Games of pure conflict - PowerPoint Presentation

Slide1

Games of pure conflicttwo person constant sumSlide2

Two-person constant sum game

Sometimes called zero-sum game.

The sum of the players’ payoffs is the same, no matter what pair of actions they take

.

In a two-person constant sum game, one player’s gain is the other’s loss.Slide3

Maximin strategy

One way to play a game is to take a very cautious view.

Your payoff from any action depends on other’s actions.

In a two-player game, you might assume other player always does what is worst for you.

Given that assumption, you would choose the strategy such that gives you the best

payoff available

if the other player always does what is worst for you given your strategy.Slide4

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

Is this a constant sum game?

A) Yes B) NoSlide5

Penalty Kick

.9, .1

.5, .5

.3 , .7 .8, .2

Jump Left Jump Right

Goalkeeper

Shooter

Kick Left

Kick right

Is this a constant sum game?

A) Yes B) NoSlide6

Going to the Movies

3,2

1,1

0,0

2,3

Bob

Alice

Movie A

Movie B

Movie A

Movie B

Is this a constant sum game?

Yes

No

MaybeSlide7

Mixed strategies and maximin

Suppose you are

Hider

, choosing a mixed strategy, and you believe that

Seeker

will do what is worst for you, given your mixed strategy.

This is not a silly assumption in a two-player zero sum game, because what is worst for you is best for your opponent.

The maximin player will choose her best mixed strategy given that she believes opponent will respond with the strategy that is worst for her.Slide8

Clicker questionSuppose that you are

Hider

and you choose to hide upstairs with probability .6. What strategy by

SEEKER

is worst for you?

Look upstairs with probability .6

Look upstairs and downstairs with equal probability

Look upstairs for sureLook upstairs with probability .4Slide9

Clicker questionIf you are

Hider

and hide upstairs with probability .6 and

Seeker

uses the strategy that is worst for you, what is your expected payoff?

.6

.4

.5.35Slide10

More generally

If you are

Hider

and you hide upstairs with probability p>1/2, what is the strategy for

Seeker

that is worst for you

?

Look upstairsWhat is your expected payoff if he does that?You win only if you hide downstairs. Probability of this is 1-p. Expected payoff is(1-p)x1+px0=1-pSlide11

What if you hide upstairs with p<1/2?

What is worst thing that

Seeker

can do to you?

(He’ll look downstairs for sure.)

What is your expected payoff?Slide12

Maximin for hide and seekSlide13

The pessimist’s viewSlide14

Penalty Kick

.9, .1

.5, .5

.3 , .7 .8, .2

Jump Left Jump Right

Goalkeeper

Shooter

Kick Left

Kick right

Let’s look from pessimistic shooter’s viewSlide15

Shooter’s ViewSlide16

Clicker question

If shooter randomly chooses left with probability p>4/9, what Goalie strategy is worst for shooter

Jump left

Jump right

Jump left with same probability that shooter shoots left

Jump left with probability ½, right with probability ½.Slide17

Clicker question

If Shooter shoots left with probability p, what is the best response for Goalie.

Jump left with probability p

Jump left with probability ½

Jump left for sure if p>5/9, right if p<4/9

Jump left with probability 1-pSlide18

Constant sum games and Maximin

Note that when shooter uses

maximin

strategy, his own payoff is the same for either response by Goalkeeper.

If shooter’s payoff is the same from both strategies, so is goalkeeper’s. (Why?)

If goalkeeper’s strategy is same from both strategies, goalkeeper is willing to randomize.Slide19
Slide20

Clicker Question

If Goalie jumps left with probability ½, what strategy by Shooter is worst for Goalie?

Shoot left

Shoot right

Shoot left or right with equal probabilitySlide21

Clicker question

What strategy by

Goalie

makes

S

hooter

equally well off from shooting left or right?

Jump left with probability ½Jump left with probability 2/3Jump left with probability 1/3Slide22

Summing up

In

Maximin

equilibrium:

Shooter shoots to left with probability 4/9

Goalkeeper jumps left with probability 1/3

Shooter scores with probability .663

Goalkeeper makes save with probability .366Maximin is also a Nash equilibrium in zero sum gamesSlide23

Maximin and the movies

3,2

1,1

0,0

2,3

Bob

Alice

Movie A

Movie B

Movie A

Movie B

This is not a constant sum game.

Maximin

equilibrium is not a Nash equilibrium.

Slide24

Alice’s ViewSlide25

Maximin equilibrium

Symmetric story for Bob.

In

maximin

equlibrium

each is equally likely to go to either movie.Slide26

If Alice is equally likely to go to Movie A or Movie B, what is Bob’s best response?

Randomize with probability ½

Go to Movie B

Go to Movie ASlide27

Is the maximin equilibrium for Alice and Bob a Nash equilibrium?

Yes

NoSlide28

Some more ProblemsSlide29

Rock

Paper

Scissors

Rock

0,0

-1,1

2,-2

Paper

1,-1

0,0

-1,1

Scissors

-2,21,-10,0

Advanced Rock-Paper-Scissors

Are there pure strategy Nash

equilibria

?

Is there a symmetric mixed strategy Nash equilibrium?

What is it? Slide30

Rock

Paper

Scissors

Rock

0,0

-1,1

2,-2

Paper

1,-1

0,0

-1,1

Scissors

-2,21,-10,0

Finding Mixed Strategy Nash Equilibrium

Let probabilities that column chooser chooses rock, paper, and scissors be

r

,

p

, and

s

=1-p-r

Row chooser must be indifferent between rock and paper

This tells us that -p+2(1-p-r)=r-(1-p-r)

Row chooser must also be indifferent between rock and scissors.

This tells us that –p+2(1-p-r)=-2r+p

We have 2 linear equations in 2 unknowns. Let’s solve.

They simplify to 4r+4p=3 and 4p=2.

So we have

p

=1/2 and

r

=1/4. Then

s

=1-p-r=1/4.Slide31

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 if

Player 1 plays bottom with positive probability, player 2 gives zero

Probability 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?)Slide32

More mechanically

Suppose player 1 goes middle with probability m and bottom with probability 1-m.

Then expected payoffs for player 2 are:

1m+3(1-m) for playing left

3m+2(1-m) for playing middle

1m+2(1-m) for playing right

We see that playing right is worse than playing middle if m>0.

So let’s see if there is a mixed strategy Nash equilibrium where Player 2 plays only left and middle and Player 1 is willing to play a mixed strategy. Slide33

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

and right but does not kick to center?Slide34

If there is a Nash equilibrium where kicker

never kicks middle but mixes between left and

right, Goalie will never play middle but will

mix left and right (Why?)

If Goalie never plays middle but mixes left

and right, Kicker will kick middle. (Why?)

So there can’t be a Nash equilibrium where

Kicker never kicks Middle. (See why?)

Slide35

Problem 4: For what values of x is there a mixed strategy Nash equilibrium in which the victim might resist or not resist and the Mugger assigns zero probability to showing a gun? Slide36

Mugger’s Game

If there is a Nash equilibrium in which mugger does not show gun and both mugger and victim have mixed strategies, it must be that the mugger’s payoff in this equilibrium is at least as high as that of showing a gun.Slide37

Mixed strategy equilibriumwith no visible gun

Resist

Don’t resist

No Gun

2,6

6,3

Hidden Gun

3,2

5,5

Note that there is no pure strategy N.E.If Victim resists with probability p then

Mugger’s expected payoff from having no gun is

2p+6(1-p)=6-4pMugger’s expected payoff from having a hidden gun is 3p+5(1-p)=5-2p

Mugger will use a mixed strategy only if 6-4p=5-2p, which implies p=1/2.

If p=1/2, the expected payoff from not showing a gun is 4.Slide38

Mugger’s GameIf mugger shows gun, he is sure to get a payoff of x.

If victim’s strategy is to resist with probability 1/2 if he doesn’t see a gun, then expected payoff to mugger from not showing a gun is

3x1/2 +5x1/2=4

.

So there is a mixed strategy N.E. where mugger doesn’t show gun if x≤ 4.Slide39

EntryN players consider entering a market. If a firm is the only entrant its net profit is 170.

If more than one enter each has net profit 30.

If a firm stays out it has net profit 60.

Find a symmetric Nash equilibrium.

In symmetric N.E. each enters with same probability p. Slide40

Equilibrium

Let q=1-p. If a firm enters, the probability that nobody else enters is q

N-1

If nobody else enters, your profit is 170. If at least one other firm your expected profit is 10. So if you enter, your expected profit is

170q

N

-

1+10(1-qN-1)

If you don’t enter your expected profit is 60.So there is a mixed strategy equilibrium if 170qN-1+10(1-q

N-1)=60, which implies that160qN-1=50 and q=(5/16)

1/N-1Then p=1-q=1-(5/16)1/N-1Slide41

Saddam and UN(Let’s Pretend Saddam had WMD’s)

Part a) Saddam is hiding WMDs in location X, Y, or Z. UN can look either in X AND Y or in Z.

All Saddam cares about is hiding. All UN cares about if finding.

This reduces to a simple hide and seek game. Only trick: Saddam has more than 1 N.E. mixed strategySlide42

Saddam and UN

Part b) Saddam is hiding WMDs in location X, Y, or Z. UN can look in any two of these places.

Think of UN’s strategy as “where not to look”.

In N.E. probability of each strategy will be equal. (Why?)

Also in N.E. Saddam’s strategy of hiding missiles in each place is the same. (Why?)Slide43

See you on Thursday…Slide44

Hints on some more problems from Chapter 7 Slide45

Problem 9.

Each of

3

players is deciding between the pure strategies go and stop. The payoff to

go is

120/m ,

where m is the number of players that choose go, and the payoff to stop is 55 (which is received regardless of what the other players do). Find all Nash equilibria in mixed strategies.

Let’s 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=0Slide46

What about equilibria where one guy is in for sure and other

two enter with identical mixed strategies?

For mixed strategy guys who both

Enter with probability p, expected payoff from entering is

(120/3)p+(120/2)(1-p). They are indifferent about entering or not if

40p+60(1-p)=55. This happens when p=1/4.

This will be an equilibrium if when the other two guys enter with

Probability ¼, 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.Slide47

Problem 7.7, Find mixed strategy Nash

equilibriaSlide48

c dominates a and y dominates z

A mixed strategy N.E. strategy does not give positive probability

To any strictly dominated strategy

Look at reduced game without these strategiesSlide49

A Nash equilibrium is

any

strategy pair in which the defense defends

against

the outside run with probability .5 and the offense runs up the

middle

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=.25

will be N.E. probabilities for the offense.Problem 8

, Chapter 7