Midterm Scores - PowerPoint Presentation

Slide1

Midterm Scores

181 points total

Grade RangeScoreNumber of students in this rangeA160-18116B120-15916C90-11916D70-899F<404

Median Score: 120

Top Score 181Slide2

For those who did well

…

f0rmtySlide3

And keep climbing…Slide4

For those who did poorly…

Slide5

Fair warning…

Course builds on what you have learned.Material becomes harder.

If you are lost now, your chances of catching up are almost zero.Last day to drop class is Feb 1.Slide6

What is a strategy?

Most of you showed that you know. A strategy is NOT a possible “history of play.”

(i.e. Not, he did this, then he did that)It is a contingency plan. What will player do at each information set that he might reach.If you didn’t list strategies correctly wherever required, Study the textbook! Read pages pp 36-38 carefully and thoughtfully.Slide7

Mixed StrategiesSlide8

Mixed strategies

If a player’s strategy is to do something for sure at each information set, we say it is a pure strategy. A player who ``randomizes’’ between ``pure strategies’’, assigning a specific probability to taking each possible pure strategy is said to use a mixed strategy.

Why might you do that? Slide9

Simple hide and seek

1,0

0, 10,1 1,0 Look Upstairs Look Downstairs Player 2 (Seeker)Player 1 (Hider)Hide upstairs ( Hide downstairs

pSlide10

The game of hide and seek has

Two pure strategy Nash

equilibria One pure strategy Nash equilibrium No pure strategy Nash equilibrium 0,1 1,0 1,0 0, 1Hide upstairsYour Text HereHide downstairs

Look upstairs

Look downstairsSlide11

Best response

Suppose Hider randomizes and hides upstairs with probability 2/3.

What is Seeker’s best response? Look upstairs for sure Look downstairs for sureRandomize with Probability of upstairs 2/3.Randomize with Probability of downstairs 2/3 Slide12

Could there be a Nash equilibrium strategy profile in which Hider is more likely to hide upstairs than downstairs?

Yes

NoMaybeSlide13

In Hide and Seek, is there a Nash equilibrium in which one of the two players uses a pure strategy?

Yes

NoMaybeSlide14

Mixed strategy as best response

In a two-player, two-strategy game playing a mixed strategy is a best response to what the other player is doing only if your payoffs from the two pure strategies are equal.

A) TrueB) FalseSlide15

Mixed strategies for Hide-and-Seek

Let p be the probability that hider hides upstairs. For what value of p would seeker be willing to use a mixed strategy?

Let q be the probability that seeker looks upstairs. For what value of q would hider be willing to use a mixed strategy?Slide16

A Fundamental Theorem

Some 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. Slide17

Advanced Hide and Seek

1,-1

-1, 1-3,3 1,-1Plains ForestLarge Attacking Army Smaller Retreating Army Plains ForestSlide18

Mixed 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.Slide19

Example: Advanced Hide and Seek

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

Where p is probability Retreater is in the plains For the attackerPayoff to Plains is p3+(1-p)(-1)=4p-1.Payoff to Forest is -1p +(1-p)1=1-2p4p-1=1-2p if and only if 6p=2, p=1/3.Slide20

How about Retreater?

When does Retreater have a mixed strategy best response?Where q is the probability that Attacker comes through the plains

For the retreater, the expected payoffs arego through plains -3q+1x(1-q) =1-4qgo through forest q-1x(1-q)=2q-1Payoffs are same if 1-4q =2q-1 which implies that q=1/3.Slide21

The mixed strategy Nash equilibrium

Retreating army goes to the forest with probability 2/3 and plains with probability 1/3.

Attacking army goes to the forest with probability 2/3, plains with probability 1/3.Probability both go to plains =1/9Probability both go to forest=4/9.Probability they go to different places 4/9.Expected payoff For Attacking army is 4/9x1+1/9x3+4/9x(-1)=1/3For Retreating army is =1/3.Slide22

Best response Mapping

0

11p= Probability 1 chooses Plainsq=probability2 chooses Plains1/3

1/3

Player 2’s Reaction Function (in Red)

Player 1’s

Reaction

Function

(in Green)Slide23

Chicken Game

0

, 01, 00, 1-10, -10Swerve Don’t Swerve SwerveDon’t Swerve

Player 1

Player 2

q 1-q

P

1-p

Two Pure Strategy Nash

equilibriaSlide24

Mixed 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.Slide25

A Dark Tale from the Big AppleSlide26

The Kitty Genovese Case

In 1964, as she returned home from work late at night, Kitty Genovese was assaulted and murdered, near her apartment in Queens, New York City

.According to a story in the New York Times For more than half an hour, 38 respectable, law-abiding citzens in Queens watched a killer stalk and stab a woman in three separate attacks.  . . . Not one person telephoned the police during the assault” Slide27

Pundits’ Reactions

Pundits found this “

emblematic of the callousness or apathy of life in big cities, particularly New York.”The incident was taken as evidence of ``moral decay’’ and of “dehumanization caused by the urban environment.”Slide28

In Defense of New Yorkers?

Sociologists, John Darley and Bibb

Latane suggested an alternative theoryCity dwellers might not be “callous” or “dehumanized.” They know many are present and believe that it is likely that someone else will act. Darley and Latane called this the “bystander-effect.” They found this effect in lab experiments: Someone pretended to be in trouble, When subjects believed nobody else could help, they did so with probability .8.When they believed that 4 others observed the same events, they helped with probability .34.Slide29

Volunteer’s Dilemma Game

Andreas

Diekmann, a sociologist, created a game theoretic model, the “Volunteer’s Dilemma” N-player simultaneous move game: Strategies Act or Not. All who act pay C. If at least one acts, those who acted get B-C.. Those who didn’t act get B. If nobody acts, all get 0.In symmetric mixed strategy Nash equilibrium, as N increases, it less likely that any one person calls. In fact, it is more likely that nobody calls.Slide30

Volunteers’ 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.Slide31

Mixed 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-1Then (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.Slide32

Defense of New Yorkers

In what sense does the Volunteer’s dilemma game suggest that New Yorkers may not be

“callous and dehumanized”?Slide33

Further defense of New Yorkers

Less interesting for theory, but facts deserve respect.

Fact-checkers later found the journalists’ story partly fabricated (albeit by NYC-based journalists). No evidence that 38 people knew what was going on. It was 3 am on a cold night. Windows were closed. One person tried to help. Slide34

Battle of Sexes

3,2

1,1 0,0 2,3BobAliceMovie AMovie BMovie AMovie B

BR

A

(A)=A

BR

A

(B)=B

BR

B

(A)=A

BR

B

(B)=BSlide35

Does this game have pure strategy Nash

equilibria?

Yes, there is exactly oneYes, there are twoNo there are no pure strategy Nash equilibriaSlide36

Mixed 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 is3(1-q)+ q1=3-2q.Expected payoff to Movie B for Alice is 2q+(1-q)0=2qPayoffs are the same if 3-2q= 2q, so q=3/4. Slide37

Similar 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. Slide38

Have a nice weekend!