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Exercise Set WS 12/13 Game Theory Exercise Set Question 1 (GE with production) In an economy, there are two consumers (i = 1; 2), two firms (j = 1; 2) and two goods (labor/leisure and rice). Each consumer is endowed with 1 unit of leisure but no rice. Denote xi as consumer i's consumption of rice and li leisure. (Note that if consumer i consumes li amount of leisure, she has (1 − li) amount of labor available to sell.) The utility they derive from the consumption bundle (xi; li) is ui(xi; li) = ln(xi) + ln(li). The two firms use labor to grow rice. Let yj denote firm j's output of rice and Lj firm j's input of labor. Firm 1's production function is y1 = AL1 where A is a positive real. 1 Firm 2's production function is y2 = (L2) 2 . Firm 1 is owned by consumer 1 and firm 2 is owned by consumer 2. The earnings of a firm are paid lump-sum to the owner. a) Define a competitive equilibrium for this economy. b) Assume both firms produce positive output. Calculate the competitive equilibrium price vector and allocation, as a function of the parameter A. (Hint: Normalize the price of rice to 1.) c) For what values of the parameter A will we have a corner solution, where one of the firms produces zero output? Question 2 (GE with uncertainty) In a village, there are 150 consumers and one physical commodity (rice) per state of nature. For i = 1; :::; 150, consumer i is a von Neumann-Morgenstern expected utility maximizer with (Bernoulli) utility of certain consumption given by ui(xi) = ln(xi). For i = 1; :::; 100, consumer i lives on the highlands, and has an endowment of 1 in all states of nature. For i = 101; :::; 150, consumer i lives on the flood plain. When a flood occurs, all con- 1 sumers living on the plain receive an endowment of 0. The probability of a flood is 10 . 9 When a flood does not occur (probability 10 ), all consumers living on the plain receive an endowment of 1. a) If the entire village trades state-contingent commodities, define a competitive equili- brium for this economy. b) Calculate the competitive equilibrium price vector and allocation. (Hint: Normalize p2 to 1. (Hint: treat rice in the flood state and in the no flood state as two different goods.) 1 WS 12/13 Game Theory Question 3 (Nash-Equilibrium in a three player game) To prevent humans from being bitten, the government offers one unit of banked blood per day to every vampire for free. The blood bank of the capital has three units of banked blood of different conditions. Suppose that three vampires want to go out for lunch and are interested in these blood units. The manager of the blood bank makes a special offer: with a small "contribution" of 20 Dollars a vampire can increase the probability of recei- ving a good unit. Dracula, one of the three vampires, just like his two colleagues, doesn't want to pay the contribution. Because of different expected quality, the vampires value the best unit with 150 Dollars, the second best unit with 100 Dollars and the worst with 50 Dollars. The allocation of the units occurs according to the following: If everyone or no one pays, the blood units will be randomly assigned, and Dracula assumes that the probabilities of getting any of the three blood units are the same. If two and only two vampires pay the contribution, these two will get the best two units with the same probability. The one who didn't pay will get the worst unit. If one and only one pays, that vampire gets the best unit and the other two have equal probabilities receiving the worst unit. Model the game in normal form. Assume that the vampires have common valuation of the banked blood, what can you recommend to Dracula? Question 4 (Dividing Money) Two people have 10 Dollars to divide between themselves. They use the following procedure. Each person names a number of Dollars (a nonnegative integer), at most equal to 10. If the sum of the amounts that the people name is at most 10, then each person receives the amount of money she named (and the remainder is destroyed). If the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the remaining money. If the sum of the amounts that the people name exceeds 10 and the amounts named are the same, then each person receives 5 Dollars. Determine the best response of each player to each of the other player's actions; plot them in a diagram and thus find the Nash-Equilibria of the game. 2 WS 12/13 Game Theory Question 5 (Tullock Contest) Assume that there are only two producers on the market for refuelling aircrafts: Bobus and Airing. The Air Force of Tasmania is using old aircrafts made in the late 50's. Accor- dingly, the minister of defence wants to buy new aircrafts, either from Airing or Bobus. The revenue of the contract is the same for both producers. Accordingly, it is denoted by V . But only one producer can get the contract. To convince the minister, both firms invest in marketing and bribery. The expenditures of Airing are given by xA and those of Bobus are given by xB. The probability that the minister chooses one firm is given by ( xi for xA + xB > 0 xA+xB pi = 1 i = A; B: 2 for xA + xB = 0 a) What is a strategy of a producer in the given game? What is the expected revenue for Airing and for Bobus? Is it possible that xA + xB = 0? (Suppose that there is no communicationn between Airing and Bobus.) b) Calculate the Nash-Equilibrium (xA and xB) as well as xA +xB! What does a producer earn in expectation? Suppose that Bobus can cooperate with a local producer. If the local producer is contrac- ted by Bobus, then the local producer invests in convincing the minister. Bobus does only pay a share of V if Bobus can sell the aircrafts. If Airing can sell the aircrafts, then Bobus pays nothing, but the local partner has to bear his effort xL. c) What share ∗ does Bobus offer to the local partner? Assume that Bobus first contracts the local partner and, afterwards, Airing and the local partner invest in convincing the minister simultaneously. (Hint: Use backward induction.) d) Assume that an unit of effort made by the local partner is worth two units of effort made by Airing. Calculate the share ∗∗ that Bobus is offering his local partner, now. 3 WS 12/13 Game Theory Question 6 (War of Attrition) The game known as the War of Attrition elaborates on the ideas captured by the game Hawk-Dove. It was originally posed as a model of a conflict between two animals fighting over prey. Each animal chooses the time at which it intends to give up. When an animal gives up, its opponent obtains all the prey (and the time at which the winner intended to give up is irrelevant). If both animals give up at the same time then each has an equal chance of obtaining the prey. Fighting is costly: each animal prefers as short a fight as possible. The game models not only such a conflict between animals, but also many other dis- putes. The "prey" can be any indivisible object, and ”fighting" can be any costly activity - for example, simply waiting. To define the game precisely, let time be a continous variable that starts at 0 and runs indefinitely. Assume that the value party i attaches to the object in dispute is vi > 0 and vi the value it attaches to a 50% chance of obtaining the object is 2 . Each unit of time that passes before the dispute is settled (i.e. one of the parties concedes) costs each party one unit of payoff. Thus if player i concedes first, at time ti, her payoff is −ti (she spends ti units of time and does not obtain the object). If the other player concedes first, at time tj, player i's payoff is vi − tj (she obtains the object after tj units of time). If both players 1 concede at the same time, player i's payoff is 2 vi − ti, where ti is the common concession time. a) Find the best response function Bi(ti); i 6= j, for player i; j = 1; 2 and give a graphical description. b) Determine from a) all Nash-Equilibria of the game! What are the features common to all equilibria? 4 WS 12/13 Game Theory Question 7 (Second-price sealed-bid auction) Denote by vi the value player i attaches to the object; if she obtains the object at the price p her payoff is vi−p. Assume that the players' valuations of the object are all different and all positive; number the players 1 through n in such a way that v1 > v2 > ::: > vn > 0. Each player i submits a (sealed) bid bi. If player i's bid is higher than every other bid, she obtains the object at a price equal to the second-highest bid, say bj, and hence receives the payoff vi − bj. If some other bid is higher than player i's bid, player i does not obtain the object, and receives the payoff of zero.
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