Transcript Part 5 - Nash's Theorem

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…
Prisoner’s Dilemma
Confess
Not
Confess
Confess
(2,2)
(4,1)
Not
Confess
(1,4)
(3,3)
Chicken
Swerve
Swerve
Not
Swerve
Not
Swerve
(3,3)
(2,4)
(4,2)
(1,1)
Prisoner’s Dilemma
Prisoner #2
Confess
Not
Confess
Confess
(2,2)
(4,1)
Not
Confess
(1,4)
(3,3)
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
Prisoner #2
Confess
Not
Confess
Confess
(2,2)
(4,1)
Not
Confess
(1,4)
(3,3)
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
Equilibrium
point
Prisoner #2
Confess
Not
Confess
Confess
(2,2)
(4,1)
Not
Confess
(1,4)
(3,3)
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 – Example 2: an arms race
Country #2
Continue
to Arm
Respect
Treaty
Continue (2,2)
to Arm
(4,1)
Respect
Treaty
(3,3)
Country
#1
(1,4)
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
Continue
to Arm
Respect
Treaty
Continue (2,2)
to Arm
(4,1)
Respect
Treaty
(3,3)
Country
#1
(1,4)
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.
Chicken
Player #2
swerve
swerve
do not
swerve
(3,3)
(2,4)
(4,2)
(1,1)
Player
#1
do not
swerve
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.
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).
Chicken
Equilibrium
point #1
Player #2
swerve
swerve
do not
swerve
(3,3)
(2,4)
Equilibrium point #1
Player 1 – swerve
Player 2 – do not swerve
(4,2)
(1,1)
And …
Player
#1
do not
swerve
The two pure strategy
equilibrium points in this game
are:
Chicken
Player #2
swerve
swerve
do not
swerve
(3,3)
(2,4)
Equilibrium point #1
Player 1 – swerve
Player 2 – do not swerve
(4,2)
(1,1)
And
Player
#1
do not
swerve
The two pure strategy
equilibrium points in this game
are:
Equilibrium point #2
Player 1 – do not swerve
Player 2 – swerve
Equilibrium
point #2
Chicken
Player #2
swerve
swerve
do not
swerve
(3,3)
(2,4)
(4,2)
(1,1)
Player
#1
do not
swerve
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
swerve
swerve
(3,3)
do not
swerve
(2,4)
Player
#1
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
swerve
swerve
do not
swerve
(3,3)
(2,4)
(4,2)
(1,1)
Player
#1
do not
swerve
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
back
down
back
down
proceed
(3,3)
(2,4)
(4,2)
(0,0)
U.S.
proceed
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.