Nash Equilibrium: How to Solve Games by Hand

Normal-form games and dominant strategies

A normal-form game is a triple G = (N, S, u): a set of players N, a strategy set Sᵢ for each player, and a payoff function uᵢ giving each player's payoff for every strategy combination — for two players this is exactly the payoff matrix behind the Prisoner's Dilemma. Strategy sᵢ' strictly dominates sᵢ if it gives strictly higher payoff against every strategy the opponent might choose. A player with a strategy that dominates every alternative has a dominant strategy — rational play regardless of what anyone else does. In the Prisoner's Dilemma, Defect strictly dominates Cooperate for both players, which is exactly why (Defect, Defect) is such a robust prediction even though it's collectively worse.

Iterated elimination and best responses

Even without a single dominant strategy, repeatedly removing dominated strategies can narrow the solution — if all strategies eliminate down to one cell, that cell is the game's unique equilibrium. The more general tool is the best-response function BRᵢ(s₋ᵢ), the strategy that maximises player i's payoff given the others' choices:

BRᵢ(s₋ᵢ) = { sᵢ ∈ Sᵢ : uᵢ(sᵢ,s₋ᵢ) ≥ uᵢ(sᵢ',s₋ᵢ) for all sᵢ' ∈ Sᵢ }

In a matrix: underline each row's best response per column,
underline each column's best response per row.
A cell with BOTH underlined = Nash equilibrium.

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Battle of the Sexes: two equilibria, no unique prediction

A couple prefers different events but both prefer being together. Marking best responses in the payoff matrix reveals two cells where both payoffs are underlined — (Opera, Opera) and (Football, Football) are both pure-strategy Nash equilibria. This is a genuine coordination problem: the model has no single prediction, exactly the phenomenon it was built to illustrate.

Mixed strategies and the indifference principle

When no pure equilibrium exists (matching pennies, rock-paper-scissors), players randomise. The key technique is the indifference principle: in a mixed equilibrium, each player is exactly indifferent between the strategies they mix over, or they would deviate. For Battle of the Sexes, solving Player 2's indifference condition gives Player 1's optimal mixing probability p = 2/3 for Opera; by symmetry Player 2 mixes toward Opera with probability q = 1/3. Note the counter-intuitive twist: each player's mixing probability is set by the other player's indifference condition, not their own preference — a detail that trips up newcomers constantly.

Frequently asked questions

What is a Nash equilibrium?

A Nash equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing their own strategy, given what every other player is doing. It is found where every player's chosen strategy is simultaneously a best response to everyone else's.

How do you find a Nash equilibrium in a payoff matrix by hand?

Underline each player's best response to every strategy the other player might choose — for each column, mark the best payoff in that column for the row player, and for each row, mark the best payoff for the column player. Any cell where both payoffs are underlined is a pure-strategy Nash equilibrium, because neither player can do better by deviating alone.

How do you solve for a mixed-strategy Nash equilibrium?

Use the indifference principle: in a mixed equilibrium each player must be exactly indifferent between the pure strategies they mix over, otherwise they would deviate to the strictly better one. To find how you should mix, solve the equation that makes your opponent indifferent between their own options — not your own payoffs.

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