HW Consider the following game:

Transcription

1 HW 1 1. Consider the following game: 2.

2 HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child, I C (A), and income for the parent, I P (A). Second, the parent observes the incomes I C and I P and then chooses a bequest, B, to leave to the child. The child s payoff is U(I C +B)= log(i C +B); the parent s is U(I P -B)+kU(I C +B) = log(i P - B)+klog(I C + B), where k > 0 reflects the parent s concern for the child well-being. Assume: bequest B can be positive or negative, the income functions I C (A) and I P (A) are strictly concave and are maximized at A C >0 and A P > 0, respectively. Verify the Rotten Kid theorem: in the backwards induction outcome, the child chooses the action that maximizes the family s aggregate income, I C (A)+I P (A), even though only the parent s payoff exhibits altruism.

3 HW 3 A. Read Chapter 5 of the text book. B. Refer the problem with players Karl, Rosa and Ernesto that was discussed in class. Find a Nash equilibrium (that is not subgame perfect) other than the one we found in class. C. Do the following problems: 1. There is a selection committee with three members (1, 2 and 3) who have to choose one of four candidates (a, b, c and d). The preferences of the members among the candidates are as follows u 1 (d) < u 1 (c) < u 1 (b)< u 1 (a) u 2 (c) < u 2 (d) < u 2 (a)< u 2 (b) u 3 (c) < u 3 (a) < u 3 (b)< u 3 (d) where u i is the utility function of person i. The following procedure is used to choose among the candidates. First, Member 1 vetoes one of the candidates, thereby knocking him out of contention. Then Member 2 vetoes one of the three remaining candidates, thereby knocking him out of contention. Finally, Member 3 vetoes one of the two remaining candidates. The candidate not vetoed by any of the members is chosen. Each member wishes to maximize his utility. Who will Member 1 veto? Who will Member 2 veto? Who will Member 3 veto? Justify your answers. (Hint: From the point of view of any particular committee member, maximization of his utility requires him to look forward and predict the behavior of members who will exercise their veto after that particular committee member. In particular, you are asked to find subgame perfect outcome of the game) Also answer the following questions. a) Write down the set of players, set of terminal histories, player function, and players preferences for the game. b) How many strategies does each player have? c) Find a Nash equilibrium of the above game which is not Subgame perfect. 2. Three friends take part in the following pizza splitting game. In period 1, friend 1 is asked to split the pizza of size 1 into at most two parts (i.e. he is free to not split the pizza at all). Let the two resulting fractions of pizza be p 1 and 1-p 1. In period 2, friend 2 is asked to make a single split on any one of the existing pizza parts (again he is free to make no split at all). Once the two friends have made their splitting decisions, period 3 comes and friend 3 is asked to choose one of possible many pizza parts on the table. This is followed by friend 2 choosing one of the remaining parts. Whatever is left is consumed by friend 1. If you were friend 1 (and you and your friends all liked pizza), what would your choice of p 1 be?

4 HW 4 A. Do the following problems: 1. Consider a market in which there are two firms, both producing the same good. Firm i s cost of producing q i units of the good is C i (q i ) = 8q i for each i {1, 2}; the price at which output is sold when the total output is Q is P d (Q) = max{24 Q, 0}, where Q = q 1 + q 2. Each firm s strategic variable is output and the firms make their decisions sequentially: one firm chooses its output, then the other firm does so, knowing the output chosen by the first firm. a) Model this situation as an extensive game. Players: Terminal Histories: Player function: Preferences: b) Find the subgame perfect equilibrium and outcome of Stackelberg s duopoly game. c) Consider a variant of the situation in which firm 2 has some fixed cost of production (f > 0): C 2 (q 2 ) = f + 8q 2 for q 2 > 0 = 0 for q 2 = 0 Find the subgame perfect equilibrium and outcome of Stackelberg s duopoly game in each of the following cases: i) f = 1 ii) f = 9 iii) f = 25

5 2. Consider a market in which there are three firms, all producing the same good. Firm i s cost of producing q i units of the good is C i (q i ) = 0 for each i {1, 2, 3}; the price at which output is sold when the total output is Q is P d (Q) = max{32 Q, 0}, where Q = q 1 + q 2 + q 3. Each firm s strategic variable is output and the firms make their decisions sequentially: one firm chooses its output, then the second firm does so, knowing the output chosen by the first firm, and finally the third firm makes its choice of output knowing the output chosen by the first two firms. a) Model this situation as an extensive game. Players: Terminal Histories: Player function: Preferences: b) Find the subgame perfect equilibrium and outcome of Stackelberg s duopoly game. Strategy of firm 3 c) Can you suggest a Nash Equilibrium strategy profile which results in an outcome different from the one which results from subgame perfect equilibrium above? Strategy of firm 3

6 3. A legislature has k members, where k is an odd number. Two rival bills, X and Y, are being considered. The bill that attracts the votes of a majority of legislators will pass. Interest group X favors bill X, whereas interest group Y favors bill Y. Each group wishes to entice a majority of legislators to vote for its favorite bill. First interest group X gives an amount of money (possibly zero) to each legislator, then interest group Y does so. Each interest group wishes to spend as little as possible. Group X values the passing of bill X at $V X > 0 and the passing of bill Y at zero, and group Y values the passing of bill Y at $V Y > 0 and the passing of bill X at zero. (For example, group X is indifferent between an outcome in which it spends V X and bill X is passed and one in which it spends nothing and bill Y is passed.) Each legislator votes for the favored bill of the interest group that offers her the most money; a legislator to whom both groups offer the same amount of money votes for bill Y (an arbitrary assumption that simplifies the analysis without qualitatively changing the outcome). For example, if k = 3, the amounts offered to the legislators by group X are x = (100, 50, 0), and the amounts offered by group Y are y = (100, 0, 50), then legislators 1 and 3 vote for Y and legislator 2 votes for X, so that Y passes. (In some legislatures the inducements offered to legislators are more subtle than cash transfers.) We can model this situation as the following extensive game. Players The two interest groups, X and Y. Terminal histories The set of all sequences (x, y), where x is a list of payments to legislators made by interest group X and y is a list of payments to legislators made by interest group Y. (That is, both x and y are lists of k nonnegative integers.) Player function P( ) = X and P(x) = Y for all x. Preferences The preferences of interest group X are represented by the payoff function where bill Y passes after the terminal history (x, y) if and only if the number of components of y that are at least equal to the corresponding components of x is at least 0.5 (k + 1) (a bare majority of the k legislators). The preferences of interest group Y are represented by the analogous function (where V Y replaces V X, y replaces x, and Y replaces X). Find the subgame perfect equilibrium and outcome of the above game in each of the following cases: i) V X = 900, V Y = 600 and k = 5 Equilibrium: Strategy of group X Strategy of group Y ii) V X = 1200, V Y = 600 and k = 3 Equilibrium: Strategy of group X Strategy of group Y

7 4. (Multiple choice question) There is a pile of 17 matchsticks on a table. Players 1 and 2 take turns in removing matchsticks from the pile, starting with player 1. On each turn, a player has to remove a number of sticks that equals the square of a positive integer, such that the number of matchsticks that remain on the table equals some non-negative integer. The player, who cannot do so, when it is his /her turn, loses. Which of the following statements is true? a) If player 2 plays appropriately, he/she can win regardless of how 1 actually plays. b) If player 1 plays appropriately, he/she can win regardless of how 2 actually plays. c) Both players have a chance to win, if they play correctly. d) The outcome of the game cannot be predicted on the basis of the data given. 5. Each of n 2, i = 1,..., n can make contributions s i [0, w] (w > 0) to the production of some public good. The contributions are made sequentially, that is, beginning with individual 1, each individual i makes the contribution after 1,, i - 1 have contributed and before i + 1,., n make their contributions. Assume that each individual i observes the contributions of the preceding i - 1 contributors before choosing her contribution. Their payoff functions are given by π i (s 1,.., s n ) = n(min{s 1,.., s n }) s i. a) Model this situation as an extensive game. Players: Terminal Histories: Player function: Preferences: b) Find the subgame perfect equilibrium and outcome of the game in question. Equilibrium: Strategy of individual 1 Strategy of individual 2 Strategy of individual 3.. Strategy of individual i B. Read Chapter 5 and 6 of the text book and try all the problems.

8 HW 5 1. Consider the following game played between players 1 and 2. Player 1 moves first and plays either L or R. This is observed by both the players. If L is played, players 1 and 2 simultaneously choose from the sets {l, r} and {t, d} respectively and the game ends with payoffs (3, 1), (0, 0), (0, 0) and (1, 3) if (l, t), (l, d), (r, t) and (r, d) are played respectively. If Player 1 plays R, players 1 and 2 simultaneously choose from the sets {m, n} and {q, s} respectively and the game ends with payoffs (2, 1), (1, 2), (1, 2) and (2, 1) if (m, q), (m, s), (n, q) and (n, s) are played respectively. (i) (ii) Model this game as an extensive form game with perfect information and simultaneous moves. Compute all subgame perfect Nash equilibria (SPNE) in the game. 2. Players 1 and 2 choose an element of the set {1,...,K}. If the players choose the same number, then player 2 pays 1 to player 1; otherwise no payment is made. Find all pure and mixed strategy Nash equilibria of this game. 3. Define a Bayesian game. Formulate problems 4 through 8 as Bayesian games (i.e. specify players, states, actions, signals, beliefs, payoff functions) 4. Consider a Cournot duopoly game with incomplete information. Suppose that demand is given by p = max{1 Q, 0}, where Q is the total quantity produced in the industry. Firm 1 selects a quantity q 1, which it produces at zero cost. Firm 2 s cost of production is private information (selected by nature). With probability ½, firm 2 produces at zero cost. With probability ½, firm 2 produces with a marginal cost of ¼. Call the former type of firm 2 L and the latter type H (for low and high cost, respectively). Firm 2 knows its type, whereas firm 1 knows only the probability that L and H occur. Let q 2 H and q 2 L denote the quantities selected by the two types of firm 2. Then when firm 2 s type is L, its payoff is given by u 2 L = (1 q 1 q 2 L )q 2 L. When firm 2 s type is H, its payoff is u 2 L = (1 q 1 q 2 H )q 2 H (q 2 H )/4. As a function of the strategy profile (q 1 ; q 2 L, q 2 H ), firm 1 s payoff is u 1 = ((1 q 1 q 2 L )q 1 )/2 + ((1 q 1 q 2 H )q 1 )/2 Solve for Bayesian Nash Equilibrium. (Ans. (q 1 ; q 2 L, q 2 H )= (3/8; 5/16, 3/16)) 5. Consider a simple simultaneous-bid poker game. First, nature selects numbers x 1 and x 2. Assume that these numbers are independently and uniformly distributed between 0 and 1. Player 1 observes x 1 and player 2 observes x 2, but neither player observes the number given to the other player. Simultaneously and independently, the players choose either to fold or to bid. If both players fold, then they both get the payoff 1. If only one player folds, then he obtains 1 while the other player gets 1. If both players elected to bid, then each player receives 2 if his number is at least as large as the other player s number; otherwise, he gets 2. Compute the Bayesian Nash equilibrium of this game. (Hint: Look for a symmetric equilibrium in which a player bids if and only if his number is greater than some constant α. Your analysis will reveal the equilibrium value of α.) 6. Consider the following game. Nature selects A with probability ½ and B with probability ½. If nature selects A, then players 1 and 2 interact according to matrix A. If nature selects B, then

9 the players interact according to matrix B. Suppose that, before the players select their actions, player 1 observes nature s choice. (That is, player 1 knows which matrix is being played.) Player 2 does not observe nature s choice. These matrices are pictured here. Solve for Bayesian Nash equilibrium. A B 1 2 L R 1 2 L R U 2, 1 0, 0 U 1, 2 0, 0 D 0, 0 1, 2 D 0, 0 2, 1 7. Suppose that a public good is provided to a group of people if at least one person is willing to pay the cost of the good. Assume that the people differ in their valuations of the good, and each person knows only her own valuation. Denote the number of individuals by n, the cost of the good by c > 0, and individual i s payoff if the good is provided by v i. If the good is not provided then each individual s payoff is 0. Each individual i knows her own valuation v i. She does not know anyone else s valuation, but knows that all valuations are at least v L and at most v H, where 0 v L < c < v H. She believes that the probability that any one individual s valuation is at most v is F(v), independent of all other individuals valuations, where F is a continuous increasing function. Suppose F(v) = max{0, min{(v v L )/ (v H v L ), 1}} i.e. uniform distribution from v L to v H. The following mechanism determines whether the good is provided. All n individuals simultaneously submit envelopes; the envelope of any individual i may contain either a contribution of c or nothing (no intermediate contributions are allowed). If all individuals submit 0 then the good is not provided and each individual s payoff is 0. If at least one individual submits c then the good is provided, each individual i who submits c obtains the payoff v i c, and each individual i who submits 0 obtains the payoff v i. Solve for the Bayesian Nash equilibrium when n = 2 and n = Consider a Coumot duopoly operating in a market with inverse demand P(Q) = max{a Q, 0}, where Q = q 1 + q 2 is the aggregate quantity on the market. Both firms have total costs c(q i ) = cq i, but demand is uncertain: it is high (a = a H ) with probability θ and low (a = a L ) with probability 1 θ. Furthermore, information is asymmetric: firm 1 knows whether demand is high or low, but firm 2 does not. All of this is common knowledge. The two firms simultaneously choose quantities. Suppose a H = 100, a L = 80, θ = 0.5, and c = 0. What is the Bayesian Nash equilibrium of this game? 9. Consider the first-price sealed-bid auction, but each bidder i observes only his own valuation v i. The valuation is distributed uniformly and independently on [0, v*] for each bidder. Derive symmetric (pure strategy) Bayesian Nash equilibrium of this auction if there are two bidders. Suppose that bids can only be non-negative. (Look for an equilibrium in which bidder i s bid is a linear function of his valuation.)

10 HW 6 1. Consider an all-pay auction with two players (the bidders). Player 1 s valuation v 1 for the object being auctioned is uniformly distributed between 0 and 1. That is, for any x [0, 1], player 1 s valuation is below x with probability x. Player 2 s valuation is also uniformly distributed between 0 and 1, so the game is symmetric. After nature chooses the players valuations, each player observes his/her own valuation but not that of the other player. Simultaneously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids. That is, if player i bids b i, then his/her payoff is b i if he/she does not win the auction; his/her payoff is v i b i if he/she wins the auction. Calculate the Bayesian Nash Equilibrium strategies (bidding functions). (Hint: The equilibrium bidding function for player I is of the form b i (v i ) = kv i 2 for some k.) 2. Consider a common-value auction with two players, where the value of the object being auctioned is the same for both players. Call this value Y and suppose that Y = y 1 + y 2, where y 1 and y 2 are both uniformly distributed between 0 and 10. That is, for any x [0, 10], we have y 1 < x with probability x/10. Suppose that player 1observes y 1 (but does not observe y 2 ) and that player 2 observes y 2 (but does not observe y 1 ). The players simultaneously submit bids, b 1 and b 2. If player i bids higher than does player j, then player I wins the auction and gets the payoff Y b whereas player j gets 0. a) Suppose player 2 uses the bidding strategy b 2 (y 2 ) = 3 + y 2. What is player 1 s best-response bidding strategy? b) Suppose each player will not select a strategy that would surely give him/her a negative payoff conditional on winning the auction. Suppose also that this fact is common knowledge between the players. What can you conclude about how high the players are willing to bid?

11 HW 7 In a 2 x 2 signaling game, there can be any or all of the following Weak Sequential Equilibria (WSE): both types of Player 1 may play pure strategies in equilibrium (if they play the same strategy, we say it is a pooling equilibrium; if they differ, we say it is a separating equilibrium); one type of Player 1 may play a pure strategy while the other plays a mixed strategy (leading to a semi-separating equilibrium); or both types of Player 1 may play mixed strategies. (We won t deal with the latter two cases.) When looking for a WSE Decide whether you re looking for a separating or pooling equilibrium. 2. Assign a strategy (a message for each type) to Player 1; make sure it is not strictly dominated. 3. Derive beliefs for Player 2 according to Bayes rule at each information set reached with positive probability along the equilibrium path. Set arbitrary beliefs for Player 2 at information sets that are never reached along the equilibrium path. 4. Determine Player 2 s best response. 5. In view of Player 2 s response, check to see whether Player 1 has an incentive to deviate from the strategy you assigned her in any state of the world (in other words, for all types of Player 1). If she does not, you have found a WSE. If she does, this is not an equilibrium - return to step 2 and assign Player 1 a different strategy. 6. Once you have exhausted all possible strategies within an equilibrium subset, return to step 1 and select a different type of WSE. Find WSE for the following game: A signaling game in which there are two types of player 1, strong and weak, the probabilities of these types are 0.9 and 0.1 respectively, the set of messages is {B, Q} (the consumption of beer or quiche for breakfast), and player 2 has two actions, F(ight) or N(ot). Player 1's payoff is the sum of two elements: she obtains two units if player 2 does not fight and one unit if she consumes her preferred breakfast (B if she is strong and Q if she is weak). Player 2's payoff does not depend on player 1's breakfast; it is 1 if he fights the weak type or if he does not fight the strong type.