Big words

Every term on the site, in alphabetical order, said the way a mathematician says it. Each one links to the chapter where it first shows up and gets used for real.

Arrow's impossibility theorem: Kenneth Arrow's 1951 proof that no way of turning everyone's rankings into one group ranking can satisfy a short list of reasonable rules at the same time, once there are three or more choices. Ch. 14
Backward induction (in-DUCK-shun): Solving a game with turns by starting at the very end, labeling the last positions, then the ones before them, all the way back to the start. Ch. 5
Best response: The choice that gives you the biggest payoff against one particular choice by the other player. Found by looking down a column (for you) or across a row (for Robo). Ch. 2
Binary (base 2): Writing numbers with places worth 1, 2, 4, 8, 16, each holding only a 0 or a 1. 13 is 1101. The secret of Nim lives here. Ch. 6
Braess's paradox: Adding a road to a network can make every driver's trip slower, because each driver's selfish best choice clogs the new road. Ch. 19
Common knowledge: Something everyone knows, and everyone knows that everyone knows, and so on without end. Different from everyone merely knowing it, and the difference can change what people do. Ch. 18
Conditional probability: The chance of something given that you know something else: the fraction of the cases you are now in that also have the thing you care about. Counting cases is the honest way to find it. Ch. 17
Condorcet winner and Condorcet's paradox (kon-dor-SAY): A Condorcet winner beats every other candidate head to head. The paradox: majorities can go in a circle, A over B, B over C, C over A, so there may be no such winner. Ch. 14
Dominant strategy: A choice that is at least as good as every other choice no matter what the other player does. Grab, in the Cookie Game. Ch. 2
Dominated strategy: A choice that is worse than some other choice no matter what the other player does. A sensible player never uses it, so it can be crossed out. Ch. 2
Envy-free: A way of sharing where nobody would rather have anyone else's piece, by their own measure. Stronger than proportional, and harder to reach with three or more people. Ch. 12
Equilibrium (Nash equilibrium) (ee-kwi-LIB-ree-um): A combination of choices where nobody wants to switch, because each player is already playing a best response to the others. Where a game comes to rest. Ch. 3
Evolutionarily stable strategy (ESS): A strategy that, once almost everyone uses it, cannot be invaded by a few players trying something else. Evolution finds it without anyone calculating. Ch. 16
Expected value: The average payoff you would get if you played a huge number of times: each outcome times its probability, added up. Ch. 4
Game tree: A drawing of every move and every reply, branching out from the current position until every branch ends. Solving it is minimax. Ch. 7
Grundy value (GRUN-dee): The single number that tells you everything about a position in a game like Nim: 0 means losing for the player to move, and the value of several games played at once is the XOR of their values. Ch. 10
Hawk-Dove game: Two animals meet over a prize: fight (Hawk) or display and back off (Dove). When fighting is costly, a population settles at a mix of both, with the hawk fraction equal to prize divided by cost. Ch. 16
Law of large numbers: The more times you repeat a chance experiment, the closer the fraction of each outcome gets to its probability. Ch. 1
Median voter: The voter in the exact middle, with half the crowd on each side. Two competing candidates (or ice cream carts) both end up there. Ch. 21
Mex: Minimum excluded value: the smallest whole number not in a set. The mex of {0, 1, 3} is 2. The engine behind Grundy values. Ch. 10
Minimax (MIN-ee-max): Choose the move whose worst outcome is the best available, assuming the other player plays their best. How a computer plays tic-tac-toe perfectly. Ch. 7
Misère (mee-ZAIR): The version of a game where taking the last move loses instead of wins. It shifts the losing positions by one. Ch. 5
Mixed strategy: Choosing at random with deliberately chosen odds, so that you cannot be predicted. Heads half the time. Ch. 4
Monte Carlo method: Estimating something by running many random trials and counting. Robo plays Hex this way: imagine hundreds of random finishes, pick the move that won most often. Ch. 9
Nim and nim-sum: The many-pile stone game, and the number found by writing the piles in binary and adding the columns without carrying. Nim-sum 0 means the player to move is losing. Ch. 6
Payoff: What a player gets at the end: points, cookies, a win, a loss. Written as a function, u(your choice, their choice). Ch. 1
Payoff grid (payoff matrix): A table with one row for each of your choices and one column for each of theirs, showing both payoffs in every box. The whole game, written down. Ch. 1
Player: Anyone who makes a choice in a game. Two people, two animals, two computer programs, or four thousand drivers. Ch. 1
Price of anarchy: How much worse the selfish equilibrium is than the best coordinated plan, as a ratio. In the traffic network it is 80 over 65. Ch. 19
Prisoner's Dilemma: A game where each player has a dominant strategy and both playing it leaves both worse off than if neither had. The Cookie Game. Ch. 2
Private value: In an auction, the most a bidder would pay and still be happy. Known to the bidder, secret from everyone else. Ch. 13
Proportional (fair division): A way of sharing among n people where each thinks they got at least 1/n by their own measure. Cut-and-choose guarantees it for two. Ch. 12
Pruning: Skipping branches of a game tree that cannot change the answer, because one reply already ruled a move out. Same answer, far less work. Ch. 7
Public good: Something everyone benefits from whether or not they paid for it, which is exactly why each person is tempted not to pay. Ch. 20
Repeated game: The same game played many times by the same players, who remember. It makes cooperation possible, because a grab today can be answered tomorrow. Ch. 11
Second-price (Vickrey) auction: Sealed bids, highest wins, but pays the second-highest bid. Bidding your true value is a dominant strategy. Ch. 13
Strategy: A complete plan for what you will do in every situation the game can throw at you. "Always grab" is a strategy. So is "copy what they did last round." Ch. 1
Strategy-stealing argument: A proof that the first player can win by showing that any winning plan for the second player could be stolen. It proves the win exists without finding it. Ch. 8
Sum of games: Two or more games on the table at once, where each turn you move in exactly one of them. Its Grundy value is the XOR of the parts. Ch. 10
Tit for Tat: Share first, then do whatever the other player did last round. Four lines long, and it won the famous tournaments. Ch. 11
Tragedy of the commons: When many people share something that regrows, each person's best move is to take a little more, and the sum of those moves can destroy it. Ch. 20
Ultimatum game: One player proposes how to split a pile; the other says yes or no; no means nobody gets anything. Theory says offer one coin. People say otherwise. Ch. 15
Value of a game: In a zero-sum game, the payoff each player can guarantee with the right mixed strategy. Von Neumann proved it always exists. Ch. 4
Winning and losing positions (W and L): W: the player whose turn it is can force a win. L: they cannot. A position is W if any move leads to an L. Ch. 5
XOR (EX-or): Adding binary numbers column by column without carrying: a column is 1 if it holds an odd number of 1s. Written ⊕ by mathematicians. Ch. 6
Zero-sum game: Whatever one player wins, the other loses; the payoffs in every box add to zero. Chess, Matching Pennies, tic-tac-toe. The Cookie Game is not one. Ch. 4