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Econ 106D: Notes for Lecture 14 Marek Pycia May 16, 2013 1 Model We consider a setting with one good (or object) to be auctioned and n bidders, each of whom would like to obtain it. Each bidder i = 1, ..., n has value vi for the auctioned good. The values vi are independently drawn from a distribution F on some interval [v, v¯]. For instance, each vi may be independently drawn from the uniform distribution on [v, v¯]; we denote the uniform distribution U [v, v¯]. In particular, a uniform distribution on [0, 1] is denoted U [0, 1]. Each bidder knows her or his value but does not know the values of others. The bidders only know that the values of others were independently drawn from distribution F . (Each bidder i also knows that each other bidder j knows j’s value but only knows that values of bidders other than j come from the distribution F , and bidder i also knows that bidder j knows the extent to which i knows j’s value, etc. In other words, bidders understand the strategic situation, they just do not know the values others drew from the distribution F ). This model of bidders’ values is known as the ”independent private values” model. The bidders participate in an auction. We will consider several auction formats below. In each of the formats the bidders submit bids; a bid is a number from [v, v¯]. Then one of the bidder wins the good, and the bidders make payments to the auctioneer. Who wins the good, and who pays what depends on the bids the bidders submitted. If a bidder wins the good and pays price t then his or her profit is v − t. If the bidder does not win the good and pays t then her or his profit is −t (in most, but not all, auction formats we will discuss the payment is zero if the bidder does not win). Each bidder would like to maximizes his or her expected profits. If buyer i with value vi has probability pi of winning the good and makes expected payment t then the buyer’s expected profit is pivi − t. Note that another phrase for expected profits is average profits. As you see in this part of the course we will rely on some concepts from probability 1 (distribution, expected value): I am posting a supplementary note briefly outlining these and other probability concepts. 2 Nash equilibrium We will analyze the auction by finding a Nash equilibrium — another concept we did not use before, but one most of you saw in Econ 101. To define this concept we need to ask ourselves what are bidders’ strategies in an auction game. Since bidder i submits just one bid bi his strategy seems to be this one number bi. However, this number might depend on the bidder’s value vi. Thus, when other bidders think about i’s strategy they recognize that the bid bi might depend on the value vi which they do not know. To take this into account, we define the strategy of each bidder i to be a profile of bids bi(vi), one bid for each possible value vi. In other words, bidder i’s strategy is a function from values to bids. This allows us to define: Definition 1 A set of bidding strategies is a Nash equilibrium if each bidders strategy choice maximizes his payoff given the strategies of the others. Note that this is the same definition of Nash equilibrium as in Econ 101, except that in Econ 101 you typically looked at simpler games (eg 2x2 games such as a prisoner’s dilemma). To account for the added complexity that stems from each bidder having some information (about his or her own value) that other bidders do not have, this equilibrium concept is sometimes called ”Bayesian-Nash Equilibrium”). Let us now look at several canonical examples of auctions. 3 Second-Price Sealed-Bid Auction The auction proceeds as follows: • Bidders ‘simultaneously’ submit sealed bids. • Bidder who submitted the highest bid wins the good • Winner pays the second highest bid. If two or more bidders submitted the same bid, and this is the highest bid, then we need some additional rule to decide who gets the object; such a rule is called the tie-breaking rule. 2 We will assume that in case of such a tie, the object is allocated at random to bidders who submitted the highest bid. In our analysis, tie-breaking will not play much of a role, because ties will almost never happen. How should one bid? Following the long tradition in economics we will address this question by finding a Nash equilibrium. In the second-price auction things are simple as it turns out that bidding one’s true value, bi = vi is a dominant strategy — a bidder wants to do it, no matter what others are doing (recall that we talked about dominant strategies in the context of matching and no-transfer allocation). In particular, bidding my true value maximizes a bidder’s payoff if other bidders bid their true values; that is bidding true value is a Nash equilibrium. Proposition 1 In the second price auction bidding one’s true value is a dominant strategy, and a Nash equilibrium. Before proving this result (the proof as always in this class is optional), let us look at an example. Example. Assume there are three bidders with values drawn uniformly on [0, 1]. Suppose the bidders drew values v1 = .2, v2 = .5, and v3 = .7. The bidders then submit bids b1 = .2, b2 = .5, and b3 = .7. Bidder 3 wins the auction and pays the price t = b2 = .5. The price paid by the winning bidder becomes the revenue of the seller. Proof of Proposition 1. We want to show that bidding your true value v is at least as good as any other bid b. First let us check that you cannot be better off by bidding b > v than by bidding your value v. • If the highest opposing bid is less than v, or higher than b, it makes no difference whether you bid b or v. • If the highest opposing bid is between v and b; you win if you bid b while you would lose bidding v, but you are better off losing in this case as by winning you pay the second highest bid which is more than your value. Second, let us check that you cannot be better off by bidding b < v than by bidding your value v. As before, the outcome of the two bids only differs if the highest opposing bid is between b and v. And, in this case bidding v is better you win and pay less than your value. Note that intuitively the same argument applies to the ascending auction (also known as English auction) we run in the previous lecture. Recall that in an ascending auction price starts at zero, and rises. At each moment, buyers indicate their willingness to continue 3 bidding or decide to exit. The auction ends when just one bidder remains (if the last few bidders exit at the exact same moment, we need a tie-breaking rule as in the second-price sealed bid auction). The last bidder remaining in the auction wins the good, and pays the price at which the second remaining bidder dropped out. The ascending auction is however more difficult to properly analyze than the second- price sealed-bid auction : notice that when deciding when to exit bidders can condition their decision one when other bidders’ exited. In other words, bidders’ strategies can be much more complex than in the second-price sealed-bid auction. We will thus abstain from a carful analysis of the ascending auction. 4 First-Price Sealed-Bid Auction The auction proceeds as follows: • Bidders simultaneously submit bids. • Bidder who submitted highest bid wins (if two or more bidders submitted the same bid which is higher than other bids, then we randomize, that is we use the same tie-breaking rule as in the second-price sealed-bid auction). • The winning bidder pays his own bid. Notice that this auction resembles the descending price auction (also known as Dutch auction) we run in class. In a descending auction price starts high and drops continuously. At any point in time, a bidder can claim the good at the current price and pay that price (in case of ties, we again use the same tie-breaking rule as above). Auction ends as soon as some bidder claims the good. The two auctions are strategically equivalent: in both the strategy is just a single number (the bid, or at what price to stand up). When bidders bid bi in the first-price sealed-bid auction they get exact same chances of winning, and pay the same amount, as if they planned to stand up at prices ti = bi in the descending auction. Notice that in the descending auction the price at which to stand up cannot be con- ditioned on other bidders’ behavior (just as the bid in the sealed bid auctions cannot be conditioned on other bidders’ bids). In this sense, the resemblance between the first-price sealed bid auction and the descending auction is closer than between the second-price sealed- bid auction and the ascending auction. 4 How should one bid? Again we will answer this question by finding a Nash equilibrium of the bidding game. As we discussed after we run the class experiment with the descending auction, in the first-price auction it does not make sense to submit the bid equal to one’s value. Doing it would earn the bidder zero payoff whether he wins or looses. The bidders want to submit bids lower than their true value, but how much lower? There is a trade-off: the higher the bid the higher the chance to win the good, but the lowest profit margin from winning. The probability of winning by bidding b depends on what other bidders bid. And, thus what to bid depends on the strategies of other bidders. Let us compute the equilibrium in the following example: Example. There are two bidders and their values drawn independently from U [0, 1], the uniform distribution of [0, 1]. Let us try to find an equilibrium in which both bidders bid linearly in their values, bi = βvi where β > 0 is constant. Notice that we postulate that both bidders use the same bidding strategy. In general in this analytical development we focus on equilibria in which all bidders use the same strategy; we call such equilibria symmetric equilibria. Assuming that Bidder 2 bids b2 = βv2 what is the best response of Bidder 1 with value v1? Bidder 1’s expected profit is (v1 − b1)P[b1 ≥ b2] (notice that what happens when the two bids are equal, b1 = b2 does not matter as given bidder 2 strategy, such an event has probability zero). Let us compute P[b1 ≥ b2] = P[b1 ≥ βv2] = P[b1 β ≥ v2]. The formula on the right-hand side of the above equation is just the cdf (cumulative distri- bution) of the value of bidder 2 at b1 β (see the supplementary probability note). Notice also that b1 ≤ β because by bidding b1 = β Bidder 1 wins with probability 1 and so there is no point to bid more and to pay more. We thus have P[ b1 β ≥ v2] = b1 β . Plugging all of this into Bidder 1’s maximization problem we see that this bidder wants to set b1 to maximize the expected profit of (v1− b1) b1β . The maximum is achieved for b1 = 12v1. Why? We could give a geometric argument based on the symmetry of the expected profit 5 function around b1 = 1 2 v1 (as in class), or we can use the machinery you studied in your calculus classes and in Econ 101: differentiating the objective with respect to b1 we get the first order condition −2b1 + v1 = 0 which gives us b1 = 12v1; by differentiating the objective function again we then check that the second order condition is also satisfied −2 < 0. We thus see that if Bidder 2 bids linearly, b2 = βv2 then Bidder 1 best response is to bid linearly as well b1 = 1 2 v1. The same argument applies to Bidder 2: if Bidder 1 bids linearly then an analogous computation to the one above shows that b2 = 1 2 v2 (in fact, the computation is not even needed as the situation is fully symmetric). We have thus established that if one of the bidders bids half his value, then the other bidder also optimally chooses to bid half his value. In other words, bidders’ strategies of bidding half the value are in Nash equilibrium. On Tuesday we will ask the question what are the seller’s revenues and the bidders’ profits (payoffs) in the second-price and first-price auctions, and we will then use it to establish a methodology to find the equilibria which does not require us to guess how the equilibrium looks like. 6