A Nash equilibrium is one of those concepts that is easier to find than to define. Once you can run the best-response check on a payoff matrix, you can solve every 2×2 game your introductory microeconomics course is going to throw at you. This guide builds the definition, runs it on three different games, and points out the standard traps.
The Definition, In One Line
A Nash equilibrium is a combination of strategies — one for each player — such that no player can do better by unilaterally changing their own strategy, given the strategies of the others.
The "unilaterally" matters. Players might jointly do better with a different combination, but if any single player would lose by switching alone, the current cell still counts as a Nash equilibrium. That gap — collectively suboptimal, individually stable — is exactly what makes the prisoner's dilemma a Nash equilibrium even though both players prefer the cooperative outcome.
How to Read a Payoff Matrix
A 2×2 payoff matrix lists each player's strategies on one axis and shows the resulting payoffs in each cell, by convention as (Row payoff, Column payoff).
Consider this game between two competing app developers, Alpha and Beta, each choosing between Free and Paid pricing:
| Beta: Free | Beta: Paid | |
|---|---|---|
| Alpha: Free | (3, 3) | (5, 1) |
| Alpha: Paid | (1, 5) | (4, 4) |
If both pick Free, each earns 3. If Alpha picks Free and Beta picks Paid, Alpha earns 5 and Beta earns 1. And so on. Each cell is a possible outcome; the Nash equilibrium is a cell with a specific stability property.
Best-Response Checking (the Procedure)
The single most useful technique for finding Nash equilibria in a 2×2 game is best-response checking. The procedure:
- Fix the column player's strategy. Underline (or circle) the row player's higher payoff. That marks the row player's best response.
- Repeat for the other column.
- Now fix each row in turn. Underline the column player's higher payoff in that row. Those are the column player's best responses.
- **A cell where both payoffs are underlined is a Nash equilibrium.**
Apply it to the Alpha/Beta matrix. Holding Beta = Free: Alpha gets 3 from Free vs. 1 from Paid → Alpha's best response is Free. Holding Beta = Paid: Alpha gets 5 from Free vs. 4 from Paid → Alpha's best response is Free again. (Free is, in fact, dominant for Alpha.) By symmetry, Free is Beta's best response in either row.
Both payoffs underlined in the (Free, Free) cell. So (Free, Free) is the unique Nash equilibrium, even though both players would prefer (Paid, Paid) at (4, 4). Same logic as the prisoner's dilemma — for the canonical setup, see the prisoner's dilemma.
Two Other Patterns Worth Knowing
Best-response checking handles all the standard 2×2 cases. Two more patterns to recognize.
Coordination Games — Multiple Nash Equilibria
| B: Format A | B: Format B | |
|---|---|---|
| A: Format A | (4, 4) | (0, 0) |
| A: Format B | (0, 0) | (3, 3) |
Run the procedure. Holding B = A: A prefers 4 to 0 → best response is A. Holding B = B: A prefers 3 to 0 → best response is B. By symmetry, B's best responses are A in column A and B in column B. So two Nash equilibria: (A, A) and (B, B). Both are stable — once both players pick the same format, neither wants to switch alone, even though one of the equilibria is plainly better than the other. This is the standard coordination story; in real markets it shows up as competing technical standards (Blu-ray vs. HD-DVD, VHS vs. Betamax).
Matching Pennies — No Pure-Strategy Equilibrium
| B: Heads | B: Tails | |
|---|---|---|
| A: Heads | (1, −1) | (−1, 1) |
| A: Tails | (−1, 1) | (1, −1) |
Apply best-response checking. In every cell, exactly one player is winning and the other has a strict incentive to switch. No cell ends up with both payoffs underlined — there is no Nash equilibrium in pure strategies. The standard solution introduces mixed strategies: each player randomizes 50/50 between Heads and Tails, and the resulting probability distribution is the Nash equilibrium. Mixed-strategy equilibria are usually beyond an introductory class, but knowing they exist (and that some games require them) saves you from looking forever for a pure-strategy answer that isn't there.
Dominant Strategies vs. Nash Equilibria
A dominant strategy is a player's best choice no matter what the rival does. If every player has a dominant strategy, the resulting cell is automatically a Nash equilibrium. The converse is not true: most Nash equilibria are not in dominant strategies (the coordination game above has two Nash equilibria but no player has a dominant strategy — each player's best response depends on what the other does).
Order of operations: check for dominant strategies first (they make the answer obvious). If none, do best-response checking. If best-response checking yields no pure-strategy equilibrium, the answer lives in mixed strategies.
For the next layer — repeated games, oligopoly applications, and cartel breakdown — see oligopoly and game theory.
Conclusion
Nash equilibrium basics rest on a single idea — no player wants to deviate alone — and a single procedure — best-response checking on the payoff matrix. Apply it row by row and column by column, mark each player's best response in each conditional, and the cells where both are marked are the equilibria. A game can have one, several, or, in pure strategies, none. Read carefully which case you're in, and the rest of the analysis falls out.
