Strategy

Strategy: Game Theory

Dominant strategies, Nash equilibrium, and the logic of competition.

1. 1. Prisoner's dilemma

   medium

   Two firms choose Cooperate (keep prices high) or Defect (cut prices). Payoffs (profit): both Cooperate $(10,10)$; both Defect $(4,4)$; if one Defects while the other Cooperates, the defector gets $12$ and the cooperator $2$. Find each firm's dominant strategy and the equilibrium.

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   Take Firm 1's view. If Firm 2 Cooperates: Defect gives $12$ vs Cooperate $10$ → Defect. If Firm 2 Defects: Defect gives $4$ vs Cooperate $2$ → Defect. So Defect is dominant for Firm 1, and by symmetry for Firm 2.

   The unique Nash equilibrium is (Defect, Defect) with payoffs $(4,4)$ — worse for both than $(10,10)$. That gap is the dilemma: individually rational choices give a collectively poor outcome.

2. 2. Two pure equilibria

   hard

   Two firms decide whether to adopt technology standard $A$ or $B$. If they match on $A$ they get $(3,3)$; if they match on $B$, $(2,2)$; if they mismatch, both get $0$. Find all pure-strategy Nash equilibria.

   Show solution

   Check each cell for a profitable unilateral deviation.

   $(A,A)=(3,3)$: deviating to $B$ gives $0$ — no one deviates → equilibrium. $(B,B)=(2,2)$: deviating to $A$ gives $0$ — no one deviates → equilibrium. $(A,B)$ and $(B,A)$ give $0$; either firm can switch to match and gain → not equilibria.

   There are two pure-strategy Nash equilibria: $(A,A)$ and $(B,B)$. This is a coordination game — both prefer to agree, and $(A,A)$ is the Pareto-superior one, but $(B,B)$ is still self-enforcing.

Go deeper

• Casebook: Apple 1997 →
• Atlas: History of Economic Thought →