M20.L04 — Mixed Strategies and Existence of Nash Equilibrium

Module: Module 20 — Monopoly and Game Theory (Advanced) Lesson: L04 of 06 Duration: ~30 minutes Level: Intermediate Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics

---

Learning Objective

Compute mixed strategy Nash equilibria using indifference conditions.

---

Mixed Strategy Equilibrium

Player 1 plays U with probability p, D with (1-p) Player 2 plays L with probability q, R with (1-q)

Equilibrium requires: 1. Player 2's mix makes Player 1 indifferent: q×u₁(U,L) + (1-q)×u₁(U,R) = q×u₁(D,L) + (1-q)×u₁(D,R) 2. Player 1's mix makes Player 2 indifferent: p×u₂(U,L) + (1-p)×u₂(D,L) = p×u₂(U,R) + (1-p)×u₂(D,R)

Australian example: Tennis serve (Federer vs Djokovic at Australian Open):

	Returner: L	Returner: R
Server: L	80, 20	30, 70
Server: R	20, 80	70, 30

---

Worked Example

[Australian context]

Find mixed NE in this AFL penalty kick game:

	Goalie: L	Goalie: R
Kicker: L	0.7, 0.3	0.9, 0.1
Kicker: R	0.8, 0.2	0.5, 0.5

Solution: 1. Goalie indifferent when: 0.7p + 0.8(1-p) = 0.9p + 0.5(1-p) → p = 0.6 2. Kicker indifferent when: 0.3q + 0.2(1-q) = 0.1q + 0.5(1-q) → q = 0.375 NE: Kicker aims L 60%, Goalie dives L 37.5%

---

Common Misconception

"Mixed strategies mean players randomize because they're uncertain"

Players randomize precisely to make opponents indifferent - a strategic requirement, not a reflection of uncertainty.

---

Key Takeaways

• Mixed NE requires making opponents indifferent
• Probability weights balance expected payoffs
• Nash proved all finite games have at least one NE (possibly mixed)
• Applications in sports, security, auctions

---

Practice

1. Find mixed NE in: | | L | R | |------|---|---| | U | 2,1 | 0,0 | | D | 0,0 | 1,2 |
2. Prove p=2/3, q=1/3 is NE in the above
3. Explain why matching pennies has no pure NE

---

Further Resources

• 📺 Mixed Strategy Nash Equilibrium
• 📚 Advanced Game Theory

---

Provenance: Curtis and Irvine | Utah Economics