Experiments on Auctions

Transcription

1 Experiments on Auctions Syngjoo Choi Spring, 2010 Experimental Economics (ECON3020) Auction Spring, / 25

2 Auctions An auction is a process of buying and selling commodities by taking bids and assigning commodities to winning bidder(s). Auctions have been widely used, ranging from the sale of perishable commodities like sh and owers, to the transfer of assets from public to private hands, such as timber rights and o -shore oil leases, to the sale of rights to use electro-magnetic spectrum for communication. An interesting historical event is that the Praectorian guard sold o the entire Roman Empire by means of an auction in 193 A.D., after having killed the Emperor Pertinax (for more information, go to Experimental Economics (ECON3020) Auction Spring, / 25

3 Some Common Auction Forms The open ascending-price auction (or English auction) is the oldest and perhaps most prevalent auction form. In one variant, the sale is conducted by an auctioneer who begins by calling out a low price and raises it as long as there are at least two interested bidders. The auction stops if there is only one interested bidder. The object is sold to this bidder at the price at which the second-last bidder dropped out. In the open descending-price auction (or Dutch auction) an auctioneer calls out prices in a descending order until one bidder claims the item at a given price. In the sealed-bid rst-price auction, bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object and pays what he bid. In the sealed-bid second-price auction (or Vickery auction), bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object but pays not what he bid, but the second highest bid. Experimental Economics (ECON3020) Auction Spring, / 25

4 Valuations: Private Value vs. Common Value An inherent feature of auctions is the uncertainty regarding values facing both sellers and buyers. Auctions are used precisely because the seller is unsure of the values that bidders attach to the object being sold (the maximum willingness to pay). One situation regarding the valuations of the object is that each bidder knows the value of the object to him-/herself at the time of bidding. This is called one of private values. In this situation no bidder knows with certainty the values attached by other bidders and knowledge of other bidders values would not a ect how much the object is worth to a particular bidder. The assumption of private values is most plausible when the value of the object to a bidder is derived from its consumption or use alone. Experimental Economics (ECON3020) Auction Spring, / 25

5 In another situation values are unknown to each other at the time of the auction and may be a ected by information available to other bidder. Such a situation is called one of interdependent values. A special case of this is a situation in which the value, though unknown at the time of bidding, is the same for all bidders, called one of a pure common value. A common value model is most appropriate when the value of the object being auctioned is derived from a market price that is unknown at the time of the auction. Experimental Economics (ECON3020) Auction Spring, / 25

6 Equivalence of Auctions Four auction formats (two open auctions and other two sealed-bid auctions) di er in the way that they are implemented in the real world. Open auctions usually require that the bidders collect in the same place. In contrast sealed bids may be sumbitted by mail. A bidder may observe the behavior of other bidders in open auctions but not in sealed-bid auctions. However, some of these practical di erences are irrelevant for rational decision makers. Experimental Economics (ECON3020) Auction Spring, / 25

7 Observation 1: The open ascending-price auction (or Dutch auction) is strategically equivalent to the sealed-bid rst-price auction. A bidder s strategy in the rst-price auction maps his private information into a bid. The only relevant information available in the Dutch auction is that some bidder has agreed to buy at the current price; but that causes the auction to end. Bidding a certain amount in a sealed-bid rst-price auction is equivalent to o ering to buy at that amount in a Dutch auction. Thus, for every strategy in a rst-price auction there is an equivalent strategy in the Dutch auction and vice versa. Experimental Economics (ECON3020) Auction Spring, / 25

8 Observation 2: When values are private, the open ascending-price auction (or English auction) is equivalent to the sealed-bid second-price auction. The English auction o ers information about when other bidders drop out; and by observing this, it may be possible to infer something about their privately known information. With private values, however, this information is of no use. With interdependent values, seeing some other bidder drop out early may bring bad news that may cause a bidder to reduce his own estimate of the value of the object. With private values, in an English auction, it clearly cannot be optimal to stay in after the price exceeds the value or to drop out before the price reaches the value. Likewise, in a second-price auction it is best to bid the value. Experimental Economics (ECON3020) Auction Spring, / 25

9 Revenue Equivalence The performance of di erent auction formats is usually evaluated on two grounds: revenue and e ciency. From the perspective of the seller, a natural criterion in comparing di erent auction forms is the revenue (or expected selling price) that they can obtain. Revenu Equivalence Theorem: If all bidders are risk neutral and have independent private value for the good, all four auction formats (for the single unit of the good) yield the same expected revenue. From the perspective of the society as a whole, e ciency (that the object end up in the hands of the person who values it the most ex post) may be more important. Experimental Economics (ECON3020) Auction Spring, / 25

10 A Classroom Experiment Go to Session name is sjc. Experimental Economics (ECON3020) Auction Spring, / 25

11 A First-Price Private Value Auction The auctions you just played is a version of the sealed-bid rst-price auction with private values in which (i) there are two bidders in a group or (ii) there are more than two bidders. To understand the behavior of bidders, we consider an equilibrium strategy, as a benchmark, when there are two risk-neutral bidders. Let V i denote a value that player i = 1, 2 assigns to the object. Assume that each V i is independently and identically uniformly distributed on [0, 10]. Each bidder submits a sealed bid of b i. Given a realization v i of the value and the choice of bid b i, player i s payo is given by u i (b i, b j jv i ) = vi 0 b i if b i > b j if b i < b j. Experimental Economics (ECON3020) Auction Spring, / 25

12 Equilibrium Strategy Let bidder 1 assume that bidder 2 uses an increasing strategy, b 2 (v 2 ) = a v 2, where a > 0. The expected payo for bidder 1 is given by (v 1 b 1 ) Pr (b 1 > b 2 (v 2 )) = (v 1 b 1 ) Pr v 2 < b 1 a b1 = (v 1 b 1 ). 10 a Experimental Economics (ECON3020) Auction Spring, / 25

13 Taking the derivative of this and arranging it, we can nd b 1 = v 1 2. Thus, an equilibrium strategy for each player i is just bidding half of his/her realized value. In fact, the equilibrium strategy can be easily extended into N risk-neutral bidders: (N 1) b (v) = v. N As the number of bidders become larger (i.e., more competition), the (N 1) ratio, N, becomes closer and closer to 1. Experimental Economics (ECON3020) Auction Spring, / 25

14 Experimental Economics (ECON3020) Auction Spring, / 25

15 Explanations of Overbidding As we noticed, the bidders in an experiment have a tendency of overbidding relative to an equilibrium bidding strategy of a risk-neutral bidder. One possible explanation of overbidding in a rst-price auction is risk aversion. Consider an individual s utility function has the form of power function: (Expected payo ) = (v b) 1 r b. 10 a Experimental Economics (ECON3020) Auction Spring, / 25

16 Taking the derivative and rearranging it, we can nd that b (v) = v 2 r. When r = 0 (risk neutrality), we have the same expression as before. As r (< 1) increases from zero (more risk averse), the bidding strategy becomes more conservative. An alternative explanation is based on a judgemental error of winning probability (a systematic underestimation of winning probability). Experimental Economics (ECON3020) Auction Spring, / 25

17 A Classroom Experiment Go to Session name is sjc. Experimental Economics (ECON3020) Auction Spring, / 25

18 A First-Price Common Value Auction You just participated in a rst-price common value auction in which (i) there are only two bidders in a group or (ii) there are more than two bidders. Two bidders observe separate components of a common prize value, i.e., bidder 1 observes v 1 and bidder 2 observes v 2. The monetary value of the prize to the winning bidder is an average of these two values: (v 1 + v 2 ). 2 The signals are private in that each bidder only knows his/her own estimate, but the bidders know that the other bidder s estimate is the independent realization of a random variable whose distribution is uniform on the interval, [0, 10]. Experimental Economics (ECON3020) Auction Spring, / 25

19 Equilibrium Strategy To understand the behavior of bidders, we consider an equilibrium strategy, as a benchmark, when there are two risk-neutral bidders. Let bidder 1 assume that bidder 2 uses an increasing strategy, b 2 (v 2 ) = a v 2, where a > 0. Given this, the probability of winning with a bid of b 1 is given by Pr (b 1 > b 2 (v 2 )) = Pr v 2 < b 1 = b 1 a 10 a. The expected payo conditional on winning with a bid of b 1 is v 1 + E v 2 jv 2 < b 1 a = v 1 + b 1 2a. 2 2 Experimental Economics (ECON3020) Auction Spring, / 25

20 Equilibrium Strategy The expected payo for bidder 1 is the product of the probability of winning and the di erence between the conditional expected payo and the bid: 2 Pr (b 1 > b 2 (v 2 )) 4 v E v 2 jv 2 < b 1 a b " # b1 v1 + b 1 2a = b a 2 Experimental Economics (ECON3020) Auction Spring, / 25

21 By taking the derivative and rearranging it, we have b 1 = a 4a 1 v 1. In a symmetric equilibrium, the equilibrium strategy becomes b (v) = v 2. In fact, risk-averse bidders have the same equilibrium strategy in this setup (see Holt and Sherman (2000)). It implies that a pattern of overbidding can not be explained by risk aversion if players are rational. Experimental Economics (ECON3020) Auction Spring, / 25

22 Experimental Economics (ECON3020) Auction Spring, / 25

23 Winner s Curse We again observe the general pattern of overbidding in this common value auction and the empirical bid/value relationship is signi cantly atter than the symmetric Nash equilibrium. In contrast to the private value auction, the fact that a bidder wins the auction has a relevant information: having the highest bid means that others value estimates were relatively low. A bidder does not realize this may bid too high and end up paying more than the prize is worth. This possibility due to judgemental error of overestimating the true value of the object is called Winner s Curse. Rational bidders recognize that winning is informative and make correct inferences from the equilibrium bids of others. Experimental Economics (ECON3020) Auction Spring, / 25

24 One Explanation of Winner s Curse: Limited-liability e ect One approach to explain the winner s curse is to investigate whether losses are caused by artifacts of the experimental design. Especially, this may occur due to the inability of the experimenter to actually collect losses from laboratory subjects, called a limited-liability e ect. In our previous experiment, subjects enjoyed limited liability as they could not lose more than their starting cash balances. With downside losses eliminated the only constraint on more aggressive bidding is the opportunity cost of bidding more than is necessary to win the object. In exchange, higher bids increase the probability of winning the object and making positive pro ts. The net e ect could be an incentive to bid more than the Nash equilibrium prediction. One possiblity of avoiding this so-called bankruptcy problem is combining this experiment with another experiment in which subjects earn some amount of money for certain. Experimental Economics (ECON3020) Auction Spring, / 25

25 Naive Bidding The winner s curse can occur if a bidder does not realize that winning the auction implies the other bidder s value estimate is relatively low. The simplest naive model is obtained by using the unconditional expected value of the other bidder s value estimate, which is 5 when the values are uniform on [0, 10]. Since 5 is typically larger than the conditional value, v 1 /2, this bias will tend to produce higher bids, even with risk neutral bidders, and thus will give rise to a winner s curse. Experimental Economics (ECON3020) Auction Spring, / 25