Spring 07 Final Exam ECONS : Strategy and Game Theory Tuesday May, :0 PM - 5:0 PM irections : Complete 5 of the 6 questions on the exam. You will have a minimum of hours to complete this final exam. No notes, books, or phones may be used during the exam. rite your name, answers and work clearly on the answer paper provided. Please ask me if you have any questions. Problem (0 Points) [From Lecture Slides]. Consider the following game between two competing auction houses, Christie s and Sotheby s. Each firm charges its customers a commission on the items sold. Customers view each of the auction houses as essentially identical. For this reason, which ever house charges the lowest commission charge will be the one most of the customers want to use. However, if they can cooperate they might be able to make more money without undercutting each other. The stage-game for the competition between the auction houses is shown below. Sotheby s 7% 5% % 7% 7, 7, 0 -, Christie s 5% 0,,, %, -, 0, 0 (A) (.5 points) List all pure strategy Nash equilibria of the stage-game. (B) (.5 points) List the preferred cooperative outcome of the stage-game and determine the best payoff a player gains by unilaterally deviating from this outcome. (C) (5 points) escribe the Grim-trigger strategy for the infinitely repeated game between Sotheby s and Christie s. () (0 points) If both auction houses have a common discount factor 0 δ, find the condition on this discount factor that will allow the Grim-trigger strategy to be sustained as a Subgame Perfect Nash equilibrium of the infinitely repeated game. Solution: Using ISS it becomes clear that both auction houses have a dominant strategy of 5%. Hence, for part (A) there is a single Nash equilibrium of (5%, 5%). For part (B), the preferred cooperative outcome is (7%, 7%) where both auction houses collude to keep their commission rates high and not undercut each other. hen both auction houses play this
strategy, they each receive a payoff of 7. However, each auction house has an incentive to deviate to 5%, getting 0, given the other auction house plays 7%. The grim trigger strategy is for both players to start by playing the cooperative outcome of 7% and then for each following period play 7% if everyone played 7% previously and play 5% forever if anyone cheated the previous round. Given δ, the grim-trigger can be sustained as a SPNE of the infinitely repeated game if 7 δ 0 + δ δ 7 0 0δ + δ 0δ δ 0 7 6δ δ 6 =
Problem (0 Points) [From Lecture Slides]. Consider the following simultaneous-move prisoner s dilemma stagegame. Assume that b > a and that d > c. Players will repeat the game an infinite number of times. Player Cooperate efect Player Cooperate a, a, c, b efect b, c d, d (A) (5 points) erive the inequality that must hold to sustain cooperation through a Grimtrigger strategy as a SPNE of the infinitely repeated game. (B) (5 points) erive the condition on the common discount factor δ that allows the Grimtrigger strategy to satisfy the inequality from part (A). (C) (5 points) escribe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b a) increases. escribe intuitively what the value (b a) is and why it has the relationship with δ that you identified. () (5 points) escribe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b d) increases. escribe intuitively what the value (b d) is and why it has the relationship with δ that you identified.
Problem (0 Points) [From Lecture Slides]. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6 There are pure strategy profiles for the simultaneous game of incomplete information above. The stranger s strategies are listed first, then yatt Earps. ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) ( G / C, ) (A) (5 points) hat is the Gunslinger type s dominant strategy? hich of the pure strategy profile s above does this eliminate from being possible BNEs? (B) (5 points) Find the pure strategy Bayes-Nash equilibria of the game. Solution: Each player has only two typical strategies, ait = and raw =. An equilibrium will specify a strategy for yatt Earp and a strategy for each type of the stranger (Gunslinger, Cowpoke). Suppose that yatt Earp plays at his single information set. Then what is the best response for each type of the stranger? e can highlight Earp s strategy in blue in the figure. Given this behavior, we can highlight the Gunslinger and Cowpoke s best response. Both types would like to choose if Earp is choosing. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6
e now give Earp a chance to deviate from given the s behavior of /. e compute the expected payoffs of choosing each strategy given the specified strategy for the. EU() = (/) + 5(/) = 6/ + 5/ = / EU( ) = (/) + 6(/) = / + 6/ = 9/ () Since EU() = / > 9/ = EU( ) yatt Earp cannot do strictly better by deviating and since both players (and their types) are best responding to the other player s strategy, we have a Bayes-Nash equilibrium of this simultaneous game of incomplete information. But we are not done. Suppose that yatt Earp instead plays wait. Then we highlight the best response for each type of the stranger. Nature Gunslinger Cowpoke yatt Earp yatt Earp 5 6 hen yatt Earp chooses to wait, the Gunslinger best responds with and the Cowpoke best responds with. oes Earp want to deviate given this best response? The expected payoffs are EU( ) = (/) + (/) = / + / = / EU() = (/) + (/) = 6/ + / = 0/ Because Earp cannot do strictly better by deviating to under these probabilities, is a best response to the stranger s strategy /. Hence (/, ) is also a BNE. 5
Problem (0 Points) [From Problem Set ]. Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bit, first price. Let v and b denote your valuation and bid, respectively, and let ˆv and ˆb denote the valuation and bid of your opponent. Your payoff is v b if it is the case that b ˆb. Your payoff is 0 otherwise. Although you do not observe ˆv, you know that ˆv is uniformly distributed over the interval [0, ]. That is, v is the probability that ˆv < v. You also know that your opponent bids according to the function ˆb(ˆv) = ˆv. (A) (5 points) rite your expected payoff function from bidding a value b. (B) (5 points) rite the probability of winning the auction as a function of your bid b. (C) (5 points) erive the optimal (best response) bidding rule (Hint: Take derivative, set equal to zero, etc). () (5 points) Suppose your value is v = /, what is your optimal bid? 6
Problem 5 (0 Points) [From Practice H]. Consider the signaling game below. There are players, P and P. Player has types, N and S. Nature chooses N and probability / and S at probability /. After learning their type, P chooses either Left L or right R. Player observes player s strategy and then updates beliefs about player s type before choosing a strategy up U or down. (, 0) L P R P γ U (, ) N (, 0) Nature (, 0) L P S R γ P U (, 0) (, ) The separating pure strategy profiles for this game are (L N /R S, U) (R N /L S, ) (R N /L S, U) (R N /L S, ) The pooling pure strategy profiles for this game are (L N /L S, U) (L N /L S, ) (R N /R S, U) (R N /R S, ) (A) (0 points) Can any of the separating strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs. (B) (0 points) Can any of the pooling strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs. Solution: For part A we check the separating strategies. Suppose (L N /R S ), then γ = 0 and P chooses. hen P chooses, the N type doesn t deviate, but the S type does ( ). So not a PBE. Suppose (R N /L S ), then γ = and P chooses U. hen P chooses U, the N type won t deviate ( > ) and the S type won t deviate ( > ). So (R N /L S, U) is a PBE. Part (B) we look at the pooling strategies. Suppose (L N /L S ), then P has arbitrary beliefs γ and the expected payoffs are, EU(U) = γ EU() = ( γ) EU(U) EU() γ γ γ γ 7
So we have two cases. hen γ / P plays U. The N type wants to deviate in this case ( > ). hen γ < /, P chooses. In this case, the both types do strictly worse by deviating to R. So we have a PBE of (L N /L S, ) γ < The other pooling strategy is (R N /R S ), then γ = /. In this case we have EU(U) = = EU() = So P chooses U. hen P chooses U the N type doesn t want to deviate, but the S type can do strictly better by deviating to L ( > ). So this isn t a PBE (which is obvious since L is strictly dominate for the S type).
Problem 6 (0 Points) [From Practice H]. Consider the scene from The Princess Pride where Prince Humperdinck discovers esley alive in a bedroom with Princess Buttercup after having killed him earlier that day. Prince Humperdinck suspects that esley has no strength (is eak) but isn t sure. He also knows that if esley is strong there is no way he could take him. esley therefore has two types: Strong and eak. Prince Humperdinck believes the probability that he is strong is only / (after all esley has been mostly-dead all day). esley has two strategies, he can get out of bed O (act tough, bluff, etc.) or he can stay in bed, B. After observing esley s action, Prince Humperdinck can choose to surrender S or fight F. esley s payoffs are listed first. (, 0) S F µ B esley O γ S F (, 0) (0, ) (, 0) S Humperdinck µ F B esley Strong Nature eak O Humperdinck γ S F (0, ) (, 0) (, ) (, ) Find all Perfect Bayes Nash equilibria (PBE) of the game. Solution: Consider the separating strategy (B, O ). Then µ = and γ = 0. H chooses S if B and F if O. Given H s choices, the strong type doesn t want to deviate, but the weak type wants to deviate to B, so (B, O ) cannot be part of a PBE. Consider the separating strategy (O, B ). Then µ = 0 and γ =. H chooses F if B and S if O. Under this policy the strong type doesn t want to deviate and the weak type does want to deviate - going from to /. Consider the pooling strategy (B, B ), then µ = / and γ is arbitrary. H s choice of B is EU(S) = 0 EU(F ) = + = = So H will choose F if B. On the other side, we have EU(S ) = 0 EU(F ) = γ + ( γ) = γ 0 Two cases γ 9
. γ / choose F.. γ > / choose S. In case, the strong type cannot do strictly better by deviating. The weak type doesn t want to deviate either since Humperdinck fights no matter what. In case, the strong type wants to deviate to O since they will get instead of 0. So we have a PBE (B/B, F/F ) when γ The last strategy is (O, O ) which makes γ = / and µ arbitrary. EU(S) = 0 EU(F ) = µ + µ = µ µ hen µ / then H chooses F if B. On the other information set, EU(S ) = 0 EU(F ) = + = > 0 So when both types choose O, H chooses F. In this case, the strong type is getting 0 and by deviating gets 0, so no strict benefit. The weak type is getting / and by deviating gets so the weak type wants to deviate. In the other case when µ > / then H chooses S if B. ere this the case, the strong type gets by switching and so not sustainable as PBE. 0