In Class Exercises Problem 1 A group of n students go to a restaurant. Each person will simultaneously choose his own meal but the total bill will be shared amongst all the students. If a student chooses a meal of price p and contributes x towards paying the bill, then his payoff is p x. Compute all pure strategy Nash equilibria of this game. Is the equilibrium unique? Symmetric? Discuss the limiting cases of n = 1 and n. Problem 2 Each of n 2, i = 1,..., n can make contributions s i [0, w] (w > 0) to the production of some public good. Their payoff functions are given by π i (s 1,.., s n ) = n(min{s 1,.., s n }) s i. Find all pure strategy Nash equilibria in the game. Problem 3 Consider an n firm quantity setting game where the cost function for firm i is given by C i (x i ) = c i.x i where c i 0. The inverse demand function is given P(X) = a bx where a, b > 0 and X = x 1 +... + x n. The payoff function for firm i is therefore π i (x 1,.., x n ) = (a b( j x j )x i c i.x i. (i) Compute a (pure strategy) Nash equilibrium for the game. Compute also the equilibrium price. (ii) What happens to the equilibrium quantity choice of firm j if there is an increase in firm i s cost; i.e in c i? What happens to equilibrium price? Problem 4 Players 1 and 2 are bargaining over how to split one rupee. Both players simultaneously name shares they would like to have, s 1 and s 2, where 0 s 1, s 2 1. If s 1 + s 2 1, then the players receive the shares they named; if s 1 + s 2 > 1, then both players receive zero. What are the purestrategy Nash equilibria of the game?
a. Write down the set of players, set of terminal histories, player function, and players preferences for the game above b. Represent the above game in normal form and suggest the Nash Equilibria. c. How many subgames does the game above has? Find all Nash Equilibrium of each subgame of the game above. d. Now, Find subgame perfect Nash Equilibrium. What is the outcome if SPNE is played? EXERCISE 154.2 (Examples of extensive games with perfect information) a. Represent in a diagram the two-player extensive game with perfect information in which the terminal histories are (C, E), (C, F), (D, G), and (D, H), the player function is given by P( ) = 1 and P(C) = P(D) = 2, player 1 prefers (C, F) to (D, G) to (C, E) to (D, H), and player 2 prefers (D, G) to (C, F) to (D, H) to (C, E). b. The political figures Rosa and Ernesto each has to take a position on an issue. The options are Berlin (B) or Havana (H). They choose sequentially. A third person, Karl, determines who chooses first. Both Rosa and Ernesto care only about the actions they choose, not about who chooses first. Rosa prefers the outcome in which both she and Ernesto choose B to that in which they both choose H, and prefers this outcome to either of the ones in which she and Ernesto choose different actions; she is indifferent between these last two outcomes. Ernesto s preferences differ from Rosa s in that the roles of B and H are reversed. Karl s preferences are the same as Ernesto s. Model this situation as an extensive game with perfect information. (Specify the components of the game and represent the game in a diagram.) c. Write down the set of players, set of terminal histories, player function, and players preferences for the game in Figure below: d. Represent the above games in normal forms carefully listing the strategies and the payoffs, solve for Nash Equilibrium and Sub-game perfect Nash Equilibrium. EXERCISE 161.2 (Voting by alternating veto) Two people select a policy that affects them both by alternately vetoing policies until only one remains. First person 1 vetoes a policy. If more than one policy remains, person 2 then vetoes a policy. If more than one policy still remains,
person 1 then vetoes another policy. The process continues until only one policy has not been vetoed. Suppose there are three possible policies, X, Y, and Z, person 1 prefers X to Y to Z, and person 2 prefers Z to Y to X. Model this situation as an extensive game and find its Nash equilibria. EXERCISE 171.4 (Burning a bridge) Army 1, of country 1, must decide whether to attack army 2, of country 2, which is occupying an island between the two countries. In the event of an attack, army 2 may fight, or retreat over a bridge to its mainland. Each army prefers to occupy the island than not to occupy it; a fight is the worst outcome for both armies. Model this situation as an extensive game with perfect information and show that army 2 can increase its subgame perfect equilibrium payoff (and reduce army 1 s payoff) by burning the bridge to its mainland, eliminating its option to retreat if attacked.
EXERCISE 183.2 (Dividing a cake fairly) Two players use the following procedure to divide a cake. Player 1 divides the cake into two pieces, and then player 2 chooses one of the pieces; player 1 obtains the remaining piece. The cake is continuously divisible (no lumps!), and each player likes all parts of it. a. Suppose that the cake is perfectly homogeneous, so that each player cares only about the size of the piece of cake she obtains. How is the cake divided in a subgame perfect equilibrium? b. Suppose that the cake is not homogeneous: the players evaluate different parts of it differently. Represent the cake by the set C, so that a piece of the cake is a subset P of C. Assume that if P is a subset of P not equal to P (smaller than P ) then each player prefers P to P. Assume also that the players preferences are continuous: if player i prefers P to P then there is a subset of P not equal to P that player i also prefers to P. Let (P 1, P 2 ) (where P 1 and P 2 together constitute the whole cake C) be the division chosen by player 1 in a subgame perfect equilibrium of the divide-and-choose game, P 2 being the piece chosen by player 2. Show that player 2 is indifferent between P 1 and P 2, and player 1 likes P 1 at least as much as P 2. Give an example in which player 1 prefers P 1 to P 2. EXERCISE 188.1 (Stackelberg s duopoly game with fixed costs) Suppose that the inverse demand function is given by and the cost function of each firm i is where c 0, f > 0, and c < α. Show that if c = 0, α = 12, and f = 4, Stackelberg s game has a unique subgame perfect equilibrium, in which firm 1 s output is 8 and firm 2 s output is zero. 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). EXERCISE 193.1 (Three interest groups buying votes) Consider a variant of the model in which there are three bills, X, Y, and Z, and three interest groups, X, Y, and Z, who choose lists of payments sequentially. Ties are broken in favor of the group moving later. Find the bill that is passed in any subgame perfect equilibrium when k = 3 and (a) V X = V Y = V Z = 300, (b) V X = 300, V Y = V Z = 100, and (c) V X = 300, V Y = 202, V Z = 100. EXERCISE 193.4 (Sequential positioning by three political candidates) Consider a further variant of Hotelling s model of electoral competition in which the n candidates choose their positions sequentially and each candidate has the option of staying out of the race. Assume that each candidate prefers to stay out than to enter and lose, prefers to enter and tie with any number of candidates than to stay out, and prefers to tie with as few other candidates as possible. Model the situation as an extensive game and find the subgame perfect equilibrium outcomes when n = 2 (easy) and when n = 3 and the voters favorite positions are distributed uniformly from 0 to 1 (i.e. the fraction of the voters favorite positions less than x is x) (hard).
1. 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.) 2. Consider the following strategic situation. Two opposed armies are poised to seize an island. Each army s general can choose either attack or not attack. In addition, each army is either strong or weak with equal probability (the draws for each army are independent), and an army s type is known only to its general. Payoffs are as follows: The island is worth M if captured. An army can capture the island either by attacking when its opponent does not or by attacking when its rival does if it is strong and its rival is weak. If two armies of equal strength both attack, neither captures the island. An army also has a cost of fighting, which is s if it is strong and w if it is weak, where s<w. There is no cost of attacking if its rival does not. Identify all pure strategy Bayesian Nash equilibria of this game. 3. 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 the players interact according to matrix B. These matrices are pictured here. A B 1 2 V W 1 2 V W X 6, 0 4, 1 X 0, 0 0, 1 Y 0, 0 0, 1 Y 6, 0 4, 1 Z 5, 1 3, 0 Z 5, 1 3, 0 a) Suppose that, when the players choose their actions, the players do not know which matrix they are playing. That is, they think that with probability ½ the payoffs are as in matrix A and that with probability ½ the payoffs are as in matrix B. Solve for Bayesian Nash equilibrium. b) Now 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. Solve for Bayesian Nash equilibrium. c) In this example, is the statement A player benefits from having more information true or false? 4. Two players have to simultaneously and independently decide how much to contribute to a public good. If player 1 contributes x 1 and player 2 contributes x 2 then the value of the public good is 2(x 1 +x 2 +x 1 x 2 ), which they each receive. Assume that x 1 and x 2 are positive numbers. Player 1 must pay a cost (x 1 ) 2 of contributing; thus player 1 s payoff in the game is u 1 = 2(x 1 +x 2 +x 1 x 2 ) - (x 1 ) 2. Player 2 pays the cost t(x 2 ) 2 so that player 2 s payoff is u 2 = 2(x 1 +x 2 +x 1 x 2 ) - t(x 2 ) 2. The number t is private information to player 2; player 1 knows that t equals 2 with probability ½ and it equals 3 with probability ½. Compute the Bayesian Nash equilibrium of this game. 5. Consider the following static game of incomplete information. Nature selects the type (c) of player 1, where c = 2 with probability 2/3 and c = 0 with probability 1/3. Player 1 observes c (he knows his own type), but player 2 does not observe c. Then the players make
simultaneous and independent choices and receive payoffs as described by the following matrix. 1 2 X Y A 0, 1 1, 0 B 1, 0 c, 1 Compute the Bayesian Nash equilibrium. 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 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. A B 1 2 L R 1 2 L R U 2, 2 0, 0 U 0, 2 2, 0 D 0, 0 4, 4 D 4, 0 0, 4 Solve for Bayesian Nash equilibrium. 7. Consider a Cournot duopoly game with incomplete information. Suppose that demand is given by p = max{10 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 2. 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 = (10 q 1 q 2 L )q 2 L. When firm 2 s type is H, its payoff is u 2 L = (10 q 1 q 2 H )q 2 H (2q 2 H ). As a function of the strategy profile (q 1 ; q 2 L, q 2 H ), firm 1 s payoff is u 1 = ((10 q 1 q 2 L )q 1 )/2 + ((10 q 1 q 2 H )q 1 )/2. Solve for Bayesian Nash Equilibrium.