Economics 410-3 Spring 2015 Professor Jeff Ely Midterm Exam Instructions: There are three questions. You have until 11:00 to complete the exam. This is a closed-book and closed-notebook exam. Please explain all of your answers carefully. 1. Here is a two-player game of complete information in strategic form. Each player’s action set is the unit interval, i.e. A1 = A2 = [0, 1]. The payoffs are as follows. If a1 + a2 ≤ 1 then player i earns payoff ai. If a1 + a2 > 1 then each player’s payoff is zero. (a) Say that a Nash equilibrium is efficient if the sum of the payoffs equals 1. What is the set of all efficient Nash equilibria? Any pair (a1, a2) such that a1 + a2 = 1 is a pure-strategy Nash equilibrium and these are all the efficient equilibria. (b) Does there exist an inefficient Nash equilibrium? If so give an example, if not prove there is none. There are inefficient mixed equilibria. For example suppose each player plays ai = 1/3 and ai = 2/3 each with equal probability. Then if the opponent plays a−i = 1/3 his payoff is 1/3 (for sure) and if the opponent plays a−i = 2/3 he obtains expected payoff of 1/3 (becuase with probability 1/2 the sum of the two actions is no greater than 1 and in that case he gets a payoff of a−i =2/3). Any a−i > 2/3 would give a payoff of zero, any a−i ∈ (1/3, 2/3) would earn expected payoff a−i/2 < 1/3 and any a−i < 1/3 would give payoff a−i. Thus this is a mixed-strategy Nash equilibrium and with positive probability the two players get zero, hence it is inefficient 2. The 2 bidder all-pay single-object auction works like this. Each bidder i submits a sealed bid bi (any non-negative real number). The bids are simultaneous and the winner is the high bidder. (Ties will be broken by a fair coin flip.) The winner gets the object. Both bidders pay their bid (this is what puts the “all” in all-pay). That is, if bidder i has private value vi then his payoff is vi − bi if he wins with bid bi and his payoff is −bi if he loses. This question is about interim Bayesian Nash equilibrium of the all-pay auction with incomplete information and private values. The two bidders’ private values (v1 and 1 v2) are independently uniformly distributed on the unit interval. An equilibrium is efficient if the winner is always the bidder with the highest value. (a) Momentarily putting aside the all-pay auction, explain why with this type space the efficient interim Bayesian Nash equilibrium of the second-price sealed-bid 2 auction gives interim expected payoff vi /2 to a bidder with value vi. Each player bids his value, wins with probability vi and makes expected payment (conditional on winning) of vi/2. Thus his interim expected payoff is vi(vi − 2 vi/2) = vi /2. (b) The revenue equivalence principle says that, given this type space, any effi- cient equilibrium of any auction format will generate the same interim expected payoffs for the bidders. Applying the revenue equivalence principle and the ob- servation in the previous part find an efficient interim Bayesian Nash equilibrium of the all pay auction. Prove that it is an interim Bayesian Nash equilibrium and that it is efficient. If the equilibrium is efficient then bidder i wins whenever his value exceeds the other’s. This happens with probability vi. Let bi(vi) be the bid submitted by type vi. Bidder i’s interim expected payoff would be vi · vi − bi(vi) since he receives value vi only in the event he wins but he pays bi(vi) whether or not he wins. 2 2 Since this must equal vi /2 we find that bi(vi)= vi /2. Now we verify that this is indeed an efficient interim Bayesian Nash equilibrium. There are many ways to do this, but here is the simplest. Consider any type vi and a bid bi ∈ [0, 1/2]. When bidder −i uses the strategy specified above, the interim expected payoff type vi would earn from bidding bi is given by Prob ({b−i(v−i) < bi}) vi − bi. Since the strategy b−i is strictly increasing, it has an inverse. In particular there 2 is a unique type v ∈ [0, 1] of bidder 2 that also bids bi, and indeed bi = v /2. This means that the probability i wins with bid bi is just the probability that the type of bidder −i is less than v. These observations enable us to rewrite the interim expected payoff as follows 2 v · vi − v /2. Let us factor out v and write v(vi − v/2). And now let’s interpret this formula in a different way. Consider the dominant strategy equilibrium of a second-price auction. If type vi were to instead bid v, 2 he would win with probability v. This is because the opponent is bidding his value and therefore a bid of v wins exactly when the opponent’s value is less than v, an event which has probability v. And conditional on winning, type vi of bidder i would earn expected payoff vi − v/2 because v/2 is the expected value of the opponent’s bid conditional on that bid being below v. That is, the formula above is the interim expected payoff of type vi bidding v in the second-price auction when the opponent bids according to his dominant strategy. Since bidding one’s true value is an interim Bayesian Nash equilibrium of the second-price auction (indeed it is the dominant strategy), it follows that setting v = vi maximizes this formula. Recalling the definition of v this means 2 that bidding bi = vi /2 maximizes the interim expected payoff to type vi in the all-pay auction, at least among all bids bi ∈ [0, 1/2]. Clearly any bid higher than 1/2 is strictly worse since it wins with probability 1 but leads to a larger payment than bidding 1/2 (which also wins with probability 1). We have shown that the derived strategy is an interim Bayesian Nash equilib- rium. To show that it is efficient note that since each bidder is using the same strictly increasing strategy, the high bidder will always be the bidder with the high value. 3. Consider the following game of persuasion with verifiable communication. Player 1 is called the sender. He privately observes the state of the world θ from the binary set Θ= {L, H} and then sends a message to the receiver. This is the only role the sender plays. Player 2 is called the receiver. The receiver does not observe the state but obtains information about it from the sender’s message. Before receiving a message from the sender, the receiver has a prior belief about the state. Specifically, it is common knowledge that the receiver’s prior belief assigns probability 2/3 to state L. After seeing the message from the sender, the receiver chooses an action a from the binary set A = {L, H}. This ends the game. There is a conflict of interest. Regardless of the state, the sender prefers the receiver to choose action H. Specifically, 1 if a = H u1(a, θ)= (0 otherwise On the other hand, the receiver’s payoff depends on the action as well as the state. In particular the receiver would like her action to match the state and she has the 3 following payoff function 3 if a = θ u2(a, θ)= (0 otherwise Note that neither player’s payoff depends on the message sent. In a game of verifiable communication, the set of messages available to the sender depends on the state (by contrast to a cheap talk game.) In this game there are two possible messages. The message ∅ represents “silence” and can be sent in either state. The message h is proof that the state is H: it is only available when the true state is H. (a) This is an extensive-form game with imperfect information in which the first move is a move by “Nature” who chooses the state to be either L or H with exogenous probabilities 2/3 and 1/3 respectively. The extensive form for this game is depicted below. However, the information sets have been left out. Draw the information sets that represent the imperfect information described above. Which nodes are singleton information sets, and which nodes are grouped into non-singleton information sets? See below. The two nodes after player 1’s choice of ∅ are grouped into an in- formation set belonging to player 2. All other nodes are singleton information sets. ✏✏r 1,0 H✏✏ Player 1 r ∅ r✏✏ ✟✟ ♣PP ✟ ♣ PP L (2/3) ✟ ♣ PP ✟ ♣ L r 0,3 ✟ ♣ ✟ 2 ♣ ✟✟ ♣ ✏r ❜❍ ♣ H ✏✏ 1,3 Nature ❍ ♣ ✏ ❍ r♣✏✏ ❍ ✟PP ❍ ∅ ✟✟ PP H (1/3) ❍❍ ✟ PPr ❍ ✟ L 0,0 ❍❍r✟ Player 1 ❍❍ ✏r ❍ H ✏✏ 1,3 ❍ ✏✏ h ❍Prr✏ PP 2 PP L Pr 0,0 4 (b) This part of the question is about the strategic-form representation of the game. i. Explain why the receiver has four possible strategies. He has to choose one of two actions after each of two possible messages received. ii. The ex ante payoff profile associated with a strategy profile (σ1, σ2) is found by first deriving the probability distribution over terminal nodes induced by the profile (please note that Nature’s move plays a role here) and then computing the expected value of the two players’ payoffs.