebra says where.
Chance, crowds, and clever puzzles
Games against chance, games with thousands of players, and puzzles where the whole trick is figuring out what everybody else knows.
- Chapter 17
The three doors
A bicycle, two goats, and a host who knows where the bicycle is. Switching wins twice as often as staying, and the whole reason is what the host knows.
- Chapter 18
What everybody knows that everybody knows
Muddy foreheads, a number-guessing crowd, and three hats: puzzles where the whole trick is knowing what everybody else knows.
- Chapter 19
Traffic
This chapter is still being written.
- Chapter 20
The pond that everyone owns
Four fishers share a pond that regrows. Each one's best move is to take a bit more, and together they can empty it. Real communities know the way out.
- Chapter 21
Ice cream carts
This chapter is still being written.