ECON 2G03 Lecture Notes - Lecture 8: Nash Equilibrium, Cooperative Game Theory

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limewolf4

16 Jan 2019

School

McMaster University

Department

Economics

Course

ECON 2G03

Professor

Krishnakali Sen Gupta

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Document Summary

Game- situation in which players (participants) make strategic decisions that take into account each other"s action and response. Strategy- rule or plan of action for playing a game. Optimal strategy- strategy that maximizes a player expected payoff. Non-cooperative versus cooperative games: cooperative game- game in which participants can negotiate binding contracts that allow them to plan joint strategies, non-cooperative game- game in which negotiation and enforcement and binding contracts are not possible. Dominant strategy- strategy that is optimal no matter what an opponent does: suppose firm a and b sell competing products and are deciding whether to undertake advertising campaigns. Each firm will be affected by the competitor"s decision. Equilibrium in dominant strategies- outcome of a game in which each firm is doing the best it can regardless of what its competitors are doing. Dominant strategies- i"m doing the best i can no matter what you do. You"re doing the best you can no matter what.

Related Questions

Objectives:

ÃÂ· Learn how individuals contribute to teamwork

ÃÂ· Experience some of the features of group work and teamwork

ÃÂ· Understand what managers and organizational developers do to transform

ÃÂ· groups into teams

ÃÂ· Articulate the tangible benefits (both quantitative and qualitative) of

ÃÂ· high-performing teams

ÃÂ· Finish with an interest in learning more about these concepts and

ÃÂ· techniques to apply what you learn

Background: For this assignment, you will plan and play a game with your family or friends, or at work based on the idea of the classic prisoner's dilemma. If you have had a class on game theory, you will be well aware of this concept. It forms the basis of many TV game shows. The prisoner's dilemma was illustrated in Truman Capote's book, "In Cold Blood" concerning the 1959 robbery of a Kansas farmhouse by Perry Smith and Dick Hickock, who murdered their victims in order to eliminate the witnesses. After the men were captured, the police interrogated them separately. To get a confession, the police offered the men a reduced sentence for cooperating. Failure to cooperate would result in a death penalty charge for both.

In the prisoner's dilemma, if both parties cooperate they are mildly punished; if one betrays another, one is severely punished while the other goes free; and if both betray one-another, both are moderately punished. Can you think of settings where you work in which the organizational structure has created a prisoner's dilemma? Competition can (but does not necessarily) bring out conflict.

In game theory, there are non-cooperative and cooperative games. A non-cooperative zero-sum game has a definite winner and loser. For one to win, one must lose. We often think of politicking as an element of that. A cooperative game is where everyone who plays is better off for having played than not having played the game. That is not to say that everyone is equally well off, but simply better off than they were before playing. Hence, even in a cooperative game, some people will likely benefit more than others.

Instructions

Review in your textbook, the Big Five personality dimensions, pp. 82Ã¢ï¿½ï¿½84, and then consider the 3.1 Personality Insights inventory, pp. 89Ã¢ï¿½ï¿½91
Select four or five friends, coworkers, or family members and have them take the personality inventory
Plan and play a game with the participants
Write a 5-10 page paper that addresses the assignment questions below

As you plan and play your game, you will negotiate the type of game you want to play and your role in it. Strive to determine some personal characteristics of the game participants a head of time, or by observing them. Consider how the personal characteristics of the individuals in the game manifest themselves in the informal roles they assume during the planning and execution of the game.

For the purpose of this assignment, you will want both a cooperative and a competitive element. To do that, you can form several groups to compete and cooperate with each other, or you can have individuals in one group compete and cooperate. You can use rewards and/or punishments to create the competitive/cooperative motivations.

Evaluation: When groups are formed, we want to consider how organizational structures, processes, and situations impact on group motivation, politics, and goals achievement. Alignment of all of these structural, personal, and interpersonal components will help to ensure that the group will meld into a team. A defining feature of teamwork (versus group work) is cooperation!

One of the factors that affect cooperation is conflict. Conflict is defined as when someone perceives that his/her interests are opposed or negatively impacted by another. Conflict can at times produce positive results depending on the basis for the conflict and how the conflict is handled (functional conflict). The worst kind of dysfunctional conflict produces betrayal.

Note that team performance can be observed in how well the team coordinates their interdependencies, works efficiently, communicates, and accomplishes their mission. Consequently, as you implement your game, assess your players based on these factors and the informal roles they play.

After you have played the game, reflect on the personality assessments your participants took, and then consider how they planned and played the game.

Write a (5- 10) page analysis to address the following questions:

1. How well did the personality assessments reflect in the behaviors participants displayed during the game?

2. What do these assessments tell us about how we perceive ourselves versus how others perceive us?

3. What informal roles did participants assume during the game, and were they different from the planning portion from the playing portion of the exercise?

4. What did I learn about myself? For example, did the exercise show that you have characteristic ways of relating to others that are distinctive, or similar, to those the others? Did the exercise show that in a particular type of situation you acted in a particular way, or that when others acted in a particular way, you felt happy or anxious or angry, etc.?

5. What did I learn about someone else in the exercise? Did you see something new about them, such as "they seemed more capable than I imagined," or "inattentive when things were stressful?"

6. In playing your game, what could you have done to increase cooperative behaviors?

7. Based on your game experience, what are some of the key factors involved in gaining cooperation from people in a business setting?

8. How would you go about getting cooperation from a "difficult" co-worker? Give an example or scenario.

Please provide a unique type of game theory preferably instead of Mafia Game.

Text:

Organizational Behavior and Management 10th Edition

John Ivancevich, Robert Konopaske, Michael Matteson

cyanfox187

Problem 1

In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).

PrisonerÃ¢â¬â¢s Dilemma

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	10,80
	defect	80,10	40,40

Stag Hunt

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	5,40
	defect	40,5	40,40

Chicken

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	50,80
	defect	80,50	40,40

1. For each game, draw a graph with player 1Ã¢â¬â¢s best response function (choice of p as a function of q), and player 2Ã¢â¬â¢s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.

2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.

3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?

Problem 2

Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability ÃÂ± to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other personÃ¢â¬â¢s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:

	Y	F		Y	F
Y	0, 0	0, 1	Y	0, 0	0, 1
F	1, 0	-1, 1	F	1, 0	1, -1
Player 2 is strong (ÃÂ±)			Player 2 is weak (1-ÃÂ±)			Player 2 is strong (ÃÂ±)

After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of ÃÂ± (ÃÂ± Ã¢â°Â¤ 1/2 or ÃÂ± Ã¢â°Â¥1/2).

jadebuffalo465

(Please show as much work as possible)

1. You are bidding in a second-price auction for a painting that you value at $800. You estimate that other bidders are most likely to value the painting at between $200 and $600. Which of these is likely to be your best bid?

a. $1,000

b. $800

c. $600

d. $400

2. Which of the following is true about different ways of conducting a private-value auction?

a. A first-price auction is strategically equivalent to a second-price auction.

b. A first-price auction is strategically equivalent to an English auction.

c. A second-price auction is strategically equivalent to an English auction.

d. None of the above

3. Suppose that five bidders with values of $500, $400, $300, $200, and $100 attend an oral auction. Which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

4. In the above auction, if the bidders with the first- and third-highest values ($500 and

$300) collude, which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

5. If a seller is concerned about collusion among bidders, which of the following changes to the auction, should the seller make?

a. Hold frequent, small auctions instead of infrequent large auctions.

b. Conceal the amount of winning bids.

c. Publically announce the name of each auction's winner.

d. Hold a second-price instead of a first-price auction.

6. You're holding an auction to license a new technology that your company has developed. One of your assistants raises a concern that bidders' fear of the winner's curse may encourage them to shade their bids. How might you address this concern?

a. Release your analyst's positive scenario for the technology's future profitability.

b. Release your analyst's negative scenario for the technology's future profitability.

c. Use an oral auction.

d. All of the above

7. In a first-price auction, you bid ________ your value, and in a second-price auction you bid _________ your value.

a. at; above

b. below; above

c. below; at

d. below; below

8. You hold an auction among three bidders. You estimate that each bidder has a value of either $16 or $20 for the item, and you attach probabilities to each value of 50%. What is the expected price? If two of the three bidders collude, what is the price?

9. In Sweden, firms that fail to meet their debt obligations are immediately auctioned off to the highest bidder. (There is no reorganization through Chapter 11 bankruptcy.) The current managers are often high bidders for the company. Why?

10. When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, representatives of other museums that have no chance of winning are actively wooed by the auctioneer to attend anyway. Why?

11. The deities Mars and Venus often do battle to create the weather conditions on Earth. Venus prefers extreme temperatures (especially heat), while Mars prefers temperate conditions. The payoffs (expressed in Points of Wrath) are given below.

		Venus
		Warm	Chill
Mars	Warm	20 , 0	0 , 10
Chill	0 , 90	20 , 0

What is the unique mixed-strategy equilibrium of the above game?

(Let p be the probability of "Warm" for Mars, and q the probability of "Warm" for Venus.)

a) p=9/10, q=1/2

b) p=1/2, q=1/10

c) p=1/2, q=1/2

d) p=1/10, q=1/10

Player 2

		H	D
Player 1	H	0 , 0	4 , 1
D	1 , 4	2 , 2

12. The above game is the title of the hawk-dove game and used by evolutionary biologists to describe evolutionary processes. It is also used to model how a business should grow. In the above game, what is the Nash equilibrium in pure strategies and mixed strategies.?

Assume the cost of producing the goods is zero and that each consumer will purchase each good as long as the price is less than or equal to value. Consumer values are the entries in the table.

	Good 1	Good 2
Consumer A	$2,300	$1,700
Consumer B	$2,800	$1,200

13. Suppose the monopolist only sold the goods separately. What price will the monopolist charge for good 1 to maximize revenues for good 1?

a. $2,300

b. $2,800

c. $1,200

d. $1,700

14. What is the total profit to the monopolist from selling the goods separately?

a. $4,500

b. $6,300

c. $7,000

d. $6,000

15. What is a better pricing strategy for the monopolist? At this price, what are the total profits to the monopolist?

a. Bundle the goods at $2,800; Profits = $5,600

b. Bundle the goods at $4,000; Profits = $8,000

c. Charge $2,800 for good 1 and charge $1,700 for good 2; Profits = $4,500

d. Charging the lowest price for each good individually is the best pricing strategy; Profits = $7,000

16. The prisoners' dilemma is an example of

a. a sequential game.

b. a simultaneous game.

c.a shirking game.

d. a dating game

17. Nash equilibrium

a. is where one player maximizes his payoff, and the other doesn't.

b. is where each player maximizes his own payoff given the action of the other player.

c.is where both players are maximizing their total payoff.

d. is a unique prediction of the likely outcome of a game.

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ECON 2G03 Lecture Notes - Lecture 8: Nash Equilibrium, Cooperative Game Theory

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