Mixed Strategy Problems

8/10/2019 Mixed Strategy Problems

1/18

Working Some Problems


2/18

Telephone Game

How about xexed strategies?Let Winnie call with probability p and wait with probability 1-p.For what values of w is Colleen indifferent?Expected payoff for Colleen from calling is????Expected payoff for Colleen from waiting is???

C

Are there any pure strategy equilibria?


3/18

Listing the N.E Let p be probability that Winnie calls and 1-p the probability

that she waits. Let q be the probability that Colleen calls and1-q the probability that she waits.

Nash equilibria are strategy profiles. The mixed strategy equilibria include:a) p=0 and q=1

b) p=1 and q=0c) p=1/4 and q=1/4


4/18

A Duel

Fastball Curveball

Fastball .35,.65 .3,.7

Curveball .2,.8 .5,.5

Pitcher throws

Batterprepares for

Does this game have any pure strategy equilibria?

A)YesB)No


5/18

A Duel

Fastball CurveballFastball .35,.65 .3,.7


Pitcher throws

Batterprepares for

In Nash equilibrium if Batter has positive probabilityof using each strategy, what is the probability thatPitcher throws a fastball?A) 1/3

B) 2/3C) D) 4/7

E) 3/5


6/18

Lets go figure In N.E, Batter will play each strategy with positive

probability only if the two strategies have thesame expected payoff for him.

Suppose Pitcher throws a fastball with probability pand a curveball with probability 1-p.

Batters expected payoff from Prepare for Fastball is.35p+.30(1-p)

Batters expected payoff from Prepare for curveball

is .2p+.5(1-p). These payoffs are equal if .35p+.30(1-p)=.2p+.5(1-p). Solve this equation for p.


7/18

A Duel

Fastball Curveball

Fastball .35,.65 .3,.7


Pitcher throws

Batterprepares for

In a mixed strategy Nash equilibrium what is theprobability that Batter prepares for a fastball?A) 4/7

B) 3/7C) D) 6/7

E) 4/5


8/18

Does this game have a Nash equilibrium in which Kicker mixes left

and right but does not kick to center?


9/18

If there is a Nash equilibrium where kickernever kicks middle but mixes between left andright, Goalie will never play middle but willmix left and right (Why?)

If Goalie never plays middle but mixes leftand right, Kicker will kick middle. (Why?)

So there cant be a Nash equilibrium where Kicker never kicks Middle. (See why?)


10/18

Problem 4: For what values of x is there a mixed strategy Nashequilibrium in which the victim might resist or not resist and theMugger assigns zero probability to showing a gun?


11/18

Problem 7.7, Find mixed strategy Nash equilibria


12/18

c dominates a and y dominates z

A mixed strategy N.E. strategy doesnot give positive probability

To any strictly dominated strategy

Look at reduced game without these strategies


13/18

Problem 7.7 Find mixed strategy Nash equilibia

For player 1, Bottom strictly dominates Top. Throw out Top

Then for Player 2, Middle weakly dominates Right. Therefore ifPlayer 1 plays bottom with positive probability, player 2 gives zeroProbability to Right.There is no N.E. in which Player 1 plays Bottom with zero probability, (Why?)(If he did, what would Player 2 play? Then what would 1 play?)


14/18

A Nash equilibrium is any strategy pair in which the defense defendsagainst the outside run with probability .5 and the offense runs up themiddle with probability .75. No matter what the defense does,The offense gets the same payoff from wide left or wide right,So any probabilities pwl and pwr such that pwl+pwr=.25will be N.E. probabilities for the offense.

Problem 8, Chapter 7


15/18

Problem

10.

Each of three players is deciding between the purestrategies go and stop. The payoff togo is 120, where m is the number of players thatchoose go, and the payoff to stop is 55 m

(which is received regardless of what the other playersdo). Find all Nash equilibria in mixed strategies.

Lets find the easy ones.

Are there any symmetric pure strategy equilibria?

How about asymmetric pure strategy equilibria?

How about symmetric mixed strategy equilibrium?

Solve 40p^2+60*2p(1-p)+120(1-p) 2=55

40p2

-120p+65=0


16/18

What about equilibria where one guy is in for sure and othertwo enter with identical mixed strategies?

For mixed strategy guys who bothEnter with probability p, expected payoff from entering is(120/3)p+(120/2)(1-p). They are indifferent about entering or not if40p+60(1-p)=55. This happens when p=1/4.

This will be an equilibrium if when the other two guys enter withProbability , the remaining guy is better off entering than not.

Payoff to guy who enters for sure is:

40*(1/16)+60*(3/8)+120*(9/16)=92.5>55.


17/18

Rock Paper Scissors

Rock 0,0 -1,1 2,-2

Paper 1,-1 0,0 -1,1

Scissors -2,2 1,-1 0,0

Advanced Rock-Paper-Scissors

Are there pure strategy Nash equilibria?

Is there a symmetric mixed strategy Nash equilibrium?

What is it?


18/18

Remember: Midterm on Thursday

Chapters 2-5 and 7No need to bring bluebooks.Calculators and phones not allowed

Mixed Strategy Problems

Documents

Transcript of Mixed Strategy Problems