How do you find a Nash equilibrium in an oligopoly payoff matrix?

To find a Nash equilibrium, identify each firm's best response: the choice that gives it the highest payoff for a given choice by its rival. A cell is an equilibrium when both firms are choosing a best response. Neither can improve its own payoff by changing its decision while the other keeps its choice unchanged.

Suppose two competing firms, Alba and Beacon, each independently choose a regular price or a discount for the same sales period. In this matrix, the payoffs are profits in thousands of currency units, with Alba's profit first and Beacon's second.

Start with Alba. Hold Beacon's decision fixed and compare the first numbers within that column:

• If Beacon keeps its regular price, Alba compares 18 with 25. Alba prefers to discount.
• If Beacon discounts, Alba compares 7 with 11. Alba again prefers to discount.

Discounting is therefore Alba's strictly dominant strategy in this particular game: it produces more profit whichever action Beacon takes.

Now hold Alba's decision fixed. Compare Beacon's second numbers across each row:

• If Alba keeps its regular price, Beacon compares 16 with 23. Beacon prefers to discount.
• If Alba discounts, Beacon compares 6 with 10. Beacon again prefers to discount.

Both best responses meet at (discount, discount), with profits (11, 10). Check it directly: with Beacon still discounting, Alba would lose 4 by switching to its regular price. With Alba still discounting, Beacon would lose 4 by switching. Neither firm benefits from changing its decision alone.

That equilibrium does not maximise combined profit. If both charged their regular prices, total profit would be $18+16=34$, compared with $11+10=21$ at the equilibrium. However, neither has a reason to stick to the regular-price outcome in this one-period game: either firm can gain by discounting while the other keeps its price unchanged. A larger combined profit does not make an outcome an equilibrium.

This method explains each firm's incentives; it is not advice to coordinate prices. If the firms make repeated decisions, face binding rules or have different payoffs, their incentives could change.

Avoid comparing Alba's number with Beacon's number in the same cell; each firm compares its own possible payoffs. Also, not every game has a dominant strategy. The best-response test still works. A matrix can have several pure-strategy equilibria, where each firm chooses a definite action rather than randomising, or none.

Explore the oligopoly payoff-matrix question in the first Economics HL mock-exam set. Compare each firm's options while holding its rival's choice fixed, then use those comparisons to explain why their decisions depend on each other.