Chapter 27

Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bid, first price. Let \(v\) and \(b\) denote your valuation and bid, respectively, and let \(\hat{v}\) and \(\hat{b}\) denote the valuation and bid of your opponent. Your payoff is \(v-b\) if it is the case that \(b \geq \hat{b}\). Your payoff is 0 otherwise. Although you do not observe \(\hat{v}\), you know that \(\hat{v}\) is uniformly distributed over the interval between 0 and 1 . That is, \(v^{\prime}\) is the probability that \(\hat{v}

Complete the analysis of the second-price auction by showing that bidding one's valuation \(v_{i}\) is weakly preferred to bidding any \(x

Suppose that a person (the "seller") wishes to sell a single desk. Ten people are interested in buying the desk: Ann, Bill, Colin, Dave, Ellen, Frank, Gale, Hal, Irwin, and Jim. Each of the potential buyers would derive some utility from owning the desk, and this utility is measured in dollar terms by the buyer's "value." The valuations of the 10 potential buyers are shown in the following table. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text { Ann } & \text { Bill } & \text { Colin } & \text { Dave } & \text { Ellen } & \text { Frank } & \text { Gale } & \text { Hal } & \text { Irwin } & \text { Jim } \\ 45 & 53 & 92 & 61 & 26 & 78 & 82 & 70 & 65 & 56 \\ \hline \end{array} $$ Each bidder knows his or her own valuation of owning the desk. Using the appropriate concepts of rationality, answer these questions: (a) If the seller holds a second-price, sealed-bid auction, who will win the auction and how much will this person pay? (b) Suppose that the bidders' valuations are common knowledge among them. That is, it is common knowledge that each bidder knows the valuations of all of the other bidders. Suppose that the seller does not observe the bidders' valuations directly and knows only that they are all between 0 and 100 . If the seller holds a first-price, sealed-bid auction, who will win the desk and how much will he or she have to pay? (Think about the Nash equilibrium in the bidding game. The analysis of this game is a bit different from the [more complicated] analysis of the first-price auction in this chapter because here the bidders know one another's valuations.) (c) Now suppose that the seller knows that the buyers' valuations are 45,53 , \(92,61,26,78,82,70,65\), and 56 , but the seller does not know exactly which buyer has which valuation. The buyers know their own valuations but not one another's valuations. Suppose that the seller runs the following auction: She first announces a reserve price \(p\). Then simultaneously and independently the players select their bids; if a player bids below \(\underline{p}\), then this player is disqualified from the auction and therefore cannot win. The highest bidder wins the desk and has to pay the amount of his or her bid. This is called a "sealed-bid, first price auction with a reserve price." What is the optimal reserve price \(p\) for the seller to announce? Who will win the auction? And what will the winning bid be?

Consider the two-bidder auction environment discussed in this chapter, where the bidders' values are independently drawn and distributed uniformly on the interval \([0,900]\). Demonstrate that, by setting a reserve price, the auctioneer can obtain an expected revenue that exceeds what he would expect from the standard first- and second-price auctions. You do not have to compute the optimal reserve price or the actual equilibrium strategies to answer this question.

Consider an "all-pay auction" with two players (the bidders). Player 1's valuation \(v_{1}\) for the object being auctioned is uniformly distributed between 0 and 1. That is, for any \(x \in[0,1]\), player 1's valuation is below \(x\) with probability \(x\). Player 2's valuation is also uniformly distributed between 0 and 1, so the game is symmetric. After nature chooses the players' valuations, each player observes his/her own valuation but not that of the other player. Simultaneously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids. That is, if player \(i\) bids \(b_{i}\), then his/her payoff is \(-b_{i}\) if he/she does not win the auction; his/her payoff is \(v_{i}-b_{i}\) if he/she wins the auction. Calculate the Bayesian Nash equilibrium strategies (bidding functions). (Hint: The equilibrium bidding function for player \(i\) is of the form \(b_{i}\left(v_{i}\right)=k v_{i}^{2}\) for some number \(k\).)

Suppose that you and two other people are competing in a third-price, sealed- bid auction. In this auction, players simultaneously and independently submit bids. The highest bidder wins the object but only has to pay the bid of the third-highest bidder. Sure, this is a silly type of auction, but it makes for a nice exercise. Suppose that your value of the object is 20 . You do not know the values of the other two bidders. Demonstrate that, in contrast with a second-price auction, it may be strictly optimal for you to bid 25 instead of 20 . Show this by finding a belief about the other players' bids under which 25 is a best-response action, yet 20 is not a best-response action.

Consider the common-value auction described in this chapter, where \(Y=y_{1}+y_{2}\) and \(y_{1}\) and \(y_{2}\) are both uniformly distributed on \([0,10]\). Player \(i\) privately observes \(y_{i}\). Players simultaneously submit bids and the one who bids higher wins. (a) Suppose player 2 uses the bidding strategy \(b_{2}\left(y_{2}\right)=3+y_{2}\). What is player 1's best-response bidding strategy? (b) Suppose each player will not select a strategy that would surely give him/her a negative payoff conditional on winning the auction. Suppose also that this fact is common knowledge between the players. What can you conclude about how high the players are willing to bid? (c) Show that for every fixed number \(z>0\), there is no Bayesian Nash equilibrium in which the players both use the bidding strategy \(b_{i}=y_{i}+z\).

Consider a first-price auction with three bidders, whose valuations are independently drawn from a uniform distribution on the interval \([0,30]\). Thus, for each player \(i\) and any fixed number \(y \in[0,30], y / 30\) is the probability that player \(i\) 's valuation \(v_{i}\) is below \(y\). (a) Suppose that player 2 is using the bidding function \(b_{2}\left(v_{2}\right)=(3 / 4) v_{2}\), and player 3 is using the bidding function \(b_{3}\left(v_{3}\right)=(4 / 5) v_{3}\). Determine player 1's optimal bidding function in response. Start by writing player 1 's expected payoff as a function of player 1's valuation \(v_{1}\) and her bid \(b_{1}\). (b) Disregard the assumptions made in part (a). Calculate the Bayesian Nash equilibrium of this auction game and report the equilibrium bidding functions.