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Mixed Strategies

Slide2What is a mixed strategy?

A player chooses to ``randomize’’ between ``pure strategies’’, assigning a specific probability to taking each possible pure strategy.

If the other player knows your strategy, when would you want it to be a mixed strategy with non-zero probabilities of more than one action?

Slide3Matching Pennies:or Simple hide and seek_

1,-1 -1, 1

-1,1 1,-1

Heads Tails

Player 2 (Seeker)

Player 1

Heads (Hider) 1-pTails

q

p

1-q

Slide4The game of matching pennies has

two pure strategy Nash

equilibria

One pure strategy Nash equilibrium

One mixed strategy Nash equilibrium and no pure strategy Nash

equilibria

Two mixed strategy Nash

equilibria

and no pure strategy Nash

equilibria

One mixed strategy Nash equilibrium and two pure strategy Nash

equilibria

.

Slide5Nash equilibrium in Matching Pennies

Suppose Player 1 randomizes and plays Heads 2/3 of the time, what is Player 2’s best response?

Heads for sure

Tails for sure

Randomize with Probability of Heads 2/3.

Randomize with Probability of Tails 2/3

Slide6Best response Mapping

0

1

1

p= Probability 1 chooses H

q=probability

2 chooses H

1/2

1/2

Player 2’s Reaction Function (in Red)

Player 1’s

Reaction

Function

(in Green)

Slide7Nash equilibrium in Mixed Strategies

Intersection of Reaction Functions

Each is doing best response to other’s strategy

Slide8A Fundamental Theorem

S

ome games have no equilibrium in pure strategies: Examples: matching pennies; rock, paper scissors

Every game in which there is a finite number of pure strategies has at least one mixed strategy equilibrium.

Slide9Advanced Hide and Seek

1,-1 -1, 1

-3,3 1,-1

Plains Forest

Seeker’s Choice

Hider’sChoice

Plains 1-pForest

q

p

1-q

Slide10Mixed strategy equilibrium

In a mixed strategy equilibrium, all strategies that are assigned positive probability have equal expected value.

You can use this fact to find mixed strategy Nash

equilibria

.

Slide11Example: Advanced Hide and Seek

When does Seeker have a mixed strategy best response. The payoffs to looking in the plains and looking in the forest must be the same.

Where p is probability Hider is in the plains,

Payoff to Plains is p3+(1-p)(-1)=4p-1.

Payoff to Forest is -1p +(1-p)1=1-2p

4p-1=1-2p if and only if 6p=2, p=1/3.

Slide12Best response Mapping

0

1

1

p= Probability 1 chooses Plains

q=probability

2 chooses Plains

1/3

1/3

Player 2’s Reaction Function (in Red)

Player 1’s

Reaction

Function

(in Green)

Slide13Expected Utility Theory of Choice Under Uncertainty

Suppose that you face random outcomes. You assign a “utility” to each possible outcome in such a way that your choices among uncertain prospects are those that maximize “expected utility”.

Slide14Expected utility Example: Utility of money

Suppose you have a lottery that will with probability 1/4 win 10 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.

Would you sell it for 2.5 million dollars?

Yes

No

Slide15Expected utility Example: Utility of money

Suppose you have a lottery that will with probability 1/4 win 10 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.

Would you sell it for 1 million dollars?

Yes

No

Slide16Expected utility Example: Utility of money

Suppose you have a lottery that will with probability 1/4 win 100 million dollars and with probability ¾ will be worthless. You get just one chance to sell your ticket.

Would you sell it for 500 thousand dollars?

Yes

No

Slide17Construct a utility scale

Let u(10 million)=1 Let u(0)=0.

Then ask question. How much money X for sure would be just as good as having a ¼ chance of winning 10 million and ¾ chance of 0?

Then assign u(X)=(3/4)u(0)+(1/4)u(10,000,000)=

(3/4)0+(1/4)1=1/4.

Slide18

Assigning utility to any income

Lets choose a scale where u(0)=0 and u(10 million)=1.

Take any number X. Find a probability p(X) so that you would just be willing to pay $X for a lottery ticket that pays 10 million with probability p(X) and 0 with probability 1-p(x).

Assign utility p(X) to having $X.

Slide19Field Goal or Touchdown?

Field goal is worth 3 points.

Touchdown is worth 7 points.

Which is better? Sure field goal or probability ½

of touchdown?

Slide20Finding the coach’s von Neumann Morgenstern utilities

Set utility of touchdown u(T)=1

Set utility no score u(0)=0

The utility

of a gamble in which you get a touchdown with probability p and no score with probability 1-p is

pu

(T)+(1-p)u(0).

What utility u(F) to assign to a

sure field

goal?

Let p* be the probability such that the coach is indifferent between scoring a touchdown with probability p* (with no score with

prob

1-p*) and

having a sure field goal.

Then u(F)=p*u(T)+(1-p*)u(0)=p*x1+(1-p*)x0=p*.

Slide21Volunteers’ Dilemma

N people observe a mugging. Someone needs to call the police. Only one call is needed. Cost of calling is c. Cost of knowing that the person is not helped is T. Should you call or not call?

T>c>0. Many asymmetric pure strategy

equilibria

.

Also one symmetric mixed strategy equilibrium.

Slide22Mixed strategy equilibrium

Suppose everybody uses a mixed strategy with probability p of calling.

In equilibrium, everyone is indifferent about calling or not calling if expected cost from not calling equals cost from calling.

Expected Cost of

of

not calling is

T(1-p)

N-1

Expected cost of calling is c.

Equilibrium has c=

T(1-p)

N-1

so 1-p=(c/T)

1/N-1

Then (1-p)

N

=(c/T)

N/N-1

is the probability that nobody calls. This is an increasing function of N.

So the more

People who observe, the less likely that someone calls.

Slide23Chicken Game

0

, 0

1, 0

0, 1

-10, -10

Swerve Don’t Swerve

SwerveDon’t Swerve

Player 1

Player 2

q 1-q

P1-p

Two Pure Strategy Nash

equilibria

Slide24Mixed Strategy

When is Player 1 indifferent between the two strategies, Swerve and Don’t Swerve?

Expected payoff from Swerve is 0.

Expected payoff from Don’t Swerve is

q-10(1-q).

So Player 1 will use a mixed strategy best response only if 0=11q-10 or q=10/11.

Similar reasoning

inplies

that in Nash equilibrium p=10/11.

Crash occurs with probability 1/121.

Slide25Battle of Sexes

3,2 1,1 0,0 2,3

Bob

Alice

Movie A

Movie B

Movie A

Movie B

BRA(A)=ABRA(B)=B

BR

B

(A)=A

BR

B

(B)=B

Slide26Mixed Strategy Equilibrium

Let p be probability Alice goes to movie A and q the probability that Bob goes to movie B.

When is there a mixed best response for Alice?

Expected payoff for Movie A for Alice is

3(1-q)+ q1=3-2q.

Expected payoff to Movie B for Alice is

2q+(1-q)0=2q

Payoffs are the same if 3-2q= 2q, so q=3/4.

Slide27Similar for Bob

From the symmetry of the game, we see that a mixed strategy is a best response for Bob if p=3/4.

In a symmetric mixed strategy, each goes to his or her favorite movie with probability ¾.

Probability that they get together at Movie A is 3/4x1/4=3/16. Probability that they get together at Movie B is also 3/16.

Probability that they miss each other is 5/8.

Probability that each goes to favorite movie is 9/16. Probability that they each go to less preferred movie is 1/16.

Slide28Have a nice weekend!

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