ESSAYS ON ALMOST COMMON VALUE AUCTIONS
DISSERTATION
Presented in Partial Ful…llment of the Requirements for the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Susan L. Rose, M.A.
*****
The Ohio State University
2006
Dissertation Committee: Approved by
John H. Kagel, Adviser Dan Levin Adviser Lixin Ye Graduate Program in Economics c Copyright by
Susan L. Rose
2006 ABSTRACT
In a common value auction, the value of the object for sale is the same to all bidders. In an almost common value auction, one bidder, the advantaged bidder, values the object slightly more than the other, regular bidders. With only two bidders, a slight advantage is predicted to have an explosive e¤ect on the outcome and revenue of an auction. The advantaged bidder always wins and revenue decreases dramatically relative to the pure common value auction. Ascending auctions, which reduce to two bidders, are thought to be particularly vulnerable to the explosive e¤ect, which may discourage entry. My dissertation investigates the explosive e¤ect in experimental
English clock auctions.
The …rst essay, “An Experimental Investigation of the Explosive E¤ect in
Almost Common Value Auctions,” uses a two-bidder wallet game to test these pre- dictions. I …nd the e¤ect of an advantage to be proportional, not explosive, con…rming past studies. I develop a behavioral model that predicts the proportional e¤ect and test it against the data. The model has two types of bidders: naïve and sophisticated.
Naïve bidders use a rule of thumb bidding function while sophisticated bidders are fully rational and account for the probability that a rival is naïve or sophisticated when best responding. I was able to classify subjects as naïve or sophisticated, and those classi…ed as sophisticated do have a better understanding of the game. However,
ii all subjects su¤ered from the winner’s curse, which may have masked the explosive e¤ect and been exacerbated by the structure of the wallet game.
The second essay, “Bidding in Almost Common Value Auctions: An Experi- ment,”moves the analysis to a four bidder auction to directly test the entry predic- tions. I used a more intuitive common value structure and controlled for the winner’s curse by using subjects with prior experience in common value auctions. I found that although subjects did not su¤er from the winner’s curse, there is no evidence of an explosive e¤ect. Advantaged bidders won no more auctions than predicted by chance.
Entry and auction revenue were una¤ected by the presence of advantaged bidders.
iii To my parents
iv ACKNOWLEDGMENTS
I could not have completed this work without the encouragement and support of many people, particularly my committee members. I am forever indebted to John
Kagel for his guidance and his moral and …nancial support. I am grateful to Dan
Levin for his enthusiastic encouragement, support and suggestions. I thank Lixin Ye for his advice and comments.
I would also like to thank John Ham, Stephen Cosslett and Ji Tao for their pa- tient discussions of econometrics. I thank Johanna Goertz for the use of her data, and Mashiur Rahman for his programming skills. Finally, I’d like to thank Mark
Owens, Kirill Chernomaz, Serkan Ozbeklik, Hankyoung Sung and Asen Ivanov for their comments, moral support and help testing software.
v VITA
May 22, 1969 ...... Born - Roseau, Minnesota
2001 ...... B.A. Business Economics, Moorhead State University 2002 ...... M.A. Economics, The Ohio State Uni- versity 2002-Present ...... Graduate Teaching and Research Asso- ciate, The Ohio State University
FIELDS OF STUDY
Major Field: Economics
Studies in: Industrial Organization Experimental and Behavioral Economics
vi TABLE OF CONTENTS
Page
Abstract...... ii
Dedication...... iv
Acknowledgments...... v
Vita...... vi
ListofTables...... ix
ListofFigures ...... xi
Chapters:
1. INTRODUCTION ...... 1
2. AN EXPERIMENTAL INVESTIGATION OF THE EXPLOSIVE EF- FECT IN ALMOST COMMON VALUE AUCTIONS ...... 4
2.1 Introduction ...... 4 2.2 TheWalletGame...... 7 2.3 Summary of the Avery & Kagel Paper ...... 8 2.4 TheModel ...... 9 2.4.1 The Symmetric Auction ...... 10 2.4.2 The Asymmetric Auction ...... 14 2.5 Experimental Design ...... 23 2.6 Experimental Hypotheses ...... 25 2.7 Experimental Results ...... 27 2.8 Summary and Conclusions ...... 46
vii 3. BIDDING IN ALMOST COMMON VALUE AUCTIONS:AN EXPERI- MENT...... 49
3.1 Introduction ...... 49 3.2 Theoretical Considerations ...... 52 3.2.1 Pure Common Value Auctions ...... 52 3.2.2 Almost Common Value Auctions ...... 54 3.3 Experimental Design ...... 56 3.3.1 General Design ...... 56 3.3.2 Experimental Procedures ...... 58 3.4 Results ...... 59 3.5 Summary and Conclusions ...... 65
Appendices:
A. PROOF OF PROPOSITION 1 ...... 69
B. PROOF OF PROPOSITION 2 ...... 72
C. INSTRUCTIONS FOR TWO PLAYER AUCTIONS ...... 76
C.1 INSTRUCTIONS -SYMMETRIC ...... 76 C.2 INSTRUCTIONS-ASYMMETRIC ...... 79
D. INSTRUCTIONS FOR THE FOUR PLAYER AUCTIONS ...... 83
Bibliography ...... 87
viii LIST OF TABLES
Table Page
2.1 Summary of Experimental Sessions ...... 23
2.2 Test of Nash and Expected Value Models in the Symmetric Auctions 29
2.3 Estimates for the Asymmetric Auctions ...... 29
2.4 Comparison of Bidding in the Symmetric Auctions with Regular Bid- ders in the Asymmetric Auctions ...... 30
2.5 Comparison of Sealed Bid and English Clock Mechanism ...... 32
2.6 Means of the Clustering Variables ...... 36
2.7 Criteria for Classifying Subjects in the Asymmetric Auctions . . . . . 38
2.8 Symmetric Auctions by Type ...... 40
2.9 Sophisticated Players of Symmetric Auctions over Regions I & III . . 40
2.10 Compare Actual to Predicted Percentages Won by Region ...... 41
2.11 Asymmetric Auctions with Classi…cation Dummies ...... 42
2.12 Asymmetric Auctions with Advantage Dummy by Type ...... 44
2.13 Compare Actual to Predicted Percentgaes Won By Region ...... 44
2.14 Compare Bidding in Symmetric and Asymmetric Auctions by Classi- …cation...... 45
ix 3.1 First Price Auctions (standard error of the mean in parentheses) . . . 60
3.2 Pure Common Value English Auctions ...... 62
3.3 Bidding in Almost Common Value Auctions ...... 63
3.4 Predictions of Behavioral Bidding Model for Almost Common Value Auctions...... 65
x LIST OF FIGURES
Figure Page
2.1 Sophisticated Best Responses to Naive Bidding Function ...... 12
2.2 Sophisticated Player’sBest Response to an Opponent of Unknown Type 15
2.3 Sophisticated Regular versus Naive Advantaged ...... 18
2.4 Sophisticated Advantaged versus Naive Regular ...... 20
xi CHAPTER 1
INTRODUCTION
The use of auctions to sell publicly owned resources (such as spectrum rights) is growing. Understanding how auctions work both theoretically and practically is crucial to the proper design of such auctions.
Many resources being sold through auctions(…shing, mineral or spectrum rights) can be modeled using a common value auction. In a common value auction, the value of the object for sale is the same to all the bidders, but the bidders are uncertain of the exact value. For example, when companies bid on an oil lease, they don’tknow exactly how much oil is under the ground. Instead, the conduct geological surveys to estimate the amount of oil beneath the ground and bid based on their estimate.
The standard common value auction model assumes that all the bidders involved are symmetric. An almost common value auction relaxes this assumption slightly.
One of the bidders (the advantaged bidder) places a slightly higher value on the object than the other regular bidders. In the oil lease example, one of the bidders may have a lower cost structure than its competitors. This di¤erence represents a private value advantage. Bikhchandani (1988) has shown that when there are only two bidders present, even a small epsilon advantage causes an explosive e¤ect on the outcome
1 and revenue of an auction. The advantaged bidder always wins and seller revenue is dramatically decreased.
The intuition behind this result is clear. In a pure common value auction with two bidders, if a tie occurs each party is indi¤erent between paying their own bid and winning the object, and losing the object. Such indi¤erence is impossible in the almost common value case. If such a tie occurs, the advantaged bidder prefers to increase her bid to insure winning, while the regular bidder prefers to lower his bid and lose the object.
It is not unlikely that one bidder may have a known private value advantage.
Even in auctions with more than two bidders, the private value advantage may have a serious impact on an auction. The existence of private value advantages have been blamed for the less than anticipated revenue raised by the Los Angeles PCS license auction1, and for the seemingly low takeover price in the Glaxo-Wellcome merger2.
Klemperer (1998) raises serious concerns about the use of ascending auctions in cases where a private value advantage exists. First, a known advantage may cause other bidders to not participate in an auction, thus reducing competition and driving prices down. Second, ascending auctions always reduce to just two bidders, the case in which the explosive e¤ect occurs.
Avery and Kagel (1997) tested the explosive e¤ect in an experiment using a two bidder wallet game. They found the e¤ect of the advantage to be proportional, not explosive. The …rst essay of this dissertation presents and tests a model that explains the proportional e¤ect. In the model, there are two types of players: naive and
1 Paci…c Telephone was widely believed to have had a private value advantage due to their famil- iarity with the region and their existing customer database. See Klemperer (1998) page 760. 2 See Klemperer (1998) page 763.
2 sophisticated. Naive players use a rule of thumb bidding function and bid the their signal plus the unconditional expected value of their opponent’ssignal. They ignore the adverse selection e¤ect inherent in winning in a common value auction. The sophisticated players are the standard Bayesian rational agents of economic theory who adjust their behavior to re‡ect the proportions of naive and sophisticated players in the population. The model was tested in an experiment using the wallet game. I was able to classify the players into naive and sophisticated groups. I found that the two groups did use di¤erent bid functions in the symmetric and asymmetric auctions.
However, neither group matched the predicted bid functions for their type.
The second essay examines the e¤ects of an advantaged player when there are more than two bidders in an auction. This paper directly addresses the concerns raised by
Klemperer, regarding the use of ascending auctions. In addition, the nature of the common value is changed from the wallet game to a more natural approach which uses a random variable as the common value. I used subjects that were experienced in similar auctions and had already overcome the winner’s curse. Despite this, I found no explosive e¤ect. Advantaged bidders won no more auctions than predicted by chance alone. These results suggest that common value auctions may be more robust to small asymmetries in practice than the theory suggests.
3 CHAPTER 2
AN EXPERIMENTAL INVESTIGATION OF THE EXPLOSIVE EFFECT IN ALMOST COMMON VALUE AUCTIONS
2.1 Introduction
In a common value auction, the value of the object for sale is the same to all bidders. The typical example is an oil lease, where the amount of oil under the ground is the same regardless of who wins the lease. In an almost common value auction, one of the bidders is advantaged and values the object slightly more than the other regular bidders. In the oil lease example, one bidder may have developed a new, lower cost drilling method, giving that bidder a cost advantage. If this advantage is very small, it is not obvious that it will have an impact on the outcome or revenue generated by the auction as the advantaged bidder does not necessarily have the highest estimate
(signal) of the object’svalue. But, in a startling result, Bihkchandani (1988) showed that in a two-bidder second-price auction even a very small epsilon advantage has an explosive e¤ect on the allocation and revenue of an auction. The bidder with the advantage always wins and seller revenue is dramatically reduced.
The possibility that one bidder (often with a known identity) may have an ad- vantage over other bidders is not an unlikely scenario and these theoretical results
4 may have important implications for auction design. For example, Klemperer (1998) raises serious concerns about the use of ascending auctions for common value items since ascending auctions always reduce to two players in the end. Thus, the slightly advantaged bidder will always win, deterring the regular bidders from even entering when participation is (even slightly) costly. His concerns are supported by anecdotal evidence such as the lower than expected revenue in the Los Angeles PCS license.
Paci…c Telephone is widely believed to have had an advantage due to its customer database and familiarity with the region. However, Avery & Kagel (1997), AK from now on, conducted an experiment on almost common value auctions with a sealed bid mechanism and found the e¤ect of the advantage to be proportional, not explosive.
The subjects in the AK study fell prey to the winner’scurse even in the pure common value auctions. Speci…cally, they appeared to be employing a rule of thumb bidding function equivalent to bidding the unconditional expected value of the item without accounting for the adverse selection e¤ect inherent in winning. Given that subjects failed to follow Nash equilibrium bidding in the symmetric game, perhaps it is not too surprising that the explosive e¤ect failed to materialize in the "almost" common value game.
This paper has two main objectives: the …rst is to present a behavioral model of the two-bidder almost common value auction. Bidders may be one of two types; naive or sophisticated, with each type representing a di¤erent level of rationality. In the model, the e¤ect of a small advantage is proportional, not explosive. The second objective is to move the analysis to an English clock auction.
The AK study documented deviation from the Nash equilibrium predictions by the bidders. My …rst objective is motivated by a desire to explain the deviations. I
5 test the model with an experiment designed to allow me to identify the percentages of naive and sophisticated players in the study. The second objective is motivated by the desire to address Klemperer’sconcerns regarding ascending auctions and the observation that behavior in ascending auctions is typically much closer to equilibrium outcomes than in corresponding sealed bid auctions.3
The use of auctions to sell publicly owned resources, such as spectrum rights, is growing. Spectrum auctions have raised billions of dollars worldwide, with some auctions, such as the British 3G auctions, far exceeding expectations and others, such as the Swiss UMTS Spectrum auctions, falling far short of expectations. (See
Binmore & Klemperer, 2002 and Wolfstetter, 2001) Understanding the role that small asymmetries play in common value auctions has important mechanism design implications. Sorting out which bidders have an advantage, and determining the size of an advantage present in an auction is very di¢ cult when using real-world data.
This makes the problem ideal for a laboratory study where these variables can be controlled. I am unaware of any studies prior to this one that use an English clock auction to study this issue.
The structure of this chapter is as follows: Sections 1.2 and 1.3 explain the wallet game and summarize the AK …ndings. Section 1.4 presents our model. Section 1.5 explains the experimental design I used, while Section 1.6 describes the experimental hypotheses. I report the results in Section 1.7 and Section 1.8 concludes.
3 See the survey in Kagel and Levin’s"Common Value Auctions and the Winner’sCurse," Prince- ton University Press, 2002
6 2.2 The Wallet Game
The wallet game is a pure common value auction where each of two players ob- serves an independent random variable. Player 1 observes X and Player 2 observes
Y: The two players then bid in a second-price auction for an object with value V , where V = X + Y . The unique, symmetric Nash equilibrium predicts that players bid twice their value.4 The Nash equilibria of the wallet game, like all second-price auctions, rests on the willingness to pay argument. Players bid so that, in the event of a tie they are indi¤erent between winning and paying their own bid and losing the auction. In equilibrium, the winning bidder never makes negative pro…ts.
In the almost common value version of the wallet game, one of the bidders has a private value advantage of K, where K is some positive number. The object is worth
V = X + Y + K to the advantaged bidder, and worth V = X + Y to the regular bidder. There is no bid price at which both players are content to tie. The regular bidder knows that the advantaged bidder values the object slightly more, and since a tie implies that the advantaged bidder is unwilling to pay any more the regular bidder must be losing money at that bid price. This creates an adverse selection problem for the regular bidder who thus wants to break the tie by lowering his bid.
In contrast, the advantaged bidder, knowing that the regular bidder is not willing to pay the tieing bid, wants to break the tie by raising her bid. It is this that leads to the explosive e¤ect demonstrated by Bihkchandani. The Nash equilibrium predicts
4 The wallet game has an in…nite number of asymmetric equilibria (such as bully-sucker). See Milgrom (1981) for details regarding equilibria in common value second-price auctions.
7 that the advantaged bidder will win the auction, and that the regular bidder will bid
much lower than in the pure, symmetric common value version of the wallet game.5
2.3 Summary of the Avery & Kagel Paper
AK used the wallet game as the basis for their experiment with a sealed-bid
second-price auction. X and Y were distributed independently and uniformly over
the range [1; 4]. They used two treatments. In the symmetric case, K was set to
zero for both players. In the asymmetric case, K was set to 1 for one of the bidders.
Along with the Nash equilibrium predictions, AK also tested a behavioral model that may best be called the expected value model of bidding. For the symmetric case, this model predicts that players will bid their expected value given their signal. In the asymmetric auctions the expected value model predicts that advantaged players simply add the advantage to their bid. The bid function of the regular player in the asymmetric case remains unchanged relative to the symmetric case.
AK found that the expected value model provides a better …t of the data in both the symmetric and asymmetric cases than the Nash equilibrium predictions. The e¤ect of the advantage on winning and revenue was proportional instead of explosive.
The expected value model, however, does not fully explain the data. It is clear from the data that there is a great deal of heterogeneity among the players and AK found evidence to suggest that some of the players were best responding, as in Nash theory and in contrast to the expected value model.
5 See Klemperer (1998) and Avery & Kagel (1997) for a more complete discussion of this equilib- rium in the wallet game.
8 In the next section I present a model that explicitly accounts for the heterogene-
ity of the players and allows me to further explore the possibility that players best
respond.
2.4 The Model
The AK study documented the players deviations from the theoretical predictions,
and showed that the explosive e¤ect does not occur if all players use a naive expected
value bidding function. In addition, their results provide an example of heterogeneity
among experimental subjects. Failure to achieve Nash equilibrium and heterogene-
ity among subjects are well documented phenomena in the experimental economics
literature and have spurred e¤orts to develop models that account for heterogeneity
through di¤ering levels of sophistication.6 Stahl (1993) models heterogeneity as a
hierarchy of ever increasing levels of intelligence. At the bottom are unintelligent
players that blindly follow a single …xed strategy, while at the very top are the stan-
dard, fully rational Bayesian agents with an in…nite number of types in between.
Although there are likely to be many player types in the population, I focus on the
two extremes, which I call naive and sophisticated. I then consider how allowing
these two types changes the behavioral predictions.
The wallet game is used as the basis of the model. X and Y are drawn iid from a uniform distribution on [0; 1]. Let X be the signal received by the sophisticated player and Y be the signal received by the naive player. The value of the wallet is
V = X + Y . Naive type players ignore the inference of information contained in the
1 event of winning. Thus, they always use a ‘naive’form of bidding, BN (Y ) = 2 + Y .
6 For a survey of examples, see Camerer, 2003 Chapter 5.
9 This is the player’ssignal, Y; plus the unconditional expected value of the opponent’s signal. I chose this bid function for the naive player for two reasons. First, it is a reasonable rule of thumb. The naive player simply fails to take the next logical step and consider what winning implies for the value of the opponent’s signal. Second, this bid function is supported by past experiments and thus seems a reasonable bid function to expect players to use. The sophisticated type player is the standard,
Bayesian rational agent. The sophisticated player adjusts her behavior to account for the possibility that her opponent is naive. How the behavior adjusts depends on the probability of facing a naive or sophisticated rival, and thus on the proportion of naive and sophisticated types in the population. This proportion is captured by the parameter [0; 1]. There are naive types and (1 ) sophisticated types in the 2 population. I model behavior from the point of view of the sophisticated player, as a naive player is ‘hard wired’to bid BN (Y ), and consider how behavior changes with respect to :
2.4.1 The Symmetric Auction
First, I calculate the best response bid function of the sophisticated player when she knows her opponent is naive ( = 1.) The naive player’s bid function is
1 BN (Y ) = 2 + Y . The sophisticated player conditions her bid on winning, and considers her opponent’s bid function. With a bid of B, the sophisticated player
1 1 wins whenever BN (Y ) = + Y B, or Y B . Thus, the sophisticated player 2 2 who observes X = x solves:
B 1 2 1 Max [(x + t) ( + t)]dt (2.1) B 2 Z0
10 Di¤erentiating with respect to B yields,
1 1 1 1 1 [x + (B )] [ + (B )] = x 0; as x (2.2) 2 2 2 2 Q Q 2
3 1 I know that when Y = 1, BN (Y ) = . If X then the expected value of 2 2 the wallet to the sophisticated player is greater than the naïve type player’s bid:
V = X + Y Y + 1 . The sophisticated player earns a positive pro…t of X 1 . 2 2 Therefore, when the sophisticated player receives a signal X [ 1 ; 1] she bids to ensure 2 2 she wins using the following bid function:
3 1 Bs(X) = if X (2.3) 2 2 Conversely, when the sophisticated player receives a signal X [0; 1 ), her 2 2 expected value of the wallet is less than the naive player’s bid. The naive player’s behavior creates a winner’scurse problem for the sophisticated player in this range.
The sophisticated player does not want to win for such signals, and bids low enough to be sure she loses. Therefore,
1 Bs(X) = X if X < 2 (2.4) The sophisticated player switches between wanting to lose and wanting to win when her expected pro…t switches from negative to positive.
When the sophisticated player knows that her opponent is also sophisticated ( =
0), she uses the symmetric Nash bid function BS(X) = 2X. The naive player’sbid function and the sophisticated player’s best responses to players of known type are shown in Figure 2.1.
11 Figure 2.1: Sophisticated Best Responses to Naive Bidding Function
1 I now present the equilibrium of the game when (0; 1) and I let = . 2
Proposition 1 The unique symmetric equilibrium is given by7:
1 2X if x 4 1 1 1 1 1 1 1 2X + 1 [ 2 ] (1 2X) 1 if 4 < X 2 BS(X) = 8 1 1 1 1 1 3 (2.5) 2X [ ] (2X 1) if X < > 1 1 2 1 2 4 < 2 X if 3 X 1 4 > 7 As mentioned before,:> I do not discuss the equilibrium bid function of the naive player since the 1 naive player is hard-wired to bid B(Y ) = 2 + Y:
12 and illustrated in Figure 2.2.
Proof. See Appendix A
1 First note that: BS( 2 ) = 1; the lim BS0 (X) = lim BS0 (X) = 0;and the 1 1 L’Hopital’srule I derive for = 2 ( = 1) that 2X + (1 2X) ln(2 4X) if 1 X 1 B (X) = 4 2 (2.6) S 2X + (1 2X) ln(4X 2) if 1 X 3 2 4 Next, I sketch the economic reasoning behind the proof. If the sophisticated player’s signal falls within Region I or Region III of Figure 2.2, (X [0; 1 ] or X [ 3 ; 1]), 2 4 2 4 then the sophisticated player knows that a tieing bid can only come from another so- 1 phisticated player. This is because the naive player never bids less than 2 in Region 3 I, and never bids more than 2 in Region III. Using the maximum willingness to pay argument, the sophisticated player uses the symmetric Nash equilibrium bid function B(X) = 2X in those regions: When the sophisticated player’ssignal falls within Re- gion II, a tieing bid could come from either a naive player or another sophisticated 1 1 player. If the signal falls in the range ( 4 ; 2 ) the possibility of a tie with a naive player causes the sophisticated player to bid more cautiously than if she knew her opponent was sophisticated. This is illustrated as the increasing curve in Region II of Figure 2.2. The sophisticated player bids cautiously because if the tieing bid comes from a naive player, the value of the object will be less than 2X, where X is the sophisticated player’s signal. The possibility of a tie coming from another sophisticated player, however, causes the sophisticated player to bid more aggressively than if she knew 13 1 3 her opponent was naive. If the signal is in the range ( 2 ; 4 ), the possibility of tieing with a naive player makes the sophisticated player bid more aggressively than against another sophisticated player, but less aggressively than against a known naive type player. This is illustrated as the decreasing curve in Region II of Figure 2.2. The sophisticated player bids more aggressively than against a known sophisticated oppo- nent because if the tie is from a naive player the value of the object is greater than 2X. The possibility that the tie does come another sophisticated player disciplines the bidding, and makes it less aggressive than against a known naive player. How 1 1 cautious a sophisticated player is for signals in the range ( 4 ; 2 ), and how aggressive 1 3 a sophisticated player is for signals in the range ( 2 ; 4 ) will depend on the value of : When is small, the curves move toward the Nash equilibrium, when is large, the curves become ‡atter. This analysis implies that sophisticated players will lose more 1 frequently than naive players for signals below 2 , and win more frequently than naive 1 players for signals above 2 . 2.4.2 The Asymmetric Auction Now suppose that one player has a private value advantage of K, where K [0; 1 ). 2 2 The value of the wallet to the advantaged player is VA = X +Y +K, while the value to the regular player remains VR = X+Y . The situation is much more complicated than the symmetric case because the sophisticated player may be advantaged or regular and her opponent may be naive or sophisticated. This yields four possible cases that must be considered: a sophisticated regular player versus a naive advantaged player; a sophisticated advantaged player versus a naive regular player; a sophisticated regular 14 Figure 2.2: Sophisticated Player’sBest Response to an Opponent of Unknown Type player versus a sophisticated advantaged player and a sophisticated advantaged player versus a sophisticated regular player. As in the symmetric case, I …rst discuss a sophisticated player’sbest response to an opponent of known type ( = 1 or = 0), and then discuss what happens when (0; 1). 2 15 If the sophisticated player’s opponent is also sophisticated ( = 0), the result is the explosive Nash equilibrium. The advantaged player wins all the auctions and increases her bid by more than the amount of the advantage. The regular sophisticated player reduces his bid relative to the symmetric auction. Below, I …rst discuss the case in which a sophisticated player is regular, and then turn my attention to the case in which the sophisticated player is advantaged. I assume that the naive player is "hard wired" to use the expected value model. 1 When the naive player is regular, he bids B(Y ) = 2 +Y: When advantaged, the naive 1 player adds the value of K to his bid, resulting in a bid of B(Y ) = 2 + Y + K: Case 1 - Sophisticated Regular versus Naive Advantaged A The naive advantaged player adds the private value component to his bid; BN (Y ) = 1 + Y + K. For the sophisticated regular player to win, BR(X) BA (Y ) must be 2 S N true. Thus BR(X) 1 + Y + K and Y BR(X) 1 K. The sophisticated S 2 S 2 regular player’sproblem is: B 1 K 2 1 Max [(x + t) ( + t + K)]dt (2.7) B 2 Z0 Di¤erentiating with respect to B yields: 1 1 1 1 1 [x + (B K)] [ + (B K) + K] = x K 0; as x + K (2.8) 2 2 2 2 Q Q 2 A 3 When the naive player is advantaged, I know that if Y = 1, then BN = 2 + K. 1 When X > 2 + K the value to the sophisticated regular bidder is V = X + Y > 1 + K + Y . The sophisticated regular player earns a positive pro…t of X ( 1 + K). 2 2 16 Therefore, from the previous discussion of the symmetric case, the bid function of the sophisticated regular player is X if X [0; 1 + K) BR(X) = 2 (2.9) S 3 + K if X 2 [ 1 + K; 1] 2 2 2 In contrast to the case with two sophisticated players, the regular player no longer wants to lose the auction for all signal values. In the symmetric case (K = 0), the sophisticated regular player wants to lose whenever X [0; 1 ), because expected 2 2 1 pro…t is negative. (See Figure 2.1.) For signal values above X = 2 , the expected pro…t of the sophisticated player becomes positive. A positive value of K raises the point at which the sophisticated player’s expected pro…t switches from negative to 1 positive. I illustrate this in Figure 2.3 using K = 3 . The e¤ect of the advantage on the outcome of the auction is proportional to the size of K, not explosive. As K 1 increases to 2 the point at which the sophisticated player’sexpected pro…t switches from negative to positive moves to the right and the region in which the sophisticated regular player wants to win shrinks to nothing. When K 1 , the explosive e¤ect is 2 restored; the advantaged bidder wins all the auctions and revenue decreases relative to the symmetric case with two sophisticated players. Case 2 - Sophisticated Advantaged versus Naive Regular R 1 The naive regular player’s bid function is BN (Y ) = 2 + Y (the hard-wired bid function). The sophisticated player wins when BA(X) BR (Y ) = 1 + Y or B 1 S N 2 2 Y . The sophisticated advantaged player’sproblem becomes: B 1 2 1 Max [(x + t + K) ( + t)]dt (2.10) B 2 Z0 17 Figure 2.3: Sophisticated Regular versus Naive Advantaged Di¤erentiating with respect to B yields 1 1 1 1 1 [x + (B ) + K] [ + (B )] = x + K 0; as x K (2.11) 2 2 2 2 Q Q 2 It is clear from the previous discussions that the sophisticated player will bid X + K if X [0; 1 K] BA(X) = 2 (2.12) S 3 if X 2 ( 1 K; 1] 2 2 2 18 This is the reverse of the sophisticated regular versus naive advantaged case. The value of K lowers the point at which the expected pro…t of the sophisticated advan- taged player switches from negative to positive relative to the symmetric case. Even though the sophisticated advantaged player values the object more than the naive regular player, there is a region of signals where the sophisticated advantaged player 1 does not want to win the object. I illustrate this in Figure 2.4 using K = 3 . When 1 K < 2 , the e¤ect of the advantage on the outcome of the auction is proportional. As K increases the point at which expected pro…t switches from negative to positive moves to the left and this region becomes smaller. When K 1 , the region in which 2 the sophisticated advantaged player wants to lose disappears, and the explosive e¤ect is restored. Cases 3 & 4 - Sophisticated versus Sophisticated When a sophisticated advantaged player faces a sophisticated regular player, I am back in the case described by AK in their paper and I paraphrase their argument. Here, the advantaged player wins all the auctions. If this were not so, there must be a A R case in which BS (X) = BS (Y ). But, the advantaged player knows the object is worth K more to her than to her opponent. Thus, if the regular player is willing to pay A R BS (X) = BS (Y ), the advantaged player wants to increase her bid to insure winning. The regular player also knows that the object is worth K more to the advantaged player and so wants to reduce his bid to break the tie. Thus the sophisticated advantaged player bids more than the maximum possible value to her opponent and the sophisticated regular player bids his minimum possible value. With signals drawn from the [0; 1] interval, the sophisticated advantaged player’sbid function is 19 Figure 2.4: Sophisticated Advantaged versus Naive Regular 20 3 + X BA(X) X + 1 (2.13) 2 S The sophisticated advantaged player’s bid function is bounded from above by 3 2 + X since any higher bid is weakly dominated. The sophisticated regular player’s bid function is R BS (Y ) = Y (2.14) as any bid smaller than this is dominated. Sophisticated Player is Uncertain of Opponent’sType Now I turn my attention to what happens when (0; 1). As noted above, 2 this is far more complicated than the symmetric case. Here, all four possible types of players (sophisticated advantaged, sophisticated regular, naive advantaged, naive regular) have distinct bid strategies that are functions of both their type and signal. Thus, the four cases must be solved simultaneously yielding a system of di¤erential equations. The strategies of the naive types (regular or advantaged) are "hard- wired". Therefore, I discuss the solution solely from the point of view of the two sophisticated types. R A R A Denote by BN (X);BN (X);BS (X); and BS (X) the bidding strategies of a naive regular, naive advantaged, sophisticated regular, and sophisticated advantaged bidder respectively. 1 Proposition 2 The equilibrium bidding with K = 2 used in our design, is given by: R 1 A BN (X) = 2 + X; BN (X) = 1 + X; 21 BR(X) = X; and 3 BA(x) ( 3 + x) . See the appendix for discussion of the S 2 S 2 more general case of K [0; 1 ) 2 2 Proof. See Appendix B As K decreases, the region in which the advantaged (regular) player wants to lose (win) becomes larger. As K increases, the region becomes smaller. Both the size of this region and the e¤ect on the outcome of an auction are proportional to the size of the advantage. The e¤ect of an advantage on both winning and revenue is proportional for K [0; 1 ), and only becomes explosive when K 1 . 2 2 2 This is a second price auction, and as discussed earlier, the Nash equilibrium solution relies on a willingness to pay argument. Consider a case in which the two bidders tie. The regular player knows that the other player is advantaged and that the object is thus worth more to his opponent. If the tieing bid is the most the advantaged player is willing to pay for the object, then the regular player cannot be willing to pay that amount and wants to lower his bid. Conversely, the advantaged bidder realizes that if the regular player is willing to pay the tieing bid price, then the object must be worth that amount to the regular player. This means the object is worth even more to the advantaged player and she would like to increase her bid to break the tie. Thus, in the almost common value case, there can be no tie between the regular and advantaged players. This is what causes the explosive e¤ect. In this model, the sophisticated player’suncertainty about her opponent’s type mitigates the explosive e¤ect for ‘small’K. The design used in our experiment restores this explosive e¤ect by using a ‘large’value of K: 2.5 Experimental Design 22 This experiment uses the "wallet game" model. The private signals of the bidders, X and Y , are distributed independently and uniformly over [1; 4]. These parameters were deliberately chosen to allow comparison to the sealed bid experiment done by 1 AK. For the asymmetric case I set K = $1:50. This is equivalent to setting K = 2 when the signals are distributed over [0; 1]. This level of K restores the explosive e¤ect by insuring that there is no region of signals in which a sophisticated advantaged player would want to lose, or a sophisticated regular player would want to win, such as those shown on Figures 2.3 & 2.4. Using such a large advantage creates a stark contrast between advantaged and regular players and should help players understand that winning without the advantage leads to losses. This should aid in classifying the players as naive or sophisticated in the asymmetric auctions. I ran a total of four sessions. Two sessions of the standard symmetric auctions and two of the asymmetric auctions8. See Table 2.1. Subjects were recruited by e-mail from a list of students registered for economics undergraduate courses at The Ohio State University in the past year. They were told the sessions would last approximately two hours, and guaranteed a minimum payment of at least $6.00. Session Treatment Subjects 1 Symmetric 17 2 Symmetric 16 3 Asymmetric 16 4 Asymmetric 17 Table 2.1: Summary of Experimental Sessions 8 When there was an odd number of players, 1 player in each round was randomly selected by the computer to sit out. 23 The subjects were given a $12.00 starting balance which included their $6.00 par- ticipation fee. Pro…ts and losses incurred during the session were added or subtracted from their balances and paid to them in cash at the end of the session. Average pay- ments in the symmetric sessions were $16.37 and $27.33 in the asymmetric sessions. All subjects received more than the minimum payment of $6.00. In each session I conducted three practice auctions to allow subjects to become familiar with the software and to give them an opportunity to ask questions regarding the procedures. This was followed by twenty-…ve auction rounds played for cash. To account for pos- sible trial and error at the beginning of the sessions, I dropped the …rst …ve rounds of each session from the analysis. At the beginning of each round, the computer software randomly and anonymously paired the subjects and generated their private signals. In the asymmetric auctions, the computer also randomly determined which player in each pair was advantaged. Playing both roles allows players to see the problem from both sides and past studies (see, for example, Kagel and Levin, 1999) have shown that it facilitates learning. I chose to use a random assignment rather than having players switch back and forth to avoid the possibility of a supergame developing. With only two players in the auction, players knew that if they were not advantaged their partner was. Further, if players switched each period, advantaged players would know that their partner had been advantaged in the last auction and vice versa. I did not want players to bid passively when regular because the experimental procedures encouraged them to hope others would reciprocate, rather than understanding the implications of winning when regular. The random assignment of the advantage means that players were not necessarily advantaged or regular an equal number of times during the experiment. 24 The players participated in an English clock auction with the common value de- termined by the sum of their private signals. In all cases, the new signal along with the possible value range of the object based on the signal appeared on the players’ screens 15 seconds prior to the start of each auction. The clock started at $2:00 and counted up by pennies. The players were instructed to drop out of the auctions when the clock reached the highest price they were willing to pay9. As soon as one player in each pair dropped out of the auction, the auction ended. The players in the asymmetric auctions knew the value of K and knew if they were advantaged or regular in each auction round. At the end of each auction, the results were displayed on the players’computer screens. They were given the following information for auctions in which they par- ticipated: the common value of the object, the signals of both players, the drop out price, and the high bidder’s pro…t. The high bidder’s bid was displayed as XXX since the actual value the player was willing to bid was unknown. The value of K for a particular bid, either zero or $1.50, was included in the results given to the players in the asymmetric auctions and displayed next to the signals. Players were given 30 seconds to review the results before new information for the next round was displayed. Players could, if they wished, review all of their past auctions by scrolling through a history section displayed on their computer screens. 2.6 Experimental Hypotheses This experiment is designed to answer two questions. First, does switching to the English clock mechanism reduce the winner’s curse and generate the explosive 9 See Appendix C for the instructions read aloud at the start of each session. 25 e¤ect in the almost common value auctions? Second, if the explosive e¤ect is not present in the aggregate data, does my model better explain the data than the Nash equilibrium or expected value model? In the symmetric auctions, the Nash equilibrium predicts that bidders bid twice their signals, B(X) = 2X. The expected value model predicts that players bid their signals plus the unconditional expected value of their opponent’s signal, or B(X) = 2:5 + X for the parameters used in our design: My naive/sophisticated model predicts that naive type players use the expected value bidding function, while the sophisticated type players best respond. In the symmetric auctions, this implies that the slope of the bid function for naive players is one, while the slope of the sophisticated players’bid function is two in Regions I & III shown on Figure 2.2. The slope of the sophisticated player’sbid function in Region II of Figure 2.2 depends on the signal and the proportion of types in the population. Our model also makes predictions regarding the outcomes of the symmetric auctions. Sophisticated players should lose more frequently than naive players when receiving 1 a signal below 2.5 ( 2 on the [0; 1] interval) and should win more frequently than naive players when receiving a signal above 2.5. In the asymmetric auctions, the Nash equilibrium model predicts that the advan- taged bidder will win all the auctions, and the regular bidder will lower his bid relative to the symmetric auction. The expected value model predicts that the regular play- ers will bid their signals plus the unconditional expected value of their opponent’s signal. The advantaged players simply add the value of the advantage to their regular bid, or B(X) = 2:5 + X + 1:5 for the parameters used in our design. Which player 26 (advantaged or regular) wins the auction in the expected value model depends on the realization of the private signals. The model predicts that sophisticated players will win all auctions in which they are advantaged and lose all auctions in which they are regular. Naive players will use the expected value bid function, and may win or lose auctions depending on the signals received and whether or not their opponent is sophisticated or naive. Sophisticated players will lower their bids relative to the symmetric auction, but naive players will not. 2.7 Experimental Results Aggregate Results As a …rst step in the analysis, I test the data against the benchmark models of Nash equilibrium and the expected value hypothesis and then look for the explosive e¤ect. I am using an English clock mechanism with just two players and the auction ends as soon as one player drops out. Thus, I do not know the high bidder’sexact bid, only that it is higher than the drop out price, and this drop out price varies by auction. I accommodated the varying drop out prices by specifying a variable threshold Tobit model with random e¤ects throughout the analysis. The drop out price of the low bidder forms the threshold for the high bidder. Since I have just two bidders in each auction, this value is exogenous to the high bidders and the Tobit model is appropriate.10 For the symmetric auctions, I speci…ed: Bit = 0 + 1Xit + it i = 1; 2; :::N t = 1; 2; :::T (2.15) f g f g 10 In the case of two bidders, the log likelihood function of the Tobit model with variable threshold is identical to that of estimating the minimum of two random variables. 27 where i indicates the ith subject and t indicates the auction period. The error term it is comprised of two components; it = i + it. The term i is the error speci…c to individual i, while the term it is the error associated with the auction period.11 The results for all the auctions and for the last 10 rounds as well as the tests of 0 = 2:5, 1 = 1 and 0 = 0; 1 = 2 are shown in Table 2.2. In the asymmetric auctions, the explosive Nash equilibrium predicts that the advantaged players will win 100% of the auctions. Given the actual signals used in the experiment, the expected value model predicts that advantaged bidders will win 85.7% of the auctions. Instead, the advantaged players won only 75.6% of the auctions. Under the expected value hypothesis, the slope of the bid function does not vary between advantaged and regular players; only the intercept changes. To test this I speci…ed Bit = 0 + 1Xit + 2Dit + it (2.16) Where Dit = 1 if the player is advantaged and Dit = 0 otherwise. The expected value model predicts that 2 = 1:5, while the Nash equilibrium predicts that 2 = 3. The results are shown in Table 2.3. I can clearly reject both the Nash equilibrium and the expected value model for both the symmetric and asymmetric auctions. In both cases, however, the results are clearly closer to the expected value hypothesis than to Nash equilibrium. 11 See Green, page 294. 28 Test of Test of 0= 0; 0= 2:5; a b Rounds Estimated Bid Function LL 1= 2 1= 1 Obs B(X) = 2:99+ 0:93X 640 All -422.70 946.59 86.88 (0:10) (0:04) (320) B(X) = 2:98+ 0:93X 352 Last 10 -227.82 468.65 24.6 (0:14) (0:05) (176) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. Table 2.2: Test of Nash and Expected Value Models in the Symmetric Auctions Rounds Estimated Bid Functiona LL Obsb B(X) = 3:10+ 0:81X+ 1:10D 640 All -523.62 (0:14) (0:05) (0:10) (320) B(X) = 2:83+ 0:91X+ 1:16D 352 Last 10 -242.98 (0:18) (0:06) (0:11) (176) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. Table 2.3: Estimates for the Asymmetric Auctions Although advantaged bidders did not win 100% of the auctions (or even the per- centage predicted by the expected value hypothesis) and advantaged bidders did not increase their bids even as much as predicted by the expected value hypothesis, I performed one last test of the explosive e¤ect relative to the symmetric auctions. The explosive e¤ect predicts that the regular bidders will dramatically reduce their bids relative to the symmetric auctions. I pooled the data from the symmetric and asymmetric auctions and compared the behavior in the symmetric auctions with that of the regular bidders in the asymmetric auctions. I speci…ed 29 Bit = 0 + 1Xit + 2Tit + it (2.17) where Tit = 1 indicates the asymmetric auctions. The Nash equilibrium predicts that 2 < 0, while the expected value model predicts that 2 = 0. The results are reported in Table 2.4. The treatment dummy is signi…cant over all the rounds and has the correct sign, but it is small. Regular players’bids decrease by only $0.32 for a mean signal of 2.5. The treatment dummy is not signi…cant over the last 10 rounds. I have established that the explosive e¤ect does not exist in the aggregate data of the asymmetric auctions and that the results are closer to the expected value model than to the Nash equilibrium model. Rounds Estimated Bid Functiona LL Obsb B(X) = 3:10+ 0:90X 0:32T 960 All -720.79 (0:10) (0:03) (0:11) (398) B(X) = 2:95+ 0:92X 0:20T c 480 Last 10 -366.73 (0:14) (0:04) (0:14) (194) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. c: Not signi…cantly di¤erent from zero. Table 2.4: Comparison of Bidding in the Symmetric Auctions with Regular Bidders in the Asymmetric Auctions My contention that the results of both the symmetric and asymmetric auctions are closer to the expected value model is further supported by examining the pro…ts of the bidders. Winning bidders lost money in 35.31% of the symmetric auctions. This is inconsistent with the Nash equilibrium in which bidders never lose money. It is consistent with the expected value model as those bidders with signals less than 2.5 30 are expected to overbid and fall prey to the winner’scurse, while those with signals above 2.5 are expected to underbid and earn positive pro…ts. Those bidders that won with a signal less than 2.5 lost money 84.52% of the time, while those with signals above 2.5 lost money only 17.8% of the time. This lends further support to my conclusion that the results are more consistent with the expected value model than with the Nash equilibrium model. In the asymmetric auctions, overall winning bidders lost money in 16.6% of the auctions. The regular bidders lost money 56.4% of the time and advantaged bidders lost money in 3.7% of the auctions. That bidders lost money clearly indicates that the subjects were overbidding and su¤ering from the winner’s curse. I chose to use an English clock auction in part because, as noted above, it has helped mitigate the worst e¤ects of the winner’scurse in past experiments and I hoped to get a better test of the explosive e¤ect. I combined the symmetric auction data with the AK inexperienced data to see if switching to the English clock mechanism reduced the winner’s curse for the subjects. I again speci…ed a variable threshold Tobit model with random e¤ects as follows: Bit = + Xit + Mit + it i = 1; 2; :::N t = 1; 2; :::T (2.18) 0 1 2 f g f g where Mit = 1 for the English clock mechanism and Mit = 0 for the sealed-bid second-price mechanism used by AK. If the English clock mechanism is e¤ective in reducing the winner’s curse, the coe¢ cient for the mechanism dummy should be negative. The results are shown in Table 2.5. The mechanism dummy variable has the wrong sign, is very small and is statistically insigni…cant. I next speci…ed: Bit = + Xit + MXit + it i = 1; 2; :::N t = 1; 2; :::T (2.19) 0 1 2 f g f g 31 where MXit is an interaction term created by multiplying Mit by the signal Xit. The expected sign of the coe¢ cient is positive, as the English clock mechanism should make the players more responsive to their signal. The interaction term is positive, but small and statistically insigni…cant as shown in Table 2.5. The …nal speci…cation included both the mechanism dummy and the interaction term as follows: Bit = + Xit + Mit + MXit + it i = 1; 2; :::N t = 1; 2; :::T (2.20) 0 1 2 3 f g f g Neither term is statistically signi…cant, nor are they jointly signi…cant. The results are also shown in Table 2.5. Joint Speci…cation Estimated Bid Functiona LL Obsb test Mech. B(X) = 2:79+ 1:06X+ 0:03M c 926 -933.96 Dummy (0:15) (0:04) (0:13) (320) Interaction B(X) = 2:81+ 1:05X+ 0:01MXc 926 -957.37 Term (0:11) (0:05) (0:03) (320) Mech. Dummy & B(X) = 2:70+ 1:10X+ 0:19M c 0:06MXc 920 -959.39 0.76 Interaction (0:18) (0:07) (0:23) (0:09) (320) Term a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. c: Not signi…cantly di¤erent from zero. Table 2.5: Comparison of Sealed Bid and English Clock Mechanism There are two possible explanations for the failure of the English clock mechanism to mitigate the winner’s curse in the experiment. First, the structure of the game used requires players to bid above their signal value. This may be confusing and 32 counterintuitive to the subjects, causing them to overbid. Second, English clock auctions with three or more bidders reveal more information than their sealed bid counterparts. Knowing the drop out prices of other bidders can allow the remaining bidders to update their beliefs about the value of the object and adjust their bids accordingly. My design used only two bidders in each auction. Thus, as soon as one bidder drops out, the auction ends. In this case, the English clock mechanism does not reveal any additional information relative to the sealed bid auction. Classifying the Players Overview of Cluster Analysis In most situations it is easy to de…ne groups and classify observations on the basis of cuto¤ points or intervals (e.g. age or income). But, suppose I would like only a few distinct groups. If I classify individuals based on socioeconomic variables, some individuals may have membership in more than one group. It would be better to group the individuals based on their overall similarity, but deciding which individuals (or observations) are most similar may be di¢ cult to determine, especially if a large number of characteristics are used. This is where cluster analysis is useful. Cluster analysis is a set of techniques that are designed to identify groups and classify observations into those groups based on similarity of characteristics12. If I am classifying individuals on the basis of three characteristics, I could represent each individual as a point in a three dimensional space, with each dimension re‡ecting a di¤erent characteristic. (For example, age, income, and education). After I have plotted all the individuals in this space, clusters of individuals may emerge, and I could simply eyeball the data and say this is group A and that is group B. Cluster 12 See Lorr, 1983 for a survey of cluster analysis techniques. 33 analysis formalizes this process and forces me to de…ne the criteria used to assign group membership. The goal is to minimize the di¤erences within a group and maximize the di¤erences between groups. A subset of cluster analysis (and the type used in this study) is partitioning. The researcher chooses the number of clusters to be formed, and also chooses a distance measure (e.g. euclidean). The data is divided into the chosen number of groups, randomly or through some predetermined method and the means of the relevant characteristics for each group are calculated. Next, each observation is examined and its distance from each group is calculated using the distance measure. The observation is then reassigned to the group it is nearest. Means are recalculated, and the process repeats until no observations are reassigned. I used Stata’s "cluster kmeans" command with the euclidean distance measure to classify the subjects. The Stata algorithm splits the observations randomly into the speci…ed number of groups; in my case, two. It calculates the group means and then compares each observation to the means and reassigns it to the group with the closest means. After all the observations have been examined and reassigned, Stata recalculates the group means. Recalculating the group means after reassignment ensures that the order in which the observations are examined does not matter. The process repeats until it completes a pass without reassigning any observations. (Stata Reference Manual, pg. 265) Classifying the Symmetric Auction Subjects The model de…nes players as naive or sophisticated based on their bidding behavior. Naive players use the ex- pected value bid function, while sophisticated players use the bid function speci…ed 34 in Proposition 1. The sophisticated player’sbid function is complicated and depends on the region from which the signal is drawn as well as : But, if the signal is drawn 1 3 from Region I or Region III (x < 4 or x > 4 ) then the sophisticated bid function is the same as the Nash bid function (as shown on Figure 2.2). Moreover, the Nash bid function can serve as a linear approximation of the bid function for signals drawn from Region II of Figure 2.2. I have between 17 and 20 observations for each indi- vidual subject. Rather than estimating two regressions (one for signals drawn from Regions I & III and one for signals drawn from Region II of Figure 2.2) I used the Nash bid function as a proxy for the sophisticated bid function when classifying the bidders, and will later focus the analysis on behavior in Regions I and III where the Nash bid function and the sophisticated bid function coincide. I used a variable threshold Tobit regression to estimate each individual subject’s bid function. The goal is not to do hypothesis testing using these coe¢ cients; they are used solely to classify the subjects as naive or sophisticated. I speci…ed: Bt = 0 + 1Xt + t (2.21) I used the estimated coe¢ cients and the standard errors of the coe¢ cients as the clustering variables. There are 10 subjects in one group and 23 in the other. I then used the group means of these variables (shown in Table 2.6) to determine which group was naive and which was sophisticated. I have classi…ed the group with 10 subjects, or 30%, as sophisticated. It is important to note that cluster analysis can not tell us how many groups there should be, this must be decided based on theory. Cluster analysis only tells us that if there are two groups, this is the makeup of those groups. Further, although 35 Intercept Standard Slope Standard Group Intercept Error Slope Error Naive 3.33 0.38 0.80 .017 Sophisticated 1.67 0.72 1.53 0.34 Table 2.6: Means of the Clustering Variables cluster analysis has identi…ed two groups, it does not mean that one group is truly sophisticated and the other truly naive. I must do additional testing to determine how well each group …ts my de…nition of naive or sophisticated. Classifying the Asymmetric Auction Subjects I take a di¤erent approach to classifying the subjects in the asymmetric auctions. Rather than estimating individual bid functions and comparing them to naive and sophisticated bid functions, I use a strict criteria based on winning and losing auctions when advantaged or regular. I made this change for two reasons. First, the sophisticated advantaged bid function is distinctly di¤erent from the sophisticated regular bid function. This means I would need to run two separate regressions for each individual; one for advantaged play and one for regular play. I have 17 to 20 observations for each subject. But, subjects were not advantaged or regular an equal number of time. The number of observations for advantage bidding ranges from 4 to 13 for an individual subject, too few for regressions. Second, advantaged players won 75.6% of all the auctions. This mean that most of the observations for advantaged players are censored. The need for two separate regressions, the small number of observations, and the censoring of winning bids all combine to make it extremely di¢ cult to estimate bid functions for comparison with 36 naive and sophisticated bid functions. However, I deliberately chose K = $1:50 to insure that sophisticated players are expected to win all auctions when advantaged and lose all auctions when regular. I used this as the criteria for classifying the subjects. I …rst looked for subjects that met the strict criteria of winning all auctions when advantaged and losing all auctions when regular. No subjects met this criteria. I relaxed the criteria to require that subjects win only 90% of the auctions when advantaged and win no more than 10% of the auctions when regular. Two subjects or 6% satis…ed this criteria. I relaxed the criteria further and de…ned the 80% criteria as winning 80% of the auctions when advantaged and winning no more than 20% of the auctions when regular. Seven subjects or 21% met this criteria. I relaxed the criteria one more time to winning 70% when advantaged and winning no more than 30% when regular. Seventeen subjects or 51% met the criteria.13 This is summarized in Table 2.7. 13 I did not want to punish individuals for winning or losing an auction because of highly irrational behavior by an opponent. For example, sophisticated regular bidders should stay in the auction until the clock reaches X + 1. This is the minimum possible value to the regular bidder. But, suppose the advantaged opponent dropped before the clock reached X; the regular bidder’s signal. The sophisticated regular bidder was following the correct bid function, but lost the auction because the advantaged opponent behaved in a highly irrational way. Likewise, an advantaged bidder could lose an auction because the regular opponent stayed in the auction beyond any reasonble point. So, I looked for such situations. I found three auctions where a regular bidder won because the advantaged bidder dropped before the clock reached the minimum possible value to the regular bidder. Two cases a¤ected our classi…cation of bidders. In one case, a bidder that would have been classi…ed as sophisticated by the 80% criteria is now classi…ed as sophisticated by the 90% criteria. In the second case, a bidder that would be sophisticated by the 70% criteria is now sophisticated by the 80% criteria. The third case had no e¤ect on the classi…cation. I also found seven cases where the regular bidder stayed in the auction beyond the maximum possible value to the regular bidder, but the advantaged players won anyway in all seven cases. Thus, these cases had no e¤ect on the classi…cation of subjects. 37 Percentage of Auctions Won Number of Subjects Criteria Advantaged Regular Meeting Criteria Strict 100% 0% 0 90% Criteria 90% or more 10% or less 2 80% Criteria 80% or More 20% or less 7 70% Criteria 70% or more 30% or less 17 Table 2.7: Criteria for Classifying Subjects in the Asymmetric Auctions The 80% criteria is not very strong. If players strictly followed the naive (ex- pected value) bid function, then given the actual signals realized in the experiment, advantaged players should have won 85.7% of the auctions. I found it surprising that so few subjects met the 90% or 80% criteria. I expected to see more subjects classi…ed as sophisticated in the asymmetric auctions that in the symmetric auctions. Achieving the correct bid function in the symmetric auction requires understanding how to calculate the expected value conditional upon winning. By contrast, the asymmetric problem appears much easier to solve; win when advantaged, lose when regular. I also expected that using such a large advantage would reinforce this idea. It is di¢ cult to win an auction when regular in our design and earn positive pro…ts. Players that won when regular lost an average of $0:17 per auction, while advantaged winners averaged pro…ts of $1:80 per auction. This had no a¤ect on player behavior in the auctions as the percentage of auctions won by regular bidders remained stable over the last ten rounds and winning regular bidders did not appear to improve over time by losing less money. Average losses for winning regular bidders over the last ten rounds were $0:34 per auction. 38 Comparing Naive & Sophisticated Bidders Once the subjects in the symmetric and asymmetric auctions were classi…ed, I tested to see if the groups are truly di¤erent, and if they di¤er in the ways predicted by our model. Symmetric Auctions First, I speci…ed a variable threshold Tobit model with ran- dom e¤ects for each type as follows: Bit = 0 + 1Xit + it i = 1; 2; :::N t = 1; 2; :::T (2.22) f g f g and tested each group against the expected value and Nash equilibrium models. The results are shown in Table 2.8. I reject both models for both groups, but the sophisticated group does come closer to the expected value model. When I examine the di¤erences between all the auctions and the last 10 rounds, it appears that the sophisticated players are learning over time, while the naive players are not. The sophisticated player’s bid function is not linear in Region II of Figure 2.2. So, I also tested the sophisticated group by dropping observations from Region II and running the regression only on Regions I & III and comparing the results to the Nash equilibrium predictions. These results are shown in Table 2.9. Again,I must reject the Nash equilibrium bidding model. Our naive/sophisticated model predicts that naive players bid too aggressively relative to the Nash equilibrium for signals low signals, and not aggressively enough for high signals. I calculated predicted wins and losses using the sophisticated bid function against a naive player for Regions I & III. Table 2.10 shows the actual percentage of auctions won for each type as well as the predicted percentage won for 39 Test of Test of Naive Nash Rounds Estimated Bid Functiona LL Obsb Bid Bid All B(X) = 1:81+ 1:42X 195 -123.53 27.68 93.77 Sophisticated (0:23) (0:10) (109) B(X) = 3:37+ 0:77X 445 All Naive -258.71 64.70 1082.27 (0:11) (0:04) (211) Last 10 B(X) = 1:24+ 1:61X 98 -56.54 17.59 18.00 Sophisticated (0:36) (0:15) (62) B(X) = 3:47+ 0:72X 222 Last 10 Naive -126.07 48.60 662.84 (0:14) (0:05) (98) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. Table 2.8: Symmetric Auctions by Type Test of Rounds Estimated Bid Functiona LL Obsb Nash Bid B(X) = 1:81+ 1:42X 91 All -56.90 60.70 (0:23) (0:10) (53) B(X) = 1:33+ 1:55X 45 Last 10 -23.60 11.63 (0:40) (0:17) (30) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. Table 2.9: Sophisticated Players of Symmetric Auctions over Regions I & III 40 naive or sophisticated players. Note that although both types win too many auctions in Region I, only the naive types win too few auctions in Region III. Region I Region III Type Actual Naive Soph Actual Naive Soph Naive 22.3 12.5 0 78.6 88.4 100 Sophisticated 18.4 10.5 0 88.2 86.3 100 Table 2.10: Compare Actual to Predicted Percentages Won by Region Thus, while I cannot say that the sophisticated group is truly sophisticated, I can say that they are more sophisticated than their naive counterparts. The players classi…ed as sophisticated seem to understand that they should bid aggressively when they have a high signal, but they do not seem to understand that they should bid cautiously when they have a low signal. Asymmetric Auctions I tested the asymmetric auctions using the following vari- able threshold Tobit speci…cation. Bit = 0 + 1Xit + 2Ci + it i = 1; 2; :::N t = 1; 2; :::T (2.23) f g f g Again, Ci = 1 for the sophisticated players and 0 otherwise. Because sophisti- cated players are expected to behave very di¤erently when advantaged than when regular, I ran separate regressions for advantaged players and regular players. I ran the regression using the classi…cations from the 80% criteria (7 subjects classi…ed as sophisticated and 26 as naive). I did not include an interaction term as the slope of the predicted bid functions for both types of players is one. The classi…cation 41 Rounds Estimated Bid Functiona LL Obsb All B(X) = 3:47+ 2:12X+ 2:79C 320 Advantaged -238.86 (0:99) (1:05) (1:05) (242) B(X) = 2:85+ 0:86X+ 0:08Cc 320 Regular -302.34 (0:14) (0:04) (0:15) (78) Last 10 B(X) = 3:95+ 0:91X+ 0:83C 160 Advantaged -66.62 (0:36) (0:17) (0:38) (126) B(X) = 2:77+ 0:91X+ 0:02Cc 160 Regular -149.60 (0:18) (0:06) (0:19) (34) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. c: Not signi…cantly di¤erent from zero. Table 2.11: Asymmetric Auctions with Classi…cation Dummies dummy should have a positive sign for the advantaged players and a negative sign for the regular players. These results are shown in Table 2.11. The classi…cation dummy has the correct sign and is signi…cant over all the rounds and over the last 10 rounds for the advantaged bidders. However, the classi…cation is not statistically signi…cant for the regular bidders over all the rounds or over the last 10 rounds. I also tested the e¤ect of the advantage within each type of players. I speci…ed: Bit = 0 + 1Xit + 2Dit + it i = 1; 2; :::N t = 1; 2; :::T f g f g where Dit = 1 if the player is advantaged and 0 otherwise. These results are shown in Table 2.12. As expected from our classi…cation criteria, the sophisticated players respond much more strongly to the advantage than the naive players. I tested the bid functions of each group against the naive and sophisticated bid functions. Interestingly, I must reject both bid functions for the naive players, but I cannot reject the naive bid function for the sophisticated players. 42 These results indicate that players classi…ed as sophisticated bid more aggressively than their naive counterparts when advantaged, but do not reduce their bids (relative to naive players) when regular. Table 2.13 shows the actual percentage of auctions won as well as the naive and sophisticated predictions for the three regions of signals noted in Figure 2.2. If a sophisticated player truly understands the implications of an advantage, they should win when advantaged even with a low signal, and lose when regular even with a high signal. Table 2.13 shows that the players I have classi…ed as sophisticated only partially understand this. Sophisticated, advantaged players won 73% of the auctions in Region I, unfortunately, sophisticated, regular bidders also won 64% of the auctions in Region III. Clearly, they do not understand that they should lose when regular even with a high signal. However, given that the sophisticated players only satisfy the very weak 80% criteria, it is not surprising that I see little e¤ect in the bid functions or in the win/loss criteria in Regions I & III. The sophisticated players appear to understand that they should be aggressive when they have an advantage, but they do not appear to understand that they should be cautious when regular. Comparing the Groups Across Auctions As a last test of the classi…cations, I compared the behavior of bidders in the symmetric and asymmetric auctions by classi…cation. I speci…ed a variable threshold Tobit with random e¤ects: Bit = + Xit + Tit + it i = 1; 2; :::N t = 1; 2; :::T (2.24) 0 1 2 f g f g where Tit = 1 for the asymmetric auctions and 0 otherwise. The results are shown in Table 2.14. 43 Test of Test of Naive Nash Rounds Estimated Bid Functiona LL Obsb Bid Bid All B(X) = 2:60+ 0:96X+ 1:61D 140 Soph -66.45 0.95 437.89 (0:21) (0:08) (0:18) (71) B(X) = 3:14+ 0:78X+ 0:94D 500 Naive -426.62 35.37 1125.56 (0:16) (0:06) (0:11) (249) Last 10 B(X) = 2:69+ 0:89X+ 1:86D 70 Soph -43.26 2.16 147.68 (0:36) (0:16) (0:32) (34) B(X) = 2:96+ 0:84X+ 1:01D 250 Naive -176.04 20.13 847.93 (0:21) (0:07) (0:12) (126) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. c: Not signi…cantly di¤erent from zero. Table 2.12: Asymmetric Auctions with Advantage Dummy by Type Region I Region II Region III Type Actual Naive Soph Actual Naive Soph Actual Naive Soph Regular Naive 16.2 0 0 26.9 13.9 0 49.2 47.7 0 Soph 4.3 0 0 16.0 9.3 0 63.6 54.5 0 Advantaged Naive 49.2 59.0 100 71.8 87.4 100 88.7 100 100 Soph 73.3 60.0 100 90.7 86.1 100 100 100 100 Table 2.13: Compare Actual to Predicted Percentgaes Won By Region 44 Type Rounds Estimated Bid Functiona LL Obsb B(X) = 3:36+ 0:80X 0:40C 690 Naive Regular All -508.89 (0:09) (0:03) (0:12) (277) B(X) = 3:25+ 0:79X 0:23Cc 344 Last 10 -247.27 (0:14) (0:04) (0:16) (127) B(X) = 2:20+ 1:24X 0:26Cc 270 Soph Regular All -191.05 (0:21) (0:07) (0:22) (121) B(X) = 1:64+ 1:42X 0:07Cc 136 Last 10 -97.28 (0:31) (0:11) (0:28) (67) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted b: Number of censored observations are shown in parentheses. c: Not signi…cantly di¤erent from zero. Table 2.14: Compare Bidding in Symmetric and Asymmetric Auctions by Classi…ca- tion I compared the naive regular bidders in the asymmetric auctions with the naive bidders in the symmetric auctions. The treatment dummy variable is signi…cant but small over all the rounds. Over the last 10 rounds, the treatment dummy is small and statistically insigni…cant. This is consistent with the predictions of the model. Naive regular players do not alter their bidding in the asymmetric auctions relative to the symmetric auctions. I also compared the sophisticated regular bidders of the asymmetric auctions to the sophisticated bidders in the symmetric auctions. Here, the e¤ect of the treatment dummy is insigni…cant for all the rounds and over the last 10 rounds. This is inconsistent with the model, which predicts that regular sophisticated players bid signi…cantly lower in the asymmetric auctions compared to the symmetric auctions. It is however, consistent with my earlier results showing that the classi…cation dummy was not signi…cant for the regular bidders in the asymmetric auctions. It is also consistent with my results from the symmetric auctions showing 45 little di¤erence in the percentages of auctions won by naive and sophisticated players for low signal values. 2.8 Summary and Conclusions I do not …nd any evidence of the explosive e¤ect in my experiment and the players clearly su¤ered from the winner’s curse, as evidenced by the frequency with which they lost money. When tested against the data, the Nash equilibrium model and the expected value hypothesis are rejected for both the symmetric and asymmetric auctions, although the expected value hypothesis provides a better …t than the Nash model. I developed a behavioral model with two types of players that provides another possible explanation for the lack of an explosive e¤ect in the asymmetric auctions. 1 In the behavioral model, the e¤ect of a ‘small’ advantage (K < 2 ) is predicted to be proportional to the size of K. It is only when K 1 , that the explosive 2 e¤ect is restored. My model de…nes players as naive or sophisticated based on their bidding behavior. The naive players are hard wired to bid the unconditional expected value while the sophisticated players best respond. Using cluster analysis based on individual bid functions, I classi…ed 10 or 30% of the subjects as sophisticated in the symmetric auctions. I used a strict win/lose criteria for classifying the subjects in the asymmetric auctions. Sophisticated players should win all auctions in which they are advantaged and lose all auctions in which they are regular. No subjects met the strict criteria of winning 100% of advantaged auctions and winning 0% of regular auctions. Even 46 after relaxing our criteria to 80%, only 7 subjects or 21% of the subjects met the criteria. I was surprised by how few subjects met the 80% criteria. If players had strictly followed the expected value bid function, advantaged players should have won 85.7% 1 of the auctions. I used a large value of K, equal to 2 on the [0; 1] interval to restore the explosive e¤ect in the asymmetrical auctions, expecting this to simplify the problem for the subjects. Rather than calculating the expected value of the item conditional on winning, they simply needed to win when advantaged and lose when regular. While the groups clearly di¤er in their behavior, I cannot term their behavior sophisticated or naive as de…ned by the model. The model predicts that in the symmetric auctions naive players will win more frequently for low signals than the sophisticated players, and naive players will lose more frequently than sophisticated players for high signal values. While the group I call sophisticated does win more frequently than those I call naive for high signal values, there is no di¤erence between my sophisticated players and my model’s naive predictions. In the asymmetric auctions, the model predicts that sophisticated players will be more aggressive (bid higher) than naive players when advantaged, but will be less aggressive (bid lower) than naive players when regular. Sophisticated players only partially understand this. They did bid higher for any given signal than the naive players when advantaged, but they did not bid lower their relative to the naive players when regular. The model also predicts that sophisticated players should lower their bids relative to the symmetric auctions when regular, but they do not. The …rst goal of this study was to enrich the existing literature on almost common value auctions. My model does predict the proportional e¤ect found in the previous 47 AK study. However, my data analysis suggests that behavior is not neatly captured by this simple model. The second goal was to address Klemperer’sconcerns regarding the use of ascending auctions. I gave the explosive e¤ect its best chance, both by moving the analysis to the English auction and by increasing the size of the advantage. In spite of these e¤orts, the explosive e¤ect did not materialize. My results and my use of the English clock auction should, combined with the results of AK, help to mitigate some of Klemperer’sconcerns. 48 CHAPTER 3 BIDDING IN ALMOST COMMON VALUE AUCTIONS:AN EXPERIMENT 3.1 Introduction In a pure common value auction, the value of the object for sale is the same to all the bidders, but is unknown. Instead, the bidders receive signals about the object’s value which they use to form an estimate of the common value. In the symmetric risk neutral Nash equilibrium for this game the bidder with the highest estimate of the object’svalue wins the auction (Wilson, 1977; Milgrom, 1981). An almost common value auction di¤ers from a common value auction in the following way: One bidder, the advantaged bidder, values the object more than the regular bidders. That is, in addition to its common value, the advantaged bidder places an added (private) value on the item; e.g., in the regional air wave rights auctions Paci…c Telephone was widely believed to place a higher value on the West Coast regional area than their potential rivals because of their familiarity with the region and their existing customer base (Klemperer, 1998). This private value, called the private value advantage, can have a signi…cant impact on equilibrium outcomes in both second-price sealed-bid auctions and, more importantly, in ascending price (English) auctions. 49 Bikhchandani (1988) shows that when there are only two bidders, even an epsilon private value advantage has an explosive e¤ect on the outcome in a second-price sealed-bid auction. The advantaged bidder always wins as regular bidders bid very passively due to the heightened adverse selection e¤ect, and seller revenue decreases dramatically. Even in an auction with more than two bidders, the private value advantage may have a serious impact on the auction outcomes. In this regard, Klemperer (1998) raises concerns about the use of ascending price (English) auctions in cases where a private value advantage exists. First, a known advantage may cause other bidders to not participate in the auction, thus reducing competition and driving prices down. Second, ascending price auctions always reduce to just two bidders, the case in which the explosive e¤ect occurs. Although these e¤ects clearly do not exist in all ascending price auction environments, they do hold in a wide variety of settings.14 The present experiment compares bidding in a pure common value English auction with an almost common value English auction using a within subjects design. All subjects had previous experience with a series of …rst-price sealed bid auctions. As such, they had learned to overcome the worst e¤ects of the winner’s curse, earning positive average pro…ts equal to a large share of the predicted pro…ts. These same subjects also do well in the pure common value ascending price auctions, again earning a respectable share of predicted pro…ts. However, when placed in an almost common value auction the explosive e¤ect fails to materialize. Advantaged bidders win only 27% of the auctions (versus 25% predicted by chance factors alone) and there is no signi…cant reduction in seller revenue compared to the pure common value auctions. 14 See Levin and Kagel (2005) for an example of an auction environment in which the explosive e¤ect is not present in an ascending price auction. 50 A behavioral model where the advantaged bidders simply add their private value advantage to their information signal about the common value, and proceed to bid as if in a pure common value auction, better organizes the data than the explosive equilibrium. Two previous experimental studies have investigated the explosive e¤ect predicted under the Bikhchandani (1988) and Klemperer (1998) models. Avery and Kagel (1997) look for it in a two person, second-price sealed bid "wallet auction." They failed to …nd an explosive e¤ect. Rose and Levin (2004) extended the analysis to two person English clock auctions, since the clock auctions are known to yield outcomes closer to equilibrium than second-price sealed bid auctions. They too failed to …nd an explosive e¤ect. In both cases, however, bidders were subject to a winner’scurse in the corresponding pure common value auctions. Thus, these failures to …nd an explosive e¤ect could be attributed to the bidders lack of experience and the fact that they still su¤ered from a clear winner’scurse, in which case the initial conditions that the Nash bidding model requires to get an explosive e¤ect are clearly not satis…ed.15 This experiment di¤ers from these earlier experiments in two ways. First, I use subjects with previous common value auction experience who have clearly learned to overcome the worst e¤ects of the winner’s curse. Second, in using an auction with four bidders, I can directly address Klemperer’s concerns regarding the use of ascending auctions on bidders reluctance to enter the bidding process in the …rst place. 15 Part of the problem here might be the nature of the wallet auction itself which requires bidders to bid above their signal values, something most bidders are reluctant to do. 51 This chapter is organized as follows: Section 3.2 presents the theoretical back- ground. The experimental procedures and subjects prior experience are discussed in Section 3.3. Section 3.4 gives the results and Section 3.5 concludes. 3.2 Theoretical Considerations 3.2.1 Pure Common Value Auctions My experimental design uses an irrevocable exit English "clock" auction with the true value V drawn from a uniform distribution over [V ; V ]: The private signals xi are drawn iid from a uniform distribution over [V ; V + ]: Levin, Kagel and Richard (1996) derive the risk neutral Nash equilibrium (RNNE) for this design. Here, I summarize their results, tailored to this four bidder model. Bidders remain active in the auction until the price reaches the point at which they are indi¤erent between winning the object and paying that price or losing the object. I refer to this price as their reservation price. Bidders determine their reservation prices using their own private signals and the information released during the auction through the drop-out prices of other bidders. I order the private signals from lowest to highest and denote them as x1 < x2 < x3 < x4. Let d1 < d2 < d3 denote the sequential drop out prices. (Note that the auction ends when only one active bidder remains. Thus, there are only three drop out prices in a four bidder auction.) Let I1 denote the information released to the remaining active bidders when the …rst drop out occurs. I2 denotes the information released after the second drop out occurs. No new information is released when the third bidder drops out as the auction ends. In the symmetric RNNE the low signal holder drops out of the auction at his private signal value. He assumes that all the other bidders have the same private 52 signal that he holds. If everyone else drops at his private signal, the low bidder is indi¤erent between losing the object and winning the object and paying his signal value. Why doesn’tthe low signal holder stay active in the auction longer since more information is revealed as the auction continues? Given a uniform distribution, if the low signal holder stays active in the auction past his signal value and wins, then the expected value of the item conditional on winning is the average of his signal and d3+x1 the highest drop out price (E(V ) = 2 < d3), which is less than the price he must pay. The low signal holder can gain information by staying active in the auction, but the information comes too late to be of use and the low signal holder can expect to lose money if he wins. Now, let 1(x; I1) be the reservation price of a bidder with private signal x who remains active after the …rst drop out, and 2(x; I2) be the reservation price of an active bidder after the second drop out occurs. Then (x; Ii) = E(V i(x; Ii)) for i = 1; 2 i j where 1(x; I1) denotes the event that xi = x for all remaining i, i = 2; 3; 4 and f g 2(x; I2) denotes the event that xi = x for all i, i = 3; 4 remaining after the second f g drop occurs. That is, the bidders assume that all the remaining bidders have the same private signal that they do and di = (x; Ij); for j = 1; 2; and i = 2; 3; 4 j f g f g In what follows I focus on the signals falling in the range V + < x1 < xj < V (called Region 2). Given the uniform distribution of signal values around V , conditional on having the high signal value (x1 +xi)=2 provides a su¢ cient statistic for V , so that bidders determine their reservation prices based on their own signals and 53 the …rst drop out price; i.e., they ignore the information contained in the additional drop out prices.16 So that: x1 + xi di = for i = 2; 3; 4 2 f g The high bidder’sexpected pro…t for the pure common value English auction is 42 1 E() = [ ][ ] n + 1 V V (n + 1)(n + 2) 3.2.2 Almost Common Value Auctions The almost common value auctions are exactly the same as the pure common value auctions with the exception that the value of the item to one bidder, the advantaged bidder, is VA = V + k where k > 0. The value to the three remaining (regular) bidders is VR = V : As in the pure common value auctions, bidders remain active in the auction until the price reaches the point at which they are indi¤erent between winning the object and paying that price or losing object. Let d1 < d2 < d3 denote the sequential drop out prices and assume that all signals are within region 2. Proposition 3 In the explosive symmetric equilibria of the almost common value auction, the advantaged bidder wins the auction with probability one. The advantaged bidder remains active in the auction until the price reaches BA(x; dj) x + j = 0; 1; 2; 3 (3.1) the regular bidders remain active in the auction until the price reaches 16 See Appendix A of Levin, Kagel, and Richard (1996) for a more complete derivation of the equilibrium using uniform distributions. See Milgrom and Weber (1982) for a discussion of the general solution. 54 i B (x; dj) x j = 0; 1; 2; 3; i = 1; 2; 3 (3.2) R where dj represents the drop out prices. Proof. Note, since the reasoning here does not depend on the drop out prices, I i i simplify the notation to BA(x) and BR(x): First, I show that BR(x) is a best response i given BA(x);B (x) : Suppose that regular bidder i remains active until the price f R g reaches B(x) > BA(x): Given that the advantaged bidder remains active until the price at least reaches x + , and that the signal x is always within of the true value V , such a bid would insure that the regular bidder loses money. Therefore, the regular bidder would prefer to drop out of the auction earlier and lose the item. Any i bid that insures losing is an optimal response; bidding BR(x) is one such bid. i Next, I show that given BR(x);BA(x) is optimal. Suppose the advantaged bidder remains active in the auction until the prices reaches B(x) < Bi (x) = x . The R signal x is never more than away from the true value V ; thus the regular players’ i bids of B (x) V : Since the value to the advantaged player is VA = V + k, the R advantaged bidder would prefer to remain active longer and win the auction. Given i BR, any bid that wins is an optimal bid, raising the bid further does not matter, and since V x + , bidding BA(x) is one such bid. To show that revenue declines in the almost common value auctions, I compare the expected pro…t of the advantaged bidder to the expected pro…t of the high bidder in the pure common value auction. The advantaged bidder wins the auction with probability one (earning V + k) and pays the bid of the highest signal holder among the three regular bidders. For signals falling in Region 2 (V + < x1 < xj < V ), 55 the advantaged bidder’sexpected pro…t is EA() E(V ) + k (E((xR;h V ) ) (3.3) j where xR;h is the highest signal among the regular bidders: Since the private signals are drawn from a uniform distribution over [V ; V + ] I have 3 EA() E(V ) + k (E( 2 + V ) ) (3.4) 4 After simplifying, the expected pro…t of the advantaged bidder is 1 EA() k + (3.5) 2 For the parameters used in our design (n = 4, = $12; [V ; V ] = [$50; $350]; and k = $2), average pro…t per auction in the almost common value auction should, at a minimum, be almost three times larger than in the pure common value auctions ($8.00 versus $2.35), with seller revenue $3.65 less than in the pure common value auctions. An explosive equilibrium generates two hypotheses for the e¤ect of the private value advantage: (1) The advantaged bidder should win all the auctions, and (2) average bidder pro…ts should be substantially higher, and seller revenue substantially lower, than in the pure common value auctions. 3.3 Experimental Design 3.3.1 General Design In the experiment [V ; V ] = [$50; $350], = $12 and k = $2. The distribution of V ; the value of , the number of bidders in the auction, and the size of k were all 56 common knowledge, as this information was included in the instructions which were read aloud at the beginning of the session17. At the beginning of each auction, subjects were randomly matched into groups of four. In the almost common value auctions one bidder, chosen at random, was designated to be the advantaged bidder.18 It was common knowledge that there was a single advantaged bidder present in each auction, and bidders always knew their own status, as an advantaged or regular bidder. In each auction period a new V and a new set of private signals, xi were drawn. Each bidder’sprivate signal was displayed on his or her computer screen along with the range of possible values for V based on xi. After a pause to allow bidders to review the information, the clock ‡ashed red three times and began counting up in increments of $1.00. Bidders were considered to be actively bidding until they pressed a key to drop out of the auction. Once they dropped out, they could not re-enter the auction. When a bidder dropped out of the auction, the clock paused and the drop out price was displayed on the screens of the remaining bidders. At the end of the pause, the clock resumed counting upwards, this time in increments of $0.50. This process repeated for the second dropout, but with price increments of $0.25 following the pause. When the third dropout occurred, the auction ended and the signals were revealed next to each bidder’s drop-out price. The winning bidder’s signal was also displayed with the drop-out price shown as XXX. In the almost common value auctions, the value of k (0 or $2.00) was also revealed next to the signals after the auction ended. In 17 See Appendix D for the instructions. 18 I randomly determined the advantaged bidder since the equilibrium predicts passive bidding (and zero pro…ts) for the regular bidders. Switching bidder roles in this way has been employed before in common value auctions with insider information (Kagel and Levin, 1999). It is commonly assumed that switching roles speeds up the learning process as regular bidders get to see the problem from the point of view of the advantaged bidder and vice versa. 57 addition, V , the price paid, and the winning bidder’s pro…t or loss was calculated and displayed to all the bidders in the relevant auction market. Bidders were given time to review this information before the next auction. When the new information for the next auction was posted, the results of the last auction were moved to the history section. A bidder could always see the results of his last three auctions, with the most recent at the top of the screen. Earlier auctions could be reviewed by using a scroll bar. 3.3.2 Experimental Procedures Bidders were given a $15.00 starting capital balance (which included their $6.00 show up fee) to allow for experimentation and protect against bankruptcy. Pro…ts and losses earned during the session were added or subtracted from this balance. They were paid their end of the session balances in cash. Payments varied from a low of $25.10 to a high of $86.25 with an average payment of $46.74. No one went bankrupt. The experiment began with two practice, pure common value English auctions to familiarize subjects with the auction procedures. Subjects were encouraged to ask questions both during the instructions and the two practice rounds. Then 15 pure common value English auctions were played for cash. At the end of these auctions, a brief set of instructions were read out loud describing the almost common value auction structure, followed by 15 almost common value English auctions played for cash. There were a total of twenty-eight subjects in the experiment, so that seven four- bidder auctions were conducted simultaneously throughout. All the subjects had 58 participated in two prior four-bidder …rst-price sealed-bid auction sessions with the same underlying structure.19 I chose to use experienced bidders because past experiments have shown that in- experienced bidders in …rst-price sealed-bid and ascending price clock auctions fall prey to the winner’scurse20. That is, they tend to overbid and earn negative average pro…ts with considerable numbers of bankruptcies as a consequence. In contrast, experienced bidders (even those who have participated in just one experimental ses- sion) typically have learned to overcome the worst e¤ects of the winner’s curse. In using twice experienced bidders any failure to observe an explosive e¤ect in the almost common value auctions can not be attributed to unfamiliarity with common value auctions or to a gross winner’scurse. In the experiment the clock speed was 0.25 seconds per tick. That is, the clock increased by $1.00 per quarter second prior to the …rst drop out and by $0.50 per quarter second after the second drop out etc. The pause after every drop out was 3 seconds21. 3.4 Results The analysis focuses on the last 10 auctions for each treatment, thereby dropping periods during which subjects were adjusting to the clock format and the change in 19 The only di¤erence was the support for V which was [50; 950] in the …rst-price auctions. The reason for the change is that since the price clock needed to start at 50, it would have taken an inordinately long period of time for each auction had the same support been used. 20 See Kagel and Levin (2002) for a review of the literature. 21 Drop outs occurring during the pause were counted as dropping out at the same price, but as dropping later than the initial drop out. If additional bidders dropped out during the pause, the pause was extended for another 3 seconds. 59 treatment between pure and almost common value auctions.22 Further, the analysis is limited to draws in the interval (V + < xi < V ) for which there is little or no end-point information regarding V to impact on bidding. This yields a total of 60 pure common value English auctions, 63 almost common value auctions and 50 …rst-price sealed-bid auctions. Performance measures for the last 10 …rst-price sealed-bid auctions from the expe- rienced subject sessions are shown in Table 3.1. The high signal holder won 94% of the auctions, indicating a high degree of symmetry in bidding. Predicted pro…ts un- der the RNNE based on the random draws used in the experiment averaged $4.84 per auction compared to average actual pro…ts of $3.18 per auction, 65% of the predicted pro…t. Although this is a statistically signi…cant shortfall compared to predicted pro…ts, the results are quite comparable to those found in …rst-price common value auctions with even more experienced bidders (see, for example, Kagel and Richard, 2001). Thus, I conclude that subjects in this experiment had overcome the worst e¤ects of the winner’scurse and were earning a respectable share of pro…ts predicted under the RNNE in the …rst-price auctions. Percentage of Auctions Predicted Auction Won by Predicted minus High Signal Holder Pro…ts Actual Pro…ts Actual Pro…ts 94:0% $4:84 $3:18 $1:67 (3:39) (0:62) (0:72) (0:38) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted Table 3.1: First Price Auctions (standard error of the mean in parentheses) 22 The qualitative results are robust to including the …rst 5 auctions for both treatments. 60 Table 3.2 reports the results for the pure common value English auctions. Average actual pro…ts are positive, averaging $3.45 per auction. This is signi…cantly higher than the pro…ts predicted under the RNNE ($2.42 per auction).23 The proximate cause for this is that on average the …rst drop-out in each auction occurred before the predicted dropout ($3.63 below x1 on average), and there was inadequate adjustment to this fact on the part of the remaining bidders.24 To account for this I also compute predicted pro…ts assuming that higher signal holders employ the actual low-drop out price, averaging it with their own signal to determine when to drop out, just as in the equilibrium characterized in Section 3.2.1, but without any adjustment for the fact that lower signal holders were persistently dropping out too soon. Pro…ts predicted under this model are referred to as Nash2 in what follows. As shown in Table 3.2, bidders are earning pro…ts that are marginally lower than those predicted under 25 the Nash2 model. Finally, note that only 63.3% of the auctions are won by the 23 The statistical analysis employed here might be objected to on the grounds that since I have a single session, even with subjects being randomly rematched following each auction, I have "only a single observation." Such a claim essentially asserts that whatever session level e¤ects might be present in experiments of this sort totally dominate how subjects bid (see Frechette, 2005). There is no empirical basis for such a claim. In this respect, it is worthwhile noting that tests for session level e¤ects in a series of …rst-price sealed-bid common value auctions using the same procedures as those employed here report essentially no evidence for such neighborhood e¤ects (Ham, Kagel, and Tao, 2005). Further, statistical tests designed to distinguish whether individual subject bidding errors within a given English clock auction are better modeled as totally independent or totally dependent across rounds (drop-outs) come out in favor of the former assumption (Levin, Kagel, and Richard, 1996). 24 Bikhchandani and Riley (1991) note that reservation prices for other than the two highest signal holders in the pure common value English auction are not unique. That is, there exist symmetric RNNE in which the low signal holder and the second lowest signal holder are indi¤erent between dropping out as described in the text or at lower prices. However, the expected pro…t calculated in Table 3.2 remains the same in their model as higher signal holders are predicted to adjust to these lower dropouts. 25 Pro…ts were also calculated under a variant of the Nash2 model that permits bidders to be a bit more sophisticated. Bidders know that all signals must be within 2 of each other, so that they should ignore a drop price that is too far from their own signal (i.e. a drop price that is less than x 2.) Predicted pro…ts using this alternative measure are indistinguishable from the Nash2 predictions. 61 high signal holder, which is surprisingly low, particularly given the high percentage won by high signal holders in the …rst-price sealed bid auctions. However, Monte Carlo simulations assuming the existence of stochastic bidding errors, in conjunction with independent error draws between successive rounds of each auction, can readily account for this low percentage (see Levin, Kagel and Richard, 1996). Given that the actual pro…ts earned are higher than those predicted under the symmetric RNNE, but below those predicted under Nash2, I conclude that subjects have overcome the worst e¤ects of the winner’scurse in the pure common value English auctions, albeit with inadequate adjustment to the fact that, on average, the …rst dropout consistently occurred several dollars below the low signal value. Percentage of Predicted Predicted minus Auctions Won by Pro…ts Actual Pro…ts High Signal Holder Nash Nash2 Actual Pro…ts Nash Nash2 63:3% $2:42 $4:30 $3:45 $1:03 $0:85 (6:27) (0:52) (0:58) (0:64) (0 :45) (0:45) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted Table 3.2: Pure Common Value English Auctions The performance of the bidders in the almost common value auctions is sum- marized in Table 3.3. The explosive Nash equilibrium predicts that the advantaged bidders will win all of the auctions, regardless of whether or not they are the high signal holder. However, advantaged bidders won only 27.0% of the auctions, little more than one would expect based on chance factors alone (25.0%). By contrast, bidders with the high private information signals won 62.0% of the auctions, which is 62 not signi…cantly di¤erent from the 63.3% frequency in the pure common value Eng- lish auctions (Z < 1.0). The net result is no signi…cant di¤erences in seller revenue between the pure common value and almost common value auctions: average seller revenue was $0.87 lower in the pure common value auctions (t < 1.0). Finally av- erage pro…t per auction was $3.12, well below predicted pro…ts of $7.30 under the explosive equilibrium (t = -6.71, p < 0.01).26 Percentage of Percentage Predicted Predicted Auctions Won Won by Pro…t Minus by High Advantaged Advantaged Actual Actual Signal Holder Bidder Bidders Pro…t Pro…t 62:0% 27:0% $7:30 $3:12 $4:18 (6:17) (5:64) (0:54) (0:66) (0:62) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted Table 3.3: Bidding in Almost Common Value Auctions The di¤erences between the predicted outcomes under the explosive equilibrium and the actual outcomes can be attributed to both advantaged and regular bidders not playing their part of the predicted equilibrium. What would have happened if advantaged bidders had stepped up to play their part of the equilibrium and bid xi + ? Would this have resulted in lower or higher pro…ts than actually achieved given how regular bidders were actually bidding? In this case advantaged bidders would have earned an average of $2.61 per auction, compared to the $1.63 per auction actually earned.27 Thus, the advantaged bidders failed to take advantage of relatively 26 Predicted pro…ts are based on the actual sample of draws here. 27 Bidding xi + advantaged bidders would have won 60 of the 63 auctions. Note, in these calculations I take the bids of regular bidders as given. To account for the censoring of winning bids 63 pro…table unilateral deviations in the direction of the explosive equilibrium. I return to the issue of why the explosive equilibrium did not emerge in the concluding section of the chapter. The data in Table 3.3 show that subjects were clearly not following the explosive Nash equilibrium. In what follows I construct a behavioral model that takes signi…cant steps towards organizing their behavior. I start by assuming that advantaged bidders are simply adding their private value advantage to their signals, and proceeding to play according to the equilibrium outlined in Section 3.2.1; i.e., as if they were in a pure common value English clock auction but with a signal value equal to xi + k.I look at two variations of this model: (i) in which I assume that the bidder with the lowest signal value drops out at that value (called MPureCV - mistaken pure common value) and (ii) in which bidders use the observed …rst drop-out price to average with their own signal value in determining when to drop out (MPureCV2). Note that the private value advantage k is simply added to the signal of an advantaged bidder, before averaging their signal value with the …rst drop-out price, but the k is added on to whatever the common value component of the pro…ts are when they win. The results are shown in Table 3.4. Average predicted pro…ts are $2.28 and $3.85 per auction under the MPureCV and MPureCV2 models, respectively. Average actual pro…ts are not signi…cantly di¤erent from either of these predictions. Both models predict that advantaged bidders will win 33.3% of the auctions compared to the 27% actually won. Further, advantaged bidders won 12 of the 21, or 57% of the auctions they were predicted to win under both models. While far from perfect, this is a substantially better "hit rate" than the by regular bidders, I employ the bid predicted under the second variation of the behavioral model developed below. 64 Percentage of Auctions Predicted Predicted Minus Advantaged Pro…ts Actual Pro…ts Predicted Bidder Minus Wins MPureCV MPureCV2 MPureCV MPureCV2 Actual Wins 33:3% $2:28 $3:47 $0:84 $0:73 6:35% (5:99) (0:44) (0:90) (0 :53) (0:48) (5:93) a: Standard errors are shown in parentheses. All coe¢ cients are statistically signi…cant at the 95% level unless otherwise noted Table 3.4: Predictions of Behavioral Bidding Model for Almost Common Value Auc- tions explosive bidding model. Finally, advantaged bidders were the low signal holder 12 times (after adding in their private value advantage) and were the low bidder in 11 of these auctions.28 Thus, I conclude that bidding in the almost common value auctions is (i) not explosive and (ii) better organized by a model in which advantaged bidders simply add their private value advantage to their signal value and proceed to bid as if in a pure common value auction, with regular bidders behaving the same as in the pure common value auctions. 3.5 Summary and Conclusions In an experiment employing experienced subjects who were familiar with common value auctions, and had already overcome the worst e¤ects of the winner’scurse, I …nd no evidence of the explosive e¤ect of a private value advantage in an English clock auction. Advantaged bidders won only 27% of the auctions, little better than the 28 The average drop point of low bidders, relative to the low signal (plus the private value advan- tage when relevant), was statistically indistinguishable from the average in the pure common value auctions: $2.83 below in the almost common value auctions versus $3.63 in the pure common value auctions (t < 1.0). Further, there are no signi…cant di¤erences in the average drop point relative to their signal value (after adding in the private value advantage) for advantaged and regular low bidders - $2.41 versus $2.92 (t < 1.0). 65 25% predicted by chance factors alone, with no signi…cant change in average revenue compared to a series of pure common value English auctions. Further, bidders are better modeled as simply adding their private value advantage to their signal of the common value and proceeding to play as if in a pure common value auction, rather than seeking to win all the auctions as the explosive Nash equilibrium predicts. Why do bidders perform reasonably close to the predicted Nash equilibrium in the …rst-price common value auctions and in the pure common value English auctions but fail to come anywhere close to the explosive Nash equilibrium in the almost common value auctions? One explanation that comes immediately to mind is that the adjust- ments to the winner’scurse in both the sealed bid and English auctions represents a hot stove type learning - adjusting to the adverse selection e¤ect without really under- standing it. There is clear evidence to this e¤ect from past experiments: Kagel and Levin (1986) show that moderately experienced bidders earning a respectable share of predicted pro…ts in …rst-price sealed bid auctions with four bidders increase their bids in auctions with six or seven bidders, thereby succumbing once again to the win- ner’scurse. Levin, Kagel and Richard (1996) show that the close conformity to the symmetric RNNE found in pure common value English auctions can be explained by a simple signal averaging hypothesis that does not require that bidder’srecognize the adverse selection e¤ect inherent in winning the auction. As such the initial conditions that the theory speci…es as generating the explosive e¤ect are absent –bidders being fully aware of the adverse selection e¤ect inherent in winning the auction and that this will be exacerbated in the presence of a bidder with a private value advantage. 66 However, the mechanism speci…ed in the theory for producing the explosive e¤ect is not the only means to achieving it. For example, suppose that advantaged bid- ders are simply emboldened to bid more aggressively because of their private value advantage. Then in those cases where regular bidders become aggressive enough to beat them they are very likely to su¤er losses, so that they bid more passively in later auctions, which further emboldens the advantaged bidders. Why didn’t something like this happen here? I, of course, do not know why, but the fact remains that it did not happen even though such a deviation would have been pro…table for advantaged bidders even if the regular bidders did not respond with very passive bidding. What, if anything, does all of this have to say about behavior outside the lab? Here I am speculating, but with some insight. First, it’sclear that a helpful condition for producing the explosive e¤ect of a private value advantage is that both the advantaged and regular bidders understand the process. To do this it would seem helpful for bidders holding the private value advantage to announce to their rivals that they intend to top their opponents bids. This is in fact what PacTel did in the FCC major trading areas (MTAs) broadband personal communications services licenses for the Los Angeles and San Francisco licenses (Cramton, 1997).29 Second, it would seem that the advantaged bidder would have to have the resources and a su¢ ciently large private value advantage to make such an announcement credible. As such I seriously doubt the theory’s prediction that even a small private value advantage would set o¤ the explosive e¤ect, even among sophisticated bidders.30 29 Bidders in the laboratory experiment obviously had no opportunity to make such announce- ments. These announcements, which would typically have a reputational element to them, are absent in the formal theory as well. 30 There may also be some incentive under these circumstances in …eld settings for predatory bidding on the part of rivals (see, for example, Cramton, 1997 for discussion of the rich array of 67 strategic options that were available for regular bidders in the FCC spectrum auctions). This would work against the revenue reducing forces implied by the explosive e¤ect and indeed seems to have been at play in the MTA broadband sales. 68 APPENDIX A PROOF OF PROPOSITION 1 In the symmetric case, the model has two type of bidders: Naive and Sophisticated, denoted by N, and S; and let BN (x) and BS(x) respectively, denote their bidding 1 functions. As motivated in the text, BN (x) = 2 + x: Assume that there exists an interval of bids, [B; B]; such that if B [B; B]; then it could have come from a bid 2 by either the N bidder or the S bidder. In other words, B [B; B], there exists x 8 2 and y such that BN (x) = BS(y) = B: Having such a range, [B; B]; in equilibrium of a second-price-auction, implies that the S bidder must be indi¤erent, in the event of a tie, between winning and paying that (tied) bid or losing (the tie breaker). I now derive such equilibrium conditions. Consider an S bidder who has a signal x and considers bidding B [B; B]: If such a bid yields a tie with the N bidder, it implies 2 y = B 1 ; where y is the signal of the N bidder. Alternatively, if such a bid yields a 2 tie with a rival S bidder, it implies, in a symmetric equilibrium, that the rival’ssignal is also x: Thus the indi¤erence equilibrium condition is: 1 BS(x) = x + (x)[BS(x) ] + (1 (x))x; (A.1) 2 where, conditional on BS(x) yielding a tie, (x) is the probability that the tieing bid comes from a bid by a N rival and (1 (x)) is the probability that the tieing 69 bid comes from a S rival. Since I am interested in probabilities that are conditional on the event of a tie, the prior probability of facing each rival, and (1 ); are not the relevant probabilities here as I must use the posterior probabilities. Below I derive (x) but …rst I make a few simple observations. It is easy to see that equation A.1 can be simpli…ed (we do it for (x) < 1; as otherwise there cannot be a tie with the S rival.): (x) 1 BS(x) = 2x + )[x ]; (A.2) (1 (x)) 2 1 Note that any solution to equation A.2 implies BS( 2 ) = 1; as I have in our Figure 2.2. B [B; B); consider " > 0; small enough so that [B;B "] [B; B);let R stand 8 2 for ‘rival’ and consider Pr [R’sbid is N’s R’sBid [B;B "]]= j 2 (Pr[R’sBid [B;B "] R’sbid is N’s]P r R is N ) 2 j f g (P r[R’sbid [B;B "] R’sbid is N’s P r R is N + 2 j g f g P r[R’sbid [B;B "] R’sbid is S’s P r R is S ) 2 j g f g = (P r[R’sbid [B;B "] R’sbid is N’s ) 2 j g (P r[R’sbid [B;B "] R’sbid is N’s + 2 j g P r[R’sbid [B;B "] R’sbid is S’s (1 )) 2 j g 70 Given that signals are distributed uniformly on [0; 1], and that the N strategy is 1 BN (x) = 2 + x; I can simplify and rewrite the above expression as: P r[R’sbid is N’s R’sbid [B;B "] (A.3) j 2 g " = " + (1 )[(B) (B ")] = + (1 )[(B) (B ")]=" However since, lim [(B) (B ")]=" = 0(B) = 1=BS0 (x), I conclude that " 0 ! BS0 (x) (x) = B (x)+(1 )) and that S0 (x) BS0 (x) (1 (x)) ) = (1 ) : Substituting in equation A.2; I …nally derive equation 2.6 in the text: BS0 (x) 1 BS(x) = 2x + (1 ) [x 2 ] 1 1 3 3 The two initial conditions, BS( 4 ) = 2 ; BS( 4 ) = 2 , are needed to maintain conti- nuity in the bidding functions. 71 APPENDIX B PROOF OF PROPOSITION 2 The asymmetric case for a small advantage, that is, K (0; 1 ); is more compli- 2 2 cated to derive and solve than in the symmetric case. However, in the experimental design I wanted to have a “large” K. Whenever K 1 the asymmetries are large 2 enough that I can solve for equilibrium directly. In the asymmetric case I have four types of bidders: Naive-Regular, Naive-Advantaged, Sophisticated-Regular and Sophisticated-Advantaged, denoted by N R, N A, SR; and SA respectively. The bid- ding strategy for the two naive types is simple as the bid functions are by assumption. R A Let BN (x) and BN (x) denote the bidding functions for the two naive types. They are given by: 1 1 BR (x) = + x; BA (x) = + K + x: (B.1) N 2 N 2 R A Let BS (x) and BS (x) denote the bidding functions for the two sophisticated types, Regular and Advantaged and assume that they are strictly monotonic (which will be R A veri…ed in equilibrium). Let S (B) and S (B) denote the inverse of these two functions respectively. I …rst assume that there exists an interval of bids, [B; B] such 72 that if B [B; B]; then it could have come from SR or SA. (Recall that it is a 2 common-knowledge who is a Regular bidder and who is the Advantaged bidder.) In other words, B [B; B], there exists x and y such that, BR(x) = BA(y) = B: 8 2 S S But how could such an interval, [B; B]; exist with K > 0? As in Levin & Kagel, 2004, I may escape the “explosiveness”result of the two bidders almost-common-value literature due to the existence of a N R bidder who is aggressive enough to discipline the SA bidder This discipline reduces the adverse selection problem for the SR bidder who may now (in equilibrium while guarding against the winner’scurse) attempt to win and thus allowing for such a [B; B]: Having such a range, [B; B]; in equilibrium of a second-price-auction, implies that both SR and SA must be indi¤erent, in the event of a tie, between winning and paying that (tied) bid or losing (the tie breaker). I now derive such equilibrium conditions. Consider a SR (SA) bidder who has a signal x and considers bidding B [B; B]: If 2 the tie comes with N A (N R) bidder, the rival’ssignal, y, must be y = BR(x) 1 K; S 2 (y = BA(x) 1 ): Alternatively, if the tie comes with a SA (SR) bidder the rival’s S 2 A R R A signal, y, must be y = S (BS (x)); (S (BS (y))): Thus, I can write: 1 BR(x) = x + (x)[BR(x) K] + (1 (x))A(BR(x)); (B.2) S S 2 S S and 1 BA(y) = y + K + (y)[BA(x) ] + (1 (y))R(BA(y)); (B.3) S S 2 S S e e R where (x) is the probability that conditional on BS (x) yielding a tie, the tieing bid comes from a bid of the N A and where (y) is the probability that conditional on 73e A R BS (y) yielding a tie, the tieing bid comes from a bid of the N : As in the symmetric case here both (x) and (y) involve computing posteriors, using Bayes rule, and thus, R A dBS (x) dBS (y) 31 as before, the expressionse (x) and (y) contain dx and dy respectively. e With simple algebra these two equations can be simpli…ed to: (x) 1 BR(x) = A(BR(x)) + x + [x (K + )]; (B.4) S S S 1 (x) 2 and; A R A (y) 1 BS (y) = S (BS (y)) + y + K + [y + K ]: (B.5) 1 (y) 2 e Assume momentarily that there exists y [0; 1] and z [0; 1]; such that, BR(y) = 2 e 2 S A BS (z): It immediately follows from equation B.4 and equation B.5 that in such a case, (y) 1 BR(y) = A(BR(y)) + y + [y (K + )] (B.6) S S S 1 (y) 2 (y) 1 = z + y + [y (K + )] 1 (y) 2 and A R A (z) 1 BS (z) = S (BS (z)) + z + K + [z + K ] (B.7) 1 (z) 2 e (z) 1 = y + z + K + [z + Ke ] 1 (z) 2 e 31 (x) and (x) are derived in the same way I derivede them in the symmetric case. Since I solve directly for the equilibrium for the speci…c design, further details, including derivation of initial conditions aree omitted. 74 and therefore (y) [y (K + 1 )] = K + (z) [z + K 1 ]: 1 (y) 2 1 (z) 2 e e 1 (y) 1 However, for K 2 that leads to a contradictions as 1 (y) [y (K + 2 )] 0 < 1 K + (z) [z + K 1 ]: 2 1 (z) 2 e e Thus, K 1 is su¢ ciently large to rule out a common bid range, [B; B]; for 2 SA and SR: Next I show that with K 1 ; the SR does not wish to win against 2 a N A rival: If x is the signal of SR and signal y is the rival’s, N A; signal, then R A 1 upon winning, S ’svalue is (x + y) and pays BN (y) = K + 2 + y, for an earning of (x K 1 ) < 0: Thus, in equilibrium SR uses the least aggressive bidding strategy 2 R that is not dominated: BS (x) = x: (Since with a signal of x and a value of x + y > x, R R any bid BN (x) < x is weakly dominated by BN (x) = x:) Given the equilibrium bidding strategies speci…ed, it is easy to show that SA always wishes to win: With a signal of x and rival’ssignal of y; SA’svalue upon winning is (K +x+y) and she pays R 1 R A either y; if S sets the price, and it is y+ 2 , if N sets the price. In both instances S A 3 makes strictly positive payo¤s. To assure winning, S must at least bid 2 ; but may bid higher. However, since the value for a SA never exceeds ( 1 + x + y) ( 3 + x); 2 2 A 3 A 3 BS (x) > ( 2 + x) is dominated by BS (x) = ( 2 + x): I summarize now the equilibrium bidding for all four types of bidders in our 1 asymmetric case with K = 2 : BR (x) = 1 + x BA (x) = 1 + x BR(x) = x; 3 BA(x) N 2 N S 2 S 3 ( 2 + x): 75 APPENDIX C INSTRUCTIONS FOR TWO PLAYER AUCTIONS C.1 INSTRUCTIONS -SYMMETRIC This is an experiment in the economics of market decision making. The National Science Foundation has provided funds for conducting this research. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at the end of the experiment. 1. In this experiment I will create a market in which you will act as buyers of the …ctitious commodity in a sequence of trading periods. In each trading period, you will be paired randomly with another participant. A single unit of the commodity will be auctioned with the two of you as bidders. Your pairings will vary over a series of trading periods and will remain anonymous. 2. The common value V* of an item is determined randomly in each trading period as the sum of two independent component values, X and Y. X and Y are assigned randomly and will lie between $1.00 and $4.00. Each value within this interval has an equally likely chance of being drawn, and the value of X has no bearing on the value of Y. No matter what the value of X, every value between $1.00 and $4.00 is equally likely to be the value of Y. 76 Prior to bidding in each trading period you will learn one, and only one of the two component values. The bidder you are paired with will learn the other value (but not your value). Example 1: Suppose you learn that X= $3.00 and the bidder you are paired with learns that Y= $1.50. Then the common value V* = $4.50. Example 2: Suppose you learn that X= $1.25 and the bidder you are paired with learns that Y= $3.75. Then the common value V* = $5.00. Note that the values of X and Y are determined randomly and independently from auction to auction. As such, a high V* in one period tells you nothing about the likely value in the next period –whether it will be high or low. It doesn’t even preclude drawing the same V* value in later periods. 3. Market Organization: In each period the clock will begin at $2.00 (the lowest possible value for the item) and count up by pennies. When the clock reaches the price you want to bid, you should press a key to drop out of the auction. The bidder who drops out of the auction is the low bidder and sets the drop out price. . The bidder who does NOT drop out of the auction is the high bidder. The high bidder earns the item and makes a pro…t equal to the di¤erence between the value of the commodity and the drop out price . That is V*–(DROP OUT PRICE) = PROFITS for the high bidder. If this di¤erence is negative, it represents a loss. If you do not make the high bid, you will earn zero pro…ts. In this case, you neither gain nor lose money from bidding on the item. 77 4. You will be given a starting capital credit balance of $6.00. Any pro…t you earn in the experiment will be added to this sum and any losses you incur will be subtracted from it. The net balance of these transactions will be calculated and paid to you in CASH at the end of the experiment along with your $6.00 participation fee. The starting capital credit balance and whatever subsequent pro…ts you earn permit you to recoup losses in one auction in part or in total in later auctions. Should your net balance drop to zero (or less) at any time during the experiment, however, you will no longer be permitted to participate. You will be paid your participation fee in CASH and allowed to leave. 5. All participants will be bidding in pairs in each trading period. hen each auction is complete, your screen will list the signals, the drop out price and the value of the item. Your screen will also show the pro…t or loss earned by the high bidder. 6. Your signals are strictly private information and are not to be revealed to anyone else. You will not be told the value of V* until after the each auction is complete. You are not to reveal your bids or pro…ts, nor are you to speak with other subjects while the experiment is in progress. This is important to the validity of the study and will not be tolerated. Let’ssummarize the main points: (1) High bidder earns the item and earns a pro…t = value of the item –drop out price. (2) Pro…ts will be added to your starting balance of $6.00, losses subtracted from it. Your balance at the end of the experiment will be paid in cash. (3) Your private information signal is one of the two component values of V* where V* = X + Y. (4) V* always falls between $2.00 and $8.00. 78 C.2 INSTRUCTIONS-ASYMMETRIC This is an experiment in the economics of market decision making. The National Science Foundation has provided funds for conducting this research. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at the end of the experiment. 1. In this experiment I will create a market in which you will act as buyers of the …ctitious commodity in a sequence of trading periods. In each trading period, you will be paired randomly with another participant. A single unit of the commodity will be auctioned with the two of you as bidders. Your pairings will vary over a series of trading periods and will remain anonymous. 2. The common value V* of an item is determined randomly in each trading period as the sum of two independent component values, X and Y. X and Y are assigned randomly and will lie between $1.00 and $4.00. Each value within this interval has an equally likely chance of being drawn, and the value of X has no bearing on the value of Y. No matter what the value of X, every value between $1.00 and $4.00 is equally likely to be the value of Y. Prior to bidding in each trading period you will learn one, and only one of the two component values. The bidder you are paired with will learn the other value (but not your value). Example 1: Suppose you learn that X= $3.00 and the bidder you are paired with learns that Y= $1.50. Then the common value V* = $4.50. Example 2: Suppose you learn that X= $1.25 and the bidder you are paired with learns that Y= $3.75. Then the common value V* = $5.00. 79 Note that the values of X and Y are determined randomly and independently from auction to auction. As such, a high V* in one period tells you nothing about the likely value in the next period –whether it will be high or low. It doesn’t even preclude drawing the same V* value in later periods. 3. There will also be a private value element for the item –Ki (where the i stands for bidder i). In each period, one bidder (either you or the bidder you are paired with) will have Ki …xed at $1.50. The other member of the pair will have Ki …xed at $0.00. The value of the item to bidder i will be V* + Ki. In other words, in each trading period one bidder will value the item at V* and the other bidder will value it at V* + $1.50. The bidder with the positive value of Ki will be determined randomly for each pair of bidders and for each trading period. You will always know your own value of K and therefore the value of K for the bidder you are paired with as well. Example 1: Suppose you learn that X = $3.00 and that your value of K = 0 (so that the other bidder’s value of K is $1.50), and that the other bidder learns that Y = $1.50. Then the value of the item to you is $4.50 and the value to the other bidder is $6.00. Example 2: Suppose you learn that X = $1.25, your value of K is $1.50 (so that the other bidder’s value of K is 0) and that the other bidder learns that Y= $3.75. Then the value of the item to you is $6.50 and the value to the other bidder is $5.00. Note that the bidder with the positive value of K is chosen independently of the values of X and Y. So if you learn X, your value of K tells you nothing about the possible value of Y. 80 4. Market Organization: In each period the clock will begin at $2.00 and count up by pennies very rapidly. As long as you are willing to buy at the current price, you do nothing. As soon as the price on the screen rises to the maximum amount you are willing to pay, hit any key. This will automatically drop you out of the bidding in this trading period. At this point the auction stops. The bidder who drops out of the auction is the low bidder and sets the drop out price. The bidder who does NOT drop out of the auction is the high bidder. The high bidder earns the item and makes a pro…t equal to the di¤erence between the value of the commodity to him and the drop out price. That is V*–(DROP OUT PRICE) = PROFITS for the high bidder. If this di¤erence is negative, it represents a loss. If you do not make the high bid, you will earn zero pro…ts. In this case, you neither gain nor lose money from bidding on the item. 5. You will be given a starting capital credit balance of $12.00 which includes your $6.00 participation fee. Any pro…t you earn in the experiment will be added to this sum and any losses you incur will be subtracted from it. The net balance of these transactions will be calculated and paid to you in CASH at the end of the experiment. The starting capital credit balance and whatever subsequent pro…ts you earn permit you to recoup losses in one auction in part or in total in later auctions. Should your net balance drop to zero (or less) at any time during the experiment, however, you will no longer be permitted to participate. 81 6. All participants will be bidding in pairs in each trading period. When each auction is complete, your screen will list the signals, the drop out price, and the value of the item. Your screen will also show the pro…t or loss earned by the high bidder. 7. Your signals are strictly private information and are not to be revealed to anyone else. You will not be told the value of V* until after each auction is complete. You are not to reveal your bids or pro…ts, nor are you to speak with other subjects while the experiment is in progress. This is important to the validity of the study and will not be tolerated. Let’ssummarize the main points: (1) High bidder earns the item and earns a pro…t = personal value for the item –drop out price. (2) Pro…ts will be added to your starting balance of $12.00, losses subtracted from it. Your balance at the end of the experiment will be paid in cash. (3) Your private information signal is one of the two component values of V* where V* = X + Y. (4) One bidder values the item at V* and the other at V* + $1.50. (5) V* always falls between $2.00 and $8.00. 82 APPENDIX D INSTRUCTIONS FOR THE FOUR PLAYER AUCTIONS You have all participated in an auction experiment similar to the one you are about to participate in. The major di¤erence has to do with the method for auctioning o¤ the item. We will …rst summarize the main points regarding how the value of the item is determined and your information regarding its value and then go on to point out the di¤erences in the auction procedure. 1. In each auction period a single unit of the commodity will be auctioned o¤ in each market. There will be four bidders in each market. 2. The value of the auctioned item (V*) will be randomly selected from the interval whose lower bound is $50.00 and whose upper bound is $350.00. Any value in this interval has an equally likely chance of becoming V*. 3. Your private estimate of V* will be randomly drawn from the interval V* - $12 and V* +$12. Any value in this interval has an equally likely chance of being drawn as your own private estimate of V*. Note that V* can never be more than your private estimate plus $12, or less than your private estimate minus $12. 83 4. You will all be given starting capital balances of $15. Any earnings will be added to this balance, any losses subtracted from it. If your balance becomes less than zero you will no longer be permitted to bid. 5. In auctioning o¤ the item you will not enter bids as was done before. In- stead, a price clock, shown on the lower right hand side of your computer screens will determine the price. In each period, the clock will begin at $50.00, the lowest possible value for the item, and will count up by $1.00 very rapidly. 6. As long as you are willing to buy the item at the current price, you do nothing. As soon as the price on the screen reaches the maximum price you are willing to pay, hit any key. This will automatically drop you out of the bidding for the commodity in this trading period. Once you have dropped out of the bidding for a period, you cannot re-enter the auction until the next trading period. 7. After the …rst bidder drops out the auction, the clock will pause for a few seconds, and then begin counting up rapidly again with increments of $0.50. After the second bidder drops out, the clock will again pause, then count up in increments of $0.25. In all cases you will see the prices at which bidders drop out, as they drop out of the bidding. Further, any drop-outs during the pause in the price clock will be counted as dropping out at that price, but as dropping later than the bidder who …rst dropped out. 8. The auction stops as soon as there is only one active bidder. This last bidder earns the item and makes a pro…t equal to the value of the item less the price at which the next-to-last bidder dropped out. That is Pro…t =(Value of the item) –(price at which the next-to-last bidder dropped). 84 All other bidders earn zero pro…t. In case the last two bidders drop out at exactly the same price, the computer will randomly decide who earned the item. In this case, the price paid will be the drop out price. 9. In each auction there is a reserve price equal to V*-12. If the price at which the next-to-last bidder drops out is less than V*-12, the high bidder earns the item and pays V*-12, earning a pro…t of $12.00. 10. Once the bidding is completed for a period, pro…ts for the high bidder will be calculated and balances updated. I will also report back on your computer screens the value of the item, the private information signals received, and the prices at which the bidder holding each signal dropped out of the bidding. 11. Everyone will receive the $6.00 show-up fee regardless of his or her auction earnings. Are there any questions? ADDITIONAL INSTRUCTIONS 1. From now on, there will be one bidder in each market for whom the value of the item will be equal to V* + $2.00. For all other bidders in that market the value of the item will be V* just as before. 2. The bidder getting the extra value for earning the item will know who she/he is in each auction period as their computer screen will show K=$2.00 in the signal information. All other bidders will see K=$0.00. The bidder who receives the extra value for earning the item will vary randomly from auction period to auction period. 3. Once the bidding is completed for a period, pro…ts for the high bidder will be calculated, balances updated and information reported back as before. 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