Game Theory
Nash Equilibrium & Dominant Strategies
Overview
Nash Equilibrium is a stable state where no player benefits from changing their strategy unilaterally. The key insight: equilibrium is about mutual best responses. To find mixed strategy NE: you choose your mixing probabilities to make your opponent indifferent — if they're not indifferent, they should deviate, and you would exploit that. This is counterintuitive but mathematically elegant.
How to Recognize
- →Two or more players making strategic decisions simultaneously
- →'What is the optimal strategy if both players are rational?'
- →A payoff matrix is given or must be constructed
- →Questions about equilibrium in competitive markets
Step-by-Step Approach
- 1.Step 1: Eliminate strictly dominated strategies (no matter what opponent does, this is always worse)
- 2.Step 2: Look for pure Nash Equilibrium: cell where neither player wants to deviate
- 3.Step 3: If no pure NE, find mixed NE: choose p to make opponent indifferent
- 4.Step 4: For zero-sum games, minimax = maximin = value of game
Key Formulas
Worked Examples
Problem
Two players: Row can play Up or Down, Column can play Left or Right. Payoff matrix (row payoffs): (U,L)=3, (U,R)=1, (D,L)=0, (D,R)=2. Find NE.
Solution
No dominant strategies: check each row. For Row: Up better than Down vs Left (3>0), but Down better than Up vs Right (2>1). Not dominated.
No pure NE: check all 4 cells. (U,L): Row prefers U (3>0) ✓, Col prefers L over R? Col's payoffs: need column's matrix. Let's assume zero-sum so Col's payoff = negative of Row's.
Col's payoffs (zero-sum): (U,L)=−3, (U,R)=−1, (D,L)=0, (D,R)=−2.
Pure NE check: (U,R): Row prefers D (2>1), not NE. (D,L): Row prefers U (3>0), not NE. (U,L): Col prefers R (−1>−3), not NE. (D,R): Col prefers L (0>−2), not NE. No pure NE.
Mixed NE: Let Row play Up with prob p. For Column to be indifferent: E[L]=E[R]. −3p+0(1−p) = −1p−2(1−p). −3p = −p−2+2p. −3p = p−2. −4p = −2. p=1/2.
Let Col play Left with prob q. E[Up]=E[Down] for Row: 3q+1(1−q) = 0q+2(1−q). 3q+1−q = 2−2q. 2q+1 = 2−2q. 4q=1. q=1/4.
Mixed NE: Row plays (Up=1/2, Down=1/2), Column plays (Left=1/4, Right=3/4).
Answer
Mixed NE: Row plays Up with prob 1/2; Column plays Left with prob 1/4. At this equilibrium, each player is indifferent between their strategies.
Common Mistakes
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Forgetting to check column player's incentives, not just row player's.
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Confusing mixed strategy equilibrium with randomization for unpredictability. You randomize to PREVENT opponents from exploiting you, not because you're confused.
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Thinking NE is always the 'best' outcome — the Prisoner's Dilemma shows it can be inefficient.
Practice Problems
Click "Show Answer" to revealIn the Stag Hunt game: two hunters. Each can hunt Stag (requires both) or Rabbit (alone). Stag gives 5 each if both hunt it, Rabbit gives 2 to whoever hunts it regardless. What are the NE?
What is the Nash Equilibrium of Rock-Paper-Scissors, and why?