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PlayerFirst choice Second Choice Third Choice Fourth Choice Fifth Choice Boromir FrodoNobodyLegolasGimli FrodoNobodyBoromirLegolas FrodoNobodyGimliBoromir FrodoLegolasGimliBoromirFrodoNobody Preferences in a Unanimous Choice Game Is there a symmetric equilibrium? Problem 4.1Problem 4.1

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Figure PR4.2 Modified Driving Conventions Game Harrington: Games, Strategies, and Decision Making, First Edition Copyright © 2009 by Worth Publishers Problem 2: Find all of the Nash equilibria

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Figure PR4.3 Team Project Harrington: Games, Strategies, and Decision Making, First Edition Copyright © 2009 by Worth Publishers Problem 4.3: What are the Nash equilibria?

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Figure PR4.4 Harrington: Games, Strategies, and Decision Making, First Edition Copyright © 2009 by Worth Publishers Problem 4.4; A)what strategies survive IDSDS? B)What strategy profiles are Nash equilibria?

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Problem 4.13; Find all of the Nash equilibria for this 3-player game Now what do we do?

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N-Player Games

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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. Symmetric N-person games

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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.

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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?

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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.

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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.

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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.

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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.

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This time I will travel by the A) Short way B) Freeway

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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?

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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+x 2 /2. When are they minimized? Hint: Use calculus.

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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?

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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.

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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

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What are the Nash equilibria?

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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 s i Weakest link version. Payoff to player i is 20 min{s 1,s 2,…,s N }-10 s i.

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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

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Nash equilibria for Airline Security Weakest Link game No Nash equilibrium has any player choosing higher level of s i than any other player. Why? Any level of security is a Nash equilibrium. Some equilibria better for all airlines than others. Explain.

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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 s i Payoff to player i is 20max{s 1,s 2,…,s N }-10s i.

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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

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

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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

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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 N m be the number of adult males and N f the number of adult females. The average number of children for each male is C/N m 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.

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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

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Nash’s Theorem Theorem (Nash, 1951): Every finite game (finite number of players, finite number of pure strategies) has at least one mixed-strategy Nash.

Nash’s Theorem Theorem (Nash, 1951): Every finite game (finite number of players, finite number of pure strategies) has at least one mixed-strategy Nash.

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