Simon Fraser University Fall 2016 Econ 302 Final Exam Solution Instructor: Songzi Du Saturday December 10, 2016, 8:30 11:30 AM NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect Bayesian equilibrium 1. (10 points) Suppose there are three firms, each has a constant marginal cost of 1 for producing each unit of a homogeneous good. Firm 1 sets its quantity of production first, then firm 2 sets its quantity (after observing firm 1 s quantity), and finally firm 3 sets its quantity (after observing firms 1 and 2 s quantities). The inverse demand function for this homogeneous good is P (Q) = 10 Q, where P is the price and Q is the total quantity. Find the SPE in this game. Describe the SPE with if-then statements. And calculate the market price in this equilibrium. 2. (40 points) Suppose that a delivery person named Clifford (player 2) is to deliver a package to a house with a chihuahua (player 1) in the yard. The yard around the house is fenced, but the gate is unlocked. Clifford can either enter the yard and deliver the package (action D) or leave an attempted delivery notice on the gate (action A). There are two types of chihuahua that Clifford may face. The first has no teeth; call this type W for weak. The second has teeth and jaws similar to those of a steel bear trap; call this type G for gnarly. The chihuahua is equally likely to be each type. Prior to entering the yard, Clifford cannot tell which type of chihuahua he faces, though he knows there is a chihuahua and can hear whether or not the chihuahua barks. Clifford likes to deliver all packages, but he would not like to have an encounter with the gnarly type of chihuahua. If Clifford chooses action A, he receives the payoff of 0 regardless of the chihuahua s type. Delivering the package with the weak chihuahua in the yard yields a payoff of 4 for Clifford, but delivering the package with the gnarly chihuahua in the yard results in a payoff of -4 for Clifford. That is, the gnarly chihuahua will bite Clifford, but the weak one will not. Both types of chihuahua would receive a benefit of 4 if Clifford delivers the package and a benefit of 0 if not. 1
Prior to Clifford s choice (D or A), each type of chihuahua chooses whether to bark (B) or not bark (N). The gnarly chihuahua does not like to bark the steel trap mouth makes barking difficult so barking costs this type an amount c > 0. The weak chihuahua likes to bark and receives a benefit x > 0 from doing so. These costs and benefits are in addition to the chihuahua s benefit of Clifford s selection of D described earlier. Note: chihuahua is a small dog, and gnarly means difficult, dangerous, or challenging. (i) Draw the game tree. (ii) Find Clifford s best response as a function of his conditional belief about the chihuahua s type. (iii) Go through the four pure-strategy conjectures about the chihuahua. For each conjecture, find the values of x and c that make the conjecture a PBE (if the PBE is possible), and describe the PBE in complete sentences. (Aside: there is the saying barking dogs never bite, which translates to one of the conjectures, a separating strategy.) (iv) Find the values of x and c such that the gnarly type randomizes between barking and not barking in a PBE (if the PBE is possible), and describe the PBE in complete sentences. (v) Find the values of x and c such that the weak type randomizes between barking and not barking in a PBE (if the PBE is possible), and describe the PBE in complete sentences. Solution: (i): if x = c = 10, then the tree is: 2
(ii): Conditional on hearing barking, Clifford s best response is B-D if P(G B) 1/2, and is B-A if P(G B) 1/2. Conditional on no barking, Clifford s best response is N-D if P(G N) 1/2, and is N-A if P(G N) 1/2. (iii): If the gnarly type weakly prefers barking over no barking, then the weak type strictly prefers barking over no barking. If the weak type weakly prefers no barking over barking, then the gnarly type strictly prefers no barking over barking. Thus, in equilibrium we cannot have (G-B, W-N). 1. Suppose the chihuahua uses (G-B, W-B). Conditional on barking, Clifford believes P(G B) = 1/2, therefore, both B-D and B-A are best responses. Conditional on no barking, P(G N) can be anything, so both N-D and N-A can be best responses. (B-D, N-D), (B-A, N-D) and (B-A, N-A) all imply that the gnarly type strictly prefers G-N over G-B, which contradicts the initial conjecture. (B-D, N-A) implies that the initial conjecture is a best response if 4 c 0. Thus, (G-B, W-B) is a part of a PBE if c 4. In this equilibrium, the chihuahua always barks; Clifford delivers the package if he hears barking, and leaves an attempted 3
delivery notice if he hears no barking. Clifford believes the chihuahua s type is gnarly with probability 1/2 if he hears barking, and believes the chihuahua s type is gnarly with probability at least 1/2 if he hears no barking. 2. Suppose the chihuahua uses (G-N, W-N). Conditional on no barking, Clifford believes P(G N) = 1/2, therefore, both N-D and N-A are best responses. Conditional on barking, P(G B) can be anything, so both B-D and B-A can be best responses. (B-D, N-D), (B-D, N-A) and (B-A, N-A) all imply that the weak type strictly prefers W-B over W-N, which contradicts the initial conjecture. (B-A, N-D) implies that the initial conjecture is a best response if 4 x. Thus, (G-N, W-N) is a part of a PBE if x 4. In this equilibrium, the chihuahua does not bark; Clifford delivers the package if he hears no barking, and leaves an attempted delivery notice if he hears barking. Clifford believes the chihuahua s type is gnarly with probability 1/2 if he hears no barking, and believes the chihuahua s type is gnarly with probability at least 1/2 if he hears barking. 3. Suppose the chihuahua uses (G-N, W-B). Conditional on no barking, Clifford believes P(G N) = 1, and therefore, N-A is the best response. Conditional on barking, Clifford believes P(G B) = 0, and therefore, B-D is the best response. (N-A, B-D) implies that the initial conjecture is a best response if 4 c 0. Thus, (G-N, W-B) is a part of a PBE if c 4. In this equilibrium, the chihuahua barks if its type is weak, and does not bark if its type is gnarly; Clifford delivers the package if he hears barking, and leaves an attempted delivery notice if he hears no barking. Clifford believes the chihuahua s type is gnarly with probability 1 if he hears no barking, and believes the chihuahua s type is gnarly with probability 0 if he hears barking. (iv) Suppose gnarly type barks with probability p (p G-B + (1 p) G-N), and the weak type barks (W-B). Conditional on no barking, Clifford believes P(G N) = 1, and his best response is N-A. 4
Conditional on barking, Clifford believes P(G N) = p/2 = p ; he must deliver p/2+1/2 p+1 the package with probability q to make the gnarly type indifferent between barking and not barking. Thus, p p+1 = 1/2, i.e., p = 1 and this mixed-strategy PBE does not exist. Alternatively, one can say that this mixed-strategy PBE is identical to the pooling equilibrium (G-B, W-B) in (iii). (iv) Suppose gnarly type does not bark (G-N), and the weak type barks with probability p (p W-B + (1 p) W-N). Conditional on barking, Clifford believes P(G B) = 0, and his best response is B-D. Then the weak type gets x+4 from barking, and cannot be indifferent between barking and not barking. Thus, this mixed-strategy does not exist. 3. (10 points) Two players find themselves in a legal battle over a patent. The patent is worth 20 to each player, so the winner would receive 20 and the loser 0. Given the norms of the country, it is common to bribe the judge hearing a case. Each player offers a bribe (simultaneously), and the one whose bribe is the highest will be awarded the patent. (It is impossible to not bribe the judge.) If the bribes are the same amount, then each has an equal chance of being awarded the patent. If a player does bribe, then the bribe can be valued at either 9 or 20. Any other number is considered very unlucky, and the judge would surely rule against a party who offered an unlucky number. The payoff of a player is the expected amount that he receives from the patent minus the bribe. (i) Find the unique pure-strategy Nash equilibrium for this game. (ii) If the norm were different, so that a bribe of 15 were also acceptable, is there a purestrategy Nash equilibrium? (iii) Find the symmetric mixed-strategy Nash equilibrium for the game with possible bribes of 9, 15, and 20. Solution: (i) The unique pure-strategy NE is each player bribing $9. (ii) No pure-strategy NE. (iii) Suppose each player bribes $9 with probability p, bribes $15 with probability q, and bribes $20 with probability 1 p q. To be indifferent, we must have: p 9(1 p) = 5
5p 5q 15(1 p q) and p 9(1 p) = 10(1 p q), i.e., p = 1/2 and q = 1/10. (We have to assume it is impossible to not bribe the judge.) 6