Game Theory

Finding a Nash equilibrium in a 2×2 game

Two players choose simultaneously. Each has two strategies — here Cooperate (C) or Defect (D). A cell shows the payoffs (Row, Column).

• A player’s best response = the strategy giving the higher payoff, given what the other player does.
• A Nash equilibrium = a cell where both players are best-responding — nobody wants to switch unilaterally.

The classic Prisoner’s Dilemma has payoffs ranked T > R > P > S (Temptation > Reward > Punishment > Sucker). Then “Defect” is a dominant strategy for both, so the unique Nash is (D, D) — even though (C, C) would make both better off. That tension is the whole point.

Try it. Adjust the four payoffs. Best responses get underlined; a cell turns green when it’s a Nash equilibrium. Break the T > R > P > S ranking and watch the equilibrium move (or multiply).

Temptation T: 5

Reward R: 3

Punishment P: 1

Sucker S: 0

	Column: C	Column: D
Row: C
Row: D

blue = Row payoff · red = Column payoff · underline = best response · green cell = Nash

What to notice

• Default (PD): both best-respond with Defect, so (D, D) is the only green cell — even though (C, C) gives both a higher payoff. Individually rational, collectively bad.
• Make R > T (cooperating beats tempting) → cooperation becomes a best response and (C, C) can turn green. It’s no longer a dilemma.
• Some payoff settings give two Nash equilibria (a coordination game) — watch two cells go green at once.