Midterm Scores

• 20 points per question

Grade Range Score Number of students in this range

A 100-120 22

B 80-99 13

C 60-79 12D 40-59 3

F <40 4

Median Score: 90:

Answers are now posted at class website.

Fair warning

• Course builds on what you have learned.• If you are lost now, your chances of catching

up are slim.• Last day to drop class is Feb 2.

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.

Mixed Strategies

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?

Simple hide and seek

1,0 0, 1

0,1 1,0

Look Upstairs Look Downstairs

Player 2 (Seeker)

Player 1 (Hider)

Hide upstairs

( Hide downstairs

p

The game of hide and seek has

A) Two pure strategy Nash equilibria B) One pure strategy Nash equilibriumC) No pure strategy Nash equilibrium

Best response

Suppose Hider randomizes and hides upstairs with probability 2/3. What is Seeker’s best response? A) Look upstairs for sure B) Look downstairs for sureC) Randomize with Probability of upstairs 2/3.D) Randomize with Probability of downstairs 2/3

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

A) YesB) NoC) Maybe

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

A) YesB) NoC) Maybe

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) True• B) False

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?

What are the mixed strategy Nash equilibria for hide and seek?

A) There are many Nash equilibrium. All have the property that p=q where p is the probability that hider hides upstairs.

B) There are many Nash equilibrium. All have the property that p=1-q where p is the probability that hider hides upstairs and q is the probability that seeker seeks upstairs.

C) There is only one Nash equilibrium and it is the one where p=q=1/2.

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.

Advanced Hide and Seek

1,-1 -1, 1

-3,3 1,-1

Plains ForestLarge Attacking Army

Smaller Retreating Army

Plains

Forest

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.

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 attacker– 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.

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 attacker the expected payoffs are– go through plains -3q+1x(1-q) =1-4q– go through forest q-1x(1-q)=2q-1– Payoffs are same if 1-4q =2q-1 which implies that

q=1/3.

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 =1/9• Probability 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)=1/3– For Retreating army is =1/3.

Best response Mapping

0

1

1

p= Probability 1 chooses Plains

q=probability2 chooses Plains

1/3

1/3

Player 2’s Reaction Function (in Red)Player 1’s Reaction Function(in Green)

Chicken Game

0, 0

1, 0

0, 1

-10, -10

Swerve Don’t Swerve

Swerve

Don’t SwervePlayer 1

Player 2

q 1-q

P

1-p

Two Pure Strategy Nash equilibria

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.

A Dark Tale from the Big Apple

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”

Pundits’ ReactionsPundits 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.”

In Defense of New Yorkers?

• Sociologists, John Darley and Bibb Latane suggested an alternative theory– City 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.

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.

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.

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-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.

Defense of New Yorkers

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

“callous and dehumanized”?

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.

Battle 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

BRB(A)=ABRB(B)=B

Does this game have pure strategy Nash equilibria?

A) Yes, there is exactly oneB) Yes, there are twoC) No there are no pure strategy Nash equilibria

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=2q

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

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.

Have a nice weekend!