Bid goodbye to low prices:
An experimental study of first-price
right-to-choose auctions
Janine Bialecki, 11084553
MSc Economics: Behavioural Economics and Game Theory University of Amsterdam, Faculty of Economics and Business
15 ECTs
July 2016
Supervisor: Theo Offerman Second examiner: Audrey Hu
Abstract
Choosing the best method of sale can prove a challenging task for a seller with multiple heterogeneous goods. Rather than selling goods separately, the seller may opt to combine unrelated markets together via a right-to-choose (RTC) auction – forcing buyers interested in different goods to compete for the right to select their preferred good. Previous experimental research shows that RTC auctions are remarkably successful in raising additional revenue over comparable separate good auctions using both second-price sealed bid and ascending auction formats. This paper presents an experimental study of first-price sealed bid RTC auctions. Contrary to the second-price case, I find that first-price sealed bid RTC auctions perform no differently to standard auctions in terms of revenue, efficiency and bidder behaviour – in both cases, revenues far exceed the symmetric risk neutral equilibrium prediction. Given the added complexity of RTC auctions, this may have important implications for sellers considering a first-price RTC format.
2 Statement of Originality
This document is written by Janine Bialecki who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
3 Contents
1. Introduction ... 4
2. Related Literature ... 6
2.1. RTC auctions – the existing literature ... 7
2.2. First-price auctions and bidder behaviour ... 9
2.3. Contribution of this paper ... 11
3. First-price RTC auctions in theory ... 12
3.1. Revenue equivalence ... 12
3.2. Expected profit equivalence ... 16
3.3. Extension – FPSB RTC auction with quantity restriction ... 18
3.4. Best response to actual strategies ... 19
4. Experimental design and procedures ... 19
4.1. Design ... 19 4.2. Procedure ... 20 4.3. Hypothesis ... 21 5. Results ... 22 5.1. Revenue ... 22 5.2. Efficiency... 24 5.3. Bidder behaviour ... 25 5.3.1. A note on outliers ... 25
5.3.2. Bid coefficients - frequency ... 26
5.3.3. Examining bid as a function of value ... 28
5.3.4. Modelling bidder behaviour ... 30
5.3.5. ‘As if’ level of competition ... 34
6. Discussion ... 35
6.1. Explaining bidder behaviour – incentives ... 35
6.2. Comparison to second-price auctions ... 36
6.3. Experimental design issues ... 38
6.3.1. Knowledge of competitors ... 39
6.3.2. Substitutability between goods ... 39
6.3.3. Announcement of winners ... 40
7. Conclusion ... 41
References... 42
Appendix A – Outlier analysis ... 46
Appendix B – Additional regressions ... 48
4 1. Introduction
A seller with multiple heterogeneous goods faces a difficult task choosing the best method to sell those goods. In recent years, a range of new auction formats has arisen, generally
designed to maximise seller revenue and/or efficiency. Right-to-choose (RTC) auctions are part of this new breed, presenting an innovative way to sell goods in thin markets.
Unlike a standard good-by-good (GBG) auction where each good is sold individually, a RTC auction amalgamates disparate markets, forcing buyers interested in different goods to compete for the right to select their preferred good.
While the use of RTC auctions is still quite niche, they are sometimes used to sell strata title apartments (Ashenfelter and Genesove 1992), and have also been used to sell jewellery and antiques (Offerman and Onderstal 2009) and even water rights (Alevy et al. 2010).
The novel feature of RTC auctions is that rather than selling heterogeneous goods separately, all potential buyers interested in those heterogeneous goods are pooled together. This creates a situation where a prospective buyer likely has to outbid more competitors (including those who are not actually interested in the same good) in order to win her own preferred good. To provide an illustrative example, imagine a developer has a boutique suite of three
apartments available for sale – the ground floor apartment which has no stairs, the mid-level apartment which is more secure, and the penthouse apartment which has desirable water views. A range of potential buyers are interested in the apartments, but they have different lifestyle requirements and therefore do not consider the three apartments to be substitutes. The developer could sell each apartment separately, attracting only the prospective buyers interested in each apartment to each respective sale; or alternatively the developer could combine the sales via a RTC auction.
In this example, a RTC auction would comprise three phases: in phase 1, all bidders submit a bid. The highest bidder wins the right to choose her preferred apartment and pays her bid. That bidder, and any other bidders interested in the same apartment, do not participate in the remainder of the auction. In phase 2, the remaining bidders submit a bid. The highest bidder wins the right to choose her preferred apartment (of the two that remain) and pays her bid. That bidder, and any other bidders interested in the same apartment, do not participate in the remainder of the auction. In phase 3, the remaining bidders submit a bid. The highest bidder wins the last apartment and pays her bid.
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The idea behind combining markets for heterogeneous goods into one auction is to bring more bidders together, since an auction with too few bidders risks being unprofitable for the seller (Bulow and Klemperer 1994). However in theory, and with risk neutral bidders, the RTC format should not raise any additional revenue over a comparable GBG auction – a bidder in a RTC auction still need only outbid the other bidders interested in the same good. Despite this theoretical prediction, previous experimental research has shown that RTC auctions are remarkably successful in raising additional revenue over comparable GBG auctions using both a second-price sealed bid (SPSB) format (see Eliaz et al. 2008 for a lab experiment example and Alevy et al. 2010 for a field experiment example) and an ascending auction format (Goeree et al. 2004).
This raises an interesting question of how a first-price sealed bid (FPSB) RTC auction compares to a FPSB GBG auction in terms of revenue, efficiency and bidder behaviour. In a RTC context, both SPSB and ascending auctions have the characteristic that other bidders have a direct impact on the price paid by the winner – in a SPSB auction the winner pays the second highest price (regardless of whether the second highest bidder is interested in the same good); while in an ascending auction the winner can see bids ascending and must deliberately one-up all other competitors to win.
A FPSB auction differs in that other bidders have an indirect impact on the winning price – a bidder will only win by submitting the highest bid, but she will still pay her own bid,
regardless of how much or how little it exceeds the next highest bid.
Given the different ways that the pool of bidders can affect the winning price in a RTC auction, it is worth investigating whether a FPSB format will enhance competition and revenue in a similar manner to that observed in SPSB and ascending formats. Further, since the FPSB format is one of the easiest to understand and execute, this may have important ramifications for sellers looking to use a RTC auction mechanism.
To address this question, analysis is conducted using a FPSB format involving K heterogeneous goods and n potential bidders who value only that good. Per Eliaz et al. (2008), the experiment studied in this paper is based on a design with K = 4 and n = 2. In each round, subjects are given a private induced value drawn independently from a uniform distribution [0, 100]. All of this information is common knowledge among experimental subjects. A within-subjects design is used (that is, all subjects participate in four GBG and
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four RTC auctions) and values for each GBG auction are matched to the corresponding RTC auction to make comparisons between auction types direct.
From the results, we see that the FPSB RTC auction format does not perform significantly differently from the FPSB GBG format in terms of revenue, efficiency or bidder behaviour. In each case, theory predicts that subjects will bid half of their value in the GBG auction and in each phase of the RTC auction, however in practice we see bid coefficients of around 0.8 for both auction types – leading to significantly higher seller revenues than theoretically predicted.1
These findings raise the question of whether second-price RTC formats (SPSB or ascending auctions) are only so successful at boosting revenues due to the direct impact of other bidders on the winning price. A possible alternative explanation is based on the fact that no financial incentives were offered in this experiment – meaning that for both auction types, winning a particular auction round may have been a more salient incentive than the hypothetical points on offer.
In any case, both the FPSB GBG and RTC auction formats result in significant revenue above the theoretical prediction, and comparable bidder behaviour. Further work – including a similar experiment with financial incentives – would help to determine whether this outcome is expected to hold in real life auctions.
The remainder of this paper is structured as follows: Section 2 discusses related literature – both in relation to RTC auctions and FPSB auctions generally; Section 3 presents theoretical predictions for bidder behaviour and revenue in the FPSB RTC auction by analogising with the SPSB results in Eliaz et al. (2008); Section 4 describes the experimental design and procedures and formally presents a hypothesis; Section 5 presents the experimental results, including revenue, efficiency and bidder behaviour; Section 6 discusses the results in further detail, providing a possible explanation for observed bidder behaviour, a discussion of key experimental design issues and potential areas for future research; finally Section 7 concludes this study.
2. Related Literature
Auctions have existed for over 2500 years (Krishna 2002) as a mechanism to sell goods – particularly goods of unknown value. As McAfee and McMillan (1987) state: “For
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manuscripts and antiques, too, prices must be remade for each transaction... how can one discover the worth of an original copy of Lincoln's Gettysburg Address except by auction method?"
From relatively simple origins, a range of more complex auction formats now exist, including combinatorial auctions (an auction for the sale of multiple goods, where bidders can make an all-or-nothing bid for a ‘package’ of goods), and Anglo-Dutch auctions (a two-stage auction, comprising an ascending-price first stage with a premium awarded to the high bidder and a descending-price second stage which determines the actual winner). RTC auctions are just one of a number of innovative new auction formats designed to extract maximum revenue under a particular set of circumstances.
2.1. RTC auctions – the existing literature
One of the first experimental investigations into RTC auctions was by Goeree et al. (2004), who note that the RTC format is a popular way to sell multiple real estate properties (such as land parcels, time-shares and strata title apartments), and discuss the behavioural advantages of using the RTC format, most notably that it “can create competition between bidders who are interested in different items for sale.”
Goeree et al. (2004) directly compare simultaneous ascending auctions with ascending RTC auctions from both a theoretical and experimental perspective. The authors show that with risk neutral bidders both auction types should theoretically raise the same revenue, but that risk averse bidders may bid more in a RTC auction, substantiating the conventional wisdom of real estate agents who choose to sell property using a RTC format. Experimental results confirm this prediction, with the RTC auction generating significantly more revenue than an equivalent ascending auction.
A key strength of the paper is that it matches theoretical predictions with experimental
methods. By recognising that risk averse bidders may prefer to win early, the authors are able to explain aggressive bidder behaviour in the RTC auction. Further, subjects in the
experiment do not know exactly how many other bidders are interested in the same good – a very realistic assumption in most real life auction environments, which likely further
enhances overbidding among risk averse bidders (as Cassady (1967) points out, “of the large number of would-be buyers assembled at a typical auction, a relatively small proportion are likely to be interested in a single item”). Facing an unknown number of competitors in a RTC
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auction format is further explored by Burguet (2007), who notes that in ascending auctions a bidder’s behaviour reveals information about willingness to pay, but not about what her preferred good is. This makes the bidding environment uncertain and creates a situation where “risk averse bidders are willing to pay a premium in order to secure their winning in the first round.”
It is worth noting that the experiment of Goeree et al. (2004) involved only 24 subjects in total – 16 subjects in the ascending RTC treatment and eight subjects in the simultaneous ascending auction treatment. In addition, with only four bidders per group, there are few expected phases of the RTC auction, leaving limited opportunity to observe whether bidders become less aggressive as RTC phases progress.
Further work – including an experimental comparison between SPSB RTC auctions and equivalent GBG auctions – is conducted by Eliaz et al. (2008). Due to the strategic
equivalence of ascending and SPSB auctions (including that bidders have a weakly dominant strategy to bid up to their own value (Vickrey 1961)), we should expect the results of Eliaz et al. (2008) to be consistent with the findings of Goeree et al. (2004). Indeed, the authors predict and find that the SPSB RTC auction raises significantly more revenue than an equivalent GBG auction, and more revenue than an equivalent GBG auction with optimal reserve price. Eliaz et al. (2008) attribute this to bidders overestimating the level of
competition they face by failing to realise that they need only concentrate on beating bidders who value the same good they do.
Finally, Salmon and Iachini (2007) study ‘pooled’ auctions, where multiple related yet non-identical goods are auctioned simultaneously, with bidders winning the right to choose from the pool in order based on their bids. Like RTC auctions, the pooled auction format is most commonly used to sell real estate, however it is more applicable when bidders’ values for goods are highly correlated (for example, an investor looking to buy one apartment may be interested in any one of eight apartments available, though she does have some order of preference). The authors refer to previous work of Menezes and Monteiro (1998) to show that for risk neutral bidders the pooled auction is theoretically revenue equivalent to a sequential auction. Once again, the authors test whether the theory holds in an experimental setting and find that the pooled auction generates substantially more revenue than a corresponding ascending auction. In practice, the authors find that subjects bid very aggressively in the pooled auction in an attempt to win their most preferred good, but often incur losses due to
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winning a lesser good. In fact, over many auction rounds, losses persisted, indicating subjects did not really learn from their previous mistakes. An interesting extension – which is
unfortunately not included – would be to ask losing subjects to describe their bidding strategy after each round in order to better understand bidder assumptions and behaviour.
The authors consider various explanations for subject behaviour, including responding to disequilibrium strategies of other subjects, and a ‘love of winning’, but ultimately conclude that subject behaviour is best described by an ‘attentional bias’, with bidders focussing their attention only on their most preferred few goods and failing to properly account for the possibility of winning lesser goods.
2.2. First-price auctions and bidder behaviour
While the theoretical equilibrium for FPSB auctions has long been established (Vickrey 1961), FPSB auctions generally do not conform well to the predicted theory when studied experimentally. A key reason for this may be the impact of risk preferences. While a SPSB auction has an equilibrium based on weakly dominant strategies (meaning that individual risk preferences should not influence bidding strategies), risk aversion may play an important role in FPSB auction bidding strategies, as risk averse buyers place a lower marginal valuation on large gains (Maskin and Riley 1983).
Studied experimentally in detail by Cox, Roberson and Smith (1982) and Cox, Smith and Walker (1983a, 1983b, 1985, 1988), FPSB auctions are subject to persistent overbidding vis-à-vis the symmetric risk neutral Nash equilibrium (RNNE), which the authors argue is due to bidder risk aversion. Smith (1987) further amalgamates the outcomes of over 1500 single-item auction experiments and finds that FPSB auctions have appreciably higher prices than Dutch, English or SPSB auctions, which he again attributes to risk aversion. Interestingly, in addition to raising additional revenue for sellers, FPSB auctions may also be preferred by a certain type of buyer – because bidders do not closely adhere to the theoretical equilibrium, weaker bidders are afforded a positive chance of winning (Vickrey 1961, appendix III and Klemperer 2002).
Although an equilibrium has been established for both symmetric risk neutral bidders and risk averse bidders with constant relative risk aversion (Cox et al. 1988), experimental bidding behaviour tends to be quite heterogeneous and therefore inconsistent with models that assume identical risk averse bidders (Cox et al. 1988). Crucially, a bidder’s strategy
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depends not just on her own preferences, but should be based on beliefs about the bidding strategy of other players (Hendricks and Paarsch 1995). With real life subjects, this becomes a difficult problem to solve.
Somewhat controversially, Harrison (1989) starts from the premise that statistically significant deviations from the symmetric RNNE bid in FPSB auctions do not necessarily lead to significant deviations in expected payoffs. Put a different way, Harrison argues that in many experiments the payoff function is too flat, giving subjects little incentive to find their best response, since expected payoffs from non-optimal decisions are close to expected payoffs from optimal decisions (Merlo and Schotter 1992).While criticised in some circles for his own assumptions (see, for example, Cox et al. 1992, Friedman 1992, and Kagel and Roth 1992), Harrison’s (1992) response to critics reiterates his belief that observed
‘anomalies’ can be reconciled with expected behaviour, as they “reflect theoretically consistent behavior under conditions where misbehavior is virtually costless.” In essence, Harrison (1992) argues that most FPSB auction experiments fail to satisfy the precepts of nonsatiation and/or dominance of payoffs.
In response to Friedman’s (1992) criticism of Harrison’s work, particularly that Harrison’s explanation relies on an asymmetric loss function, Goeree et al. (2002) incorporate insights that experimental overbidding can be described if loss functions are asymmetric with a quantal response equilibrium model to partly substantiate Harrison’s claim.
Although the various researchers did not ultimately reconcile their disparate views, the debate arising from Harrison’s critique provided interesting new insights into the reasons for bidder behaviour in FPSB auctions.
More recently, Dorsey and Razzolini (2003) and Armantier and Treich (2009) study the causes of overbidding in FPSB auctions via experimental methods including pairing a FPSB auction with an equivalent lottery, and jointly eliciting choices and subjective probabilities. Contrary to previous conventional wisdom, both papers conclude that biased probabilistic beliefs (rather than risk aversion) are the key force which drives overbidding.
Finally, a novel study by Neugebauer (2004) examines how the RNNE strategy performs in a practical experimental setting. Neugebauer asks experienced experimental subjects to
formulate a profit maximising strategy, with those strategies then programmed into automata which compete in a series of sequential first-price auctions. Neugebauer finds that the RNNE
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bidding strategy performs very poorly when competing against other strategies. This
reinforces the point that the RNNE is an equilibrium only in the symmetric case, and that this theoretically optimal strategy may not perform well at all when competing against
disequilibrium strategies. As Hendricks and Paarsch (1995) point out, a player’s “optimal strategy depends upon his beliefs concerning the bidding rules of his opponents…” Therefore, if an experimental subject knows that her counterpart is unlikely to choose the symmetric RNNE bidding strategy, she should also deviate, creating a situation where very few (if any) subjects are expected to play the symmetric RNNE bidding strategy in practice.
2.3. Contribution of this paper
A major contribution of this paper is to fill a gap in the extant literature by examining RTC auctions using a FPSB format.
An important difference between this study and the work of Goeree et al. (2004) and Eliaz et al. (2008) is that bidders do not have a dominant strategy in the standard FPSB auction. In contrast, both SPSB and ascending auctions involve a weakly dominant strategy where each subject bids up to her own value.
It is perhaps not so difficult to envisage that bidders who are aware of the weakly dominant strategy in a standard SPSB or ascending auction may simply adopt the same strategy in a RTC context, failing to recognise that they will quite likely pay the bid of a subject interested in a different good (discussed in Eliaz et al. 2008). This may – to a great deal – account for the additional revenue raised in SPSB and ascending RTC auctions above comparable GBG auctions.
On the other hand, the theoretical RNNE of a FPSB auction only holds in the symmetric case, and most experimental analysis of FPSB auctions finds that subjects tend to significantly overbid relative to the equilibrium. Given this behaviour, it is quite possible that the revenue gap between a standard FPSB auction and corresponding RTC auction will be considerably smaller.
This may have important practical implications. In particular, because the FPSB format is easy for both prospective buyers and sellers to understand and execute, the results of this experimental study could have ramifications for sellers looking to use a RTC auction mechanism in practice.
12 3. First-price RTC auctions in theory
This paper draws heavily on the analysis and results of Eliaz et al. (2008) to determine the theoretical equilibrium for the FPSB RTC auction. Although the following analysis is not a formal mathematical proof, it uses tools including revenue equivalence and a priori expected bidder profits to find the equilibrium bidding strategy. Per Eliaz et al. (2008), this section focuses on the symmetric Bayesian Nash equilibrium, where all bidders use the same
monotonic bid function in each phase. While risk preferences were less pertinent for Eliaz et al. (2008) (because bidding one’s own value is the weakly dominant strategy in a standard SPSB auction), this paper relies on the assumption that bidders are risk neutral to make analysis more tractable.
We begin by introducing some terminology and constraints which apply to both the
theoretical and experimental set up. In each case, subjects value one and only one good in the RTC auction, and each good is desired by an equal number of subjects. Each subject is given a private induced value for her preferred good drawn independently from a uniform
distribution [0, 100], and has an implied value of zero for all other goods.
The RTC auction consists of K goods, and n risk neutral bidders interested in each good, with N = nK the total number of bidders present. In the general case, there are K phases of the RTC auction – that is, every good is sold in successive phases. In an extension (discussed in Section 3.3), there are K − q < K phases, where q represents a quantity restriction. Therefore, the RTC auction takes the form RTC(K, K - q, n), meaning that there are K goods, K - q phases (with q equal to zero in the baseline case) and n bidders per good.
3.1. Revenue equivalence
At the outset, it is possible to directly compare the FPSB RTC(4, 4, 2) with the SPSB RTC(4, 4, 2). Because in both cases values are independent and identically distributed over the uniform support [0,100], it is possible to build an argument for the FPSB RTC
equilibrium using expected values and revenue equivalence.
Revenue equivalence is an important economic result first identified in Vickrey’s celebrated work (1961), as well as in later and more general work by Riley and Samuelson (1981) and Myerson (1981). Broadly speaking, the revenue equivalence theorem states that under certain conditions, the expected revenue and bidder profits for a broad class of auctions (including
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FPSB, SPSB, English and Dutch auctions) will be the same provided that bidders use equilibrium strategies.
For FPSB and SPSB auctions where each of n risk neutral bidders has a private value independently drawn from a common distribution, revenue equivalence relies on two key conditions: both mechanisms should lead to the same allocation in equilibrium (in practice, this means that the highest value bidder should win the item); and the lowest value bidder should expect zero surplus. If these conditions hold, then the expected revenue should be the same in both mechanisms, and a bidder with value v should make the same expected payment (see, for example, Klemperer 1999).
Following Vickrey’s (1961) work, for the general GBG setup in this paper, we know that a FPSB auction is theoretically revenue equivalent with a corresponding SPSB auction. Further, Eliaz et al. (2008) discuss revenue equivalence between the SPSB GBG and RTC auctions, noting that “From a theoretical standpoint, there is no basis for [the] conventional wisdom [that a RTC auction has the ability to generate higher selling prices] as [the RTC] auction format is revenue equivalent to an auction in which each good is sold separately.” Combining these two findings, it is then possible to state that the FPSB RTC auction should be revenue equivalent with the SPSB RTC auction. Further, because bidders are assumed to rely on the same monotonic bid function in each phase, bidders with equivalent value order statistics should win equivalent phases of the first-price or second-price RTC auction, leading to revenue equivalence in each individual phase (that is, the bidder with the highest value should win her preferred good in phase 1; of the remaining bidders, the bidder with the highest value should win her preferred good in phase 2 etc.)
For the RTC(4, 4, 2), a comparison of revenues based on expected values is provided in Table 1.
14 Table 1 – Revenue equivalence comparison RTC(4, 4, 2)2
Phase Second-price First-price
1 From Eliaz et al. (2008), the bid coefficient is 4/7.
Player Expected value Bid
8 88.889 50.794 7 77.778 44.444 6 66.667 38.095 5 55.556 31.746 4 44.444 25.397 3 33.333 19.048 2 22.222 12.698 1 11.111 6.349
The player with the highest value statistic should win. Using revenue equivalence, the winning bid should equal the price paid in the second-price auction. Solving to find a bid coefficient for a value of 88.889 that results in a bid of 44.444, it must be that players bid ½ of their value.
2 From Eliaz et al. (2008), the bid coefficient is 3/5. In the preceding phase, player 8 wins and therefore exits. A second player who wanted the same good as player 8 also exits. This could be any of the
remaining players, each with probability 1/7. Using expected values, this leaves six bidders with values now uniformly distributed [0, 88.889].3
Player Expected value Bid
6 76.190 45.714 5 63.492 38.095 4 50.794 30.476 3 38.095 22.857 2 25.397 15.238 1 12.698 7.619
The player with the highest value statistic should win. Using revenue equivalence, the winning bid should equal the price paid in the second-price auction. Solving to find a bid coefficient for a value of 76.190 that results in a bid of 38.095, it must be that players bid ½ of their value.
2 Unless otherwise indicated, all numbers in this paper are given to three decimal places. 3
Alternatively, it is possible to calculate the expected values for the six remaining players directly by keeping the original [0, 100] distribution, assuming there is a 1/7 chance that any player leaves and determining the expected values of the remaining players using a 1/7 probability of each possible ‘world’. It is not difficult to show that this results in equivalent expected values in phase 2, however it is more tedious to calculate. A spreadsheet that shows this directly is available upon request.
Expected revenue = expected highest bid in FPSB auction Expected revenue
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Phase Second-price First-price
3 From Eliaz et al. (2008), the bid coefficient is 2/3. In the preceding phase, player 6 wins and therefore exits. A second player who wanted the same good as player 6 also exits. This could be any of the
remaining players, each with probability 1/5. Using expected values, this leaves four bidders with values now uniformly distributed [0, 76.190].
Player Expected value Bid
4 60.952 40.635
3 45.714 30.476
2 30.476 20.317
1 15.238 10.159
The player with the highest value statistic should win. Using revenue equivalence, the winning bid should equal the price paid in the second-price auction. Solving to find a bid coefficient for a value of 60.952 that results in a bid of 30.476, it must be that players bid ½ of their value.
4 From Eliaz et al. (2008), the bid coefficient is 1. In the preceding phase, player 4 wins and therefore exits. A second player who wanted the same good as player 4 also exits. This could be any of the
remaining players, each with probability 1/3. Using expected values, this leaves two bidders with values now uniformly distributed [0, 60.95].
Player Expected value Bid
2 40.635 40.635
1 20.317 20.317
The player with the highest value statistic should win. Using revenue equivalence, the winning bid should equal the price paid in the second-price auction. Solving to find a bid coefficient for a value of 40.635 that results in a bid of 20.317, it must be that players bid ½ of their value.
Similar scenario analysis for any RTC(K, K, n) indicates that symmetric risk-neutral equilibrium bidding follows the general first-price Bayesian Nash solution: for every phase of the RTC auction.
It is also possible to verify this analysis by considering bidding strategies based on a priori values.
Expected revenue
Expected revenue
16 3.2. Expected profit equivalence
Given revenue equivalence in every phase of the RTC auction, and assuming bidders use the same monotonic bid function in each phase, we should therefore also observe bidder profit equivalence across FPSB and SPSB RTC auctions.4
Using backward induction, to begin we can see that phase 4 is a regular auction with two players vying for one good. Vickrey (1961) shows that the dominant strategy in the SPSB auction is for players to bid , while the same paper also shows that the symmetric RNNE bidding strategy in a FPSB auction is to bid .
For the RTC(4, 4, 2), a comparison of a priori expected profits using backward induction for all phases is provided in Table 2. For expected values, the a priori probability of winning, expected price conditional on winning, expected profit conditional on winning and
unconditional expected profit should not vary between auction types – therefore it is a matter of solving for the first-price RTC bid coefficient in each phase which rationalises the
conditional (or unconditional) expected profit.
Table 2 – Expected profit equivalence comparison RTC(4, 4, 2)
Phase 4 3 2 1 Second-price First- price Second-price First- price Second-price First- price Second-price First- price A priori probability of winning 1/2 1/2 1/4 1/4 1/6 1/6 1/8 1/8 Expected price, conditional on winning5 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v Expected profit, conditional on winning 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v 1/2 v Unconditional expected profit 1/4 v 1/4 v 1/8 v 1/8 v 1/12 v 1/12 v 1/16 v 1/16 v Bid coefficient6 1 1/2 2/3 1/2 3/5 1/2 4/7 1/2
4 This is possible using ex ante expected values, but not necessarily for realised values. In the case of realised
values, FPSB and SPSB auctions are not theoretically isomorphic – see Kagel and Levin (2014).
5
SPSB expected price, conditional on winning, is determined by applying the Eliaz et al. (2008) bid coefficient to a priori values, then determining the expected second bid, given the expected winning bid.
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Once again, similar scenario analysis for any RTC(K, K, n) leads to the same result – bidders follow the same strategy in every phase of the FPSB RTC auction.
By taking account of the parameters of the auction and using terminology from Eliaz et al. (2008), the FPSB RTC RNNE strategy can be generalised in the form:
(Note: k is the relevant phase, and Nk is the total number of active bidders in phase k).
An interesting result is that the conditional expected profit does not vary across phases, meaning that based on a priori values, a winning bidder should always expect to receive a profit equal to half of her value. This has some intuitive appeal – we should expect a bidder with a higher value statistic to win in an earlier phase, and to receive a higher absolute profit (due to the scope for higher profit afforded by having a high value); while bidders in later phases will generally have lower values and will therefore have less scope to achieve large absolute profits.
This is somewhat analogous to Weber’s (1983) finding that the expected price in phase k of a multiple-round homogenous good auction should not vary from round to round – a finding which Ashenfelter (1989) termed the ‘law of one price’. In Weber’s example, for all but the final round, subjects are guaranteed future chances of winning. This provides an incentive for all subjects to shade their bids by a higher proportion in early rounds, despite needing to outbid more competitors in order to win, meaning that the FPSB equilibrium involves successively more aggressive bids as phases progress (and subjects face fewer chances of winning).
In the RTC auction case where each bidder desires only one good, the same two competing forces exist – in early phases a player has more expected chances of winning, however that same player must beat more competitors in order to win. The difference is that in the RTC auction, only one of each heterogeneous good is available. Therefore, instead of bidding more aggressively as phases progress, the two forces seem to balance, resulting in an equilibrium where subjects should use the same strategy in every phase.
18 3.3. Extension – FPSB RTC auction with quantity restriction
Although it is not studied experimentally in this paper, it is possible to apply the same expected bidder profit equivalence comparison to a FPSB RTC auction with quantity restriction. Eliaz et al. (2008) study a SPSB RTC(K, K-1, n) auction from a theoretical perspective and a specific SPSB RTC(4, 3, 2) auction experimentally.
For the RTC(4, 3, 2), a comparison of a priori expected profits is provided in Table 3.
Table 3 – Expected profit equivalence comparison RTC(4, 3, 2)
Phase 3 2 1 Second-price First- price Second-price First- price Second-price First- price A priori probability of winning 1/4 1/4 1/6 1/6 1/8 1/8 Expected price, conditional on winning 3/4 v 3/4 v 2/3 v 2/3 v 5/8 v 5/8 v Expected profit, conditional on winning 1/4 v 1/4 v 1/3 v 1/3 v 3/8 v 3/8 v Unconditional expected profit 1/16 v 1/16 v 1/18 v 1/18 v 3/64 v 3/64 v Bid coefficient7 1 3/4 4/5 2/3 5/7 5/8
Again, similar scenario analysis for any RTC(K, K-1, n) leads to equivalent results.
Once again, by conducting similar analysis on any combination of N players and K goods in the RTC auction with quantity restriction, we find that the following general result holds:
This is consistent with comparisons between the baseline FPSB and SPSB RTC auctions – it is only the denominator that changes from (Nk – 1) in the SPSB case to Nk in the FPSB case
to reflect the fact that bidders are expected to bid less in the FPSB auction where they pay their own bid.
7
19 3.4. Best response to actual strategies
It is worth re-emphasising that the first-price RNNE strategies discussed is this section are symmetric – that is, they only constitute a best response for a bidder if her counterpart is playing the same strategy (Neugebauer 2004). In practice, it is very likely that bidders are risk averse and/or that bidders use some kind of heuristic, such as ‘try to earn 10 points per round for high values and 5 points per round for low values’ (see, for example, Tversky and Kahneman 1974). If a rational bidder suspects her counterpart is using a non-equilibrium bidding strategy, she should change her own best response.
When assessing bidding strategies in an experimental context it is worth keeping this in mind – a player may indeed be behaving as predicted by theory, but based on a non-symmetric behavioural analysis of her fellow subjects.
4. Experimental design and procedures
4.1. Design
The experiment consisted of two treatments and a within-subjects design – that is, each subject participated in both GBG and RTC auctions. In order to directly compare results with the SPSB auctions in Eliaz et al. (2008), there were two bidders vying for each good (n = 2) and the number of goods available was four (K = 4) in both treatments. This resulted in eight subjects per auction group, with each group participating in four GBG auctions and four RTC auctions. Each GBG auction consisted of a single phase, while each RTC auction consisted of four phases. To control for order effects, an AB/BA design was used in which four groups of subjects participated in the GBG auctions first and four groups of subjects participated in the RTC auctions first.
In total, data were collected for eight groups of eight subjects. A ninth group also participated, however there was some confusion over the first auction type in that group which resulted in the wrong bidding slips being submitted by one subject. Unfortunately, this affected all other subjects in the same group, meaning the data did not accurately reflect winners, revenue, or goods available. For this reason, data from the ninth group is excluded from all analysis.
For both treatments, the value for the good desired by each subject was independently drawn from a uniform distribution [0, 100] using random.org. Values were randomly matched to
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subjects for each GBG auction. Subjects were then anonymously paired, with values from each round of the GBG auction then matched across to the corresponding RTC auction. For example, in each group subject 5 was paired with subject 7, meaning that 5’s value for round 1 of the GBG auction became 7’s value for round 1 of the RTC auction and vice versa. This ensured that each good was desired by players with the same induced values across auction types, making revenue between the two treatments directly comparable.
The number of subjects in each group, number of goods available, number of bidders vying for each good, number of rounds of each auction type and value distribution were common knowledge among all subjects.
The experiments were performed over the course of two weeks in May 2016, using both Bachelor and Master students recruited from the University of Amsterdam and Erasmus University Rotterdam.
The experiment itself lasted just under one hour. Subjects were not paid, but each auction was couched in terms of points, with points awarded only to the winner of each auction in the form: Points earned = value – bid. Subjects were asked to participate seriously, as if points represented real money.
Before the start of each auction type, subjects answered control questions to ensure they understood the specific rules of the auction. It total, data were recorded for 64 subjects.
4.2. Procedure
Experiments were generally carried out in a classroom environment, with one group
participating in the author’s living room. The experiment was paper-based, obviating the need for all subjects to have access to a computer; however this did make the experiment
considerably slower than a comparable computer-based experiment. General instructions were read aloud at the beginning of the experiment, with specific instructions for each auction type read aloud immediately before the first auction of that type. Instructions for the experiment with the GBG auction played first are at Appendix C. Each auction round began with subjects turning over a bidding slip to find information on the good they valued (either good A, B, C, or D) and their own private value for that good for the round. Subjects wrote their bid on the same bidding slip, which was then collected before the winner/s for the round or phase were announced.
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Each round of the GBG auction comprised only one phase, with each subject bidding on her preferred good, and facing only one other subject bidding on the same good. Auctions for the four goods occurred simultaneously, but subjects were expressly informed that the auctions for the four goods were completely separate. Winners of each of the four goods were announced at the end of each round.
Each round of the RTC auction comprised four phases, with a winner announced at the end of each phase. In phase 1, all eight subjects submitted bids, with the highest bidder winning her preferred good. That bidder and the other subject who preferred the same good were
instructed not to submit bids for subsequent phases in that round. In phase 2, the remaining six subjects submitted bids, with the highest bidder winning her preferred good. That bidder and the other subject who preferred the same good were instructed not to submit bids for subsequent phases in that round. Phases 3 and 4 followed in the same manner, comprising four and two bidders respectively.
For both the GBG and RTC auctions only the winner was announced, with no information revealed on the winning bid or any other bids.
Points were recorded for each round, with subjects privately informed of their final point score at the end of the experiment.
4.3. Hypothesis
Although the theoretical RNNE prediction is that subjects will always bid half of their value in both FPSB GBG(4, 2) and RTC(4, 4, 2) auctions, it is unlikely that this equilibrium will hold in practice. In particular, Section 2.2 of this paper demonstrates quite clearly that subjects tend to significantly overbid in FPSB auction experiments. This is in contrast with SPSB auction experiments, where subjects also tend to overbid, but on a much smaller scale (see, for example, Kagel and Levin 1993).
Given the expected high bids in the FPSB GBG auction, it may be difficult for bids in the FPSB RTC auctions to exceed GBG bids by a significant amount. On the other hand, it is possible that the RTC auction will result in bidders overestimating the level of competition for their preferred good; and for risk averse bidders to bid even more aggressively than normal in early RTC phases in order to win their preferred good sooner.
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Therefore, I hypothesis that there will be a revenue difference between FPSB GBG and RTC auctions, but that this difference will be much smaller than for the SPSB case studied in Eliaz at al. (2008).
5. Results
From the theory discussion in Section 3, we should expect no difference in the bidding strategy, revenue or bidder profits between auction types, including no differences based on the order of treatment. That is, based on the symmetric RNNE, subjects should always bid half of their value, regardless of auction type or phase. Table 4 presents the equilibrium expected outcomes.
Table 4 – Equilibrium bid coefficients and expected revenues
Auction Phase 1 Phase 2 Phase 3 Phase 4 Expected revenue theory
Expected revenue values
GBG(4, 2) 1/2 N/A N/A N/A 133.333 133.094
RTC(4, 4, 2) 1/2 1/2 1/2 1/2 133.333 133.094
‘Expected revenue theory’ is based on the theoretical expected revenue if bidder values exactly match expected values, while ‘expected revenue values’ is based on the theoretical expected revenue using the actual values drawn in the experiment. Because induced values were directly matched across auction types, ‘expected revenue values’ is the same for both the GBG and RTC auctions.
5.1. Revenue
From the outset, the most pertinent question this paper seeks to answer is how FPSB GBG and RTC auctions compare in terms of revenue. Table 5 presents the mean revenue generated for each auction type by experiment group (based on each group participating in four rounds of each auction type). It also presents the symmetric RNNE revenues based on actual values drawn.
23 Table 5 – Average revenue by experimental group
Experiment Group Average revenue GBG
Average revenue RTC Average revenue RNNE A 269.5 265.25 158.5 B 192 205.75 119 C 208.75 196.25 122.25 D 206 205.75 130.5 E 236.75 256 141.5 F 198.5 188.25 129 G 238.5 248.5 137.5 H 226.5 222.75 126.5
Mean across groups 222.063 223.563 133.094
Standard deviation (25.774) (29.375) (12.642)
It is immediately clear that the revenues raised under the GBG and RTC auctions are remarkably consistent, and performing a Wilcoxon sign rank sum test to test the hypothesis that average revenues for the two auction types are equal results in a z value of -0.140.8 Therefore, we conclude that the difference in revenues between the GBG and RTC auctions is not statistically significant even at the 10% level. Further, the order of treatment does not appear to have a significant effect on revenues. Groups A, C, D and E participated in the RTC auctions first, while groups B, F, G and H participated in the GBG auctions first. In each case, the average revenue for the RTC-first groups exceeds that of the GBG-first groups by less than 1 standard deviation (and this is consistent with the higher values drawn for the RTC-first groups). The effect of treatment order on bidding strategies is also discussed in Section 5.3.4.
A Wilcoxon rank sum test comparing revenue from the GBG auctions to the equilibrium predictions results in a z value of 3.361, while a comparison of revenue from the RTC auctions with the equilibrium predictions results in a z value of -3.363.9 Therefore, revenues from both the GBG and RTC auctions are statistically different from the RNNE prediction at the 1% level.
Also of interest is how revenues from the RTC auction evolve on a phase-by-phase basis. Table 6 breaks down mean revenue from the RTC auction by phase. Data are aggregated
8 A Wilcoxon sign rank sum test is preferred for direct comparisons between GBG and RTC auctions due to the
within-subjects experimental design and use of the same induced values across auction types, meaning that data are matched pairs.
9
A Wilcoxon rank sum test is preferred for comparisons between the GBG or RTC experimental results and the theoretical RNNE prediction, as the theoretical predictions do not arise under the same experimental conditions, so the data are not matched pairs.
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across experimental groups in order to account for the fact that there are fewer data points in later phases of the RTC auction.
Table 6 - Mean RTC revenue by phase
Phase 1 Phase 2 Phase 3 Phase 4
Mean revenue - actual 77.313 63.688 50.656 31.906
Standard deviation (14.877) (16.105) (17.938) (11.917)
Mean revenue - RNNE 44 37.063 30.906 21.125
Standard deviation (5.775) (7.935) (9.085) (8.583)
Wilcoxon rank sum z score -3.361 -3.361 -3.258 -2.892
From the Wilcoxon rank sum comparisons, we see that the mean revenue raised in each phase of the RTC auction differs from the equilibrium prediction, with results for all phases
significant at the 1% level. 5.2. Efficiency
Revenue is not the only consideration for a seller looking to offer goods for sale via auction – efficiency may also be an important factor. In the economics of auctions, efficiency is
generally defined to mean that a good is won by the bidder who attaches the greatest value to it. Efficiency may be particularly important for government sellers looking to maximise welfare – for example, Offerman and Onderstal (2009) state that in certain types of government auctions, efficiency leads to the stream of future surpluses in the aftermarket being as high as possible.
Following Eliaz et al. (2008), two types of efficiency are considered:
Allocative efficiency examines whether available goods are won by the bidders with the highest induced value. Allocative efficiency is essentially the proportion of times that a good is sold to the highest value bidder; and
Cardinal efficiency examines the total welfare associated with the goods sold in the market. Cardinal efficiency is essentially a measure of the ratio of ‘realised surplus’ to ‘maximum possible surplus’.
25 Table 7 - Efficiency
Auction Ordinal efficiency (%) Cardinal efficiency (%)
Predicted Actual Predicted Actual
GBG(4, 2) 100 91.406 100 98.989
RTC(4, 4, 2) 100 92.188 100 98.435
We see that both auction types score reasonably well and reasonably consistently in terms of efficiency, with a Wilcoxon sign rank sum test showing no difference in ordinal or cardinal efficiency between the GBG and RTC auctions at the 10% level.
5.3. Bidder behaviour
While revenue and efficiency are of primary interest to potential sellers, potential buyers may be more interested in bidder behaviour – that is, understanding the behaviour of potential competitors will help a bidder to form her own strategy. Section 2.2 clearly shows that in FPSB auction experiments, bidders do not conform to the symmetric RNNE bidding strategy. Indeed, based on the similar revenue raised in the GBG and RTC auctions in this case, we expect that bidder behaviour is likely consistent between auction types, with bids far above the equilibrium prediction. This section examines bidding behaviour across auction types and through RTC phases.
5.3.1. A note on outliers
While all data from the eight experiment groups are included in the revenue and efficiency analysis, this section is focused exclusively on explaining bidder behaviour. Specifically, we are trying to understand how and why subjects bid the way they do, and extrapolate to understand how people may bid in a similar real life situation. During the experiment, there were a number of instances of subjects bidding above their value. In either the GBG or RTC auction, there is no economic explanation for this behaviour – if a subject wins after bidding above her value, she will certainly make a loss. We are then faced with the decision of whether certain bidding behaviour is so irrational that it should be excluded from further analysis. A full analysis of outliers is provided in Appendix A, but in summary, three data points are excluded from the bidder behaviour analysis (coinciding with bid coefficients of 2.5, 16 and 16).
26 5.3.2. Bid coefficients - frequency
As discussed above, while theory predicts that we should see bid coefficients of 0.5 for both the GBG(4, 2) auctions and for every phase of the RTC(4, 4, 2) auctions, the high revenues suggest subjects bid much higher in practice. From the bid data we see this is indeed true, with bid coefficients clustered around the 0.7 – 1.0 mark for both auction types.
Figure 1 – Bid coefficients by auction type
Note: In Figure 1, Figure 2 and Figure 3, the frequency for each bid coefficient marker on the horizontal axis is calculated by counting all bid coefficients less than or equal to that marker, but strictly greater than the previous marker.
Breaking the data down further, we can examine the GBG auctions on a round-by-round basis. In this case, we see bid coefficients are again clustered around the 0.7 – 1.0 mark in each round, with perhaps a tendency for subjects to choose a lower bid coefficient in round 1. Consistent bidder behaviour through rounds in the GBG auctions is not surprising, given the auction is of the same form in each round. It is conceivable that subjects who win early may decide they can shade their bid more in later rounds, and/or that subjects who lose may increase their bid in the following round, but any such effects look to be reasonably small and potentially cancel each other out.
27 Figure 2 – Bid coefficients – GBG by round
Finally, it is worth investigating how the RTC auctions progress phase-by-phase. While the theoretical RNNE predicts that bidders will not change their strategy as phases progress, bidders may perceive that they face less competition as the phases progress (provided their own preferred good is still available). Although there does not appear to be a strong effect, there is some evidence of lower bid coefficients in phase 4, with a higher proportion of bid coefficients clustered around the 0.40 – 0.65 mark. Of course, there are fewer subjects actually bidding in phase 4, so some caution should be exercised before drawing firm conclusions from this bid coefficient frequency analysis alone.
28 Figure 3 – Bid coefficients – RTC by phase
5.3.3. Examining bid as a function of value
It is also possible to examine bidder behaviour graphically using scatter plots. As a starting point, the stark difference between actual bids and the equilibrium prediction becomes apparent:
Figure 4 – Aggregate experimental bids vs equilibrium prediction
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And the similarities between bids in the GBG and RTC auctions become undeniable:
Figure 5 – Experimental bids – GBG vs RTC
We are also able to glean further insight into how bids change as phases of the RTC auction progress. While it is difficult to draw firm conclusions from the frequency graphs alone, the scatter plot shows more clearly that subjects tend to bid less aggressively as the RTC auction progresses.
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With a little investigation, it is possible to gain some insight into the cause of this tendency for subjects to bid less as RTC phases progress. Taking the average bid coefficient in the RTC auctions (mean = 0.825, median = 0.839), we can define a band for average bid coefficients as around 0.82 – 0.84, and classify subjects with an average bid coefficient of above 0.84 as ‘aggressive’ and subjects with an average bid coefficient of below 0.82 as ‘passive’. Now, creating an ‘early winning’ point system that awards 2 points to a subject who wins in phase 1 of a RTC auction and 1 point to a subject who wins in phase 2 of a RTC auction, the correlation between aggressiveness and winning early is 0.274. Therefore, there is a mild tendency for more aggressive bidders to win in early phases of the RTC auction, leaving more passive subjects in later phases. This may partly explain falling bid coefficients as the RTC phases progress.
5.3.4. Modelling bidder behaviour
Taking the scatter plot analysis further, it is possible to model subject bidding behaviour to describe bid as a linear function of value (and potentially other factors). In this experiment, the same subjects make a number of decisions over time. We begin with the knowledge that individual characteristics of subjects (such as sex, risk appetite and competitiveness) are assumed constant over the course of the experiment, and that induced values are randomly assigned. Therefore, variation across subjects should be random and uncorrelated with the major predictor variable – value. This leads to the preliminary conclusion that a random effects panel model may be appropriate. A Hausman test to compare whether random effects is preferred over fixed effects results in a Chi squared probability of 0.992, therefore we accept the null hypothesis that random effects is more appropriate than fixed effects. A further consideration is whether panel data analysis is more appropriate than pooled ordinary least squares (OLS). Pooled OLS is most appropriate where there are no significant differences across entities (that is, where there is no panel effect). In terms of this experiment, it is likely that different subjects have idiosyncratic characteristics (such as desire to win or level of boredom). This can be tested with a Breusch-Pagan Lagrange multiplier. In this case, the test gives a Chi squared probability of 0.000, meaning that random effects panel analysis is significantly more appropriate than pooled OLS due to panel effects.
Arranging the data into a panel does pose some issues: because of the AB/BA design, decisions made in each time period do not align between auction types across groups; however it is possible to address this by including a dummy variable for RTC-first.
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Therefore, in the panel, time periods progress with the GBG auctions as periods 1 - 4 and the RTC auctions as periods 5 - 20. Using standard errors clustered at the player level corrects for idiosyncratic bidder behaviour.
The random effects regression analysis can also include time invariant variables such as sex, risk attitude and competitiveness which were all included in the post-experiment
questionnaire.
Risk was measured using Gneezy and Potter’s (2007) risk elicitation method, which asks subjects to choose how much of an endowment they wish to invest in a risky option and how much to keep. In this case, subjects were asked to invest between 0 and 100 hypothetical points in a lottery which had a 2/3 chance of paying zero and a 1/3 chance of paying two and a half times the amount invested (meaning that a higher risk score is associated with greater risk seeking).
Competitiveness was measured with reference to Niederle and Vesterlund’s (2007) work which focusses on incidences of selection into a competitive environment. In this case, subjects were asked whether, for a hypothetical task, they would prefer to be compensated under a low-paying piece-rate compensation scheme, or a high-paying scheme which paid only if the subject performed at least as well as average (meaning that a competitiveness dummy score of 1 is associated with competitive behaviour). The full experimental questionnaire is included in Appendix C.
When adding dummy variables, one potential issue is multicollinearity – in this case,
collinearity between risk and competitiveness may be of particular concern. A basic analysis of the relationship between risk and competitiveness results in a correlation coefficient of 0.222, indicating that the two variables are only mildly related. Further, running a random effects panel regression of bid on value, female, risk, RTC, RTC-first and competitiveness results in variance inflation factors of less than 4 for all variables.10 Therefore, it is possible to include all post-experiment questionnaire variables without risking multicollinearity.
10 As a rule of thumb, a variable whose variance inflation factor (VIF) values are greater than 10 may merit
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Taking all of these factors into account, the random effects panel regression relating bid to value, auction type, sex, risk attitude, competitiveness and first auction type is of the form (clustered errors in parentheses):
R squared: (within = 0.890, between = 0.778, overall = 0.864), Wald Chi squared: 1690.08 We see that there is a tendency for bids to rise if the subject is female, competitive and risk seeking (this last effect is quite curious). In addition, the RTC format seems to lower bids at the outset, but increases bid as a function of value. However none of these variables are statistically significant at the 10% level, and indeed the only variable with significant predictive power is value, which is significant at the 1% level. The insignificance of factors other than value is not necessarily surprising, since value is randomly assigned and errors are clustered at the entity level. Adding time dummies (see Appendix B Table 11) does not provide additional predictive information about bidder behaviour in most cases (though it does corroborate the findings from Figure 2 that bids are lower in GBG round 1 than other GBG rounds).
Now considering the data by auction type separately and more explicitly, for the GBG auction only, bidder behaviour can be explained in the form:
R squared: (within = 0.904, between = 0.772, overall = 0.868), Wald Chi squared: 996.74 And the RTC auction can be explained in the form:
R squared: (within = 0.884, between = 0.813, overall = 0.855), Wald Chi squared: 799.37 (Regressions were also run including sex, risk attitude, competitiveness and first auction type, however none of those variables were significant at the 10% level – see Appendix B Table 12).
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Another question of interest is whether subjects change their behaviour in the second half of the experiment. Post-experiment questionnaire data indicated that a common strategy was to bid very aggressively in the first half of the experiment in order to gain a few wins, and then focus on scoring more points by bidding less aggressively in the second half.
Bidder behaviour in the first half of the experiment (the four GBG auctions for four
experiment groups, and the four RTC auctions for the other four experiment groups) can be explained in the form:
R squared: (within = 0.904, between = 0.802, overall = 0.863), Wald Chi squared: 761.26 And bidder behaviour in the second half of the experiment can be explained in the form:
R squared: (within = 0.875, between = 0.794, overall = 0.858), Wald Chi squared: 746.15 This result does corroborate subjects’ reported strategies of bidding less aggressively in the second half of the experiment; however the difference is not statistically significant, with considerable overlap between the confidence intervals of the two results. Additional regressions including dummy variables for auction type, sex, risk attitude and competitiveness lead to similar conclusions (see Appendix B Table 13).
Finally, following the frequency and scatter plot analysis, it is of considerable interest whether subjects change their bidding strategy as phases of the RTC auction progress. There are several ways this can be investigated.
Assessing all auction data, and adding phase dummies (all GBG auctions and phase 1 of the RTC auction serve as the benchmark):
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Assessing just the RTC auctions with phase dummies (phase 1 serves as the benchmark):
R squared: (within = 0.884, between = 0.816, overall = 0.856), Wald Chi squared: 1248.18 Assessing just the RTC auctions with a ‘second half’ phase dummy (phases 1 and 2 serve as the benchmark):
R squared: (within = 0.885, between = 0.816, overall = 0.856), Wald Chi squared: 1179.35 In each case, the coefficient on the phase variable goes in the direction predicted based on graphical evidence (that is, subjects bid less as phases progress), however none of these results are statistically significant at the 10% level. Again, additional regressions lead to similar conclusions (see Appendix B Table 14).
5.3.5. ‘As if’ level of competition
From the preceding analysis it is clear that bidders in both the FPSB GBG and RTC auctions bid far more aggressively than theory predicts. It is well established from a theoretical perspective that increasing the number of bidders in a FPSB auction increases average revenue (Holt 1979; Harris and Raviv 1981). Therefore, an interesting question is what level of ‘as if’ competition can describe experimental bidder behaviour. An analysis of mean squared error bids provides an understanding of how bidding strategies compare to theoretical levels of competition. Table 8 lists the mean squared error between bids and predictions for different ‘as if’ competitors.
Table 8 - Mean square error: (bid – prediction)2
As if n Combined GBG and RTC GBG RTC 2 369.876 433.520 344.298 3 146.595 165.637 138.942 4 92.735 98.051 90.598 5 80.670 80.969 80.549 6 80.727 78.773 81.512 7 84.578 81.344 85.878
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By minimising the mean squared error, we see that in the GBG auctions subjects bid as if they are in a group of six risk neutral competitors, and in the RTC auctions subjects bid as if they are in a group of five risk neutral competitors. In both cases, this far exceeds the actual group size of two subjects vying for each good.
6. Discussion
From an analysis of the results, two main conclusions arise:
The FPSB GBG and RTC auctions raise comparable revenue; and
Subjects in both auction types bid far above the RNNE prediction and often seem more focussed on winning a given round than maximising their own point score. This is in stark contrast to the findings of both Goeree et al. (2004) and Eliaz et al. (2008) who – for ascending and SPSB auctions respectively – find significant revenue and bidding differences between GBG and RTC auctions.
6.1. Explaining bidder behaviour – incentives
From Section 2.2 we certainly expect experimental subjects to bid above the symmetric RNNE prediction, however the results in this experiment are perhaps more extreme than other experimental studies of FPSB auctions. Indeed, one interesting behavioural aspect is the number of instances where subjects bid exactly their own value. Aside from subjects who draw an induced value of zero (which happened five times) there is no economic reason for a subject to ever bid her value – even if she wins the auction, she will certainly earn zero points.11 In this experiment, of 896 bid observations, the number of bids:
Above 90% of value was 337 (or approximately 37.6% of all bids)
Above 95% of value was 215 (or approximately 24.0% of all bids)
Exactly equal to value was 78 (or approximately 8.7% of all bids)
While bidding close to one’s value may be indicative of extreme risk aversion, bidding exactly equal to one’s value cannot be explained by risk aversion alone. Per the experimental findings of Grimm and Engelmann (2005), a myopic joy of winning seems to best explain this outcome.
11 Subject 8E had a value of 0 for round 4 of the GBG auction, and subject 1E had a value of 0 for all four
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Unfortunately, it is likely that the lack of real financial incentives may have played a significant role in this outcome. This is consistent with statements in the post-experimental questionnaire, where subjects indicated that they often bid in order to win the auction (particularly in early rounds), rather than with a focus on maximising points. Grimm and Engelmann (2005) make similar findings in their experiment auctioning multiple units of homogenous goods – in that case, subjects tended to use their first bid to maximise their chances of winning, before switching focus to maximise earnings in subsequent rounds. In the experiment at hand, without financial incentives, and with the announcement of the winning bidder at the end of each round, it is possible that the most salient incentive to many subjects was simply being announced as the winner (although winning subjects were only identified by number, the experimental setting meant that players could often deduce the winner in each round).
This is consistent with previous experimental findings that subjects can behave significantly differently in response to hypothetical versus real rewards (see, for example, Hertwig and Ortmann 2001). In accordance with Smith’s (1982) precepts for experimental economics, this particular experiment could be improved by ensuring that both auction types are linked to dominant and salient financial rewards.
6.2. Comparison to second-price auctions
Although the lack of incentives likely played a major role in these results, there may be other factors which help to account for the difference between these results and those of Goeree et al. (2004) and Eliaz et al. (2008).
It is worth re-emphasising that in both standard ascending and SPSB auctions, players have a weakly dominant strategy to bid up to their own value. A subject’s knowledge that she cannot lose – and indeed that she maximises her expected earnings – by choosing to bid her value in a standard SPSB auction may be a key factor in explaining why subjects appear to use the same strategy when faced with an equivalent RTC auction. That is, it might be that subjects expect the same weakly dominant strategy to hold for the SPSB RTC auction.
This seems to be a major factor behind the results in Eliaz et al. (2008). In that paper, subjects bid very close to their value for the SPSB GBG auction, with the median difference between bid and value equal to zero. Further, although bid coefficients for the RTC auctions are not reported, the actual revenues raised in each phase of the RTC auction correspond
