Abstract
Under second-price sealed bid auctions, when bidders have independent private valuations of a risky object, submitting one’s valuation is no longer dominant for non-expected utility bidders. This yields a breakdown in revenue equivalence between English auctions and second-price auctions for non-expected utility bidders. In an experimental auction market selling a single risky object, we find that an English auction yields higher seller revenue than the corresponding second-price auction. Further, the direction of revenue difference is supported by Nash equilibrium bidding behavior of betweenness-conforming non-expected utility bidders, under the additional hypothesis of bidders displaying a weak form of Allais type behavior.
The original article first appeared in the Journal of Economic Behavior & Organization 51: 443–458, 2003. A newly written addendum has been added to this book chapter.
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Notes
- 1.
In an English auction, the auctioneer gradually raises the floor price. The last remaining bidder wins at the last announced price level. In the second-price auction, the highest bidder wins and pays a price equal to the second highest bid submitted. The reader is referred to McAfee and McMillan (1987) for a survey of various types of auctions.
- 2.
The betweenness property requires the utility of a probability mixture between two lotteries to be intermediate in value between the utilities for the respective lotteries constituting the mixture. The class of betweenness- conforming preferences is axiomatized in Chew (1983, 1989), Fishburn (1983), and Dekel (1986).
- 3.
In Table 14.2, a number in the first column denotes a bidder. The last column gives the realizations of H values assigned to the corresponding bidders. For example, the object will yield for bidder 1 a resale value of $1.17 with 50 % chance and US$0.00 otherwise. Every bidder knows that H values are randomly drawn from the interval [US$0.00, US$8.00] while the L value is $0.00 for all bidders.
- 4.
In other words, no English auction is conducted in the certainty parts of Sessions B--F. It is only conducted in sessions A1 and A2.
- 5.
There was no specific instruction for bids not to be in excess of private valuation. Kagel et al. (1987) argue that this sort of instruction has a potential guiding effect, which lowers observed bids.
- 6.
Suppose that we have K groups of two sets of random variables with n k samples in each group \( {\left\{{\left\{{x}_{ki}\right\}}_{i=1}^{n_k},{\left\{{y}_{ki}\right\}}_{i=1}^{n_k}\right\}}_{k=1}^K \) , whose mean and variance are \( {\left\{{\mu}_k^x,{\mu}_k^y\right\}}_{k=1}^K \) and \( {\left\{{\sigma_k^x}^2,{\sigma_k^y}^2\right\}}_{k=1}^K \) for \( k\in \left\{1,2,\cdots, K\right\} \), and \( n={\displaystyle \sum_{k=1}^K{n}_k} \), the total number of samples. There may exist some correlation between two random variables {x ki }, {y ki }, while \( \left\{{\left\{{x}_{ki}\right\}}_{i=1}^{n_k},{\left\{{y}_{ki}\right\}}_{i=1}^{n_k}\right\} \) and \( \left\{{\left\{{x}_{hi}\right\}}_{i=1}^{n_h},{\left\{{y}_{hi}\right\}}_{i=1}^{n_h}\right\} \), \( k\ne h \), are considered independent but not necessarily i.i.d. Let \( {d}_{ki}={x}_{ki}-{y}_{ki} \). Under the hypothesis of expectation of d ki in each group being 0, i.e., \( E\Big({\displaystyle {\sum}_{i=1}^{n_k}{d}_{ki}\Big)}=0 \), by the central limit theorem, \( {\displaystyle {\sum}_{i=1}^{n_k}{d}_{ki}} \) follows the normal distribution with zero mean and variance σ d 2 k , for \( k\in \left\{1,2,\cdots, K\right\} \). Let us define the value z as,
$$ z=\frac{{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{i=1}^{n_k}{d}_{ki}}}}{\sqrt{{\displaystyle {\sum}_{k=1}^K{\sigma_d}_k^2\cdot {n}_k}}}. $$Then, under the assumption of \( E\Big({\displaystyle {\sum}_{i=1}^{n_k}{d}_{ki}\Big)}=0 \) and finite variance σ d 2 k for all \( k\in \left\{1,2,\cdots, K\right\} \), asymptotically z follows the normal distribution N(0, 1). (See for example, Shiryayev (1984).) This z is different from the corresponding statistic when we treat whole n samples as a one big data pool. Here z takes care of the possible different tendency of each group by considering each σ d 2 k . When these variances are unknown, the appropriate estimate for σ d 2 k is \( {s}_{dk}^2=\frac{{\displaystyle {\sum}_{i=1}^{n_k}{\left({d}_{ki}-{\overline{d}}_k\right)}^2}}{n_k-1} \), and we obtain the statistic z* by substituting s d 2 k for σ d 2 k in z such as
$$ z*=\frac{{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{i=1}^{n_k}\left({x}_{ki}-{y}_{ki}\right)}}}{\sqrt{{\displaystyle {\sum}_{k=1}^K{s_d}_k^2\cdot {n}_k}}}, $$which follows the standard normal distribution asymptotically. In the main text, we call the value of this statistic z*-score.
- 7.
The theoretical mean price pp from session k is given by
$$ pp=\frac{n_k-1}{n_k+1}\left({\overline{v}}_k-{\underset{\bar{\mkern6mu}}{v}}_k\right)+{\underset{\bar{\mkern6mu}}{v}}_k. $$ - 8.
The expected private value for bidder i will be \( p{H}_i+\left(1-p\right){L}_i \), where H i and L i are the respective high and low values randomly drawn from the common, public known ranges for H and L and assigned to bidder i.
- 9.
The related work can be found in the recent paper by Nakajima (2011).
- 10.
Four auction formats are English ascending-bid auction, the second-price sealed-bid auction, Dutch descending-bid auction, and the first-price sealed-bid auction.
- 11.
The favorite-longshot bias is a well observed phenomenon of higher demand for higher odds bet (= longshot) in the racetrack and other competitive gambles including sweepstakes.
- 12.
We identify risk averse subjects by those who chose a sure outcome of 0 over a lottery with half chance of getting x and half chance of getting –x.
- 13.
An individual is identified as having LSP if she chooses a longshot lottery yielding a large prize with small probability over a lottery with moderate probability of winning a moderate prize. Chew and Tan (2005) characterized LSP and showed that individuals who are risk averse can exhibit LSP under specific functional form of weighted utility and rank-dependent utility which corresponds to cumulative prospect theory for risk with objective probabilities.
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Acknowledgments
The authors would like to thank John Dinardo, Greg Engl, Cheng Hsiao, Stergios Skaperdas, Yo Sheena for their helpful comments. We acknowledge the research assistance of Vicky Bishop. Financial support from the National Science Foundation (SES 8810833) Research Grant Council (HKUST 6234/97H), and from Seimeikai 1992 is gratefully acknowledged.
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Addendum: Follow Up Research on Auction Design Under Risk and Uncertainty
This addendum has been newly written for this book chapter.
Addendum: Follow Up Research on Auction Design Under Risk and Uncertainty
Following Chew and Nishimura (2003)’s application of non-expected utility theory to explain non-equivalence between the English ascending-bid auction and the second-price sealed-bid auction for a risky object, our researches developed in the two venues; one is to reconsider a bidding behavior for a deterministic auctioned object by introducing bidders with reciprocal social preferences. The other is to take another look at risk preferences and ask about quality of risks.
The first research question arose from our study (Chew and Nishimura (2004))Footnote 9 investigating theoretically the alleged equivalence between the Dutch descending-bid auction and the first-price sealed-bid auction to see whether the bidding behavior of Allais type bidders contribute to generating any difference between the two auctions. Our model predicts that the Allais type bidders would bid higher in the Dutch auction than in the first-price auction. This is contradicted by the literature on laboratory auction experiment reporting the predicted expected price in the Dutch auction is lower than that in the first-price auction.
This led us to leave the issue of risk preferences and ask what else can account for bidders deviating from the prediction by the standard auction theory in terms of observed bidding behavior in laboratories. We revisit the revenue equivalence among four standard auction formatsFootnote 10 for a deterministic object under the IPV setting, when competing bidders have reciprocal social preferences. Nishimura et al. (2011) constructed the intention-base model of the kind of Rabin (1993) or Dufwenberg and Kirchsteiger (2004) to investigate the equivalence between English auction and the second-price auction where a bidder with lower value may choose to overbid in order to reduce the winner’s surplus by making her pay more than the second highest value. We label such a bid exceeding value as a spite bid. The standard auction theory prescribes the best response for a bidder with higher value against such a spite bid just to place a higher bid to win as long as her winning payoff is positive. In contrast, our reciprocity model allows the higher value bidder to retaliate against such a spite bid by placing a bid just below the spite bid and let her opponent win with negative payoff. Such a negative interaction between bidders is more effective in English auction than in the second-price auction, because the ascending calling price in English auction eventually reveals the spite bid, which makes it easier for the higher value bidder to counteract. Thus, the equilibrium price in the second-price auction should be bounded from below by the price in English auction.
When bidders’ values are unknown, the above negative interaction between bidders is not likely to occur since they no longer know their relative value positions. Then, the equilibrium prices in two kinds of auction should coincide in the incomplete information setting even under the negative reciprocity hypothesis. These theoretical predictions are tested experimentally, and we confirmed negative reciprocity at work in the complete information setting but not under the incomplete information setting.
Nishimura (2013) extended this approach to reinvestigate the revenue equivalence between the remaining pair of auction formats, namely the first-price sealed-bid auction and Dutch auction. It has been known in the experimental literature (Kagel (1995)) that the revenue from the first-price auction is higher than that from English auction or Dutch auction and that the observed bids in the first-price auction can be approximated by a bid function which is increasing and concave in value. Cox and Oaxaca (1996) argued that it is bidders’ constant relative risk aversion that causes the higher revenue in the first-price auction and its equilibrium bid function to be concave. The risk aversion, however, cannot explain the non-equivalence in revenue between Dutch and the first-price auction.
Nishimura (2013) proposes bidders’ reciprocal social preferences to explain the revenue non-equivalence as well as the concave bid function in the first-price auction. The crucial difference between two auctions mainly lies in that each step of the descending price ceiling in Dutch auction reveals the lesser spitefulness of one’s opponent which allows the higher value bidder to wait for the price to fall further before she makes a retaliatory bid, whereas in the first-price auction there is no such partial revelation of the spite intention. We then proceed to report the results from experimental Dutch and the first-price auctions under both complete and incomplete information settings. The theoretical predictions based on reciprocity are confirmed in the complete information setting, and the same tendency carries over to the incomplete information setting as predicted.
As to the second question to go beyond a decision making under pure individualistic risk or uncertainty, Chew and Sagi (2008) developed a new platform of preferences under risk, called source preference which allows an individual to choose one risk over the other with the same objective probabilities depending upon how the risk itself is generated. The model can describe behavioral anomalies such as familiarity bias, relating to investors’ inclination to concentrate disproportionally on investment opportunities in their own countries than elsewhere.
A strategic interaction is another important source of risk. There has been a stream of research that attempts to explain the inconsistency between choices made under non-strategic risk/uncertainty and those made under strategic risk/uncertainty, starting from Camerer and Karjalainen (1994). Fox and Tversky (1995), Fox and Weber (2002), and Eichberger and Kelsey (2011) consider such inconsistency to arise from the difference between the choices made under risk and under ambiguity, given that human behavior is intrinsically not easy to predict so that individuals perceive their opponent’s strategic choice as ambiguous. If this is so, then one should be more willing to face risks arising from non-strategic situation than from strategic situation, because ambiguity aversion is one of the most commonly observed preferential traits. In a study using neuroimaging, Chark and Chew (2013), reported that subjects accept a discount to play coordination strategically rather than randomly, and showed that their results were consistent with the predictions of source-dependent expected utility model.
Our latest study, Chew, Mao, and Nishimura (2014), turns out to be in line with Chark and Chew. We experimentally examine the demand for a sweepstake to see whether we observe a favorite-longshot bias (FLB)Footnote 11 widely reported in the racetrack betting literature. A sweepstake awards a large prize with a small probability. In particular, we focus on a variable prize sweepstake in which a single winner receives 90% of the total receipts. Then, the expected value of purchasing a ticket is negative, so that any individual who is risk averse in the usual senseFootnote 12 would not purchase a ticket. We find a significant incidence of FLB reflected in sweepstakes purchase over population sizes ranging from 2 to 141, and a greater tendency for FLB among those who exhibit longshot preference (LSP)Footnote 13 over fixed-odds lotteries. We found, however, mixed support for FLB, that is, subjects showed a greater demand for 28 population sweepstakes than for 141 population sweepstakes including those with LSP and those who are risk averse. Further and intriguingly, we observe significant demands for 2-person sweepstakes even among risk averse subjects. In other words, they are willing to take half-half chance risk generated in a two-person sweepstake market while they decline exogenously generated even-chance bets in an individual choice.
Our findings point to the notion that subjects are more willing to face risks generated in a small population sweepstakes which has tighter strategic interdependency than from a large population sweepstakes where strategic interdependency gets diluted. Our observations unveil the existence of an additional element arising from an interactive nature of sweepstake market that induces our subjects to participate in the sweepstake market along with the effect of LSP resisting the effects of risk aversion. Such an additional element may capture a recreational aspect of sweepstake demand which can only be experienced through interaction among participants. After all, the two research questions we started with at the beginning of this addendum appear to have a shared component concerning the role of intentions in a decision making that differentiates quality of the interaction among players and quality of interactive risks.
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Hong, C.S., Nishimura, N. (2016). Revenue Non-equivalence Between the English and the Second-Price Auctions: Experimental Evidence. In: Ikeda, S., Kato, H., Ohtake, F., Tsutsui, Y. (eds) Behavioral Interactions, Markets, and Economic Dynamics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55501-8_14
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