P
r
o
b
Player
l
e
m
Boromir
Preferences in a Unanimous Choice Game
4
Gimli.
1
Legolas
Frodo
First
choice
Second
Choice
Third
Choice
Fourth
Choice
Fifth Choice
Boromir Frodo
Nobody
Legolas
Gimli
Gimli
Frodo
Nobody
Boromir Legolas
Legolas
Frodo
Nobody
Gimli
Boromir
Legolas
Gimli
Boromir
Frodo
Nobody
Is there a symmetric equilibrium?
Problem 2: Find all of the Nash equilibria
Figure PR4.2 Modified Driving
Conventions Game
Problem 4.3: What are the Nash equilibria?
Figure PR4.3 Team Project
Harrington: Games, Strategies, and
Problem 4.4;
A) what strategies survive IDSDS?
B) What strategy profiles are Nash equilibria?
Figure PR4.4
Harrington: Games, Strategies, and
Problem 4.13; Find all of the Nash
equilibria for this 3-player game
Now what do we do?
N-Player Games
Symmetric N-person games
• A symmetric N-person game.
1) All players have same strategy sets
2) If you switch two players’ strategies, you switch
their payoffs and leave other players’ payoffs
unchanged.
• Special case of symmetric game—Your payoff
depends on what you do and the sum of the
actions taken by others.
A commuting game.
• You have two ways to commute from home to
work.
– The short way by narrow road
– The long way by freeway
• Commute time by freeway is always 30
minutes.
• Commute time by narrow road depends on
how many others take narrow road.
Your choice
• If N people go short way, it takes 21+N/2
minutes to make the trip.
• Freeway always takes 30 minutes
• You hate commuting and want to minimize
travel time.
• Choose your route using Clickers. We’ll do this
repeatedly, simulating a week of work days.
Your score
• You will get more points, the less your total
time spent commuting.
• You must choose one way or the other. If you
don’t click either option, you will be assessed
1 hour commuting time for that day.
Payments
• We will repeat this experiment 6 times (a 6day work week). Your score will be 150 minus
the total amount of time you spend
commuting.
• I will randomly choose one of the persons
with the highest score (least time spent
commuting) and give that person a prize of
$10.
This time I will travel by the
A) Short way
B) Freeway
Nash equilibrium
• In Nash equilibrium for this game, nobody
would want to change strategies.
• This will happen if 30=21+N/2, which implies
that N=18.
• So the Nash equilibrium is for 18 persons to
use the short way and everybody to spend 30
minutes commuting.
• Is this efficient? What would be efficient?
An efficient solution would minimize
total commuting costs.
Suppose that the class has C members.
Let x be the number of people who use short
road and C-x the number who use the freeway.
Total commuting costs are (C-x)30+x(21+x/2)=
30C-9x+x2/2. When are they minimized?
Hint: Use calculus.
Widening the short road
• What would happen if the local government
spent some money and doubled the capacity
of the short road.
• Then the time it would take to drive on the
short road when N people use it would be
21+N/4 instead of 21+N/2.
• What would the new equilibrium be?
• Is anybody better off?
What if tolls were charged?
• Suppose that all people value their time at v per
minute.
What is the equilibrium outcome with a toll of T on the
crowded road
• To equalize costs going the two ways, set
30v=(21+X/2)v+T.
This implies 30=21+X/2+T/v and X=18-2(T/v).
• If you want efficient use of the road, you would have
X=9.
Then 9=18-2(T/v) so 9=2(T/v) and T=(9/2)v.
• So, for example if if v=$1/4$, then a toll of
T=$9/8=$1.125 would get you 9 users.
Which party?
There are 4 possible parties that you could attend. One is
on Picasso Road, one is on Trigo, one is on Sabado Tarde ,
and one is on Del Playa. Your payoff is equal to the total
number of people at the party you choose so long as
there are no more than 35 people there. If more than 35
are at your party, the police will shut it down and your
payoff is 0.
Which party do you choose?
A) Picasso
B) Sabado Tarde
C) Trigo
D) Del Playa
What are the Nash equilibria?
Weakest Link Games
Example:
Airline Security Game- A weakest link game
N players—Strategy set for any player is a list of
possible levels of security {1,2,3,4, 5} action.
Player i’s action choice denoted si
Weakest link version. Payoff to player i is
20 min{s 1,s2,…,sN}-10 si.
Clicker game-weakest link
The payoff to a player who chooses effort level E
will be 20 Min -10E where E is the level of effort
chosen by that player and where Min is the
smallest effort level chosen by anyone in the
room.
My effort level is:
A) 1 B)2 C)3 D) 4 E) 5
Nash equilibria for Airline Security
Weakest Link game
• No Nash equilibrium has any player choosing
higher level of si than any other player. Why?
• Any level of security is a Nash equilibrium.
• Some equilibria better for all airlines than
others. Explain.
Best shot Games
Example:
N players—Strategy set for any player is a list of
possible levels of effort {1,2,3,4, 5}. Player i’s
action choice denoted si
Payoff to player i is
20max{s1,s2,…,sN}-10si.
Clicker game-best shot
The payoff to a player who chooses effort level E
will be 20 Max -10E where E is the level of effort
chosen by that player and where Max is the
minimum effort level chosen by anyone in the
room.
My effort level is:
A) 1 B)2 C)3 D) 4 E) 5
Equilibria
• Can’t have two players choosing more than
the minimum.
• Can’t have all players choosing minimum.
• What are the equilibria?
Clicker game Average effort
The payoff to a player who chooses effort level E
will be 20 Ave -10E where E is the level of effort
chosen by that player and where Ave is the
average effort level chosen by those in the
room.
My effort level is:
A) 1 B)2 C)3 D) 4 E) 5
Evolutionary theory of Sex Ratios
• Why do almost all mammals have essentially equal
numbers of sons and daughters?
• Every child that is born has exactly one mother and
one father. Let C be the number of children born in the
next generation. Let Nm be the number of adult males
and Nf the number of adult females. The average
number of children for each male is C/Nm and the
average number of children for each female is C/N f
• The rarer sex will have more children on average.
• If one sex is more rare, then mutations that make you
have babies of that sex will prosper.
Sex with Clickers
• Pretend that you are going to have a child and
that you seek to maximize your number of
descendants. You can choose to have either a
boy or a girl. Where B is the total number of
boys chosen and G the number of girls, the
expected payoffs are 100/B for having a boy
and 100/G for having a girl.
• Press A for Boy
• Press B for Girl