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Two neighboring homeowners, i = 1,2, simultaneously choose how many hours to spend maintaining a lawn. The AVERAGE benefit per hour for i is
(e.g., it is for homeowner 1)
And the (opportunity) cost per hour for each homeowner is 4.
(a) Give each homeowner’s (net) payoff as a function of and .
(b) Compute the Nash equilibrium.
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- A gambling game requires people to place bets on either "Big" or "Small" or "Seven" designating the result of two dice throws. Big represents a sum of 8 to 12, while small is from 2 to 6. The special space "seven" is reserved for, of course, 7. Betting on "Big" provides a 1:1 betting win ratio amount. (Ex: Any bet amount of X, will net the winner another amount X, and still get his bet amount X back. ) Betting on "Small" also provides a 1:1 betting win ratio. The space "Seven" gives a 1:5 win ratio amount: Betting 100 money units on "Seven" and having a result of 7 nets the bettor 500 money units, and still have his 100 money units returned to him. If your bet does not appear, then the player loses his/her bet. What is the net Expected value of the strategy of betting 100 money units on "Big" ? Give your answer to two decimal places.N=2 video broadcasting websites, You and Twi, must decide the number of minutes of ads to be displayed for every video that the user elects to watch. Let tY be the number of ad-minutes per video set by You, and tT the number of ad-minutes per video set by Twi. Streaming one video costs You cY=0.02, while it costs Twi cT=0.03. There are 100 million potential users, and each watches videos according to the following demand curves: qY((tY,tT) =10-2tY+tT=10-2tT+tY a- What is the cross-price elasticity between You and Twi? b- Suppose, for now, that You and Twi enter an (illegal) agreement by which they set tY=tT=t Derive the total number of users in the market as a function of t. Derive the profits for each website as a function of t. c- Now let the two platforms compete by each setting their number of ad-minutes: i. What is the best reply of You? What is the best reply of Twi? ii. Find the Nash Equilibrium of the game. iii. How many total users choose You and how many total users choose…(a) Assuming that each fishery chooses fi ∈ (0, F), to maximize its payoff function, derive the players’ best response functions and find a Nash equilibrium. (b) Is the equilibrium you found in (a) unique or not? What are equilibrium payoffs? 4 (c) Suppose that a benevolent social planner wants maximize the utility of both fisheries. In other words, the social planner solves the following problem: max w(f1,f2) w(f1, f2) = u1(f1, f2) + u2(f1, f2) =2 ln(f1) + 2 ln(f2) + 2 ln(F − f1 − f2). Find the social planner’s solution. (d) What are the fisheries’ payoffs if the quantities of fish they catch are solutions to the social planner’s problem? What can you say about the Nash equilibrium quantities of fish being caught as compared to the social planner’s solution? (e) If fishery j decides to follow the recommendation of the social planner, how much fish will firm i catch?
- Suppose two players, First and Second, take part in a sequential-move game. First moves first, Second moves second, and each player moves only once. (a). Draw a game tree for a game in which First has two possible actions (Up or Down), and Second has three possible actions (Top, Middle, or Bottom) at each node. Show which nodes are terminal/decision and write down all the (pure) strategies of each player. (b). Draw a game tree for a game in which First and Second each have three possible actions (Sit, Stand, or Jump) at each node. Show which nodes are terminal/decision and write down all the (pure) strategies of each player.(a) Assuming that each fishery chooses fi ∈ (0,F), to maximize its payoff function, derive the players’ best response functions and find a Nash equilibrium. (b) Is the equilibrium you found in (a) unique or not? What are equilibrium payoffs? (c) Suppose that a benevolent social planner wants maximize the util- ity of both fisheries. In other words, the social planner solves the following problem: max w(f1, f2) = u1(f1, f2) + u2(f1, f2) (f1 ,f2 )= 2ln(f1)+2ln(f2)+2ln(F −f1 −f2). Find the social planner’s solution. (d) What are the fisheries’ payoffs if the quantities of fish they catch are solutions to the social planner’s problem? What can you say about the Nash equilibrium quantities of fish being caught as compared to the social planner’s solution? (e) If fishery j decides to follow the recommendation of the social planner, how much fish will firm i catch?There are two students, each of whom can choose to plant flowers (F) or not to plant flowers (N). If both students plant flowers, then the (flower appreciation) benefit is 120 to each player. If exactly one student plants flowers, then the benefit is 65 to each player. The cost to either player of planting flowers is c > 0, regardless of the other player's choice. Each student starts (before any costs or benefits) with a payoff of zero. The above assumptions lead to the following game in strategic form. F N 120 – c, 120 – c 65 — с, 65 65, 65 — с 0,0 Suppose thatc< 120. This game is a prisoners' dilemma if and only if the parameter c is strictly greater than which value?
- Roger and Michelle like going out and enjoy each others’ company, but have different taste in entertainment. Michelle would like to go to a professional basketball game, while Roger prefers opera. If they fail to agree to go together to either the game or to the opera this Saturday night they will stay at home and be miserable. Both will be worse off than if they had gone together to either the game or the opera. (a) Construct a pay-off matrix for this game. (You can make up your own numbers for this problem; just make sure the numbers correctly depict the given situation.) (b) Find two Nash equilibria for this game. Explain why they are indeed Nash equilibria.Problem 4 A person has been mugged in the street by a thief, and there are n witnesses. All of the witnesses prefer someone else to chase the thief but none of them want to be the person trying to catch her. Each person gets 0 utility if the crime goes unreported, v if the crime is reported by someone else, and v −c if they report the crime. Moreover, we assume that the cost of chasing the thief is less than the utility of catching the thief: c < v. a. Define the strategic game described above. b. Find players’ best responses. c. Find the Nash equilibria of the game.Firm 1 and Firm 2 compete in the same product market by setting quantities q₁ and 92, respectively. They have the same marginal cost c₁ = C₂ = 40. The market demand is p=1240 Q, where Q = 91 +92. 19. Suppose the game is played for one period. What are the equilibrium quantities? (a) q₁ = 1000, 92 1000 (b) q₁ = 800, q2 = 800 (c) 91 = 600, q2 = 600 (d) 9₁ = 400, q2 = 400 (e) 9₁ = 200, q2 = 200 In questions 19-21, suppose that this game is repeated for infinitely many periods. Firms have the same discount rate 8. Consider the situation in which the two firms are trying to implement a collusion scheme. With this intent, they employ the following grim-trigger strategy: in the first period, each firm produces the collusion quantity; from period two onwards, the firm keep producing the collusion quantity as long as no firm has ever pro- duced any quantity other than the collusion quantity; otherwise, the firm plays the Nash Equilibrium of the stage game. As you will see, what varies from one…
- N residents witness a fire in a building. Each resident can independently choose to help or not. As long as someone helps, the fire is extinguished and every resident gets a benefit of b = 5. However, any resident stopping to help pays a personal cost of c = 4, due to the possible injury. If no resident helps the fires causes damage to the building, and the payoff to all players is 0. That is, if a player decides to help, her payoff is 5 - 4 = 1; if she does not help and at least one of the other residents helps, her payoff is 5; and, if she does not help and neither do the other residents, her payoff is 0. Solve for the symmetric mixed strategy Nash equilibrium when N = 2. What is the probability that at least one resident stops to help? Hint: The probability that at least one player stops to help is equal to the probability that either of the players stops to help or that both players stop to help. A convenient way of computing this probability is one minus the probability that none…Question 3 A city park has become a popular hotspot for food truck vendors, offering a variety of cuisines to passersby. The food trucks offer free meals to homeless groups, but university students often exploit this charity by waiting there. Both groups gain a payoff of 10 units for receiving a free meal. For homeless individuals, the cost of waiting in line is calculated as t2/320. Similarly, for college students, the cost of waiting is also t2/160. Here's the challenge: the food truck vendors cannot differentiate between homeless individuals and college students just by looking at them. This means they cannot know whether someone in line genuinely needs the free meal due to their homeless status or is a college student trying to score a complimentary meal. 1. What is the minimum wait time t to separate types? 2. After a while, the food truck vendors find they can successfully identify and turn away student students half the time. Students who are turned away receive no free meal and…Suppose that Jason and Chad each are thinking of opening up a diet coke stand on the fourth floor of this building. Suppose that potential customers are evenly spaced on a distance that is normalized to 1. Customers will buy a diet coke from whichever stand requires the least walking. If they are the same distance the customer will flip a coin. This is depicted below. 1/4 1/2 3/4 Suppose that Jason and Chad are simultaneously choosing the location of their stands, what is the Nash Equilibrium location? a. One of them puts a stand at 3/4 and the other puts a stand at 1/4 b. Chad and Jason put their stands right next to each other at 1/2 c. One of them puts a stand at 0 and the other puts a stand at 1 d. There is no Nash Equilibrium