Working Some Problems

Problem 3.10 Students 1 and 2 can each exert study levels {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. High score gets A, low 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.

What can we do with IDSDS? What is the lowest payoff you could get with effort level 1? 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

What is left? Student 2 1 2 10,8 10, 7 8, 8 9, 8 9, 7 9, 6 8, 7 8, 6 Student 1

More elimination Now 1 dominates 2 for student 1 2 10, 8 8,8 1 9, 8 9,6 Student 1

What now? What is (are) Nash equilibrium? What survives IDWDS? How do you interpret this?

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.)

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.

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

Auctioning the crown jewels

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.

Payoff matrix Sheik Sultan V is 7 2 4 6 8 10 1 0,5 0,3 0,1 0,-1 0,-3 3 5,0 5 3,0 7 1,0 9 -1,0 Sultan V is 8

Nash equilibria Sultan bids 7, sheik 6

Problem 4.15 (payoffs to firm 1) 2 3 4 never -20 -14 -2 16 40 -9 21 45 25 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..

Problem 4.15 full payoff matrix (by symmetry) 2 3 4 never -20, -20 -14, -9 -2, -2 16, 1 40, 0 -9, -14 -9, -9 3, -2 21, 1 45, 0 -2 ,-2 -2, 3 1, 16 1, 21 1, 1 25, 0 0, 40 0, 45 0, 25 0, 0 What strategies are strictly dominated?

Reduced payoff matrix 2 3 4 -9, -9 3, -2 21, 1 -2, 3 -2, -2 16, 1 -9, -9 3, -2 21, 1 -2, 3 -2, -2 16, 1 1, 21 1, 16 1, 1 Anything strictly dominant now? Are there any Nash equilibria? Describe the Nash equilibrium strategy profiles.

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?

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.

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?

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 $100

Problem 5.1c 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?

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 $100

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.

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.

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.

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≤64

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.

Ordering dinner

Diners’ Dilemma—The Menu Item Value Price Pasta $21 14 Salmon $26 21 Filet Mignon $29 30

Strategic Form Payoffs: Two diners split the bill Strategy Pasta Salmon Steak 7, 7 3.5, 8.5 -1, 7 8.5, 3.5 5 , 5 .5, 3.5 7, -1 3.5, .5 -1,-1 Diner 1

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.

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.

A lousy outcome In the only Nash equilibrium, they all order steak, even though they would all be better off What do we make of this? Does this imply that Nash equilibrium is a useless concept?

Good luck on your midterm!

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.

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