How do you find a Nash equilibrium?
Find a Nash equilibrium by checking every cell of a payoff matrix: if either player would rather switch given what the other player picked, that cell is not the equilibrium. A Nash equilibrium is a combination of choices where no player can do better by changing only their own choice.
A Nash equilibrium is a combination of choices, one for each player, where nobody can raise their own payoff by switching to a different choice while everyone else keeps playing what they already picked. It says nothing about fairness or about the best outcome for the group, only that no single player has a private reason to move away from that outcome alone.
The way to find one in a 2 by 2 game is to build the payoff matrix and check every cell in turn. Picture two food trucks, A and B, each choosing a high price or a low price for lunch, with weekly profit in thousands of dollars written as (A's profit, B's profit). Both pricing high pays out (50, 50). A high with B low pays (20, 60). A low with B high pays (60, 20). Both pricing low pays (30, 30).
Now test each cell for a player who would rather switch. At (50, 50), A could move to low pricing and earn 60 instead of 50, so A wants out and this cell is not an equilibrium. At (20, 60), A could switch to low and earn 30 instead of 20, so that fails too. At (60, 20), B could switch to low and earn 30 instead of 20, so that fails as well. At (30, 30), A moving to high would only earn 20, and B moving to high would only earn 20, so neither wants to move. Both charging low is the Nash equilibrium.
A dominant strategy makes this shortcut even quicker: it is a choice that pays a player more no matter what the other player does, so it can be spotted without checking every cell. Say two firms each choose to advertise or not, and profit in millions comes out as (4, 4) if both advertise, (9, 2) if only the row firm does, (2, 9) if only the column firm does, and (6, 6) if neither does. Advertising pays each firm more whichever way the other one moves, so both advertise, and the Nash equilibrium sits at (4, 4), even though both firms would earn more at (6, 6) if they could agree to hold back.
Not every game hands you a single answer. Suppose two firms are picking which video format to build products around, format X or format Y, and a firm only benefits if it matches its rival. Both choosing X pays (10, 10), both choosing Y pays (6, 6), and a mismatch pays (0, 0) to whichever firm gets stuck alone. Neither firm gains by switching away from a matched pair, since that only drops its own payoff to zero, so both (X, X) and (Y, Y) are Nash equilibria. Only something outside the matrix, cost, history, or the ability to signal a choice first, decides which one actually happens.
This checking method is exactly what firms in an oligopoly are doing when they weigh a price cut, an ad campaign, or a capacity expansion against how a rival is likely to respond, since only a few large players means every move gets noticed and answered. The worked oligopoly models on this site walk through pricing and output decisions the same way, cell by cell, so the payoff table becomes a tool for predicting real market behavior instead of just a classroom exercise.
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Related questions
- Is a Nash equilibrium always the best outcome for both players?
- No. The advertising example above lands both firms at (4, 4) profit even though (6, 6) is available to both if they held back, because holding back alone only invites the other firm to grab the higher payoff for itself. A Nash equilibrium describes what each player has no private reason to change, not what would make the group best off.
- Can a 2 by 2 game have more than one Nash equilibrium?
- Yes. Coordination games, like two firms picking a matching video format, often have several outcomes where neither player wants to switch alone. Both firms landing on format X is an equilibrium, and both landing on format Y is a separate equilibrium, and nothing inside the payoff matrix says which one the players will actually reach.
- What is the difference between a dominant strategy and a Nash equilibrium?
- A dominant strategy belongs to one player: it is the choice that pays that player more no matter what the other player does. A Nash equilibrium describes a full pairing of choices, one per player, where neither wants to switch given what the other picked. When both players in a game happen to have a dominant strategy, playing those two choices together is automatically the Nash equilibrium.
- Does every game have a Nash equilibrium?
- Every finite game has at least one Nash equilibrium once mixed strategies, choosing between options with set probabilities rather than picking just one, are allowed. On the AP exam the payoff matrices are small enough that checking cells for a plain, single choice equilibrium is normally all that is asked.