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M20.L03 โ€” Static Games: Nash Equilibrium and Dominant Strategies

Module: Module 20 โ€” Monopoly and Game Theory (Advanced) Lesson: L03 of 06 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

Learning Objective

Identify Nash equilibria in strategic form games using best response analysis.

Nash Equilibrium Definition

A strategy profile (sโ‚*, sโ‚‚*) is a Nash Equilibrium if:

โˆ€i, uแตข(sแตข*, sโ‚‹แตข*) โ‰ฅ uแตข(sแตข, sโ‚‹แตข*) for all sแตข โˆˆ Sแตข

Australian example: Two mining firms (BHP, Rio Tinto) choosing output:

Rio: Low Rio: High
BHP: Low 50, 50 30, 60
BHP: High 60, 30 40, 40

NE: (High, High) - mutual best responses

Worked Example

[Australian context]

Find all NE in this Sydney cafe pricing game:

Cafe B: $4 Cafe B: $5
Cafe A: $4 30, 30 50, 20
Cafe A: $5 20, 50 40, 40

Solution: 1. BR for A: If B=$4 โ†’ \(4 (30>20); If B=\)5 โ†’ \(4 (50>40) 2. BR for B: If A=\)4 โ†’ \(4 (30>20); If A=\)5 โ†’ \(4 (50>40) Only NE: (\)4, $4)

Common Misconception

"Nash equilibrium means best possible outcome"

NE only means no unilateral incentive to deviate - often suboptimal (e.g., Prisoner's Dilemma).

Key Takeaways

  • NE requires mutual best responses
  • Dominant strategies always played in NE
  • Iterated elimination finds NE in dominance-solvable games
  • Coordination games often have multiple NE

Practice

  1. Find all NE in: | | L | R | |------|---|---| | U | 3,2 | 1,1 | | D | 0,0 | 2,3 |
  2. Explain why (D,R) is NE in the above
  3. Construct a 2ร—2 game with exactly two pure NE

Further Resources

Provenance: Barreto | MIT OCW