Nash Equilibrium: Existence and Computation

What is a “stable” outcome of a game?

Imagine that all players have already announced their strategies. When does no one have an incentive to unilaterally change their strategy? This state is called a Nash equilibrium. It is not necessarily the “best” outcome for everyone—it is stable, in the sense that each player has an incentive to stick with it, if others also stick with it.

Formally, a strategy profile s* = (s₁*, ..., sₙ*) is a Nash equilibrium if for each player i:

uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*) for all sᵢ ∈ Sᵢ

Here, s₋ᵢ* means the strategies of all other players except i. In other words: given fixed opponent strategies, sᵢ* is the best response for i.

Nash Existence Theorem (1950)

Theorem: Any finite strategic game (finite number of players and strategies) has an equilibrium in mixed strategies.

The proof uses Kakutani’s fixed-point theorem. Define the best response correspondence: BRᵢ(σ₋ᵢ) = {σᵢ : uᵢ(σᵢ, σ₋ᵢ) ≥ uᵢ(σᵢ', σ₋ᵢ) for all σᵢ'}. The overall correspondence BR(σ) = ∏ᵢ BRᵢ(σ₋ᵢ) maps the simplex of strategies into itself. Kakutani’s theorem guarantees a fixed point σ* = BR(σ*)—this is the equilibrium.

What a “finite” game cannot guarantee: uniqueness of equilibrium; Pareto-optimality; equilibrium in pure strategies. For infinite games, existence requires additional conditions (upper semicontinuity of payoff functions).

Method of finding equilibrium in mixed strategies: principle of indifference

In a Nash equilibrium in mixed strategies, a player is indifferent among all strategies included in the support of their mixed strategy. Otherwise, he would concentrate probability on the superior strategy.

Numerical Example: “Heads–Tails” (Matching Pennies)

There’s no equilibrium in pure strategies (any “corner” of the matrix is unstable). Let’s seek σ₁ = (p, 1−p).

Condition of indifference for Player 2: u₂(H, σ₁) = u₂(T, σ₁) → (−1)·p + (+1)·(1−p) = (+1)·p + (−1)·(1−p) → 1 − 2p = 2p − 1 → p = 1/2

By symmetry, q = 1/2 for σ₂. The equilibrium: σ* = (1/2, 1/2) for both players. Expected payoff = 0 for each.

Cournot Oligopoly Problem: Example of Computation

Two producers choose outputs q₁, q₂ ≥ 0. The inverse demand function: P(Q) = a − bQ, Q = q₁ + q₂. Constant marginal cost c. Firm i’s profit:

πᵢ(q₁, q₂) = (a − b(q₁+q₂) − c) · qᵢ

Finding Nash equilibrium: First-order condition for firm 1:

∂π₁/∂q₁ = a − 2bq₁ − bq₂ − c = 0 → q₁ = (a − c − bq₂)/(2b)

This is a reaction function (best response): optimal q₁ for any given q₂. Symmetrically: q₂ = (a − c − bq₁)/(2b).

Solve the system (by substitution): q₁* = q₂* = (a − c)/(3b).

Equilibrium price and profit: P* = (a + 2c)/3, π* = (a−c)²/(9b).

Comparison: Competition (P = c, π = 0) < Cournot (P* = (a+2c)/3) < Monopoly (P_M = (a+c)/2). Cournot is an intermediate outcome between competitive and monopoly markets.

Check of stability (is q = (a−c)/(3b) an equilibrium?):* When q₂ = (a−c)/(3b), optimal q₁ = (a−c−b·(a−c)/(3b))/(2b) = (a−c)·(2/3)/(2b) = (a−c)/(3b) = q₂ ✓. Neither firm wants to deviate.

Multiplicity of equilibria and refinements

Most games have multiple Nash equilibria—a problem of selection. Concepts of refinement:

Subgame perfect equilibrium (Selten, 1965): Eliminates incredible threats in dynamic games. Unique for finite games with complete information.

Perfect equilibrium trembling-hand (Selten, 1975): Robust to small errors by each player. More stringent requirement.

Correlated equilibrium (Aumann, 1974): Allows for a coordinating signal (e.g., a traffic light). Includes all Nash equilibria as special cases. Easier to compute (linear programming).

Nash equilibrium in real-world politics and business

The concept of Nash equilibrium permeates modern economics and strategy. In politics, the doctrine of nuclear deterrence—Mutually Assured Destruction (MAD)—represents a Nash equilibrium: neither side initiates a first strike, knowing that a retaliatory strike would annihilate itself. This equilibrium is stable, although both players would prefer total disarmament—a classic prisoner’s dilemma on a nuclear scale. In transportation networks, Nash equilibrium appears in the “Braess paradox”: adding a new road can increase average travel time, because each driver chooses the shortest route, but their collective decisions create new bottlenecks. This explains why removing roads can sometimes speed up traffic. In finance, traders in the market make decisions based on expected actions of other participants—Nash equilibrium determines market prices through the expectations mechanism. Behavioral economics, on the other hand, shows systematic deviations of real people from Nash equilibrium predictions: people value fairness, trust, and reputation above pure material interest.

Nash equilibrium in antitrust practice and corporate strategy

Nash equilibrium has become an essential tool of antitrust analysis and corporate strategy. Antitrust regulators assess mergers by how they shift market equilibrium: if the merger of two players moves the equilibrium from more competitive (Bertrand) to less competitive (Cournot), this is grounds for prohibition even without an explicit cartel agreement. Market concentration is measured by the Herfindahl–Hirschman Index (HHI), and threshold values (1800 and 2500 points) are based on equilibrium price changes in Cournot models. In corporate strategy, airlines, mobile operators, and retailers analyze their pricing decisions, investment in capacity, and store location precisely through the lens of equilibrium. Schelling’s “focal points” concept explains how competitors, without explicit communication, arrive at coordinated decisions—for example, pricing at round numbers or “price leadership”. It is the applicability of Nash equilibrium to real competitive policy that brought microeconomics out of academic auditoriums into boardrooms.

Assignment: Three producer firms (Cournot triopoly) with P = 100 − Q, Q = q₁+q₂+q₃, c = 10. Find Nash equilibrium. Show that as n→∞, price approaches competitive (P→c). Compute welfare loss for n=3 compared to the competitive market.