Before you can solve a game you have to be able to write one down. The good news is that an astonishing range of strategic situations — from two countries deciding whether to arm, to two siblings deciding who does the dishes — fit into the same tiny, four-box picture. Learn to build and read that picture and you’ve got a universal notation for conflict and cooperation. This lesson is entirely about the notation; the next three are about what it reveals.
The three ingredients of a game
Every game, no matter how grand or petty, is built from exactly three parts. Miss any one and you don’t yet have a game you can analyse.
- Players — the decision-makers whose choices matter. Two countries, two firms, two flatmates. (Games can have many players; we’ll keep to two, because two is where every core idea already lives.)
- Strategies — the complete set of choices available to each player. “Arm” or “disarm”. “Cut price” or “hold”. “Cooperate” or “defect”. A strategy is a whole plan of action, not a single twitch.
- Payoffs — the value each player places on every possible outcome. Crucially, an outcome is a combination of choices — one from each player — and each player rates it by their own lights. Higher payoff = more preferred. Full stop.
The thing that turns these ingredients into a game rather than a solo decision is that the payoff each player gets depends on the whole combination, not just on their own move. That’s the interdependence from the last lesson, now made precise: your payoff is a function of your choice and theirs.
What a payoff is — and isn't
A payoff is just a number standing in for how much a player likes an outcome, all things considered. It already bakes in money, risk, reputation, guilt, fairness — whatever that player actually cares about. So “they’d never be that selfish” isn’t an objection to the numbers; if they wouldn’t, that is the numbers. Get the payoffs right and the player’s true preferences are already inside them.
The payoff matrix
With two players who each have two strategies, the whole game fits in a 2×2 grid called the payoff matrix. One player (“the row player”) picks a row; the other (“the column player”) picks a column; where their choices cross is the outcome, and the cell holds both their payoffs. The near-universal convention is (row player’s payoff, column player’s payoff) — row’s number first.
That’s the entire object. Four cells, eight numbers, and a complete strategic world. Here’s a real one to read. Two startups are each about to launch a gadget and must pick a connector standard — Format A or Format B. If they pick the same standard, accessories and chargers work everywhere and the whole market grows; if they split, customers are confused and both sell less. Drag any payoff and watch the analysis update — and don’t worry yet about the green “NE” cells or the “dominant strategy” line, those are lessons 3 and 2. For now, just practise reading.
Payoff matrix
The standards game: match or split?
Each cell shows (your startup's payoff, the rival's payoff). Read a cell by crossing a row with a column. Higher numbers are better outcomes for that player.
| Rival startup | ||
|---|---|---|
| Format A | Format B | |
| Your startupFormat A | NE33 | 11 |
| Format B | 00 | NE22 |
A ringed payoff is that player’s best response to the rival’s choice. A cell where both are ringed is a Nash equilibrium.
What the matrix says
Your startup — dominant strategy: none
Rival startup — dominant strategy: none
Nash equilibrium (pure): (Format A, Format A) · (Format B, Format B)
Read the top-left cell: you play Format A, the rival plays Format A, and the
cell says (3, 3) — you each get 3. Now the bottom-left: you play Format B while
they play Format A, giving (0, 0) — a total mismatch, the worst outcome for both.
The matrix has packed an entire negotiation into four boxes.
In the standards matrix above (before you edit it), you play Format B and the rival plays Format A. What does each of you get, and why is that cell bad?
Reading order matters — row first
The single most common beginner error is reading the payoffs in the wrong order. By
convention the first number is always the row player’s and the second is the
column player’s. So the cell (0, 5) means the row player got 0 and the column
player got 5 — a great outcome for the column player, a terrible one for the row
player. Get this backwards and every conclusion you draw will be mirror-flipped.
Fix the convention in your head.
Pick the right option for each blank, then check.
In a payoff matrix the player chooses between the horizontal rows, and inside each cell the number is the row player's payoff while the number belongs to the column player.
Simultaneous vs. sequential — and why we start simultaneous
There’s a hidden assumption in the matrix as we’ve drawn it: both players choose at the same time, or at least without knowing what the other has chosen. That’s a simultaneous game — think sealed bids, or two firms setting next year’s prices behind closed doors. It’s the cleanest case and where every core concept lives, so it’s where we’ll stay for this course.
The other flavour is a sequential game, where one player moves first and the other sees it before responding — chess, an opening salary offer, a country invading and waiting to see who responds. Sequential games are usually drawn as a branching tree rather than a grid, and they add rich ideas like commitment and credible threats. They’re a natural next step, but you don’t need them to understand the dilemmas that run the world. One thing worth knowing now: in a simultaneous game you can’t condition your move on what they actually did — only on what you predict they’ll do. That’s what makes it hard, and interesting.
A common trap: 'I'd just wait and see'
In a simultaneous game there is no “wait and see” — you commit without observing their move. If your plan is “I’ll do whatever they do,” you still have to choose that conditional plan blind, and so do they. Pretending you can peek is the fastest way to mis-solve a simultaneous game.
Build the vocabulary
Match each piece of the game-theory toolkit to what it actually means.
Pick a term, then click its definition.
When to reach for a matrix
The payoff matrix is your first move whenever you suspect a situation is strategic — whenever you catch yourself thinking “well, it depends what they do.” Sketching even a rough 2×2 forces you to name the players, list each side’s real options, and guess how each side ranks the four outcomes. Half the time that act alone dissolves the confusion: you discover the other side’s incentives point somewhere you hadn’t considered, or that what felt like one decision is really a pair of interlocking ones.
A 2×2 matrix is a model, and like every model it simplifies. It assumes each side has a small, clear set of options, that you can estimate the payoffs, and (in this course) that they choose simultaneously and once. Real situations can have many players, continuous choices (any price, not just “high/low”), private information, and repetition. We’ll add repetition later, and the matrix still illuminates those messier cases even when it can’t fully capture them — the skeleton it reveals is usually the thing that matters. Don’t mistake the simplification for a claim that life has only four boxes.
Where this goes next
You can now write a strategic situation down — the indispensable first step. But a matrix on its own doesn’t tell you what to do. For that we need a way to reason from the numbers to a move. The simplest such tool is the luckiest case of all: a strategy that’s best no matter what the other player does, so you don’t even have to predict them. That’s the dominant strategy, and it’s next.