Player - PowerPoint Presentation

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PlayerFirst choiceSecondChoiceThird ChoiceFourth ChoiceFifth ChoiceBoromirBoromirFrodo NobodyLegolasGimliGimliGimliFrodoNobodyBoromirLegolasLegolasLegolasFrodoNobodyGimliBoromirFrodoLegolasGimliBoromirFrodoNobody

Preferences in a Unanimous Choice Game

Is there a symmetric equilibrium?

Problem 4.1Slide2

Figure PR4.2 Modified Driving Conventions GameHarrington: Games, Strategies, and Decision Making, First EditionCopyright © 2009 by Worth PublishersProblem 2: Find all of the Nash equilibriaSlide3

Figure PR4.3 Team ProjectHarrington: Games, Strategies, and Decision Making, First EditionCopyright © 2009 by Worth PublishersProblem 4.3: What are the Nash equilibria?Slide4

Figure PR4.4Harrington: Games, Strategies, and Decision Making, First EditionCopyright © 2009 by Worth PublishersProblem 4.4; what strategies survive IDSDS?What strategy profiles are Nash equilibria?Slide5

Problem 4.13; Find all of the Nash equilibria for this 3-player game

Now what do we do?Slide6

N-Player GamesSlide7

A symmetric N-person game. All players have same strategy setsIf 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 gamesSlide8

A commuting game.You have two ways to commute from home to work. The short way by narrow roadThe long way by freewayCommute time by freeway is always 30 minutes. Commute time by narrow road depends on how many others take narrow road. Slide9

Your choiceIf N people go short way, it takes 21+N/2 minutes to make the trip.Freeway always takes 30 minutesYou hate commuting and want to minimize travel time.Choose your route using Clickers. We’ll do this repeatedly, simulating a week of work days.Slide10

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.Slide11

PaymentsWe 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.Slide12

This time I will travel by the A) Short wayB) FreewaySlide13

Nash equilibriumIn 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? Slide14

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. Slide15

Widening the short roadWhat 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?Slide16

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 roadTo 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. Slide17

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) PicassoB) Sabado TardeC) TrigoD) Del PlayaSlide18

What are the Nash equilibria?Slide19

Weakest Link GamesExample: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} action. Player i’s action choice denoted siWeakest link version. Payoff to player i is 20 min{s 1,s2,…,sN}-10 si. Slide20

Clicker game-weakest linkThe payoff to a player who chooses effort level Ewill 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) 5Slide21

Nash equilibria for Airline Security Weakest Link gameNo 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.Slide22

Best shot GamesExample: 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. Slide23

Clicker game-best shotThe payoff to a player who chooses effort level Ewill 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) 5Slide24

EquilibriaCan’t have two players choosing more than the minimum.Can’t have all players choosing minimum.What are the equilibria?Slide25

Clicker game Average effortThe payoff to a player who chooses effort level Ewill 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) 5Slide26

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. Slide27

Sex with ClickersPretend 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