1. Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that can take on values θi ∈ { 0, 1, 2 }, each value occuring with an equal probability of 1/3. The players' values are independently drawn. The seller will over the good using a second-price sealed-bid auction, but he can set a reserve price of r ≥ 0 that modifies the rules of the auction as follows. If both bids are below r then neither bidder obtains the good and it is destroyed. If both bids are at or above r then the regular auction rules prevail. If only one bid is at or above r then that bidder obtains the good and pays r to the seller.
(a) Is it still a weakly dominant strategy for each player to bid his valuation when r > 0?
(b) What is the expected revenue of the seller when r = 0?
(c) What is the expected revenue of the seller when r = 1?
(d) What is the optimal reserve price r for the seller, and what can you conclude about the value of reserve prices?
(e) Does the auction with the reserve price you found in part (d) maximize the seller's expected revenue over all mechanisms? Justify your answer briefly.
2. Two players, 1 and 2, each own a house. Player 1 values his house at v1 and player 2 values his house at v2. The values v1 and v2 are drawn uniformly and independently from the interval [0, 1]. Player 1 knows v1 but does not know v2; player 2 knows v2 but does not know v1. All of this is common knowledge. Finally, suppose the value of house 1 to player 2 is 3/2v1 and suppose the value of house 2 to player 1 is 3/2v2. (Note that Player 1 does not know 3/2v2 and Player 2 does not know 3/2v1.)
(a) Suppose the two players announce simultaneously whether they want to exchange their houses. If both players agree to an exchange, the exchange takes place; otherwise no exchange takes place. Find a symmetric Bayes Nash equilibrium of this game in pure strategies in which each player i accepts an exchange if and only if his value vi does not exceed some threshold ?.
(b) Now, suppose the value of house 1 to player 2 is 5 v1 and suppose the value of house 2 to player 1 is 5 v2. How does your answer to part (a) change?
3. Consider a seller with a single object, and two potential buyers with independent private valuations. The seller's value for the object is zero. The values of the buyers are in the interval [0, 1].
The change from the standard set-up is that the buyer values are drawn independently from di↵erent distributions. Specifically, Buyer 1's value for the object is drawn from the distribution F1(x) = x2, and Buyer 2's value for the object is distributed according to F2(x) = 2x - x2. All of this is common knowledge.
(a) Find the allocation and payments made by the buyers in the VCG mechanism. You should describe this in terms of the realized valuations (v1, v2) for the buyers.
(b) Under the VCG mechanism, find the expected payment of buyer 1 when his value is x (the expectation is with respect to the value of the other buyer). Similarly, find the expected payment of buyer 2 when his value is x.
(c) Suppose (v1, v2) = (0.7, 0.6). Describe the allocation and payments made by the buyers under the AGV mechanism for these realized valuations.
(d) Suppose (v1, v2) = (0.7, 0.6). Is the VCG mechanism individually rational in this case? What is the net payo↵ for each buyer under the AGV mechanism?
(e) Describe the optimal mechanism for this problem. You should indicate the allocation rule and the payment rule, as a function of the valuations.. (Note: there will be some situations in which you do not sell the object.)
4. Consider a seller with one good facing two risk- neutral bidders with independent private valuations. Each bidder is equally likely to be a high type (h) with value vh = 1 or a low type (l) with value vl = 0. The seller plans to use the following auction form that is parametrized by p ∈ (0, 2/3): The seller announces the "buy-it-now" price p and each bidder simultaneously chooses accept or reject; if exactly one bidder accepts, he gets the object at price p; if both bidders accept, the good is allocated to one of the bidders by a fair lottery at price p; if both bidders reject, the good is given for free (i.e., at zero price) to one of the bidders by a fair lottery. The payo↵ to a bidder with value v for getting the object at price p is v - p; the payo↵ to the seller is the revenue he collects from the sale.
(a) Identify the type-spaces T1 and T2, the strategy spaces S1 and S2, and the payoff functions of the two bidders.
(b) Show that there is an equilibrium in which bidders of type H always accept. In this equilibrium, what are the seller's expected payoff and bidders' expected payoffs?
(c) How does the seller's expected payoff change with p? Why does the revenue equivalence principle not apply when comparing various auctions that differ only in the value of p?
For parts (d) and (e), you are free to use any mechanism, not just the above auctions parametrized by p.
(d) Identify an optimal mechanism (and an equilibrium of it) that maximizes seller's expected payo↵. Justify your answer. (In this case, the good may not be sold to either bidder, if the seller so wishes.)
(e) Suppose the seller must sell the good to one of these two bidders. Does his optimal expected payo↵ increase or decrease? Find an optimal mechanism that always allocates the good to one of the two bidders and find the corresponding (optimal) expected payoff for the seller in this case.
5. Consider a relationship between a bartender and a customer. The bartender serves bourbon to the customer and chooses x ∈ [0, 1], which is the proportion of bourbon in the drink served, while 1 - x is the proportion of water. The cost of supplying such a drink (standard 4-ounce glass) is cx with c > 0. The customer, without knowing x, decides on whether or not to buy the drink at the price p. If he buys the drink his payoff is vx - p, and the bartenders payoff is p - cx. Assume that v> c and all payoffs are common knowledge. If the customer does not buy the drink he gets 0 and the bartender gets -cx. Because the customer has some experience, once the drink is bought and he tastes it, he learns the value of x, but this is only after he pays for the drink. (Thus: in the stage game, the bartender chooses x without knowing whether or not the customer will buy the drink; and the customer chooses whether or not to buy the drink at price p without knowing x.)
(a) Does this (stage) game have a unique Nash equilibrium? If so, describe it and explain why it is unique; if not, construct two different equilibria.
(b) Now assume that the customer is visiting town for 10 days, and this "bar game" will be played on each of the 10 evenings that the customer is in town. Assume that each player tries to maximize the (nondiscounted) sum of his stage payoffs. Find all the subgame-perfect equilibria of this game.
(c) Now assume that the customer is a local, and the players perceive the game as repeated infinitely many times. Assume that each player tries to maximize the discounted sum of his stage payoffs, where the discount rate is δ ∈ (0, 1). What is the range of prices p (expressed in the parameters of the problem) for which there exists a subgame perfect Nash equilibrium in which the bartender always chooses x = 1 and the customer always buys at the price p? Find also the smallest value of 6 for which such a subgame perfect Nash equilibrium exists.