A Nash equilibrium is a profile of strategies that are best responses to one another. The key comparison holds everyone else’s strategy fixed while one player considers a change. It is a mathematical consistency condition within a specified game, not a promise of fairness or a general forecast of behaviour. An invented coordination table makes the test concrete.
Begin with the unilateral-deviation condition
In Nash’s original non-cooperative-games dissertation, an equilibrium point has the property that each player’s strategy maximises their payoff against the strategies of the others. For an elementary finite table, inspect one player at a time. Changing both players’ actions simultaneously answers a different question. The condition permits ties: a player need not strictly prefer their equilibrium strategy, provided no available unilateral change gives a higher expected payoff. State the available strategies before applying the test.
Read a small coordination model
Let Row choose A or B and Column choose L or R. Assign payoffs (3,3) to (A,L), (2,2) to (B,R), and (0,0) to the other combinations. These are abstract values stipulated for this tutorial. At (A,L), Row would obtain zero by switching alone to B, while Column would obtain zero by switching alone to R. Neither improves, so the profile is a Nash equilibrium. The reasoning uses each player’s own payoff, not the sum of the two displayed values.
Test every pure action profile
| Profile | Payoffs | Possible unilateral improvement | Pure Nash equilibrium? |
|---|---|---|---|
| A, L | (3, 3) | Neither player improves | Yes |
| A, R | (0, 0) | Row can obtain 2; Column can obtain 3 | No |
| B, L | (0, 0) | Row can obtain 3; Column can obtain 2 | No |
| B, R | (2, 2) | Neither player improves | Yes |
This exhaustive four-row check establishes two pure equilibria. It does not establish that they are the only equilibria when randomisation is allowed. A mixed-strategy analysis enlarges the strategy space and needs its own calculation. Clearly label whether a reported list concerns pure strategies only.
Separate equilibrium from collective improvement
Both players prefer (A,L) to (B,R), yet (B,R) remains a Nash equilibrium because either unilateral departure produces zero. The possibility of a coordinated improvement does not contradict the unilateral condition. This example shows why equilibrium need not select the outcome that all players prefer. It also shows why an equilibrium label is not an ethical endorsement. Questions about communication, agreements, institutions or fairness need additional analysis of what changes in the game and which normative standard is being applied.
Understand what the existence theorem says
Nash’s finite-game existence result allows mixed strategies, which are probability distributions over pure strategies. It does not assert that every finite game has a pure equilibrium, that the equilibrium is unique or that an easy computation always finds it. The Yale lecture on mixed strategies explains how the best-response condition extends to expected payoffs. Keep the finite-player, finite-action setting visible when stating this elementary theorem. More general games require their own assumptions; one cannot simply remove the word finite without replacing those conditions.
Treat equilibrium selection as another question
The Yale discussion of equilibrium and randomisation illustrates that models may have multiple equilibria and that modelling details can matter. In the coordination example, identifying two pure equilibria does not determine which one will occur. A selection account might require expectations, conventions, communication or a learning process, each explicitly modelled. Observing one outcome once does not establish that participants reasoned through an equilibrium calculation. The mathematical classification and the empirical explanation should be supported separately.
Check sensitivity to omitted actions and changed payoffs
Suppose an analyst forgets an available third action that gives a player a higher payoff against the proposed profile. A profile that looked stable in the restricted table may cease to be an equilibrium in the fuller game. Likewise, an altered cost can change a best response. State how action sets and payoff rankings were obtained when applying the idea outside a classroom. An equilibrium computed perfectly from an inaccurate model remains a result about that model, not a validated description of the underlying interaction.
Equilibrium-review checklist
- Specify players, strategy sets and payoffs.
- Hold other strategies fixed during each deviation test.
- Allow ties but rule out strictly improving deviations.
- Distinguish pure from mixed equilibria.
- Keep existence, uniqueness and computation separate.
- Avoid interpreting equilibrium as fairness or efficiency.
- Document modelling assumptions and selection claims.
For the example, a reader can verify the two pure equilibria using only four payoff pairs and the deviation rule. That transparency is the useful starting point. More elaborate terminology should not obscure the basic question: whose strategy changes, what remains fixed and whether that player’s expected payoff increases.
Frequently asked questions
Can a game have more than one Nash equilibrium?
Yes. The hypothetical coordination game has two pure equilibria, and mixed strategies require additional analysis.
Does Nash equilibrium mean the best outcome for everyone?
No. It rules out improving unilateral deviations, while a coordinated change can still benefit all players.
Does the finite-game theorem guarantee a pure equilibrium?
No. It guarantees an equilibrium when mixed strategies are allowed; some finite games have no pure equilibrium.
Sources and verification
- Non-Cooperative Games, doctoral dissertation, 1950 — John Nash; Princeton University Library archive — 2026-10-23
- Game Theory, Lecture 9: mixed strategies — Open Yale Courses — 2026-10-23
- Game Theory, Lecture 8: Nash equilibrium and randomisation — Open Yale Courses — 2026-10-23