6.1 Nash equilibrium

Definition of Nash Equilibrium

A Nash equilibrium is a situation in a game where every player is making the best possible choice given what everyone else is doing. No single player can switch strategies and get a better payoff on their own. This makes it a "stable" outcome: once players land there, nobody wants to move.

This concept matters because most economic decisions don't happen in isolation. Firms set prices knowing competitors will respond. Workers negotiate wages knowing firms have alternatives. Nash equilibrium gives you a framework for predicting what happens when rational decision-makers interact strategically.

Key Components

• Strategic interdependence: each player's best move depends on what others choose
• Rational decision-making: players pick strategies that maximize their own payoff based on available information
• Mutual best responses: at equilibrium, every player's strategy is a best response to the strategies of all other players
• No incentive to deviate: no player can do better by unilaterally switching to a different strategy

Conditions for Existence

Nash's original theorem guarantees that an equilibrium exists under certain conditions:

• A finite number of players
• A finite set of strategies for each player
• Well-defined payoff functions for every possible combination of strategies
• For continuous games, the strategy spaces need to be compact and convex

If no pure strategy equilibrium exists, allowing mixed strategies (where players randomize over their options) ensures at least one Nash equilibrium in any finite game. This is a powerful result because it means you can always find an equilibrium if you're willing to consider randomization.

Nash vs. Pareto Optimality

A Nash equilibrium is strategically stable, but that doesn't mean it's socially efficient. Pareto optimality means no one can be made better off without making someone else worse off. These two concepts often diverge.

The classic example is the Prisoner's Dilemma: both players defecting is the Nash equilibrium, but both cooperating would make everyone better off (the Pareto optimal outcome). Coordination games can have multiple Nash equilibria, some Pareto superior to others. This gap between individual rationality and collective welfare is why economists often look to mechanisms, institutions, or regulations to push outcomes toward efficiency.

Mathematical Representation

Game theory uses precise notation to describe strategic interactions. This formalism lets you apply tools from optimization and fixed-point theory to prove equilibria exist and characterize their properties.

Pure Strategy Equilibria

A pure strategy Nash equilibrium is a specific strategy profile where each player picks one definite action. It's written as:

$s^* = (s_1^*, s_2^*, \ldots, s_n^*)$

The defining condition is that for every player $i$ and every alternative strategy $s_i$ available to them:

$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)$

Here, $u_i$ is player $i$'s payoff function, $s_i^*$ is their equilibrium strategy, and $s_{-i}^*$ represents the equilibrium strategies of all other players. The inequality says: player $i$ can't do better by switching away from $s_i^*$ while everyone else stays put.

Mixed Strategy Equilibria

When no pure strategy equilibrium exists, players can randomize. A mixed strategy assigns probabilities to each pure strategy:

$\sigma_i^* = (p_1, p_2, \ldots, p_k)$

where $p_1 + p_2 + \cdots + p_k = 1$. The equilibrium condition becomes:

$E[u_i(\sigma_i^*, \sigma_{-i}^*)] \geq E[u_i(\sigma_i, \sigma_{-i}^*)]$

A useful property here is the indifference principle: in a mixed strategy equilibrium, every pure strategy that a player assigns positive probability to must yield the same expected payoff. If one option gave a higher expected payoff, the player would shift all their probability to it, contradicting the idea that they're mixing.

Best Response Functions

A best response function tells you a player's optimal strategy for each possible combination of opponents' strategies:

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

A Nash equilibrium is a point where every player's strategy is a best response to the others simultaneously. Graphically, for two-player games, you can plot both players' best response curves and look for where they intersect. Each intersection is a Nash equilibrium.

Applications in Economics

Oligopoly Models

Nash equilibrium is the standard tool for analyzing markets with a small number of firms:

• Cournot competition: firms simultaneously choose quantities. Each firm's equilibrium output maximizes its profit given the other firms' output levels. For a duopoly with identical firms facing linear demand, the Nash equilibrium output per firm is lower than the competitive quantity but higher than the monopoly share.
• Bertrand competition: firms simultaneously choose prices. With identical products and constant marginal costs, the Nash equilibrium drives price down to marginal cost, even with just two firms.
• Stackelberg model: one firm moves first (the leader), and the other responds (the follower). This is solved using backward induction and yields a subgame perfect Nash equilibrium where the leader produces more than in Cournot.

Public Goods Provision

Public goods are non-excludable, meaning people can benefit without paying. This creates the free-rider problem: each person's Nash equilibrium strategy is to contribute less than the socially optimal amount, hoping others will cover the cost. Voluntary contribution games typically result in underprovision. Economists study mechanisms like Lindahl pricing and mechanism design approaches to align individual incentives with efficient provision.

Labor Market Negotiations

Bargaining between unions and firms fits naturally into a game-theoretic framework. The right-to-manage model has the union and firm negotiate wages, with the firm then choosing employment unilaterally. The efficient bargaining model has both wages and employment negotiated simultaneously. Strike threats and holdout strategies emerge as equilibrium phenomena in these models.

Finding Nash Equilibria

Several methods exist for finding Nash equilibria, and which one you use depends on the game's structure.

Dominant Strategy Method

1. For each player, check whether any strategy gives a higher payoff regardless of what opponents do
2. If a dominant strategy exists for every player, the combination of those strategies is the Nash equilibrium
3. Even weakly dominant strategies (at least as good as alternatives, strictly better against some opponent choices) can point to a unique equilibrium

This is the simplest approach and works cleanly for games like the Prisoner's Dilemma, where both players have a dominant strategy to defect.

Best Response Method

1. For each possible strategy of the opponent(s), determine each player's best response
2. In a payoff matrix, underline or circle the best response payoffs for each player
3. Find cells where both players' payoffs are marked as best responses. Those are the Nash equilibria.

For continuous strategy spaces (like Cournot duopoly), you derive each player's best response function, then solve the system of equations simultaneously.

Elimination of Dominated Strategies

1. Identify any strategy that is strictly dominated (always worse than some other strategy, no matter what opponents do)
2. Remove that strategy from the game
3. Repeat the process on the reduced game, since removing one player's strategy may cause another player's strategy to become dominated
4. If the process leaves a single strategy profile, that's the Nash equilibrium

This is called iterated elimination of strictly dominated strategies (IESDS). Be careful: elimination of weakly dominated strategies can sometimes remove Nash equilibria, so the order of elimination matters in that case.

Properties of Nash Equilibrium

Stability and Uniqueness

Not all games have a unique Nash equilibrium. Coordination games, for instance, often have multiple equilibria. When that happens, refinement concepts help narrow down which equilibrium is most plausible:

• Risk dominance selects the equilibrium that's less risky (better worst-case payoff)
• Trembling hand perfection requires that the equilibrium remain stable even if players make small random mistakes in executing their strategies

A stable equilibrium is one where small perturbations in strategies push players back toward equilibrium rather than away from it.

Efficiency Considerations

As noted with the Prisoner's Dilemma, Nash equilibria can be inefficient. The equilibrium outcome may leave "money on the table" from a social welfare perspective. This tension between individual rationality and collective welfare is central to many policy questions. Institutions, contracts, and regulations often exist precisely to shift outcomes from inefficient Nash equilibria toward Pareto improvements.

Subgame Perfection

In games where players move sequentially (extensive form games), some Nash equilibria rely on threats that a player would never actually carry out. Subgame perfect equilibrium eliminates these by requiring that strategies form a Nash equilibrium in every subgame, not just the game as a whole.

You find subgame perfect equilibria using backward induction:

1. Start at the final decision nodes of the game tree
2. Determine the optimal action at each of those nodes
3. Work backward, replacing each subgame with its equilibrium outcome
4. Continue until you reach the beginning of the game

This refinement is especially important in sequential bargaining, entry deterrence, and dynamic oligopoly models.

Extensions and Variations

Bayesian Nash Equilibrium

When players have incomplete information about each other (e.g., a firm doesn't know a rival's costs), the game becomes a Bayesian game. Each player has a type drawn from a probability distribution, and strategies map types to actions. A Bayesian Nash equilibrium requires that each type of each player maximizes expected payoff given their beliefs about others' types. Applications include auctions, signaling games, and mechanism design.

Correlated Equilibrium

A correlated equilibrium allows players to coordinate their strategies through a shared signal or mediator, without explicit cooperation. Players receive private recommendations and have no incentive to deviate from them. This generalizes Nash equilibrium and can sometimes achieve higher payoffs than any Nash equilibrium. It's relevant for analyzing tacit collusion and focal points in markets.

Evolutionarily Stable Strategies

An evolutionarily stable strategy (ESS) adapts the Nash concept to population dynamics. A strategy is an ESS if, once adopted by a population, it can't be "invaded" by a small group of mutants playing a different strategy. This provides insight into long-run behavioral stability without requiring players to be perfectly rational. Applications appear in evolutionary economics and behavioral game theory.

Limitations and Criticisms

Rationality Assumptions

Nash equilibrium assumes players are perfectly rational and that this rationality is common knowledge (everyone knows everyone is rational, everyone knows everyone knows, and so on). Experimental evidence consistently shows people deviate from Nash predictions in systematic ways. Bounded rationality models and behavioral game theory incorporate cognitive limitations and psychological biases to better match observed behavior.

Multiple Equilibria Problem

Many economically interesting games have multiple Nash equilibria, and the theory alone doesn't tell you which one will occur. This equilibrium selection problem is a genuine limitation. Refinements like trembling hand perfection and risk dominance help, but no single refinement works universally. In practice, history, culture, and focal points often determine which equilibrium players coordinate on.

Real-World Applicability

Game-theoretic models simplify reality. Specifying accurate payoff functions and information structures for real economic situations is difficult. Static Nash equilibrium also doesn't capture how players learn and adapt over time. These models work best as tools for generating qualitative insights about strategic incentives rather than precise quantitative predictions.

Nash Equilibrium in Different Game Forms

Normal Form Games

The normal form (or strategic form) represents simultaneous-move games as a payoff matrix. Each row corresponds to one player's strategies, each column to the other's, and each cell contains both players' payoffs. For 2×2 games, you can find Nash equilibria by inspection using the best response method. This representation is standard for static models like Cournot and Bertrand competition.

Extensive Form Games

The extensive form uses a game tree to represent sequential decisions. It captures the timing of moves, what each player knows when they act (information sets), and the branching structure of choices. Subgame perfect Nash equilibrium is the appropriate solution concept here. This form is natural for modeling bargaining, entry deterrence, and any situation where order of play matters.

Repeated Games

When a stage game is played multiple times, new equilibria can emerge that aren't possible in the one-shot version. Repetition allows for reputation building and punishment strategies (like tit-for-tat). The Folk Theorem states that in infinitely repeated games with sufficiently patient players, any feasible payoff that gives each player at least their one-shot Nash equilibrium payoff can be sustained as a Nash equilibrium. This explains how cooperation and tacit collusion can persist in long-term economic relationships.

Computational Approaches

Algorithms for Finding Equilibria

• Lemke-Howson algorithm: finds one Nash equilibrium in two-player games by tracing paths through the best response polytope
• Simplicial subdivision methods: extend to $n$-player games by subdividing the strategy space
• Homotopy methods: can compute all equilibria by continuously deforming a simpler game into the target game
• Approximation algorithms: used for large-scale games where exact solutions are computationally infeasible

Software Tools and Simulations

• Gambit: an open-source library for building and solving game-theoretic models
• GAMUT: a suite for generating game instances to test algorithms
• Agent-based modeling platforms (NetLogo, MASON): useful for simulating evolutionary games and learning dynamics
• Scientific computing environments like Python and MATLAB support custom implementations

Complexity Considerations

Computing a Nash equilibrium is PPAD-complete, which means it's believed to be computationally hard in the worst case (though not as hard as NP-complete problems). Finding an $\epsilon$-Nash equilibrium (where players are within $\epsilon$ of their best response) is sometimes more tractable. For specific game classes, parameterized complexity analysis can identify when efficient algorithms exist. These complexity results have practical implications for how large and detailed your economic game models can realistically be.