Nash Equilibrium | Lecture 13

1. What is Nash Equilibrium?

Nash Equilibrium Definition

A strategy profile where no player can benefit by unilaterally changing their strategy.

Named after John Nash (Nobel Prize 1994)

Formal Definition

A strategy profile $\mathbf{s}^* = (s_1^*, s_2^*, \ldots, s_n^*)$ is a Nash Equilibrium if:

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

where: $u_i$ = player $i$'s payoff, $s_i$ = player $i$'s strategy, $s_{-i}$ = all other players' strategies

Translation: Given what everyone else is doing, each player is already doing their best.

Intuition

No player has a profitable deviation
Self-enforcing agreement
Stable state of the game

Important Notes

Not necessarily the best outcome for all
A game can have multiple NE
Some games have no pure strategy NE

2. Best Response

Best Response

Player $i$'s best response to $s_{-i}$ is the strategy that maximizes their payoff given what others play.

$$BR_i(s_{-i}) = \arg\max_{s_i} u_i(s_i, s_{-i})$$

Key Insight: Nash Equilibrium = Mutual Best Response

A strategy profile is a Nash Equilibrium if and only if every player is playing a best response to everyone else's strategies.

Finding NE via Best Response:

For each player, find their best response to each possible opponent strategy
Mark the best responses in the payoff matrix
A cell where ALL players are playing best responses = Nash Equilibrium

3. Interactive: Nash Equilibrium Finder

Find Nash Equilibria

How to Find Nash Equilibria

1

Find P1's Best Responses
For each P2 column, find the row where P1 gets highest payoff

2

Find P2's Best Responses
For each P1 row, find the column where P2 gets highest payoff

3

Find Nash Equilibria
Cells where BOTH players are best responding = NE!

Select a game:

Two prisoners decide whether to cooperate or defect.

P1's Payoff | P2's Payoff

		Player 2
		Cooperate	Defect
P1	Cooperate	3, 3	0, 5
P1	Defect	5, 0	1, 1

Understanding the Highlights

Purple left border = P1's best response in that column

Cyan bottom border = P2's best response in that row

Green highlight = Nash Equilibrium (mutual best response)

Analysis Result

Click "Show Best Responses" to start the analysis, then "Find NE" to identify Nash Equilibria.

4. Multiple Equilibria & Coordination

The Coordination Problem

When a game has multiple Nash equilibria, how do players coordinate on which one to play?

Example: Driving Side

		P2
		Left	Right
P1	Left	1, 1	-1, -1
P1	Right	-1, -1	1, 1

Two NE: (Left, Left) and (Right, Right)

Coordination Mechanisms:

Focal Points: Culturally salient solutions
Communication: Pre-play negotiation
Conventions: Social norms
Randomization: Mixed strategies

Nash's Theorem (1950)

Every finite game (finite players, finite strategies) has at least one Nash Equilibrium, possibly in mixed strategies.

5. Efficiency of Nash Equilibria

Pareto Efficiency

An outcome is Pareto efficient if no player can be made better off without making another worse off.

NE ≠ Pareto Efficient

In Prisoner's Dilemma, (Defect, Defect) is the NE but NOT Pareto efficient. (Cooperate, Cooperate) Pareto dominates it!

Price of Anarchy

Measures the "cost" of selfish behavior:
PoA = (Social optimal) / (Worst NE)