n-player games

N-Player Games

• A symmetric N-person game. 1) All players have same strategy sets2) 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.

Symmetric N-person games

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 6-day 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 wayB) 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 AB) Restaurant BC) Restaurant CD) 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) PicassoB) Sabado TardeC) TrigoD) 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) PicassoB) Sabado TardeC) TrigoD) Del Playa

Modified Coordination Game

Your payoff from attending the party atX 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) PicassoB) Sabado TardeC) TrigoD) 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 gameN 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 versionN 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 versionN 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