Oligopoly: game theory and Nash equilibrium

One payoff matrix, read the same way every time, and why two firms end up cutting price when both would earn more by not.

1. Two airlines fly the same route, Belfast to London. Each sets its fare for next season without knowing what the other will choose, and each airline’s profit depends on both fares. That is interdependence, and we show it with a payoff matrix. Airline A chooses a row. Airline B chooses a column.

2. Each airline has two strategies: keep the fare high, or cut it. Label all four. High fare and low fare for A down the side, and the same for B along the top.

3. Now the payoffs: the profit each airline makes in each outcome. In every cell, A’s profit goes first, on the left, in blue. B’s goes second, on the right. If both keep fares high, they share the passengers at a good price: £6m each.

4. If A cuts and B does not, A takes most of B’s passengers: £8m to A, only £2m to B. The opposite cell is the mirror image. If both cut, they share the passengers again, but at a lower price: £4m each.

5. Now think like Airline A, one column at a time. Suppose B keeps its fare high. Cover the other column and look only at A’s numbers, in blue: £6m if A stays high, £8m if A cuts. 8 beats 6, so A cuts. Keep the circle on the 8.

6. Now suppose B cuts. A’s numbers in that column: £2m if A stays high, £4m if A cuts. 4 beats 2, so A cuts again. Keep the circle on the 4.

7. Both of A’s circles are in the low-fare row. Whatever B does, A is better off cutting. That makes cutting A’s dominant strategy.

8. Stop on that idea. A dominant strategy is the best choice for a firm whatever its rival does. Here A’s best move is the same in both columns, so A has one. Not every game has a dominant strategy. If A’s best move were to cut when B stays high, but to stay high when B cuts, A would have no dominant strategy: its best move would depend on B. Always check both columns before you say a firm has one.

9. Now think like Airline B, one row at a time, looking only at B’s numbers, in orange. Suppose A keeps its fare high: £6m if B stays high, £8m if B cuts. 8 beats 6, so B cuts. Keep the circle on the 8.

10. Now suppose A cuts: £2m if B stays high, £4m if B cuts. 4 beats 2, so B cuts again. Keep the circle on the 4.

11. Both of B’s circles are in the low-fare column, so cutting is B’s dominant strategy too.

12. Find the cell where both numbers are circled: both cut, £4m each. That is the Nash equilibrium. Given what the other airline is doing, neither can do better by changing its own fare. Ring the whole cell.

13. Here is the twist, and the mistake to avoid. The Nash equilibrium is not where they earn the most. Both keeping fares high earns £12m between them, not £8m. Each would like the other to keep fares high, but each gains by cutting, so both end up worse off. This is the prisoners’ dilemma, and it is why cartels break down: an agreement to keep fares high cannot hold when each firm gains by cheating, and in the UK it is illegal anyway.

14. Airline B keeps its fare high. Click Airline A’s best response: the payoff A should go for.

15. Now Airline B cuts its fare. Click Airline A’s best response.

16. So does Airline A have a dominant strategy?

17. Now think like Airline B, one row at a time. Airline A keeps its fare high. Click Airline B’s best response.

18. Now Airline A cuts its fare. Click Airline B’s best response.

19. Click the cell that is the Nash equilibrium.