Working Some Problems
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.
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
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.
What is left?
Student 2
Student 1
0
1
2
0
1
2
10,8
9, 8
8, 8
10, 7
9, 7
8, 7
8, 8
9, 6
8, 6
More elimination
• Now 1 dominates 2 for student 1
• 0 dominates 1 for student 2
Student 2
Student 1
0
1
0
10, 8
9, 8
2
8,8
9,6
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
V is 7
2
4
6
8
10
1
0,5
0,3
0,1
0,-1
0,-3
3
5,0
0,3
0,1
0,-1
0,-3
Sultan
5
3,0
3,0
0,1
0,-1
0,-3
V is 8
7
1,0
1,0
1,0
0,-1
0,3
9
-1,0
-1,0
-1,0
-1,0
0,-3
Nash equilibria
• Sultan bids 7, sheik 6
Problem 4.15
(payoffs to firm 1)
1
2
3
4
never
1
-20
-9
-2
1
0
2
-14
-9
-2
1
0
3
-2
3
-2
1
0
4
16
21
16
1
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..
never
40
45
40
25
0
Problem 4.15
full payoff matrix (by symmetry)
1
2
3
4
never
1
2
3
4
never
-20, -20
-9, -14
-2 ,-2
1, 16
0, 40
-14, -9
-9, -9
-2, 3
1, 21
0, 45
-2, -2
3, -2
-2, -2
1, 16
0, 40
16, 1
21, 1
16, 1
1, 1
0, 25
40, 0
45, 0
40, 0
25, 0
0, 0
What strategies are strictly dominated?
Reduced payoff matrix
2
3
4
2
-9, -9
3, -2
21, 1
3
-2, 3
-2, -2
16, 1
4
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
A) All of the Nash equilibria have 20,000 requesting
$100 and 80,000 requesting $20.
B) All of the Nash equilibria have 19,999 requesting
$100 and 81,001 reqeusting $20.
C) The Nash equilibria include all outcomes where
20,001 or more people request $100 as well as
the outcome where 19,999 demand $100.
D) 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
A) The only Nash equilibria have 19,999 requesting
$20 and no players requesting $100.
B) The only Nash equilibria have 19,999 requesting
$100 and no players requesting $20.
C) All of the Nash equilibria have 19,999 requesting
$100 and 81,001 requesting $20.
D) The Nash equilibria include all outcomes where
19,999 or more people request $100
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?
Clicker question
• With 100,000 players
A) The only Nash equilibria have 19,999 requesting
$20 and no players requesting $100.
B) The only Nash equilibria have 19,999 requesting
$100 and no players requesting $20.
C) All of the Nash equilibria have 19,999 requesting
$100 and 81,001 requesting $20.
D) The Nash equilibria include all outcomes where
19,999 or more people request $100
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 equilibrium
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?
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.
Problem 5.1c
• Suppose that all is as in Part b, except that
magazine subscription costs $19.95.
• What Is Nash equilibrium?
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
Diner 2
Diner 1
Strategy Pasta
Salmon Steak
Pasta
7, 7
3.5, 8.5 -1, 7
Salmon
8.5, 3.5 5 , 5
.5, 3.5
Steak
7,
-1,-1
-1
3.5,
.5
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?
• Do you think this effect is present in real life?
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.