## Presentation on theme: "Working Some Problems"— Presentation transcript

Slide1

Working Some ProblemsSlide2

Problem 3.10

Students 1 and 2 can each exert study levels

{0,1,2,3,4,5

}. Student 1’s exam score will be

X+1.5 with effort level x. Student 2’s score will be x with effort level x.

Student with higher

score gets A,

student with lower

score gets B.

Payoff if your effort level is x and you get an A is 10-x.

If your effort level is x and you get a B, payoff is 8-x.Slide3

What can we do with IDSDS?

What is the lowest payoff you could get with effort level

0?

What strategies are therefore strictly dominated?

Remember:

Payoff with effort

level

x is

10-

x if you get an A

Payoff with effort level x is 8-x if you get

a

B Slide4

Eliminating stuff

With effort 0, payoff will be at least 8.

With effort 5, your payoff is at most 10-5=5

With effort 4, your payoff is at most 10-4=6

With effort 3, your payoff is at most 10-3=7.

So effort levels 3, 4 and 5 are strictly dominated.Slide5

What is left?

0

1

2

0

10,8

10, 7

8, 8

1

9, 89, 79, 628, 88, 78, 6

Student 2

Student 1Slide6

More elimination

Now 1 dominates 2 for student 1

0 dominates 1 for student 2

0

2

0

10, 8

8,8

1

9, 89,6Student 2Student 1Slide7

What now?

What is (are) Nash equilibrium?

What survives IDWDS?

How do you interpret this?Slide8

Problem 3.15

10 players each choose a number from 0 to 8.

A player wins $100 if his number is exactly ½ of the average of the numbers chosen by the other 9 players.

Solve for strategies that survive IDSDS.

This should read IDWDS. (IDSDS won’t take you far.) Slide9

IDWDS

Any number bigger than 4 is weakly dominated. Why?

If nobody chooses a number bigger than 4, then 3 and 4 are weakly dominated.

If nobody chooses a number bigger than 2, then 1 weakly dominates 2.

If everybody chooses 0 or 1, then 0 dominates 1.

Why? 1 will never win. 0/2=0, so 0 will win if everybody chooses 0.

Slide10

Slight alteration to problem

Suppose payoff is $1 if you answer 1 and 1 is not half of the average. Now what survives IDSDS?Slide11

Auctioning the crown jewelsSlide12

The auction

Two bidders, Sheik and Sultan

Sultan can bid odd number 1,3,5,7,9

Sheik can bid even number 2,4,6,8

Jewels are worth 8 to Sultan, and 7 to Sheik

Bidders submit a single sealed-bid. Jewels go to the high bidder at price he bids.Slide13

Payoff matrix

2

4

6

8

10

1

0,5

0,3

0,10,-10,-335,00,30,10,-10,-3

53,0

3,0

0,1

0,-1

0,-3

7

1,0

1,0

1,0

0,-1

0,3

9

-1,0

-1,0

-1,0

-1,0

0,-3

Sultan

Sheik

V is 8

V is 7Slide14

Nash equilibria

Sultan bids 7, sheik 6 Slide15

Problem 4.15(payoffs to firm 1)

1

2

3

4

never

1

-20

-14

-216402-9-9

32145

3

-2

-2

-2

16

40

4

1

1

1

1

25

never

0

0

0

0

0

Profit in round t if alone is 10t-15.

Profit in round t if both are in is 4t-15.

If 1 enters at 1 and 2 enters at 1, Total profits to 1 are 4-15+8-15+12-15+16-15=-20

If 1 enters at 1 and 2 at 2, then profits to 1 are 10-15+8-14+12-15+15-15=-14

If 1 enters at 1 and 2 at 3, then profits to 1 are 10-15+20-15+12-15+16-15=-2

etc.. etc..Slide16

Problem 4.15full payoff matrix (by symmetry)

1

2

3

4

never

1

-20, -20

-14, -9

-2, -216, 140, 02-9, -14-9, -93, -2

21, 145, 0

3

-2 ,-2

-2, 3

-2, -2

16, 1

40, 0

4

1, 16

1, 21

1, 16

1, 1

25, 0

never

0, 40

0, 45

0, 40

0, 25

0, 0

What strategies are strictly dominated?Slide17

Reduced payoff matrix

2

3

4

2

-9, -9

3, -2

21, 1

3

-2, 3-2, -216, 1 41, 211, 161, 1

Anything strictly dominant now?

Are there any Nash

equilibria

?

Describe the Nash equilibrium strategy profiles.Slide18

Problem 5.1a

Players can request either $20 or $100. If fewer than 20% request $100, everybody gets what they asked for. If 20% or more request $100, everybody gets nothing.

If there are 100,000 players, what are the Nash

equilibria

? Slide19

Clicker question

All of the Nash

equilibria

have 20,000 requesting $100 and 80,000 requesting $20.

All of the Nash

equilibria

have 19,999 requesting $100 and 81,001

reqeusting

$20.

The Nash equilibria include all outcomes where 20,001 or more people request $100 as well as the outcome where 19,999 demand $100.The Nash equilibria include all outcomes where 19,999 or more people request $100. Slide20

Problem 5.1b

Players can request $20, request $100 or make no request. In order to make a request you have to pay $21.95. If fewer than 20% of all players request $100, everybody gets what they asked for. If 20% or more request $100, everybody gets nothing.

If there are 100,000 players, what are the Nash

equilibria

? Slide21

Clicker question

With 100,000 players

The only Nash

equilibria

have 19,999 requesting $20 and no players requesting $100.

The only Nash

equilibria

have 19,999 requesting

$100 and no players requesting $20.All of the Nash equilibria have 19,999 requesting $100 and 81,001 requesting $20.The Nash equilibria include all outcomes where 19,999 or more people request $100Slide22

Problem 5.1a

Players can request $20, request $100 or make no request. In order to make a request you have to pay $19.95. If fewer than 20% of all players request $100, everybody gets what they asked for. If 20% or more request $100, everybody gets nothing.

If there are 100,000 players, what are the Nash

equilibria

? Slide23

Clicker question

With 100,000 players

The only Nash

equilibria

have 19,999 requesting $20 and no players requesting $100.

The only Nash

equilibria

have 19,999 requesting

$100 and no players requesting $20.All of the Nash equilibria have 19,999 requesting $100 and 81,001 requesting $20.The Nash equilibria include all outcomes where 19,999 or more people request $100Slide24

Problem 5.1b

Suppose that to submit a request, you also have to pay $21.95 for a subscription to Science and you value that subscription at 0.

Find Nash equilibriumSlide25

If mag costs $19.95

N.E. has 19,999 asking for $100 and the rest asking for $20.

Do you think this is what would happen if contest were really run on these terms?

Why or why not?Slide26

What if you ask for $20?

Asking for $20 is strictly dominated.

So nobody asks for $20.

If nobody asks for $20, asking for $100 is strictly dominated.

So only remaining strategy is not ask.Slide27

Problem 5.1c

Suppose that all is as in Part b, except that magazine subscription costs $19.95.

What Is Nash equilibrium?Slide28

Problem 5.3 Commuting problem

Cost of taking the toll road is 10 +x where x is the number who take the toll road.

Cost of taking back road is 2y where y is the number who take the back road. There are 100 drivers in all and drivers must take one of these roads, so y=100-x.

Find Nash equilibrium or

equilibria

.Slide29

First cut at problem

Is there an outcome where drivers are indifferent about which road to take?

If there is, it would be a Nash equilibrium, since if you went the other way it would take longer than going the way you are going.

Indifference if

10+x=2 y =2(100-x).

x=200-2x-10, so 3x=190, x=63.33

Not an integer. Now what.Slide30

Systematic answer

In Nash equilibrium, it must also be that those who take back road are better off than if they switched to toll road.

Currently cost to a back road guy is 2(100-x).

If he took the toll road there would be x+1 people on the toll road.

So staying on the back road is best response if 2(100-x)≤10+x+1 which implies

189≤3x and hence 63≤x.Slide31

Also:

In Nash equilibrium, those who take toll road are better off than they would be if they switched to back road

. If they switched to the back road then there would be 100-(x-1)=101-x drivers on the back road.

This implies that 10+x≤2(101-x) or equivalently 3x≤192, or x≤64Slide32

So when is there a Nash equilibrium?

When x≥63 and x≤64.

Both are true if and only if x=63 or x=64.

There is a Nash equilibrium if 63 take toll road and 37 take the back road.

There is another Nash equilibrium if 64 take toll road and 36 take back road.Slide33

Ordering dinnerSlide34

Diners’ Dilemma—The Menu

Item

Value

Price

Pasta

$21

14

Salmon

$2621Filet Mignon$2930Slide35

Strategic Form Payoffs:Two diners split the bill

Strategy

Pasta

Salmon

Steak

Pasta

7, 7

3.5, 8.5

-1, 7

Salmon 8.5, 3.5 5 , 5 .5, 3.5Steak7, -1 3.5, .5

-1,-1

Diner 1

Diner 2Slide36

What if there are 4 diners?

Lets think about it in a more general way.

What does it cost me to order steak rather than pasta?

My share of the bill goes up by (30-14)/4=4.

Value to me of having steak rather than pasta is 29-21=8.

So, no matter what the other guests are doing, I am better off ordering steak than pasta.Slide37

How about ordering salmon?

If I order steak rather than salmon, my bill goes up by (30-21)/4=2.25.

The value to me of my meal goes up by 29-26=3.

This is true no matter what the other guests are ordering.

So I am better off ordering steak than either salmon or pasta.

Ordering steak is a dominant strategy for all players.

The strategy profile where all order steak is the only Nash equilibrium.Slide38

A lousy outcome

In the only Nash equilibrium,

t

hey all order steak, even though they would all be better off

What do we make of this?

Do you think this effect is present in real life?Slide39

Good luck on your midterm!Slide40

Protest game

N citizens, different ones value protesting differently.

Order them by value of protest

v1>v2>…

vN

We can draw a “demand curve” for protesting: How many people would protest if cost is p.

We also have something like a “supply curve”.

What does it cost to protest if x people are protesting.Slide41

Lets draw them

Two downward-sloping curves. Where is equilibrium? There can be more than one equilibrium.