Payoff Matrices, Dominant Strategies & Nash Equilibrium

You’ll be able to

Read a two-player payoff matrix and identify dominant strategies
Locate the Nash equilibrium of a simultaneous one-shot game
Explain why the prisoner's dilemma outcome is stable but not jointly best

Reading the matrix

Each cell holds two payoffs. The convention is (row player, column player) — the first number belongs to the player choosing rows. To analyze it, take each player in turn and ask: for each thing the other might do, what is my best response? That question, asked systematically, answers everything the exam will pose.

The two definitions

dominant strategy: best for a player NO MATTER what the other does · Nash equilibrium: a cell where NEITHER player can improve by unilaterally switching

A dominant strategy need not exist. A Nash equilibrium almost always does, and there can be more than one.

Finding a dominant strategy

Fix the opponent on their first option and note which of your options pays you more. Fix them on their second option and note again. If the same option of yours wins both times, it is your dominant strategy. If different options win, you have none — your best move depends on theirs. Do this for both players; where both have a dominant strategy, the intersection is the Nash equilibrium.

The prisoner's dilemma

The classic structure has both players with a dominant strategy to defect, producing an equilibrium that is worse for both than mutual cooperation. It is stable nonetheless, because from that cell neither can improve by switching alone. This is the model of a cartel breaking down: each member gains by cheating on the agreed output, so the agreement collapses even though all of them would prefer it to hold. Repeated play and enforceable contracts are what change the outcome.

Worked example

Two firms choose High or Low output. Payoffs in $ millions, (Firm A, Firm B): both High (2, 2); A High B Low (8, 1); A Low B High (1, 8); both Low (5, 5). Find dominant strategies and the Nash equilibrium.

1.Firm A, if B plays High: High gives 2, Low gives 1 → High is better.
2.Firm A, if B plays Low: High gives 8, Low gives 5 → High is better.
3.So High is A's dominant strategy. By symmetry, High is B's dominant strategy too.
4.Nash equilibrium is (High, High) with payoffs (2, 2).
5.But (Low, Low) would pay (5, 5) — better for both.

Answer: Both firms have High as a dominant strategy, so the Nash equilibrium is (High, High) at (2, 2). Both would prefer (Low, Low) at (5, 5), but neither can reach it alone — a prisoner's dilemma, and exactly why output cartels are unstable.

Watch out

A Nash equilibrium is not the cell with the highest total payoff. It is the cell nobody can leave profitably on their own. Confusing "best jointly" with "stable" gets the prisoner's dilemma exactly backward.

Checkpoint

A dominant strategy is one that:

Checkpoint

In a prisoner's dilemma, the Nash equilibrium is:

Checkpoint

The prisoner's dilemma explains why output cartels tend to break down because:

On the exam

Work through both players' best responses explicitly and write them down, rather than eyeballing the matrix. Game-theory rubrics award identifying each dominant strategy separately from naming the equilibrium.

Answer the 3 checkpoints as you read.