Spring 2017 Final Exam

Tuesday May 2, 3:10 PM - 5:10 PM

Directions : Complete 5 of the 6 questions on the exam. You will have a minimum of 2 hours to complete this ﬁnal exam. No notes, books, or phones may be used during the exam. Write your name, answers and work clearly on the answer paper provided. Please ask me if you have any questions.

Problem 1 (20 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% 2%

7% 7, 7 1, 10 -2, 3

Christie’s 5% 10, 1 4, 4 1, 2

2% 3, -2 2, 1 0, 0

(A) (2.5 points) List all pure strategy Nash equilibria of the stage-game.

(B) (2.5 points) List the preferred cooperative outcome of the stage-game and determine the best payoﬀ a player gains by unilaterally deviating from this outcome.

(D) (10 points) If both auction houses have a common discount factor 0 ≤ δ ≤ 1, 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 IDSDS 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. When both auction houses play this

1 strategy, they each receive a payoﬀ of 7. However, each auction house has an incentive to deviate to 5%, getting 10, 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 4δ ≥ 10 + 1 − δ 1 − δ 7 ≥ 10 − 10δ + 4δ 10δ − 4δ ≥ 10 − 7 6δ ≥ 3 3 1 δ ≥ = 6 2

2 Problem 2 (20 Points)

[From Lecture Slides]. Consider the following simultaneous-move prisoner’s dilemma stage- game. Assume that b > a and that d > c. Players will repeat the game an inﬁnite number of times.

Player 2 Cooperate Defect

Cooperate a, a, c, b Player 1 Defect b, c d, d

(A) (5 points) Derive the inequality that must hold to sustain cooperation through a Grim- trigger strategy as a SPNE of the inﬁnitely repeated game.

(B) (5 points) Derive the condition on the common discount factor δ that allows the Grim- trigger strategy to satisfy the inequality from part (A).

(C) (5 points) Describe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b − a) increases. Describe intuitively what the value (b − a) is and why it has the relationship with δ that you identiﬁed.

(D) (5 points) Describe what happens to the value of δ that will sustain the Grim-trigger strategy as an SPNE as the value (b − d) increases. Describe intuitively what the value (b − d) is and why it has the relationship with δ that you identiﬁed.

3 Problem 3 (20 Points)

[From Lecture Slides].

Nature Gunslinger 3 Cowpoke 1 4 4

Stranger Stranger

D W D W Wyatt Earp

D W D W D W D W

Stranger 3 4 1 2 2 3 1 4 Wyatt Earp 2 1 3 8 5 6 4 8

There are 8 pure strategy proﬁles for the simultaneous game of incomplete information above. The stranger’s strategies are listed ﬁrst, then Wyatt Earps.

(DG/DC ,D)(DG/W C ,D)(W G/DC ,D)(W G/W C ,D) (DG/DC ,W )(DG/W C ,W )(W G/DC ,W )(W G/W C ,W )

(A) (5 points) What is the Gunslinger type’s dominant strategy? Which of the pure strategy proﬁle’s above does this eliminate from being possible BNEs?

(B) (15 points) Find the 2 pure strategy Bayes-Nash equilibria of the game.

Solution: Each player has only two typical strategies, W ait = W and Draw = D. An equilibrium will specify a strategy for Wyatt Earp and a strategy for each type of the stranger (Gunslinger, Cowpoke). Suppose that Wyatt Earp plays D at his single information set. Then what is the best response for each type of the stranger? We 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 D if Earp is choosing D.

Nature Gunslinger 3 Cowpoke 1 4 4

Stranger Stranger

D W D W Wyatt Earp

D W D W D W D W

Stranger 3 4 1 2 2 3 1 4 Wyatt Earp 2 1 3 8 5 6 4 8

4 We now give Earp a chance to deviate from D given the Stranger’s behavior of D/D. We compute the expected payoﬀs of choosing each strategy given the speciﬁed strategy for the Stranger.

EU(D) = 2(3/4) + 5(1/4) = 6/4 + 5/4 = 11/4 EU(W ) = 1(3/4) + 6(1/4) = 3/4 + 6/4 = 9/4 (1)

Since EU(D) = 11/4 > 9/4 = EU(W ) Wyatt 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 Wyatt Earp instead plays wait W . Then we highlight the best response for each type of the stranger.

Nature Gunslinger 3 Cowpoke 1 4 4

Stranger Stranger

D W D W Wyatt Earp

D W D W D W D W

Stranger 3 4 1 2 2 3 1 4 Wyatt Earp 2 1 3 8 5 6 4 8

When Wyatt Earp chooses to wait, the Gunslinger best responds with D and the Cowpoke best responds with W . Does Earp want to deviate given this best response? The expected payoﬀs are

EU(W ) = 1(3/4) + 8(1/4) = 3/4 + 8/4 = 11/4 EU(D) = 2(3/4) + 4(1/4) = 6/4 + 4/4 = 10/4

Because Earp cannot do strictly better by deviating to D under these probabilities, W is a best response to the stranger’s strategy D/W . Hence (D/W, W ) is also a BNE.

5 Problem 4 (20 Points)

[From Problem Set 8]. 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 letv ˆ 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 observev ˆ, you know thatv ˆ is uniformly distributed over the interval [0, 1]. That is, v0 is the probability thatv ˆ < v0. You also know that your opponent bids according to the function ˆb(ˆv) =v ˆ2.

(A) (5 points) Write your expected payoﬀ function from bidding a value b.

(B) (5 points) Write the probability of winning the auction as a function of your bid b.

(D) (5 points) Suppose your value is v = 3/4, what is your optimal bid?

6 Problem 5 (20 Points)

[From Practice HW]. Consider the signaling game below. There are 2 players, P 1 and P 2. Player 1 has 2 types, N and S. Nature chooses N and probability 1/2 and S at probability 1/2. After learning their type, P 1 chooses either Left L or right R. Player 2 observes player 1’s strategy and then updates beliefs about player 1’s type before choosing a strategy up U or down D.

(3, 2) U L P1 R P2 (2, 0) γ D

1 2 N (1, 0)

Nature

1 S (1, 0) 2 U 1 − γ (2, 0) L P1 R P2 D (1, 1)

The separating pure strategy proﬁles for this game are

(LN /RS,U)(RN /LS,D) (RN /LS,U)(RN /LS,D)

The pooling pure strategy proﬁles for this game are

(LN /LS,U)(LN /LS,D) (RN /RS,U)(RN /RS,D)

(A) (10 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) (10 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 (LN /RS), then γ = 0 and P 2 chooses D. When P 2 chooses D, the N type doesn’t deviate, but the S type does (1 → 2). So not a PBE. Suppose (RN /LS), then γ = 1 and P 2 chooses U. When P 2 chooses U, the N type won’t deviate (3 > 2) and the S type won’t deviate (2 > 1). So (RN /LS,U) is a PBE. Part (B) we look at the pooling strategies. Suppose (LN /LS), then P 2 has arbitrary beliefs γ and the expected payoﬀs are,

EU(U) = 2γ EU(D) = 1(1 − γ) EU(U) ≥ EU(D) ⇐⇒ 2γ ≥ 1 − γ 3γ ≥ 1 1 γ ≥ 3

7 So we have two cases. When γ ≥ 1/3 P 2 plays U. The N type wants to deviate in this case (3 > 2). When γ < 1/3, P 2 chooses D. In this case, the both types do strictly worse by deviating to R. So we have a PBE of 1 (LN /LS,D) γ < 3 The other pooling strategy is (RN /RS), then γ = 1/2. In this case we have 1 EU(U) = 2 = 1 2 1 EU(D) = 2 So P 2 chooses U. When P 2 chooses U the N type doesn’t want to deviate, but the S type can do strictly better by deviating to L (2 > 1). So this isn’t a PBE (which is obvious since L is strictly dominate for the S type).

8 Problem 6 (20 Points)

[From Practice HW]. Consider the scene from “The Princess Pride” where Prince Humperdinck discovers Wesley alive in a bedroom with Princess Buttercup after having killed him earlier that day. Prince Humperdinck suspects that Wesley has no strength (is Weak) but isn’t sure. He also knows that if Wesley is strong there is no way he could take him. Wesley therefore has two types: Strong and Weak. Prince Humperdinck believes the probability that he is strong is only 1/4 (after all Wesley has been mostly-dead all day). Wesley has two strategies, he can get out of bed O (act tough, bluff, etc.) or he can stay in bed, B. After observing Wesley’s action, Prince Humperdinck can choose to surrender S or fight F . Wesley’s payoffs are listed first.

(1, 0) (1, 0) S S0 B Wesley O F µ γ F 0

1 Humperdinck Strong Humperdinck (0, −2) 4 (0, −2)

Nature

(1, 0) 3 Weak (− 1 , 0) S 4 S0 2 1 − µ 1 − γ 0 F B0 Wesley O0 F 3 (−1, 1) (− 2 , 1)

Find all Perfect Bayes Nash equilibria (PBE) of the game. Solution: Consider the separating strategy (B,O0). Then µ = 1 and γ = 0. H chooses S if B and F 0 if O0. Given H’s choices, the strong type doesn’t want to deviate, but the weak type wants to deviate to B0, so (B,O0) cannot be part of a PBE. Consider the separating strategy (O,B0). Then µ = 0 and γ = 1. H chooses F if B and S0 if O. Under this policy the strong type doesn’t want to deviate and the weak type does want to deviate - going from −1 to −1/2. Consider the pooling strategy (B,B0), then µ = 1/4 and γ is arbitrary. H’s choice of B is

EU(S) = 0 1 3 EU(F ) = −2 + 4 4 3 2 = − 4 4 1 = 4 So H will choose F if B. On the other side, we have

EU(S0) = 0 EU(F 0) = −2γ + (1 − γ) = 1 − 3γ ≥ 0 1 γ ≤ 3 Two cases

9 1. γ ≤ 1/3 choose F 0.

2. γ > 1/3 choose S0.

In case 1, 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 2, the strong type wants to deviate to O since they will get 1 instead of 0. So we have a PBE 1 (B/B0, F/F 0) when γ ≤ 3 The last strategy is (O,O0) which makes γ = 1/4 and µ arbitrary.

EU(S) = 0 EU(F ) = −2µ + 1 − µ = 1 − 3µ 1 µ ≤ 3 When µ ≤ 1/3 then H chooses F if B. On the other information set,

EU(S0) = 0 2 3 EU(F 0) = − + 4 4 1 = > 0 4 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 −3/2 and by deviating gets −1 so the weak type wants to deviate. In the other case when µ > 1/3 then H chooses S if B. Were this the case, the strong type gets 1 by switching and so not sustainable as PBE.