Game Theory · Nash Equilibrium
Nash Equilibrium Explained
The single most important concept in game theory — what happens when every player is doing the best they can given what everyone else is doing.
In 1950, a 22-year-old PhD student at Princeton submitted a 27-page dissertation that would eventually earn him the Nobel Prize in Economics. John Nash’s key idea was deceptively simple: in any strategic interaction, there exists a point where no single player has any incentive to change their strategy unilaterally. That point is the Nash Equilibrium — and it transformed how economists, biologists, and political scientists understand strategic behaviour.
📘 Key Term
A Nash Equilibrium is a set of strategies — one for each player — such that no player can improve their payoff by unilaterally changing their own strategy, given the strategies of all other players. Each player is choosing a best response to the strategies of others.
The Formal Condition
For a two-player game, a strategy pair (s₁*, s₂*) is a Nash Equilibrium if:
• Player 1: u₁(s₁*, s₂*) ≥ u₁(s₁, s₂*) for all strategies s₁
• Player 2: u₂(s₁*, s₂*) ≥ u₂(s₁*, s₂) for all strategies s₂
In plain English: Player 1 is doing as well as possible given Player 2’s choice, and Player 2 is doing as well as possible given Player 1’s choice. Neither player wants to deviate.
Finding Nash Equilibria: The Payoff Matrix Method
Consider two competing firms — Alpha and Beta — deciding whether to advertise. Their payoffs (profits in £m) are:
| Beta: Advertise | Beta: Don’t Advertise | |
|---|---|---|
| Alpha: Advertise | (3, 3) | (5, 1) |
| Alpha: Don’t Advertise | (1, 5) | (4, 4) |
To find the Nash Equilibrium, underline each player’s best response:
• If Beta advertises: Alpha prefers Advertise (3 > 1) ✓
• If Beta doesn’t advertise: Alpha prefers Advertise (5 > 4) ✓
• If Alpha advertises: Beta prefers Advertise (3 > 1) ✓
• If Alpha doesn’t advertise: Beta prefers Advertise (5 > 4) ✓
Nash Equilibrium: (Advertise, Advertise) with payoffs (3, 3). Note both firms could earn more (4, 4) if neither advertised — but neither can unilaterally switch without being exploited. This is exactly the Prisoner’s Dilemma structure.
Key Properties of Nash Equilibria
1. Multiple equilibria can exist. Some games have more than one Nash Equilibrium. When this happens, a coordination problem arises — players must agree on which equilibrium to play.
2. Nash Equilibria are not always efficient. As the advertising example shows, a Nash Equilibrium can leave both players worse off than an alternative outcome. It is stable, not optimal.
3. Mixed strategy equilibria always exist. Nash proved that every finite game has at least one equilibrium — in pure or mixed strategies (where players randomise).
⚠️ Common Error
A Nash Equilibrium is not the outcome where players earn the most. It is the outcome where players have no incentive to deviate. These are very different things. Students frequently confuse the socially optimal outcome with the Nash Equilibrium — do not make this mistake in exam answers.
Real-World Applications
🚦 Traffic Routing: Drivers choose routes independently. The equilibrium flow is a Nash Equilibrium — no driver can save time by switching routes given everyone else’s choices.
💰 Oligopoly Pricing: Cournot competition between firms yields a Nash Equilibrium in output quantities that lies between the competitive and monopoly levels.
🌐 International Climate Agreements: Countries choosing whether to reduce emissions face a Nash Equilibrium where each country free-rides — contributing to the tragedy of the commons.
Q1. Explain what is meant by a Nash Equilibrium and why it may not represent the best outcome for all players. [6 marks]
Answer: A Nash Equilibrium is a set of strategies where no player can improve their payoff by unilaterally changing their strategy, given the strategies of other players. It may not be best for all players because equilibria reflect individual rationality, not collective welfare. In the Prisoner’s Dilemma, both players defect at equilibrium (earning lower payoffs) even though mutual cooperation would make both better off — because neither can trust the other not to exploit a unilateral change.
References
1. Nash, J.F. (1950) ‘Equilibrium Points in N-Person Games’, Proceedings of the National Academy of Sciences, 36(1), pp. 48–49.
2. Nash, J.F. (1951) ‘Non-Cooperative Games’, Annals of Mathematics, 54(2), pp. 286–295.
3. Dixit, A. and Nalebuff, B. (1991) Thinking Strategically. W.W. Norton.
4. Osborne, M.J. (2004) An Introduction to Game Theory. Oxford University Press.
5. Tirole, J. (1988) The Theory of Industrial Organization. MIT Press.