Lesson · Games and the Social Contract · One-shot games

Nash Equilibrium in Pure and Mixed Strategies

A Nash equilibrium is a combination of strategies in which no player gains by changing alone. Some games have several such cells and some have none, and for the games with none a player may mix strategies with chosen probabilities. The mixed equilibrium of a two-by-two game is found exactly, by making each player's mix leave the other indifferent.

Open the interactive lesson Practice

Nash: no player gains by moving alone pure: a cell where both best-respond mixed: weights that leave the other even an equilibrium need not be the best outcome

Mutual best response

Core ideas

What this lesson establishes

The definition is one sentence. Everything else in the lesson is what it does and does not promise.

Equilibrium is mutual best response

A combination of strategies is a Nash equilibrium when each player's strategy is a best response to the others'. Check it cell by cell: ask of each player whether switching alone would pay.

A mixed strategy is a set of probabilities

A player who mixes picks each pure strategy with a stated probability, and a mix is a best response only if every strategy used in it is itself a best response. That is why a mixing player must be indifferent between the strategies mixed.

Equilibrium describes stability, not merit

The definition says nothing about whether the outcome is good for the pair. A game may have an equilibrium that both would swap for another cell, and the previous lesson already showed one.

The material

Pure cells, and the mix when there are none

Three games, the third of which has no pure equilibrium.

Definition · Nash equilibrium

A combination of strategies, one for each player, is a Nash equilibrium when no player can get a larger payoff by changing their own strategy while the others keep theirs. An equilibrium in pure strategies is a cell of the table; an equilibrium in mixed strategies gives each player a probability for each pure strategy.

Pure equilibria are the doubly marked cells of the previous lesson, and a game can have none, one, or several. Chicken has two. Two drivers head for each other on a narrow road and each can swerve or go straight. If both swerve nothing is lost. If one swerves and the other does not, the one who swerves gets -1 and the other gets 1. If neither swerves they crash and each gets -10.

column Swerve column Straight row Swerve 0, 0 -1, 1 row Straight 1, -1 -10, -10

The cells Swerve, Straight and Straight, Swerve are equilibria. In the first, the driver going straight gets 1 and would get 0 by swerving, and the driver who swerved gets -1 and would get -10 by going straight. Swerve, Swerve is not an equilibrium, because either driver gains by going straight. It is the one cell in which nobody is hurt, and equilibrium does not choose it.

Matching pennies is the opposite case. No cell is an equilibrium, because in every cell the player who loses the match would rather have chosen the other side. Nash's theorem says that every game with finitely many strategies has an equilibrium if mixed strategies are allowed. The theorem is stated here and not proved. What the lab does is compute the mix for a two-by-two game.

Solving for the mix

Let the column player swerve with probability q. The row player's expected payoff from swerving is 0·q − 1·(1 − q), which is q − 1. From going straight it is 1·q − 10·(1 − q), which is 11q − 10. A row player who mixes must be indifferent between the two, so the column player's probability is the one that makes them equal:

q − 1 = 11q − 10 9 = 10q q = 9/10

The column player swerves nine times in ten. The table is symmetric, so the row player does too. At that pair of mixes each driver expects -1/10, which is below the 0 each would get if both simply swerved, and the two crash with probability 1/10 · 1/10, one time in a hundred. The mixed equilibrium is an equilibrium in the full sense, since at it neither driver gains by changing alone, and it is worse for both than the cell it does not use.

The surprise is whose numbers fix whose mix. Each player's probability is chosen so that the other is indifferent. A player does not randomise to be unpredictable at the cost of expected payoff; they randomise because, at that mix, the opponent has no better strategy to move to.

Interactive

A game in a table

best responses, equilibria, dominance and efficiency, computed from the payoffs

Find the pure cells and the mix

The mixed tile gives each player's probability of the first strategy. Change a payoff and the mix moves; when it would leave the interval from 0 to 1, the tile says none.

Worked example

Row player’s strategies

Column player’s strategies

Payoffs “row,column”, rows separated by “;”

Show

Pure equilibria—

Mixed equilibrium—

Dominance—

Pareto—

Method

Solving a two-by-two game

Pure cells first, then the mix.

Mark the best responses and list the doubly marked cells

These are the pure equilibria. There may be none, one or several.

Let the column player play the first strategy with probability q

Write the row player's expected payoff from each row as a line in q.

Set the two expected payoffs equal and solve for q

That is the mix that leaves the row player indifferent. If the answer is not strictly between 0 and 1, no completely mixed equilibrium exists.

Swap the roles to find the row player's probability

Let the row player play the first strategy with probability p and make the column player indifferent.

Report all of them

Pure and mixed together are the equilibria. Chicken has three, matching pennies has one.

Pure, mixed, and good

Score 0 / 0

Choose the answer you can justify.

The mix in chicken

Let q be the chance the column driver swerves.

Row swerves: 0·q − 1·(1 − q) = q − 1 Row goes straight: 1·q − 10·(1 − q) = 11q − 10 Indifference: q - 1 = 11q - 10 Solve: 10q = 9, q = 9/10 Each driver expects 0·(9/10) − 1·(1/10) = −1/10

The same calculation for the row driver gives 9/10 again, since the table is symmetric. The lab prints this as the mixed equilibrium, with the two pure cells beside it, and it is the only completely mixed equilibrium the game has.

Common mistakes

Taking an equilibrium to be the best outcome for the groupEquilibrium is a property of each player's reply, and it says nothing about the pair. In chicken the cell where both swerve pays each 0 and is not an equilibrium, while the mixed equilibrium expects -1/10 each and crashes one time in a hundred. The previous lesson's dilemma is the same fact in starker form.

Reading a player's mix as a way of keeping the opponent guessingThe weights are fixed by the opponent's payoffs, not one's own. Drivers in chicken swerve nine times in ten because that is the weight at which the other driver is exactly as well off going straight as swerving.

Expecting one equilibrium per gameChicken has two pure equilibria and a mixed one. When a table has several, the table by itself does not say which will be played, and “Coordination, Conventions and the Stag Hunt” is about how people choose between them.

Check your understanding

✓

Finish when you can compute the mix of a two-by-two game and check it.

Given a two-by-two table, list the pure equilibria, write one player's expected payoffs as lines in the other's probability, solve for the point where they cross, and say whether that mix is an equilibrium by checking that neither player gains from a pure deviation.

Note: Chicken here is the textbook version with a crash costing ten, and its mix depends on that ten: a crash costing a hundred makes the drivers swerve more often. Try it in the payoffs box.