Game Theory

Part 5: Nash’s Theorem

Nash’s Theorem

• Any n-player variable-sum matrix game has at least one equilibrium point.

Comparing Theorems

Here are the two most important theorems of game theory…

• Minimax Theorem – Proved by Von Neumann in 1928.– States that any 2-player zero-sum matrix game has exactly one

unique equilibrium point (which is always a saddle point).

• Nash’s Theorem– Proved by John Nash in 1951.– States that any n-player variable-sum matrix game has at least

one equilibrium point.

Nash’s Theorem

• Now, because of Nash’s Theorem, given any matrix game, no matter how many players, no matter how many strategies, no matter if it’s a zero-sum game or a variable-sum game, we know there is at least one equilibrium point.

• We can say that Nash’s theorem provides a generalization of the Minimax theorem. That is, the Minimax theorem is a special case of Nash’s theorem, where n = 2 players and the game is zero-sum. In that special case, we know there is exactly one equilibrium point and in that case the equilibrium point is also a saddle point.

• According to Nash’s theorem, there is always at least one equilibrium point in any matrix game. We will see that in variable-sum games, there may be more than one equilibrium point. Also, just like with the Minimax Theorem, any equilibrium point in variable-sum games may be in mixed strategies.

Saddle Points and Equilibrium Points

• A saddle point is the combination of strategies in which each player can find the highest possible payoff assuming the best possible play by the opponent.

• An equilibrium point (also called a Nash equilibrium point) is the combination of strategies in which no player has any benefit from changing strategies assuming that the opponent (or opponents) do not change strategies.

• Every saddle point is an equilibrium point but not every equilibrium point is a saddle point.

• A saddle point occurs when each player is achieving the highest possible payoff and thus neither would benefit from changing strategies if the other didn’t also change - which is why it is also called an equilibrium point. However, there are equilibrium points in variable sum games where the players are not achieving the best possible payoff (so they aren’t saddle points) but neither will benefit by changing their strategy assuming the other doesn’t also change their strategy – which is why they are called equilibrium points.

Important Variable Sum Games

• There are two important well known 2-player variable-sum matrix games which must be discussed in any introduction to game theory…

Confess Not Confess

Confess (2,2) (4,1)

Not Confess

(1,4) (3,3)

Swerve Not Swerve

Swerve (3,3) (2,4)

Not Swerve

(4,2) (1,1)

Prisoner’s Dilemma Chicken

Prisoner’s Dilemma

Confess Not Confess

Confess (2,2) (4,1)

Not Confess

(1,4) (3,3)

Prisoner #2

Prisoner#1

The police catch two suspects. They are kept for questioning. Each has the following dilemma: to confess or not to confess. Suppose the prisoner’s are unable to communicate and must make a decision without knowing what the other decided.

What is in the best interest of each player?

Prisoner’s Dilemma

Confess Not Confess

Confess (2,2) (4,1)

Not Confess

(1,4) (3,3)

Prisoner #2

Prisoner#1

Notice that the dominant strategy for both players is to confess. Unfortunately, for them, this implies the equilibrium point is in both confessing.

Of course, both would be better of by not confessing, but not knowing what the other player’s strategy will be, forces each into playing their dominant strategy.

Prisoner’s Dilemma

Confess Not Confess

Confess (2,2) (4,1)

Not Confess

(1,4) (3,3)

Prisoner #2

Prisoner#1

Notice that the dominant strategy for both players is to confess. Unfortunately, for them, this implies the equilibrium point is in both confessing.

Of course, both would be better of by not confessing, but not knowing what the other player’s strategy will be, forces each into playing their dominant strategy.

Equilibrium point

Prisoner’s Dilemma – Example 2: an arms race

Continue to Arm

Respect Treaty

Continue to Arm

(2,2) (4,1)

Respect Treaty

(1,4) (3,3)

Country #2

Country#1

Prisoner’s Dilemma can be used to model an arms race between two countries. Suppose each sign a treaty to stop any military build-up. It could be in each country’s best interest to respect the treaty, assuming the other does as well, because, for example, instead of financing an arms-race the countries could invest in social programs or other interests. However, each could fear that the other country will break the treaty and continue to arm.

Prisoner’s Dilemma – Example 2: an arms race

Country #2

Country#1

Suppose, for example, that Country #1 chose to respect the treaty while Country #2 continued to arm. Then Country #1 would be at a disadvantage as Country #1 continues to build military strength. For this game, the equilibrium point is where each country continues to arm. Of course, this is the reason each country would expect inspections of the other’s arms as part of any acceptable treaty.

Continue to Arm

Respect Treaty

Continue to Arm

(2,2) (4,1)

Respect Treaty

(1,4) (3,3)

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

Does either player have a dominant strategy?

The answer is no – neither player has a dominant and neither has a dominated strategy.

However, there are two equilibrium points in pure strategies (and one in mixed strategies).

The name derives from a situation in which two players drive straight toward each other. Each player considers it the greatest payoff not to swerve away but to force the other to “chicken out” and swerve.

Of course, the worst payoff is if neither swerves and they crash into each other.

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

The two pure strategy equilibrium points in this game are:

Equilibrium point #1Player 1 – swervePlayer 2 – do not swerve

And …

Equilibrium point #1

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

The two pure strategy equilibrium points in this game are:

Equilibrium point #1Player 1 – swervePlayer 2 – do not swerve

And

Equilibrium point #2Player 1 – do not swervePlayer 2 – swerve

Equilibrium point #2

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

Why are these two points equilibrium points ?

If there are equilibrium points in pure strategies, even when no player has a dominant strategy, we can find these points as follows…

To find equilibrium points when there are no dominate strategies in variable sum games, consider the payoffs at each outcome:

We ask, given one player’s choice of the strategy resulting in that outcome, does the other player have any benefit in changing strategy?

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

Considering each outcome:

(3,3) is not at an equilibrium point because given player 2’s choice of swerve, player 1 would benefit by switching to the do not swerve strategy.

(4,2) is at an equilibrium point because given player 2’s choice of swerve, player 1 does not benefit by switching to swerve.

(2,4) is also at an equilibrium point because given player 1’s choice of swerve, then player 2 will not benefit from switching to do not swerve.

Finally, (1,1) is not an equilibrium point because for both player’s given the other’s choice of do not swerve, each player can benefit by switching strategy to swerve.

Chicken

Player #2

Player#1

swerve do not swerve

swerve (3,3) (2,4)

do not swerve

(4,2) (1,1)

The equilibrium points occur in this game in this way because of the way in which payoffs are distributed…

While neither player has a dominant strategy, the payoff of not swerving when the other does is higher than both swerving.

Both players risk the outcome (1,1) because of the potential of the higher payoff at (4,2) or (2,4).

The reason the equilibrium point is at (2,4) and (4,2) and not at (3,3) is because the strategies at (3,3) are not stable. Both players have an incentive to change to the do not swerve strategy in hopes of getting the higher payoff.

Chicken

USSR

U.S.

back down

proceed

back down

(3,3) (2,4)

proceed (4,2) (0,0)

The Cuban missile crisis can be modeled by the game of chicken.

In the 1960s, the USSR began supplying missiles to Cuba.

The United States began a blockade to stop the USSR.

As the crisis developed, if each had continued to proceed with their chosen strategy, the consequences could have been disastrous.

Fortunately, some last minute negotiations averted such an outcome. The U.S. did not have to back down and the USSR was able to achieve some advantage in other interests.