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Nash Equilibrium — Pure Strategy

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Nash Equilibrium — Pure Strategy

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Nash Equilibrium — Pure Strategy

🎯 Learning Objectives

Define Nash equilibrium formally and intuitively
Find pure-strategy Nash equilibria using best response underlining
Identify multiple equilibria and explain equilibrium selection
Distinguish Nash equilibrium from dominant strategy equilibrium
Apply Nash equilibrium to analyze strategic situations

📋 Prerequisites

Normal-form game representation (Week 1)
Best response analysis
Dominance concepts

2.1 Intuition: What Is a Nash Equilibrium?

A Nash equilibrium is a stable state of strategic interaction: a set of strategies (one per player) where no player can gain by unilaterally changing their strategy.

🔑 Key Insight: At a Nash equilibrium, every player is playing a best response to everyone else's strategy. Nobody regrets their choice given what others did. Think of it as a social convention that is self-enforcing: if everyone expects it to be played, everyone's best response is to play it too.

Analogy

Driving on the right side of the road: Given that everyone else drives on the right, your best response is also to drive on the right. The convention is self-enforcing.

2.2 Formal Definition

A strategy profile

s^* = (s_1^*, s_2^*, \dots, s_n^*)

is a Nash equilibrium if for every player

i

u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \quad \forall s_i \in S_i

Where

s_{-i}^*

denotes the strategies of all players except

i

. Equivalently:

s_i^* \in BR_i(s_{-i}^*)

for all

i

Nash vs. Dominant Strategy

Concept	Requirement	Strength
Dominant strategy	$s_i$ is best no matter what others do	Very strong
Nash equilibrium	$s_i^$ is best given what others actually do*	Weaker, more widely applicable

Every dominant strategy equilibrium is a Nash equilibrium, but not vice versa.

2.3 Finding Nash Equilibria: Best Response Underlining

Method

For each column (Player 2's strategy), underline Player 1's best payoff(s)
For each row (Player 1's strategy), underline Player 2's best payoff(s)
Any cell where both payoffs are underlined is a Nash equilibrium

Example 1: Prisoner's Dilemma

\begin{array}{c|cc} & \text{Silent} & \text{Confess} \\ \hline \text{Silent} & (\underline{-1}, \underline{-1}) & (-10, \underline{0}) \\ \text{Confess} & (\underline{0}, -10) & (\underline{-5}, \underline{-5}) \end{array}

Column Silent: best for P1 is 0 (Confess) → underline 0
Column Confess: best for P1 is -5 (Confess) → underline -5
Row Silent: best for P2 is 0 (Confess) → underline 0
Row Confess: best for P2 is -5 (Confess) → underline -5
Nash equilibrium: (Confess, Confess) with payoffs (-5, -5)

Example 2: Battle of the Sexes

\begin{array}{c|cc} & \text{Ballet} & \text{Football} \\ \hline \text{Ballet} & (\underline{2}, \underline{1}) & (0, 0) \\ \text{Football} & (0, 0) & (\underline{1}, \underline{2}) \end{array}

Two Nash equilibria: (Ballet, Ballet) and (Football, Football)

Example 3: Matching Pennies

\begin{array}{c|cc} & \text{Heads} & \text{Tails} \\ \hline \text{Heads} & (1, -1) & (-1, 1) \\ \text{Tails} & (-1, 1) & (1, -1) \end{array}

Column Heads: best for P1 is Heads (1 > -1)
Column Tails: best for P1 is Tails (1 > -1)
Row Heads: best for P2 is Tails (1 > -1)
Row Tails: best for P2 is Heads (1 > -1)
No cell has both payoffs underlined → no pure-strategy Nash equilibrium

This game requires a mixed strategy equilibrium (Week 3).

2.4 Multiple Equilibria and Selection

Many games have multiple Nash equilibria. How do players coordinate?

Equilibrium Selection Criteria

Criterion	Description	Example
Payoff dominance	Choose the equilibrium that gives higher payoffs to all	(Ballet, Ballet) vs (Football, Football) — depends on preferences
Risk dominance	Choose the equilibrium with lower risk of loss	In Stag Hunt, Hare is risk-dominant
Focal point (Schelling)	Use salient features (culture, history, labels)	"Meet at the Eiffel Tower"
Evolutionary stability	Which equilibrium survives evolutionary pressure	Convention that emerges over time

Stag Hunt

\begin{array}{c|cc} & \text{Stag} & \text{Hare} \\ \hline \text{Stag} & (4, 4) & (0, 3) \\ \text{Hare} & (3, 0) & (3, 3) \end{array}

Two Nash equilibria: (Stag, Stag) and (Hare, Hare)
(Stag, Stag) is payoff-dominant; (Hare, Hare) is risk-dominant
Models the trade-off between cooperation and security

2.5 Existence of Nash Equilibrium

Nash's Theorem: Every finite game (finite number of players, finite strategy sets) has at least one Nash equilibrium (possibly in mixed strategies). (Diagram)

📊 Formula Summary

Concept	Definition
Pure-strategy NE	$u_i(s_i^, s_{-i}^) \geq u_i(s_i, s_{-i}^*) \ \forall i, \forall s_i$
Mutual best response	$s_i^* \in BR_i(s_{-i}^*) \ \forall i$
Existence	Every finite game has at least one NE (Nash, 1950)

✅ Practice Questions

Q1: Find all pure-strategy Nash equilibria:

\begin{array}{c|cc} & L & R \\ \hline U & (4, 3) & (2, 1) \\ D & (1, 2) & (3, 4) \end{array}

Solution

Underline:

Column L: P1 best = 4 (U) → underline 4

Column R: P1 best = 3 (D) → underline 3

Row U: P2 best = 3 (L) → underline 3

Row D: P2 best = 4 (R) → underline 4

Both payoffs underlined in (U, L) and (D, R). Two pure-strategy Nash equilibria. Q2: Find NE:

\begin{array}{c|ccc} & L & C & R \\ \hline U & (3, 1) & (0, 2) & (1, 3) \\ M & (2, 2) & (1, 1) & (0, 0) \\ D & (1, 0) & (2, 3) & (2, 1) \end{array}

Solution

Best responses underlined:

Column L: P1 best = 3 (U)

Column C: P1 best = 2 (D) — tie with M? No, 2 > 1 > 0

Column R: P1 best = 2 (D)

Row U: P2 best = 3 (R)

Row M: P2 best = 2 (L)

Row D: P2 best = 3 (C)

Both underlined only at (M, L)? Check: P1 gets best in L column = U (3), not M (2). So (U, L): P1=3 [ok], P2=1 [ok]. (M, L): P2=2 [ok] but P1=2 not best. (D, C): P2=3 [ok] but P1=2 not best.

Only (U, L) is a Nash equilibrium. Q3: In the Stag Hunt game above, why is (Hare, Hare) sometimes called "risk-dominant"? Solution

If a player is uncertain about the other's choice, playing Hare guarantees 3 regardless. Playing Stag earns 4 if the other plays Stag, but 0 if they play Hare. A risk-averse player who assigns probability > 1/4 to the other playing Hare prefers Hare. So Hare is the safer (risk-dominant) choice. Q4: True or False: Every game has exactly one Nash equilibrium. Solution

False. Games can have zero pure-strategy NE (but then have mixed NE), one NE, or multiple NE. Nash's theorem guarantees at least one NE (allowing mixed strategies) in every finite game. Q5: Find NE:

\begin{array}{c|cc} & A & B \\ \hline A & (2, 2) & (0, 0) \\ B & (0, 0) & (1, 1) \end{array}

Solution

This is a coordination game (pure, no conflict). (A, A) and (B, B) are both Nash equilibria. (A, A) is payoff-dominant (2 > 1). Both are self-enforcing. Join Discord PreviousStrategic ThinkingNextNash Existence