Lecture 2: Advanced Nash Equilibrium — Mixed Strategies and Multiple Equilibria

The Limits of Pure Strategies

In Lecture 1, the Prisoner’s Dilemma had a clear pure-strategy Nash equilibrium. But in many games, no pure-strategy equilibrium exists.

Matching Pennies — No Pure Strategy Equilibrium
Me \ Opponent	Heads	Tails
Heads	Me +1, Opp -1	Me -1, Opp +1
Tails	Me -1, Opp +1	Me +1, Opp -1

If I choose Heads, my opponent prefers Tails. If I choose Tails, my opponent prefers Heads. No combination of pure strategies is stable. The solution to games like this is a mixed strategy.

Mixed-Strategy Nash Equilibrium

A mixed strategy means selecting each pure strategy with a certain probability.

The Intuition Behind Mixed Strategies
Concept	Explanation	Example
Mixed Strategy	Choose pure strategies probabilistically	Heads 50%, Tails 50%
Equilibrium Condition	Opponent must be indifferent to my mixed strategy	Expected payoffs equal across strategies
No Incentive to Deviate	Any pure strategy yields the same expected payoff	Unpredictability is the key

Mixed Strategy Equilibrium in Matching Pennies:
→ Both players choose Heads 50%, Tails 50%
→ Opponent cannot predict or exploit my strategy
→ Expected payoff = 0 (equilibrium)

Core Insight:
→ Making yourself unpredictable so your opponent
   cannot exploit your strategy is rational

Mixed Strategies in the Real World

Real-World Situations Involving Mixed Strategies
Situation	Role of Mixed Strategy	Problem with Pure Strategy
Soccer Penalty Kick	Kicker: randomize left/right	Always same direction → opponent predicts
Tax Audits	Tax authority: randomly audit some filers	Fixed criteria → tax evasion optimized
Military Patrols	Security forces: randomize patrol routes	Fixed routes → adversary learns pattern
Poker	Optimize bluffing frequency	Always/never bluffing → easily read

Multiple Equilibria and Coordination Games

In some games, multiple Nash equilibria exist simultaneously.

Traffic-Side Coordination Game — Two Equilibria
Me \ Opponent	Drive Right	Drive Left
Drive Right	Both +1 ✅ Equilibrium 1	Both -1 ❌
Drive Left	Both -1 ❌	Both +1 ✅ Equilibrium 2

When multiple equilibria exist, how do players select one? The answer lies in Focal Point (Schelling Point) theory. Culture, custom, law, and history coordinate people’s expectations, causing them to converge on a single equilibrium. For example, driving on the right in Korea is legally mandated, making it the dominant equilibrium.

Real-World Examples of Multiple Equilibria
Case	Equilibrium 1	Equilibrium 2	Coordination Mechanism
Currency Choice	Use the Dollar	Use the Euro	History and trade relationships
Software Standard	Windows ecosystem	Mac ecosystem	Network effects
Meeting Spot	Statue at Times Square	Grand Central Station	Cultural focal points
Social Norm	Handshake	Bow	Culture and region

Equilibrium Selection Theory

Pareto Dominance

If one equilibrium is better for all players than another, people tend to converge on the Pareto-dominant equilibrium.

Risk Dominance

Under high uncertainty, players prefer the 'safer' equilibrium — the one where the worst-case outcome is less severe.

Focal Point

Thomas Schelling's concept: even without communication, people converge on the same equilibrium through shared culture, common sense, and context.

Repetition and Learning

As the game is repeated, experience leads players to evolve toward a specific equilibrium. This is a central theme in evolutionary game theory.

Key Takeaways

Mixed Strategy: when no pure strategy equilibrium exists — use probabilistic choices to create unpredictability Equilibrium Condition: opponent is indifferent to your mixed strategy → equal expected payoffs Multiple Equilibria: Focal Points coordinate players toward one equilibrium Real Applications: penalty kicks, tax audits, and poker all rely on mixed strategies