Dominance predicts something only in the rare game where one option beats another whatever the opponent does, so the useful question is the weaker one: what is best given what the opponent is actually doing.
Best responses
Fix what everybody else is doing and the strategic problem collapses into an ordinary decision problem. If player knows the others are playing , they simply pick whichever of their own strategies pays most. That choice is the best response:
Note the plural is possible: two strategies can tie, so a best response is really a set. Note also that this is a conditional statement and predicts nothing on its own, because is not known.
On a grid, best responses are found by scanning. Go down each column and mark the row player's largest payoff in it; that row is the row player's best response to that column. Then go along each row and mark the column player's largest payoff in it. Any cell carrying both marks is a cell in which each player is doing the best they can given the other, and marking a grid this way takes a few seconds once the habit is formed.
Here is a game to mark. Player 1 chooses the row from T, M and B, player 2 the column from L, C and R:
| Player 1 | L | C | R |
|---|---|---|---|
| T | 3, 1 | 0, 2 | 5, 0 |
| M | 2, 3 | 4, 4 | 1, 2 |
| B | 1, 0 | 3, 1 | 6, 5 |
Reading down player 1's payoffs, the best reply to L is T with 3, to C is M with 4, and to R is B with 6. Reading across player 2's payoffs, the best reply to T is C with 2, to M is C with 4, and to B is R with 5. Two cells carry both marks: (M, C) and (B, R).
The definition
A Nash equilibrium is a strategy profile in which every player's strategy is a best response to the others' strategies. Written out, is an equilibrium when for every player and every alternative ,
The condition is entirely about unilateral deviations. It asks each player, one at a time, whether they could do better by changing their own strategy while everyone else holds theirs fixed. If nobody can, the profile is an equilibrium. Two players changing together is not a deviation the definition considers, which is precisely why the prisoner's dilemma has a bad equilibrium: both switching to cooperation is an improvement for both, and neither switching alone is.
That gives three ways of saying the same thing, and each is worth having. An equilibrium is a profile of mutual best responses. It is a profile at which no player regrets their own choice, given what the others turned out to do. And it is a self-enforcing agreement: if the players could talk beforehand but sign nothing, an equilibrium is an agreement each would keep out of self-interest once the talking stopped. A non-equilibrium agreement is one somebody breaks the moment it becomes binding on the others alone.
The concept is John Nash's, published as a two-page note in the Proceedings of the National Academy of Sciences in 1950 while he was a graduate student, and in full in the Annals of Mathematics in 1951. It generalised von Neumann's earlier solution of zero-sum games, which is the subject of a later lesson, to games where interests are not perfectly opposed, and it is the reason a subject that had a beautiful theory of poker acquired one of oligopoly, arms control and auctions. Nash shared the 1994 Nobel Memorial Prize in Economics with John Harsanyi and Reinhard Selten, both of whom appear later in this course.
Example. In the grid above, verify that (M, C) is an equilibrium and that (T, C) is not.
At (M, C), player 1 has 4 and could switch to T for 0 or B for 3, so staying is best. Player 2 has 4 and could switch to L for 3 or R for 2, so staying is best. Neither can gain alone, so it is an equilibrium. At (T, C), player 1 has 0 and could switch to M for 4. One profitable deviation is enough to disqualify a profile, so (T, C) is not an equilibrium, even though player 2 is perfectly happy there.
Now you. Is (B, R), paying 6 to player 1 and 5 to player 2, an equilibrium of the same game?
Answer
Player 1 at (B, R) has 6, against 5 from T and 1 from M, so no gain. Player 2 has 5, against 0 from L and 1 from C, so no gain. It is an equilibrium, and it happens to pay both players more than the other equilibrium (M, C) does. Two equilibria, one better for everyone, and nothing in the definition says which one occurs. That problem gets a lesson of its own shortly.
How it sits with dominance
The two ideas do not conflict, and the relations between them are worth stating exactly.
If every player has a strictly dominant strategy, that profile is a Nash equilibrium, and it is the only one. The prisoner's dilemma is the case in point: mutual defection is the unique equilibrium, and it is unique because a dominated strategy is never a best response to anything, so no profile containing one can qualify.
More usefully, iterated deletion of strictly dominated strategies never deletes an equilibrium strategy. Suppose it did, and consider the first equilibrium strategy deleted: it is strictly dominated in the reduced game, so it is beaten against every surviving profile of the others, including the equilibrium profile of the others, which is still present because this deletion was the first. That contradicts its being a best response. So every Nash equilibrium survives deletion, and if deletion leaves exactly one cell, that cell is the unique equilibrium of the game.
The converse fails, and it fails often: survival is much weaker than equilibrium. In the grid above nothing at all is strictly dominated, so deletion leaves all nine cells, while only two of them are equilibria. Dominance is a coarse filter; equilibrium is a fine one.
What equilibrium is not
More confusion attaches to this concept than to any other in the subject, and most of it comes from four claims that sound true and are not.
It is not the best outcome. The prisoner's dilemma equilibrium is worse for both players than the cell they both abandoned. Equilibrium is a stability condition, not a welfare condition, and the entire field of mechanism design exists because the two come apart.
It is not unique. The grid above has two, coordination games have three, and repeated games, as a later lesson shows, have infinitely many. A prediction that offers several answers is a weaker prediction, and pretending otherwise by quietly picking the nicest equilibrium is one of the standard ways an applied game-theoretic argument goes wrong.
It is not a claim about what people do on first meeting a game. Equilibrium requires each player's beliefs about others to be correct, and there is no reason a stranger's first play should satisfy that. What the concept describes is a rest point: a situation nothing in the players' own interests pushes them away from. Nash himself supplied a second reading in his 1950 thesis, the "mass action" interpretation, in which players are drawn from large populations, learn the average behaviour of the other side over time, and equilibrium is where that adjustment stops. That reading demands no cleverness of the players at all, and it is the honest one to have in mind when the theory is applied to markets, animals or traffic.
It is not a recommendation to any individual. "Play your equilibrium strategy" is good advice only if the others are playing theirs. Against a weak or predictable opponent, the equilibrium strategy is usually not the best reply, and a poker player who mixes to equilibrium against a novice is leaving money on the table in exchange for being unexploitable.
Example. Find all pure-strategy equilibria of this game by marking best responses.
| Player 1 | L | C | R |
|---|---|---|---|
| T | 5, 2 | 2, 1 | 1, 3 |
| M | 3, 0 | 4, 5 | 0, 1 |
| B | 2, 4 | 1, 2 | 6, 2 |
Player 1's best replies, column by column: against L it is T with 5, against C it is M with 4, against R it is B with 6. Player 2's best replies, row by row: against T it is R with 3, against M it is C with 5, against B it is L with 4. The only cell where the two agree is (M, C), paying 4 to player 1 and 5 to player 2. One equilibrium, and note that the largest single payoff in the table, player 1's 6 at (B, R), sits in a cell that is not one.
Now you. Find all pure-strategy equilibria of this game.
| Player 1 | L | C | R |
|---|---|---|---|
| T | 4, 5 | 1, 2 | 2, 0 |
| M | 0, 1 | 3, 4 | 5, 2 |
| B | 2, 3 | 0, 1 | 6, 2 |
Answer
Player 1's best replies are T against L (4), M against C (3), and B against R (6). Player 2's best replies are L against T (5), C against M (4), and L against B (3). Two cells agree: (T, L) paying 4 and 5, and (M, C) paying 3 and 4. Note that (B, R) is not an equilibrium despite holding player 1's best payoff in the table, because player 2 would switch to L.
When there is no equilibrium at all
Mark the best responses of this one, a tax authority choosing whether to audit and a taxpayer choosing whether to evade. Payoffs are in thousands, the taxpayer's first:
| Taxpayer | Audit | Do not audit |
|---|---|---|
| Evade | -100, 80 | 50, -50 |
| Declare | 0, -20 | 0, 0 |
Follow the best responses round. If the authority audits, the taxpayer prefers to declare, since 0 beats -100. If the taxpayer declares, the authority prefers not to audit, since 0 beats the -20 cost of a wasted audit. If the authority does not audit, the taxpayer prefers to evade, since 50 beats 0. If the taxpayer evades, the authority prefers to audit, since 80 beats -50. Four arrows, and they form a closed cycle: every cell has somebody who wants out. No cell carries both marks, so the game has no equilibrium in the strategies as defined.
This is not an artefact of the numbers. Matching pennies, in which one player wins if two coins match and the other wins if they differ, has the same cyclical structure with the smallest possible payoffs, and so does every game whose essence is that one side wants to be predicted wrongly. Penalty kicks, serve direction in tennis, bluffing, inspection, camouflage and pursuit are all of this shape, which makes the gap in the theory a serious one rather than a curiosity.
Example. Does this game have a pure equilibrium?
| Player 1 | L | R |
|---|---|---|
| T | 3, 2 | 1, 1 |
| B | 2, 0 | 0, 3 |
Player 1's best reply to L is T (3 against 2) and to R is also T (1 against 0), so T is dominant. Player 2's best reply to T is L (2 against 1). The cell (T, L) carries both marks and is the unique pure equilibrium. The cycle of the audit game does not appear here because one player has a strategy that is good regardless.
Now you. In the audit game, suppose the fine is raised so that evading under audit costs the taxpayer 300 rather than 100. Does a pure equilibrium appear?
Answer
No. The taxpayer still prefers to declare when audited, since 0 beats -300, and still prefers to evade when not audited, since 50 beats 0. The authority's preferences are untouched. The four arrows still form the same cycle, so the game still has no pure equilibrium. Raising the penalty changes how much is at stake without changing anybody's ranking in any column, and the next lesson shows the surprising thing it does change.
What a fix has to do
The cycling game exposes something specific. A player in the audit game does not want to be predictable, and every strategy considered so far is perfectly predictable, because it names one action. The obvious repair is to let a player choose a probability distribution over their actions instead, and then to ask what the equilibrium condition means when strategies are lotteries.
That repair is not a patch on a defective theory. It restores existence completely: with randomisation allowed, every finite game has at least one equilibrium, which is Nash's theorem and the reason his name is on the concept rather than only on the definition. Working out what those probabilities have to be, and reading the strange comparative statics they imply for penalties, audits and penalty kicks, is the next lesson.