Download N-Player Games

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.
Clicker Game
• Choose an integer from 1 to 5. Where X is the
average of the numbers chosen by class
members, Your payoff will be 1 if your number
is the closest number to (3/4) X. Otherwise it
will be 0.
• Press A for 1, B for 2, C for 3, D for 4, E for 5.
Iterated Deletion of Dominated
Strategies
• Would a smart player choose 5?
• If you believe that everybody in the class is
smart, would you ever choose 4?
• If you believe that everybody in the class is
smart and believes everybody else is smart,
what would you conclude?
• What if you take this another step?
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 commuter 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?
• 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.
Another congestion game
There are 4 restaurants to choose from, A, B, C, and
D. You think they are all equally good, but you
prefer to go to one that is less crowded. Your
payoff is 50-N where N is the number of people
who go to the restaurant that you choose. Which
restaurant do you choose?
A) Restaurant A
B) Restaurant B
C) Restaurant C
D) Restaurant D
A Coordination game
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 who choose
the same party that you do.
Which party do you choose?
A) Picasso
B) Sabado Tarde
C) Trigo
D) Del Playa
A Coordination game
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 who choose
the same party that you do.
Which party do you choose?
A) Picasso
B) Sabado Tarde
C) Trigo
D) Del Playa
Modified Coordination Game
Your payoff from attending the party at
X is the number of people who choose this party
plus Y(X) where Y(X) is written on the blackboard.
Which party do you attend?
A) Picasso
B) Sabado Tarde
C) Trigo
D) Del Playa
What about Nash equilibrium in these
coordination games?
• More than one Nash equilibrium. Some
equilibria may be “better than others”.
• No guarantee that the “best one” happens.
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, 6, 7}
action. Player i’s action choice denoted si
Weakest link version. Payoff to player i is
50 + 20 min{s 1,s_2,…,sN}-10 si.
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:
Airline Security Game- A best shot version
N players—Strategy set for any player is a list of
possible levels of security {1,2,3,4, 5, 6, 7}
action. Player i’s action choice denoted si
Weakest link version. Payoff to player i is
50 + 20max{s 1,s_2,…,sN}-10si.
Equilibria
• Can’t have two players choosing more than
the minimum.
• Can’t have all players choosing minimum.
• What are the equilibria?
Average Effort Games
Example:
Airline Security Game- A best shot version
N players—Strategy set for any player is a list of
possible levels of security {1,2,3,4, 5, 6, 7}
action. Player i’s action choice denoted si
Weakest link version. Payoff to player i is
20(s1+s_2+sN)/N-10si.
What are the Nash equilibria if N>2? If N=1?
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