Nash Equilibrium

Nash Equilibrium, named after John Forbes Nash Jr., is a cornerstone of Game Theory, defining a state where no player can improve their payoff by unilaterally changing their strategy, assuming others' strategies remain fixed. Introduced in Nash’s 1950 dissertation, this concept revolutionized the analysis of strategic interactions, providing a framework for predicting stable outcomes.

This guide explores Nash Equilibrium in depth, covering its definitions, pure and mixed strategy forms, detailed examples with mathematical formulations, and applications across economics, biology, and beyond, enhanced with interactive visualizations.

Definition and Types

Nash Equilibrium captures strategic stability in games with multiple players.

Basic Definition

For \( n \) players with strategy sets \( S_i \), a strategy profile \( s^* = (s_1^*, s_2^*, \ldots, s_n^*) \) is a Nash Equilibrium if:

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

Where \( u_i \) is player \( i \)’s payoff, and \( s_{-i}^* \) denotes others’ strategies.

Pure Strategy Nash Equilibrium

Each player selects a single strategy:

\[ u_i(s_i^*, s_{-i}^*) - u_i(s_i, s_{-i}^*) \geq 0 \]

Mixed Strategy Nash Equilibrium

Players randomize over strategies with probabilities \( p_i \). Expected payoff:

\[ E[u_i] = \sum_{s \in S} p_i(s_i) p_{-i}(s_{-i}) u_i(s_i, s_{-i}) \]

Equilibrium condition:

\[ E[u_i(p_i^*, p_{-i}^*)] \geq E[u_i(p_i, p_{-i}^*)] \quad \forall p_i \]

Best Response Function

Player \( i \)’s best response:

\[ BR_i(s_{-i}) = \arg\max_{s_i \in S_i} u_i(s_i, s_{-i}) \]

Equilibrium at \( s_i^* = BR_i(s_{-i}^*) \).

Existence

Nash’s theorem guarantees at least one equilibrium in finite games.

Examples

Let’s explore Nash Equilibria in classic games with calculations.

Prisoner’s Dilemma

Payoff matrix (Confess, Silent):

\[ \begin{array}{c|cc} & C & S \\ \hline C & (-5, -5) & (0, -10) \\ S & (-10, 0) & (-1, -1) \end{array} \]

Nash Equilibrium: (C, C). If Player 1 switches to S, payoff drops from -5 to -10.

Coordination Game

Payoff:

\[ \begin{array}{c|cc} & A & B \\ \hline A & (3, 3) & (0, 0) \\ B & (0, 0) & (2, 2) \end{array} \]

Equilibria: (A, A) and (B, B).

Matching Pennies (Mixed)

Payoff:

\[ \begin{array}{c|cc} & H & T \\ \hline H & (1, -1) & (-1, 1) \\ T & (-1, 1) & (1, -1) \end{array} \]

Player 1’s expected payoff for H: \( q \cdot 1 + (1-q) \cdot (-1) = 2q - 1 \). For T: \( q \cdot (-1) + (1-q) \cdot 1 = 1 - 2q \). Set equal: \( 2q - 1 = 1 - 2q \implies q = 0.5 \). Equilibrium: \( p_H = p_T = 0.5 \).

Best Response (Matching Pennies)

Intersection at \( p = q = 0.5 \) shows the mixed equilibrium.

Battle of the Sexes

Payoff:

\[ \begin{array}{c|cc} & F & O \\ \hline F & (2, 1) & (0, 0) \\ O & (0, 0) & (1, 2) \end{array} \]

Pure equilibria: (F, F), (O, O). Mixed: Player 1 plays F with \( p = 2/3 \), Player 2 plays F with \( q = 1/3 \).

Chicken Game

Payoff:

\[ \begin{array}{c|cc} & S & T \\ \hline S & (0, 0) & (1, -1) \\ T & (-1, 1) & (-10, -10) \end{array} \]

Equilibria: (S, T), (T, S).

Applications

Nash Equilibrium’s predictive power applies across diverse fields.

Economics

In the Cournot model, firms choose quantities \( q_i \):

\[ q_i^* = \frac{a - c - q_{-i}}{2} \]

Equilibrium balances profit maximization.

Evolutionary Biology

Evolutionarily Stable Strategy (ESS):

\[ u(s^*, s^*) \geq u(s, s^*) \quad \text{or} \quad u(s^*, s) > u(s, s) \]

Negotiations

Nash bargaining solution maximizes:

\[ \max_{x_1, x_2} (u_1(x_1) - d_1)(u_2(x_2) - d_2) \]

Computer Science

Congestion games model network resource allocation.

Politics

Voting equilibria predict stable voting strategies.

Cournot Equilibrium

Intersection of reaction functions shows equilibrium quantities.