Nash Equilibrium Explained: How to Find It in a Payoff Matrix

A Nash equilibrium is a combination of strategies where no player can do better by changing their own choice alone, given what everyone else is doing. It is the point where every player is simultaneously playing their best response to the others, so nobody regrets their decision once the outcome is revealed. That definition is the whole test, and once you turn it into a mechanical procedure, finding the Nash equilibrium in an AP-style 2x2 payoff matrix becomes fast and reliable. This guide gives you the best-response method, works two full examples with the arithmetic shown, draws the sharp line between a dominant strategy and a Nash equilibrium, and covers the case where no pure Nash equilibrium exists at all.

Nash equilibrium lives in the game-theory corner of oligopoly. If you want the broader tour of dominant strategies, the prisoner's dilemma, and why cartels cheat, read the game theory guide first or alongside this one. Here the focus is narrower and practical: how to find the equilibrium and how not to confuse it with a dominant strategy.

Reading a payoff matrix correctly

Everything starts with reading the grid the right way. A payoff matrix lists two players' choices, one player's options as rows and the other's as columns, and every cell holds two numbers. By convention the row player's payoff is written first and the column player's second, so a cell reading (5, 3) means the row player gets 5 and the column player gets 3.

The single most important habit: to judge a player's move, you compare only that player's own numbers, holding the other player's choice fixed. You never compare a player's payoff against their rival's payoff in the same cell. The two numbers in a cell are not competing with each other; they simply report what each player walks away with. Row player? Compare down a column. Column player? Compare across a row.

The best-response method (how to find a Nash equilibrium)

Here is the procedure that works every time.

1. For the row player, go column by column. Fix the column player's choice, look down that column, and mark the row player's largest payoff. That marked entry is the row player's best response to that column.

2. For the column player, go row by row. Fix the row player's choice, look across that row, and mark the column player's largest payoff. That is the column player's best response to that row.

3. Find the cells where both numbers are marked. Any cell in which the row player's payoff and the column player's payoff are both best responses is a Nash equilibrium. If both players are already doing the best they can given the other, neither wants to move.

This "underline the best responses" technique is foolproof because it directly encodes the definition. A game can have one such cell, several, or none, and the method finds all of them.

Worked example 1: a unique Nash equilibrium

Two firms, A and B, each decide whether to Advertise or Don't. Payoffs are annual profits in millions, with Firm A's payoff listed first.

Firm A's best responses (compare down each column).

• If Firm B Advertises (left column): Firm A earns 10 by advertising versus 8 by not. Advertise wins (10 > 8).
• If Firm B Doesn't (right column): Firm A earns 25 by advertising versus 20 by not. Advertise wins (25 > 20).

Advertise is Firm A's best response in both columns, so Advertise is Firm A's dominant strategy.

Firm B's best responses (compare across each row, using the second number).

• If Firm A Advertises (top row): Firm B earns 5 by advertising versus 4 by not. Advertise wins (5 > 4).
• If Firm A Doesn't (bottom row): Firm B earns 8 by advertising versus 6 by not. Advertise wins (8 > 6).

Advertise is Firm B's best response in both rows too, so Advertise is also Firm B's dominant strategy.

Find the equilibrium. The only cell where both players are playing a best response is (Advertise, Advertise), paying (10, 5). Confirm it directly: from that cell, if Firm A switched to Don't its payoff would drop from 10 to 8, and if Firm B switched to Don't its payoff would drop from 5 to 4. Neither wants to move, so (Advertise, Advertise) is the unique Nash equilibrium. When both players have a dominant strategy, the cell where those two dominant strategies meet is always the Nash equilibrium, and it is the only one.

Dominant strategy vs Nash equilibrium

Those two terms describe different things, and the exam tests whether you can keep them apart.

• A dominant strategy is a property of a single player's choice: an option that gives that player the highest payoff no matter what the rival does. You test it by checking it against every move the other player could make.
• A Nash equilibrium is a property of a whole outcome, an entire cell, where neither player wants to switch given the other's actual choice.

The relationship runs one way. If both players have a dominant strategy, the cell they meet in is guaranteed to be a Nash equilibrium (as in example 1). But the reverse is not true: a Nash equilibrium can exist even when nobody has a dominant strategy. That is the case people most often miss, so the next example is built to show it.

Worked example 2: Nash equilibrium with no dominant strategy

Two firms, 1 and 2, are each deciding whether to Enter a small new market or Stay Out. The market is only big enough for one firm to profit; if both enter, they split it and both lose money on the fixed cost of entry. Payoffs in millions, Firm 1 first.

Firm 1's best responses (down each column).

• If Firm 2 Enters: Firm 1 earns -10 by entering versus 0 by staying out. Stay Out wins (0 > -10).
• If Firm 2 Stays Out: Firm 1 earns 50 by entering versus 0 by staying out. Enter wins (50 > 0).

Firm 1's best move flips depending on Firm 2, so Firm 1 has no dominant strategy. By symmetry, Firm 2 has none either.

Find the equilibria. Check each cell against the definition.

• (Enter, Enter) = (-10, -10): Firm 1 would rather switch to Stay Out (0 beats -10), so this is not an equilibrium.
• (Enter, Stay Out) = (50, 0): Firm 1 is best-responding (50 beats the 0 it would get by staying out while Firm 2 stays out), and Firm 2 is best-responding (staying out gives 0, versus -10 if it entered against Firm 1). Neither wants to move. This is a Nash equilibrium.
• (Stay Out, Enter) = (0, 50): by the same symmetric logic, neither wants to move. This is also a Nash equilibrium.
• (Stay Out, Stay Out) = (0, 0): either firm would rather enter an empty market for 50, so this is not an equilibrium.

This game has two Nash equilibria and no dominant strategy for either player. It is the cleanest proof that the two concepts are not the same: whichever firm ends up entering while the other stays out is stable, because given the rival's choice, no one gains by deviating. Real markets settle these through timing, reputation, or a credible commitment to enter first.

When there is no pure Nash equilibrium

Some games have no Nash equilibrium in the simple (pure-strategy) sense the AP exam usually asks about. The textbook case is matching pennies: each player secretly shows heads or tails, and one wins when they match while the other wins when they differ.

Run the best-response check on any cell and one player always wants to switch: whoever is losing would flip their coin. No cell has both numbers as best responses, so there is no pure-strategy Nash equilibrium. Games like this resolve only in mixed strategies (randomizing your choice), which is beyond the AP scope, but knowing that a pure Nash equilibrium can simply fail to exist is worth a sentence on a free-response answer.

Practice the mechanics

The surest way to lock this in is to run the best-response check on matrix after matrix until it is automatic: mark the row player's best payoff in each column, the column player's best payoff in each row, and read off every cell where both are marked. Drill payoff-matrix reasoning in the practice sets, see how graders want the full justification written out in the FRQ answer walkthroughs, and get the wider oligopoly context, dominant strategies, the prisoner's dilemma, and collusion, in the game theory guide. Every bolded term here, from best response to dominant strategy, is defined in the economics glossary.