Advancedmicroeconomics
Nash Equilibrium: When No One Wants to Deviate
Overview
A Nash equilibrium is a set of strategies — one for each player — from which no player can profitably deviate on their own. Proposed by John Nash in 1950, it is the central solution concept of non-cooperative game theory and underpins the analysis of oligopoly, auctions, and bargaining. This lesson defines Nash equilibrium precisely, shows how to find it by underlining best responses in a payoff matrix, introduces mixed strategies for games like matching pennies, and discusses multiple equilibria and equilibrium selection.
Learning objectives
- State the formal definition of a Nash equilibrium
- Find pure-strategy Nash equilibria by identifying mutual best responses in a matrix
- Explain why every dominant-strategy equilibrium is a Nash equilibrium but not vice versa
- Compute a mixed-strategy equilibrium in a 2x2 game with no pure-strategy equilibrium
- Discuss multiple equilibria, coordination games, and equilibrium selection
The definition: mutual best responses
A Nash equilibrium is a profile of strategies, one per player, such that each player's strategy is a best response to the others' strategies. Equivalently, no player can improve their own payoff by unilaterally changing strategy while everyone else's stays fixed. The 'unilateral' qualifier is essential: players might all gain by moving together (as in the prisoner's dilemma), but the equilibrium only rules out profitable solo deviations.
To check whether a cell in a payoff matrix is a Nash equilibrium, ask two questions: given the column player's strategy, is the row player's strategy their best response? And given the row player's strategy, is the column player's strategy their best response? If both answers are yes, the cell is a Nash equilibrium. A game can have zero, one, or many equilibria.
Finding equilibria: the underlining method
The practical way to find pure-strategy Nash equilibria in a 2x2 matrix is to underline best responses. For each column (each fixed strategy of the column player), underline the row player's highest payoff in that column — those are the row player's best responses. Then for each row, underline the column player's highest payoff in that row. Any cell where both payoffs are underlined is a Nash equilibrium: a mutual best response.
Apply this to the prisoner's dilemma and only the mutual-confession cell gets both underlines — the unique Nash equilibrium. In a coordination game like choosing which side of the road to drive on, two cells get double underlines (both drive left; both drive right), giving two equilibria. The method is mechanical, reliable, and works for any finite matrix game.
Dominant strategies and Nash: the relationship
If every player has a dominant strategy, playing them is always a Nash equilibrium: each player's strategy is trivially a best response to anything the others do. The prisoner's dilemma is the canonical example. But Nash equilibrium is a broader concept — most games have equilibria without anyone having a dominant strategy.
Consider two firms choosing between standard and premium product lines, where each does best by differentiating from the rival: one goes standard while the other goes premium. Neither has a dominant strategy (the best choice depends on the rival's), yet the two differentiation outcomes are both Nash equilibria. Nash's great contribution was proving that every finite game has at least one equilibrium, possibly in mixed strategies — so the concept never comes up empty.
Mixed strategies: randomizing to stay unpredictable
Some games have no pure-strategy equilibrium. In matching pennies, each player shows heads or tails; one player wins on a match, the other on a mismatch. Whatever pure strategy you pick, the opponent wants to do the opposite — there is no mutual best response. The resolution is a mixed strategy: randomizing between actions with chosen probabilities.
The equilibrium mix makes the opponent indifferent between their pure strategies. In matching pennies, each player randomizes 50-50; then neither can exploit the other, because any predictable pattern would be punished. Penalty kicks in soccer work the same way: the kicker and goalie both randomize, and the equilibrium probabilities depend on each side's scoring chances in each direction. Mixed strategies model bluffing in poker, randomized auditing by tax authorities, and unpredictable patrol routes.
Multiple equilibria and coordination
Many important games have several equilibria, raising the selection problem: which one will players actually reach? In pure coordination games (both drive left or both drive right), any equilibrium is fine once established — the problem is coordinating on the same one, solved by conventions, communication, or focal points. In the stag hunt, mutual cooperation (hunting the stag) and mutual defection (each catching a hare alone) are both equilibria, but the cooperative one is risky: it pays best only if the other cooperates.
Economists study equilibrium selection through refinements (like subgame perfection for sequential games), evolutionary dynamics, and experiments. Real institutions — contracts, norms, legal defaults — often function as equilibrium-selection devices, steering societies toward the better of several self-enforcing outcomes.
Criticisms and limits of Nash equilibrium
Nash equilibrium is a static concept: it describes a resting point, not how players get there. It assumes common knowledge of rationality and of the game itself — strong assumptions that experiments sometimes violate, as in the ultimatum game where responders reject unfair offers. Behavioral game theory documents systematic deviations and builds richer models.
In sequential games, some Nash equilibria rely on non-credible threats (threatening an action you would not actually take if called upon). The refinement of subgame perfect equilibrium rules these out by requiring best responses at every point in the game tree, found via backward induction. Despite its limits, Nash equilibrium remains the indispensable starting point: it disciplines strategic reasoning by insisting that predictions be consistent with each player's incentives.
A Nash equilibrium need not be good, fair, or efficient — the prisoner's dilemma proves it can be Pareto inferior. Nash equilibrium predicts stability (no one wants to move), not desirability.
Key points
- A Nash equilibrium is a profile of mutual best responses: no unilateral deviation pays
- Find pure-strategy equilibria by underlining each player's best responses in the matrix
- Every dominant-strategy profile is a Nash equilibrium, but most equilibria need no dominant strategies
- Games without pure-strategy equilibria (matching pennies) have mixed-strategy equilibria
- Nash proved every finite game has at least one equilibrium, possibly mixed
- Multiple equilibria raise selection problems solved by conventions, communication, or refinements like subgame perfection
Common misconceptions
✓ Truth: It is only a stable outcome — no one gains by deviating alone. The prisoner's dilemma equilibrium is Pareto inferior, and many equilibria are unfair or inefficient.
✓ Truth: The randomization probabilities are precisely chosen to make the opponent indifferent, removing any incentive to exploit a predictable pattern. The mix is a calculated best response.
✓ Truth: Nash equilibrium applies far beyond dominant strategies. Most interesting games — coordination, differentiation, bargaining — have equilibria with no dominant strategies at all.
Practice questions
Q1. A Nash equilibrium is best described as a strategy profile in which:
- all players earn equal payoffs
- no player can improve their payoff by unilaterally changing strategy
- the total payoff is maximized
- all players cooperate
The defining property is stability against unilateral deviation: each strategy is a best response to the others. It says nothing about equality, efficiency, or cooperation — the prisoner's dilemma equilibrium is inefficient.
Q2. Using the underlining method, a cell in a payoff matrix is a Nash equilibrium when:
- it has the highest total payoff
- both players' payoffs in that cell are underlined as best responses
- one player's payoff is underlined
- the payoffs are equal
Underlining marks each player's best response to each opponent strategy. A cell with both payoffs underlined means each strategy is a best response to the other — a mutual best response, i.e., a Nash equilibrium.
Q3. In matching pennies, the Nash equilibrium requires mixed strategies because:
- players cannot communicate
- there is no cell that is a mutual best response in pure strategies
- the payoffs are zero-sum
- both players have dominant strategies
Whatever pure strategy one player chooses, the other wants to do the opposite (match vs. mismatch), so no pure profile is mutually optimal. Randomizing 50-50 makes each player indifferent and unexploitable — the mixed-strategy equilibrium.
Q4. Which statement about dominant strategies and Nash equilibrium is correct?
- They are the same concept
- A dominant-strategy profile is always a Nash equilibrium, but Nash equilibria often exist without dominant strategies
- Nash equilibria never involve dominant strategies
- Dominant strategies only exist in zero-sum games
A dominant strategy is a best response to everything, so a profile of dominant strategies is automatically mutual best responses. But most games (coordination, differentiation) have Nash equilibria where best responses depend on the opponent's choice.
Q5. A Nash equilibrium based on a threat the player would not carry out if tested is eliminated by requiring:
- mixed strategies
- subgame perfection
- dominant strategies
- zero-sum payoffs
Subgame perfect equilibrium requires optimal play in every subgame, ruling out non-credible threats. It is the standard refinement for sequential games and is found by backward induction.
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Frequently asked questions
What is the exact formal definition of Nash equilibrium?
A strategy profile is a Nash equilibrium if each player's strategy maximizes their payoff given the other players' strategies. Formally, for every player i, no alternative strategy s_i' yields a higher payoff than the equilibrium strategy s_i* when others play s_{-i}*.
Can a game have no Nash equilibrium at all?
Not if mixed strategies are allowed: Nash's theorem guarantees at least one equilibrium (possibly mixed) in every finite game. Restricting to pure strategies only, games like matching pennies have none.
What is a subgame perfect equilibrium?
It is a Nash equilibrium of a sequential game that also induces Nash equilibria in every subgame — ruling out equilibria based on non-credible threats. It is found by backward induction from the end of the game tree.
How is Nash equilibrium used in oligopoly models?
Cournot quantities, Bertrand prices, and Stackelberg outputs are all Nash equilibria of their respective games: each firm's choice is a best response to rivals' choices. Different game rules produce different equilibria, which is why the models disagree.