What Is a Game, Mathematically Speaking?

Game Theory, Part 1 of 9

During the Second World War, in a London taxi, the mathematician John von Neumann told his companion Jacob Bronowski something strange: chess is not a game. Chess, he said, is a well-defined form of computation — in any position there must, in theory, be a right move, even if no one can find it. Real games, like real life, are made of bluffing, little tactics of deception, and asking yourself what the other person thinks you mean to do. That is how Bronowski remembered the conversation in The Ascent of Man (1973) — a recollection, not a transcript.

If you love chess, that may sound almost provocative. But game theory begins with distinctions like this one. Von Neumann proved the field's first key theorem in 1928, and with the economist Oskar Morgenstern turned it into a discipline in Theory of Games and Economic Behavior (1944). In this series we'll travel from the chessboard to negotiation and trust. We start with a simple question: what does a mathematician actually call a game?

Three ingredients of every game

For a mathematician, a game is any situation where the outcome depends on the decisions of several participants — and each of them knows it. Chess fits that definition, but so do an auction, haggling over the price of a flat, and deciding whether to let a car merge at a junction.

To describe a game, you need three things:

• Players — whoever makes decisions: a person, a company, a country, a program.
• Strategies — every course of action available to each player.
• Payoffs — what each player gets for every combination of decisions: points, money, a win or a loss.

The simplest example is rock, paper, scissors. Two players, three options each, and the whole game fits in one table. Each cell shows player A's payoff: 1 is a win, −1 a loss, 0 a draw.

Player B's payoff is always the same number with the opposite sign. Games like this are called zero-sum games: whatever one player wins, the other loses. Chess, checkers and Go belong here too. Most situations in life do not — and they are what the later parts of this series are about.

A strategy is not a move

In everyday speech, a "strategy" is a general idea: "I'll play defensively." In game theory it is a complete instruction manual: what to do in every situation that could possibly arise — including the ones you're sure will never happen.

Take a game you can play with anyone right now (an instructional example). There are 10 matchsticks on the table. Players take turns removing one or two. Whoever takes the last one wins. Mathematicians call it a subtraction game, a close relative of the classic game of Nim.

Try solving it backwards:

1. One or two sticks in front of you — take them all and win.
2. Three in front of you — you've lost: whatever you take, your opponent takes the rest.
3. Four or five — leave your opponent three, and you win.
4. Six — trapped again: any move leaves your opponent four or five.

The pattern is clear: the losing positions are the multiples of 3. So with ten sticks, the first player takes one and leaves nine. From then on, they always complete the opponent's move to three: if the opponent takes one, take two; if they take two, take one. The win is guaranteed, whatever the second player does.

That is a strategy in the mathematical sense: a rule that answers every possible position. And the method we used — reasoning from the end back to the start — is called backward induction.

The same idea lies behind von Neumann's remark about chess. Chess has incomparably more positions, but the principle is the same: somewhere there is a right answer for each of them. What follows from that — and what doesn't — is the subject of the next article.

Five questions that classify any game

Chess and poker are both "games", but mathematically they live in different worlds. To see where the border runs, ask any game five questions:

1. How many players? With three or more, alliances appear and the analysis gets much harder.
2. Are the interests strictly opposed? Zero-sum — or can players win together, or lose together?
3. Can you see everything? In a game of perfect information, a player about to move knows everything that has happened so far. Cards hidden in an opponent's hand break that condition.
4. Is there chance? Dice, a shuffled deck.
5. In turn or simultaneous? A simultaneous choice, as in rock, paper, scissors, is mathematically the same as not seeing your opponent's move.

Here is how familiar games sort out:

Look at backgammon. There is chance, but no hidden information: the dice are rolled in plain view. So "chance" and "the unknown" are different things. In backgammon you don't know the future. In poker you don't know the present — what is already in your opponent's hand. That difference changes everything.

A note for the curious: game theorists also distinguish perfect information (seeing every move made so far) from complete information (knowing the rules and every player's goals). Poker has the second — everyone knows the others want to win chips — but not the first.

Why bluffing in poker is mathematics, not temperament

In chess nothing is hidden: whatever your opponent knows, you know too. So the best move doesn't depend on whether your opponent expects it. In theory you could announce your plan out loud, and the evaluation of the position would not change.

Poker doesn't work like that. If you always bet with strong cards and never with weak ones, your bet becomes a face-up card. An attentive opponent will simply fold every time you bet, and your strong hands will earn you nothing.

Von Neumann and Morgenstern had already found bluffing in a simplified poker in their 1944 book. In 1950 the mathematician Harold Kuhn proposed an even smaller model, which became the textbook example. His "simplified poker" uses only three cards — jack, queen and king. Each player gets one, the third stays face down, and only a single bet is allowed. Even in this toy game, the calculation gives a result that surprises newcomers: the optimal strategy must include bluffing.

For example, when the first player checks, the second player holding the weakest card, the jack, should bet exactly one time in three. Bluff more often and the bluff gets called; less often and the bets become too honest. And the player who acts first in this game, with perfect play on both sides, loses on average 1/18 of a chip per hand: the second player acts only after seeing the first player's decision.

This is the key difference. In a game of perfect information, the optimal strategy is a specific move in each position. In a game with hidden information, it is often a proportion: how often to do one thing or another so that your opponent cannot "read" you. Mathematicians call this a mixed strategy, and it will get its own article in this series.

This doesn't make chess "simpler". Finding the right move among an enormous number of variations is as hard as holding the right proportion of bluffs. They are simply different kinds of difficulty.

What this means at the board and beyond

This classification isn't just academic. It helps explain the order in which machines overtook humans. IBM's Deep Blue won a match against Garry Kasparov in 1997. DeepMind's AlphaGo beat Lee Sedol at Go in 2016, 4–1. But programs convincingly beat top professionals at no-limit Texas hold'em only in 2017 — most famously Libratus — and only heads-up. At a six-player table, its successor Pluribus managed it in 2019.

It's not just about computing power. A chess or Go program answers the question "what is the best move?". A poker program has to answer a different one: "how often should I make each move, knowing my opponent is watching my habits?". Machines had to learn to bluff in the right proportions.

For a human player, three questions follow — worth asking before any important decision, at the board or away from it:

• What does my opponent see? If everything, your plan has to be good even when it's been figured out. A chess trap that only works if nobody notices it is a bet on a mistake, not a strategy.
• What can't they see? If something is hidden, predictability becomes a weakness.
• Is this really a zero-sum game? On the board, yes. At work, in friendship or in negotiation it often isn't — and treating such a situation like a chess game is the costliest mistake of all.

Exercise

1. Sort these games using the five questions: dominoes, Monopoly, Mafia (also known as Werewolf), tic-tac-toe. One of them is a two-player game of perfect information with no chance at all, and it was solved long ago: with correct play from both sides, it's a draw. Which one?
2. Change the matchstick rule: now you may take one, two or three. With how many sticks on the table is the first player doomed to lose?
3. Play the matchstick game with someone who hasn't read this article. How many games does it take before they spot the pattern?

Answers are at the end.

Questions for discussion

• Think of a game you lost not because of weak moves, but because you were "read". What gave you away?
• Do you agree with von Neumann that chess is "not a real game"? What does his definition miss, seen through the eyes of a practising player?
• Which situation in your life is most like poker, and which is most like chess?

Next in the series: "Chess Has an Answer — but Nobody Knows It", on Zermelo's theorem, solved checkers, and Go, which still resists.

Answers. 1) Tic-tac-toe. Dominoes and Mafia have hidden information; Monopoly has dice and more than two players. 2) Whenever the number of sticks is a multiple of 4: the second player always completes the opponent's move to four.