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Two coffee shops share a corner. Each has to decide, without asking the other, whether to open early. What one shop should do depends entirely on what the other does. That dependence is what makes it a game.

What is a game?

A game is three things: the players, the strategies each can choose, and the payoffs they get for every combination. A payoff table lets you read all of it at a glance.

Read the table

Tap any cell. The first number is always yours, the second is theirs. Now read down your two rows. If they open late, opening early wins you the morning rush, 3 against 2. If they open early too, you have both paid staff to split the same few early customers, and you would rather have slept in: 0 against 1. There is no move that is right on its own. Which one is right depends on them.

The pieces, precisely

A game in normal (strategic) form is a tuple $G = (N, (S_i)_{i \in N}, (u_i)_{i \in N})$ : a set of players $N$ , a set of strategies $S_i$ for each player, and a payoff function $u_i$ that assigns a number to every combination of choices.

A choice by everyone at once is a strategy profile $s = (s_i, s_{-i})$ , where $s_{-i}$ is shorthand for what everyone except $i$ does. Payoffs are utilities, not necessarily dollars: they encode how a player ranks outcomes. With no uncertainty, their order is enough. Once chance or mixing enters, the differences between utility numbers matter too, up to positive affine rescaling.

Definition, strategy profile

A profile $s$ lists one strategy per player. Player $i$ evaluates it through $u_i(s)$ , holding fixed what the others, $s_{-i}$ , chose.

Work it out

Read a cell, then read a row

Start with one cell. Pick the row for your choice and the column for theirs, and read the pair. Row “you open early”, column “they open late” gives $(3, 1)$ : the first number is always yours, so you make $3$ and they make $1$ .

Now compare a whole row against the other, holding their choice fixed. Suppose they open late. Your early row pays $3$ , your late row pays $2$ , so against a late rival you want to open early. Fix their choice the other way, at early: now your early row pays $0$ and your late row pays $1$ , so you would rather open late. Comparing down a column with their choice held fixed is the move you will use in every lesson from here.

Faded. Do the same for them. If you open early, is their better reply early or late? Read their numbers (the second in each pair) along the “you open early” row: early pays them $0$ , late pays them $1$ , so they open late.

Solo. There is no row that wins for you in both columns. Say in one sentence what that means for how you should choose.

Key idea

In a game, the best move is not a property of you alone. It is a response to what everyone else is doing.

Check yourself

Check yourselfread the table

You open early and they open late. What is your payoff?

Check yourselfconcept

What information must the payoff table specify to describe this game?

Sources

Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.
von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.