Many games have no dominant strategy (Payoff Matrices and Dominant Strategies). We still need a way to predict what happens. Practice the idea in the Game Theory course's Nash Equilibrium chapter.
Key idea
A Nash equilibrium is a pair of strategies in which each player's choice is a best response to the other's: no player can do better by changing their own strategy alone. To find one, mark each player's best response to every choice of the other; a cell where both payoffs are marked is a Nash equilibrium. A game can have one, several or none (in pure strategies).
Why it works
At a Nash equilibrium, nobody has regrets given what the others did, so there is no pull to move. In a prisoner's dilemma, each player has a dominant strategy to defect, so (defect, defect) is the Nash equilibrium, even though both would be better off cooperating. That explains why cartels are hard to keep together: each member gains by cheating on the agreement, whatever the others do.
Worked example: A cartel breaks down
Two firms agree to keep output low. Payoffs (A, B) in millions: both Keep (10, 10); A Cheats and B Keeps (14, 4); A Keeps and B Cheats (4, 14); both Cheat (6, 6). Find the Nash equilibrium.
- A's best responses: against Keep, Cheat (14 > 10); against Cheat, Cheat (6 > 4).
- B's best responses are the same by symmetry.
- Only (Cheat, Cheat) has both payoffs marked: the Nash equilibrium, at 6 each, worse than the 10 each from keeping the deal.
Answer: (Cheat, Cheat).
Worked example: Two equilibria
Two friends pick a café without talking. Both at Café A: (3, 3); both at Café B: (2, 2); different cafés: (0, 0). Find the Nash equilibria.
- If the other goes to A, your best response is A (3 > 0); if the other goes to B, it's B (2 > 0).
- Both (A, A) and (B, B) are Nash equilibria: neither friend wants to switch alone.
Answer: (A, A) and (B, B).
Common mistake
Thinking a Nash equilibrium must be the best outcome for both players. It's only stable: no one can gain by changing alone. In a prisoner's dilemma, both would gain by changing together, but neither can trust the other to. Check a candidate cell by asking each player in turn: "holding the other fixed, would you switch?"
Repeated play changes things: firms that meet again and again can sometimes keep cooperating, because cheating today invites punishment tomorrow. That is the subject of the Game Theory course's repeated games chapter.