Payoff Matrix
What is Payoff Matrix?
A payoff matrix is a table listing every combination of the players' strategies and the payoff each one earns, written as (row player, column player).
Rows are one player's strategies, columns are the other player's, and each cell holds a pair of numbers. The convention is that the first number in the pair belongs to the row player and the second to the column player, though a well-labeled matrix says so directly. To solve one, find each player's best response: fix the opponent's choice, then compare only the numbers that belong to the player you are studying. A cell where both players are already playing a best response is a Nash equilibrium, and there can be none, one, or several. A payoff matrix assumes the players move at the same time or without seeing each other's move; a game where one side moves first belongs in a game tree instead.
Payoff Matrix: a worked example
Ridge Farms picks the row, Expand or Hold, and Vale Foods picks the column, with profits in millions written (Ridge, Vale). Expand against Expand gives (8, 5), Expand against Hold gives (12, 7), Hold against Expand gives (5, 9), and Hold against Hold gives (7, 6). Ridge compares down each column: 8 beats 5 under Expand and 12 beats 7 under Hold, so Expand is dominant for Ridge. Vale has no dominant strategy, preferring Hold when Ridge expands (7 over 5) and Expand when Ridge holds (9 over 6). Since Ridge will expand, Vale holds, and (12, 7) is the equilibrium: switching alone drops Ridge to 7 and Vale to 5.
The mistake students make with payoff matrix
The most common error is comparing the row player's payoffs across a row, which varies the opponent's choice instead of holding it fixed. A best response answers one question: given that the opponent has chosen this column, which of my rows pays more? So the row player compares up and down inside a single column, and the column player compares left and right inside a single row. The other slip is reading the pairs backwards and giving the row player's number to the column player, which flips the answer.
Payoff Matrix questions
Which number in a payoff matrix belongs to which player?
The first number in each cell belongs to the row player and the second belongs to the column player. Most exam matrices also label the cells directly, so read the labels rather than trusting the habit. If a label contradicts the convention, the label wins.
How do you find a Nash equilibrium in a payoff matrix?
Mark each player's best response cell by cell, then look for a cell that is a best response for both players at once. Work one column at a time for the row player and one row at a time for the column player. A cell marked twice is a Nash equilibrium, and a matrix can contain none, one, or several.
Can a payoff matrix show who moves first?
No, a payoff matrix cannot show turn order, because it assumes the players choose without seeing each other's move. A game with a first mover is drawn as a game tree and solved backward from the last decision. Forcing a sequential game into a matrix throws away the information that makes the order matter.
Related terms
Common comparisons
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