Loss aversion and sunk cost sensitivity in all-pay auctions for charity: Theory and experiments

Loss Aversion and Sunk Cost Sensitivity in All-pay Auctions for Charity: Theory and Experiments

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Loss Aversion and Sunk Cost Sensitivity in All-pay Auctions for Charity: Theory and Experiments Joshua Foster PII: DOI: Reference:

S2214-8043(19)30183-1 https://doi.org/10.1016/j.socec.2019.101486 JBEE 101486

To appear in:

Journal of Behavioral and Experimental Economics

Received date: Revised date: Accepted date:

19 April 2019 31 October 2019 31 October 2019

Please cite this article as: Joshua Foster, Loss Aversion and Sunk Cost Sensitivity in All-pay Auctions for Charity: Theory and Experiments, Journal of Behavioral and Experimental Economics (2019), doi: https://doi.org/10.1016/j.socec.2019.101486

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Highlights • We study the first-price all-pay and second-price all-pay auction for charity. • Controlled experiments consider two effects: loss aversion and sunk cost sensitivity. • Incremental bidding mechanisms rely heavily on sunk cost sensitive bidders. • A commitment device reduces contributions below theoretical predictions. • Loss aversion motivates smaller bids in sealed bid mechanisms.

Loss Aversion and Sunk Cost Sensitivity in All-pay Auctions for Charity: Theory and Experiments∗ Joshua Foster† Economics Department University of Wisconsin - Oshkosh November 14, 2019

Abstract This paper theoretically and experimentally studies the role of two behavioral biases in all-pay auctions for charity. The theory is developed to predict the effect of loss aversion in the first-price all-pay auction and sunk cost sensitivity in the war of attrition. Using controlled laboratory experiments, auction treatments are designed to test for the presence of these biases. In support of the theory, the results indicate that revenues in incremental bidding mechanisms such as the war of attrition rely on bidders who are sunk cost sensitive. It is shown that this behavioral response can be curbed significantly with a commitment device. Likewise, the results of the experimental first-price all-pay auctions find evidence of loss aversion, which reduces bidders’ average bid. These findings help explain the inconsistencies in revenues from previous all-pay auction studies and indicate a mechanism preference based on the distribution of these behavioral characteristics.

Keywords: Auctions, Market Design, Charitable Giving JEL Classification: C92, D03, D44, D64 ∗ I gratefully acknowledge financial support from the International Foundation for Research in Experimental Economics (IFREE) for this project. I would like to thank Cary Deck, Amy Farmer, Jeffrey Carpenter, Salar Jahedi, Li Hao, M. Ryan Haley, Chad Cotti, Ben Artz, and seminar participants at the University of Arkansas, ESA World Meetings, ESA North America Meetings, and the SEA Annual Meetings for their helpful comments at various stages of the development of this project. † Contact the author at [email protected].

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Many non-profit organizations today are finding charity auction events helpful in reaching their fund-raising goals. This strategy has merit, as much of the theoretical and empirical literature on charity auctions demonstrate they are superior revenue-raising mechanisms over alternative methods, such as raffles and voluntary contributions. Moreover, charitable organizations can create fund-raising opportunities by taking donations lacking mission-oriented value and converting them into cash. For instance, a human rights organization recently raised over $600,000 through a winner-pay auction when Apple CEO Tim Cook agreed to have a coffee date with the highest bidder. Attention has focused recently on all-pay auctions, which are mechanisms that collect revenue from all those who submitted a positive bid, regardless of who wins. It has been shown all-pay auctions possess properties that generate expected revenues greater than that of more traditional, winner-pay auctions (Goeree et al., 2005; Engers and McManus, 2007). However, the empirical evidence for the all-pay advantage is certainly mixed. In field studies, Carpenter et al. (2007) and Onderstal et al. (2013) find that first-price all-pay auctions under-perform relative to the first-price winner-pay in the former and the voluntary contribution mechanism (VCM) in the latter. Both of these studies conclude a lack of participation as the source of the shortfall. On the other hand, Carpenter et al. (2014) found the all-pay auction to raise more revenue than the first-price winner-pay and the second-price winner-pay in a laboratory setting. In addition, their paper also introduced a promising all-pay mechanism called the ‘bucket’ auction. In this all-pay mechanism, bidders are required to repeatedly take turns deciding whether to i) make a costly contribution in order to stay in the auction, or ii) not contribute and thus exit the auction permanently, with the last bidder choosing to contribute being the winner. At the end of the auction, the collective contributions from all bidders are converted into revenue. As a result of this bidding procedure, the bucket auction can be understood as a type of second-price all-pay auction called the war of attrition, since the auction ends when the penultimate bidder exits and thus sets the price for the winner. However, experimental evidence demonstrates it raises revenue at rates well beyond what is theoretically predicted from such a mechanism. To better understand the inconsistent performance of all-pay auctions, this paper theoretically and experimentally investigates two mechanisms in parallel, the first-price all-pay auction and the war of attrition (i.e. the bucket auction). It is hypothesized that the mechanism design of these auctions moderates distinct behavioral biases from the bidders, which in turn leads to different bidding strategies and thus different auction revenues. In the case of the war of attrition, it is hypothesized the mechanism invokes bidders’ sunk cost sensitivities through the

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mental accounting that precipitates from the incremental nature of contributions to revenue. The theoretical insight provided by this paper demonstrates this effect on bidders cuts the war of attrition two ways. While a sunk cost sensitivity leads bidders to drop out more quickly at the beginning of auctions, which decreases revenue, it can also motivate bidders to stay in far longer than they otherwise would have without sunk cost sensitivities, thus generating very large revenues in some cases. This prediction is consistent with the experimental results in this study and can, in part, explain why the revenues in wars of attrition tend to be far less or far greater than what is predicted when this bias is absent. Furthermore, this paper finds a reduction in the propensity for sunk cost sensitive bidders to make large contributions when they are provided with a commitment device capable of preemptively limiting their ability to bid beyond a certain level. In the case of the first-price all-pay auction, this paper hypothesizes that the requirement for bidders to pay their bid regardless of the auction outcome invokes loss aversion, as this process leaves most bidders to be worse off for participating. Due to the addition of this psychological cost of bidding, the theory in this paper predicts bids to be less on average than when the bias is absent. Specifically, it is shown that the symmetric mixed strategy equilibrium among loss averse bidders leads to a first order stochastically dominated distribution of bids, as compared to those submitted by bidders who are not loss averse. The experimental evidence in this paper reports differences in the bid distributions that support this hypothesis. Taken together, this paper suggests a charity’s preference over which all-pay auction to run will depend on the distribution of bidders who are sensitive to losses and sunk costs. The paper proceeds as follows: Section 1 provides an overview of the relevant literature for charity auctions; Section 2 presents a theoretical model of bidder behavior for the war of attrition and the first-price all-pay auction and establishes hypotheses to be experimentally tested; Section 3 describes the laboratory experiments conducted to test the hypotheses from Section 2; Section 4 presents the results from the experiments and summarizes the statistical evidence for (against) the hypotheses in Section 2; Section 5 provides concluding remarks.

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All-pay Auction Mechanisms for Charity Under the standard theoretical assumptions, auction bids are larger when the revenue goes

to charity than when it is for profit. This ‘charity premium’ is the result of assuming 1) bidders receive a public good benefit from auction revenues (Goeree et al., 2005), and 2) that bidders

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experience ‘warm glow’ from personally contributing to the charity (Engers and McManus, 2007). Empirically, Elfenbein and McManus (2010) compared over two thousand charity and for-profit auctions on eBay, finding that items linked to charity raised 6% more revenue on average. In a similar field experiment, Leszczyc and Rothkopf (2010) find bid levels increase by more than 40% in online charity auctions compared to otherwise identical for-profit auctions. Perhaps more importantly, the expected revenue from charity auctions can vary by mechanism under these assumptions – a deviation from the revenue equivalence theorem. As a result, the choice of auction mechanism becomes a particularly important consideration. It has been shown that all-pay auctions outperform winner-pay auctions when there is a sufficient number of bidders.1 The intuition of this rank-ordering of revenue in winner-pay and all-pay auctions rests on solving a type of free-rider problem. Goeree et al. (2005) explains that bidders in winner-pay auctions are confronted with the trade-off of winning the auction (and claiming the value of some prize less their bid) and free-riding on the benefits generated from another’s winning bid. However, this trade-off is attenuated with all-pay auctions. Furthermore, Goeree et al. (2005) demonstrates this trade-off is best managed by a last-price all-pay auction, in which the highest bidder wins and pays the lowest bid submitted. In a laboratory study on last-price all-pay auctions for charity, Damianov and Peeters (2018) suggests these predictions require a sufficient number of bidders to hold. Given that several studies have demonstrated the performances of all-pay mechanisms are sensitive to the level of competition (i.e. participation), fundraisers may prefer a mechanism that does rely on bids from many potential bidders. A relatively new mechanism introduced in Carpenter et al. (2014) called the ‘bucket’ auction has the promise of harnessing the theoretical benefits of all-pay auctions without its success requiring much at all of participation. The bucket auction is a sequential-move war of attrition, a contest in which n bidders vie for one of k < n prizes by expending resources at a constant rate through time. The contest persists until n − k

bidders choose to exit the contest, thus making it equivalent to a (k + 1)th -price all-pay auction. Standard wars of attrition have the important property of “instant exit,” in which all but k + 1 participants exit without expending any resources (Bulow and Klemperer, 1999). By imposing a charity auction environment to the war of attrition, Carpenter et al. (2014) shows that this mechanism can generate equivalent expected revenues from k + 1 bidders, while the sealed-bid alternatives require positive contributions from all n bidders. For example, in the case where

1 See Morgan (2000); Morgan and Sefton (2000); Goeree et al. (2005); Engers and McManus (2007); Isaac et al. (2008) and Carpenter et al. (2014) for a comparison of all-pay mechanisms to other mechanisms.

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there is one prize (k = 1), only two bidders are required to generate expected revenue equal to that of a second-price all-pay auction. In turn, this revenue will exceed a first-price all-pay auction in expectation. In addition to concentrating contributions to fewer bidders, the war of attrition has shown to raise revenue well beyond what is theoretically predicted in a charity setting, and simultaneously overshadowing the performances of other mechanisms. In Carpenter et al. (2014), the war of attrition raised two and a half times more revenue than the next best winner-pay mechanism and one and a half times more revenue than the first-price all-pay auction. While there is limited evidence to aid in comparison, there has been a stark difference in the level of competition reported from a laboratory study in non-charity wars of attrition. Notably, Hörisch and Kirchkamp (2010) find wars of attrition under-perform relative to theoretical predictions. The substantial empirical difference gives rise to behavioral considerations for what makes the war of attrition so inconsistent.2 Carpenter et al. (2014) reports that bidders who were sunk cost sensitive or competitive (or both) were willing to compete longer in the war of attrition relative to those who were not. One explanation for this finding that will be explored further in this paper is the appeal of the war of attrition’s sequential, incremental bidding process. Given these behavioral considerations, if bidders recognize they are prone to over-bidding, then providing them with a commitment device may curb their bidding and thus a decrease in contributions. On the other hand, it may be possible to artificially elevate contributions in a sealed-bid mechanism, such as the first-price all-pay auction, by adopting the time-based bidding properties in the war of attrition. Both of these ideas will be explored using controlled experiments. In the next section, a theoretical framework outlines predictions over bid strategies in the first-price all-pay auctions and the war of attrition. These models will provide the benchmark used to evaluate the effects of mechanism variants on bidders’ willingness to contribute.

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Theoretical Motivation This paper separately models bidding strategies in two all-pay auction mechanisms, the

sequential-move war of attrition (WoA) and the first-price all-pay auction (FPAP). It is assumed 2 A few studies of the war of attrition in specific environments have also been conducted. For instance, Bilodeau et al. (2004) studies the willingness of individuals with heterogeneous preferences to be the provider of a public good in what they call the volunteer’s dilemma. Their paper reports the willingness to provide the good matches the comparative static predictions. Oprea et al. (2013) studies the war of attrition in an industrial organizational context, in which two firms compete in an unsustainable duopoly. Their paper finds the degree of competition to be very close to the point predictions in their model.

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that in each case there are n ≥ 2 risk-neutral bidders looking to claim a single item with a pure common value v. Complete information regarding n and v is also assumed. To capture

charity auction incentives among bidders, benefits are allowed to accrue from the auction’s revenue. Following Engers and McManus (2007), it is assumed that bidders receive two constant marginal benefits from auction revenue: α) the benefit a bidder receives from each unit of revenue raised, independent of which bidder generates it; and γ) the benefit a bidder receives from personally contributing a unit of revenue to the auction, which is motivated by the notion that bidders receive “warm glow” benefits a la Andreoni (1990) from supporting the charity. In order to capture the potential for behavioral motivations in these auctions, bidders’ preferences allow for the influence of a bias that may arise from the bidding rules. In the WoA mechanism, bidders’ preferences allow for mental accounting via the sunk costs they incur through the incremental bidding process. This is represented by a constant marginal cost σ, which increases the mental account they carry with each bid submitted. In the FPAP mechanism, bidders’ preferences allow for loss aversion as they experience gains and losses about some endogenously determined reference outcome from bidding. The presence of gain-loss utility is captured by two constant marginal parameters, η and λ, with the former representing a weighting parameter and the latter representing the loss aversion parameter. A summary of the parameters that will appear in these models is provided in Table 1 below. Table 1: Summary of Model Parameters Parameter

Description

α γ λ η σ

Public good revenue benefit. Warm glow revenue benefit. Loss aversion from auction outcome. Gain-loss utility weighting parameter. Sunk cost sensitivity from auction outcome.

Analysis begins below with potentially sunk cost sensitive bidders in the WoA auction, followed by potentially loss averse bidders in the FPAP auction. While there exist several asymmetric equilibria within each mechanism, focus is given primarily to symmetric equilibria.

2.1

The Sequential-move War of Attrition The two stages of this sequential-move war of attrition are described below. Nature: the pure common value of a single auction item v, the number of potential

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bidders n, the constant marginal benefits to revenue α and γ, and the sunk cost sensitivity parameter σ are exogenously determined. Bidders are randomly assigned to a bidding sequence. Auction Stage: the auction begins and bidders choose to bid or exit sequentially in the pre-determined order. Bidding has a monetary cost of c in each cycle, and exiting is permanent. The auctioneer cycles through bidders until one bidder remains, and this bidder claims the prize. See Figure 1.

Figure 1: Sequential-move War of Attrition Start of Cycle

1

2

n

Determination of v, n, α, γ, and σ Bidding Order Revealed Pre-auction Stage

Bidder 1 bids/exits Bidder n bids/exits Cycle 1

For simplicity, attention is given primarily to Markov-perfect equilibria. This is a refinement to the subgame perfect equilibrium concept in which players employ memoryless strategies conditioned only by the current state of the game (Maskin and Tirole, 2001).3 In the symmetric case, strategies in time step t involve each of the k ≤ n remaining bidders choosing to bid

with probability ptk ∈ [0, 1] and exit with probability 1 − ptk . Equilibrium bidding strategies are

identified via backward induction in the final subgame where k = 2 bidders remain. In this   Q t subgame, the probability a bidder exits at time step t is equal to Ωtk=2 = p1t τ =1 pτ2 (1 − pt2 ), 2

which requires that they bid in the first t − 1 time steps. From this, the expected return to

3 Another natural way to model wars of attrition is establishing a perfect-Bayesian equilibrium by following Bulow and Klemperer (1999) when bidders’ valuations are distributed. The sequential-move nature of this model makes the continuous model in Bulow and Klemperer (1999) an imperfect approximation. However, it will be demonstrated that many characteristics are shared between the models.

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bidding until period t can be expressed as E[π2t ] =

t X

τ =1

Ωτ2 [v − (1 − α − γ)τ c + α(τ − 1)c]

+ 1−

t X

τ =1

Ωτ2

!

(1)

[−(1 + σ)(1 − α − γ)tc + αtc].

The first line in equation (1) represents all auction outcomes where the bidder could potentially win. Here, the probability the bidder will win the auction in some time step τ ≤ t is multiplied

by the return from winning in time step τ . This return has three terms: 1) the auctioned item’s value; 2) the cost of bidding until time step τ , which is adjusted by the revenue benefits from their bid; and 3) the revenue benefits from the other bidder that accrued through time step τ − 1. The second line in equation (1) represents all auction outcomes where the bidder loses

the auction. These outcomes require that the bidder be the first to exit at time step t, which Pt occurs with probability (1 − τ =1 Ωτ2 ). The return in this case has two terms: 1) the cost of

bidding to time step t, which is magnified by the sunk cost parameter, σ; and 2) the revenue benefits that accrued from the other bidder through time step t. For a bidder to be indifferent between bidding and exiting in time step t requires the expected payouts from both actions to be equal. Using equation (1), this condition can be expressed as pt2 [αc − σ(1 − α − γ)tc] + (1 − pt2 )[v] − (1 − α − γ)tc + α(t − 1)c = −σ(1 − α − γ)(t − 1)c − (1 − α − γ)(t − 1)c + α(t − 1)c

(2)

v − (1 − α − γ)(1 − σ(t − 1))c . ⇒ pt2 = v − (α − (1 − α − γ)σt)c

In this condition, bidders incur a sunk cost effect of exiting the auction in time step t equal to −σ(1 − α − γ)(t − 1)c, which is represented by the first term on the right hand side. This cost in-

creases as the accumulated cost (i.e. their mental account) from bidding grows. Consequently, this means the continuation profits of the auction are not constant – rather, the continuation

profits decrease with each time step. Under reasonably assumable conditions, this causes bidders to increase the probability of bidding through time. That is, when

1−γ α

>

2+σ 1+σ ,

dpt2 dt

> 0 and limt→∞ pt2 = 1

a condition that holds for the experimental data. Note, if this condition did

not hold then no bidder would ever exit the auction as pt2 would be binding at 1. Comparing this case to when σ = 0 we find the condition in (2) reduces to pt2 =

v−(1−α−γ)c , v−αc

which does not

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vary with t. In Figure 2 below, the equilibrium bid strategies in this subgame are compared in the cases where sunk costs are fully ignored (σ = 0) and when sunk costs are fully internalized (σ = 1). Relative to the rational model, a sunk cost sensitivity causes bidders to reduce the likelihood of bidding in early periods and increase the likelihood of bidding in later periods. Figure 2: Equilibrium Bid Strategies with(out) Sunk Cost Sensitivities

Probability of Bidding

p2 = 1

σ = 1 (Full Sunk Cost Sensitivity) σ = 0 (No Sunk Cost Sensitivity) p2 = 0

Time Step

In the final subgame this model closely resembles Augenblick (2015), which studies Markovperfect equilibria in for-profit penny auctions. However, an important difference is the systematic removal of bidders who exit in the war of attrition, which does not occur in penny auctions. As a result, the equilibrium bidding strategies in these models deviate for k > 2. However, it is straight forward to demonstrate that exiting in the first time step is optimal when there are more than two bidders remaining in the auction. This can be understood by comparing the expected return of bidding vs. exiting in the first time step with k ≥ 2 bidders remaining. When

k = 2, substituting the bidding strategy solution from (2) into (1) reveals the expected return in the first time step is equal to zero. However, if k > 2 then the expected return to exiting is positive. This is due to bidders costly collecting the revenue benefits generated from the last two bidders in the final subgame. As a result, the Markov-perfect equilibrium predicts bidders will exit until two bidders remain. Using the fact that ptk>2 = 0, the expected revenue from the WoA mechanism can be fully characterized with the equilibrium strategies in the final subgame. As a result, expected revenue can be expressed simply as a weighted sum over the probabilities of when the second to last bidder chooses to exit, multiplied by the amount of auction revenue generated in each case. Since the last two bidders have the opportunity to bid/exit in each time step t, there are two

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opportunities for the auction to end. If the leading bidder is the first to exit, then the auction generates (2t − 2)c in revenue, while if the trailing bidder is the first to exit, then the auction

generates (2t − 1)c in revenue. This leads to the following infinite sum describing expected

revenue, E[RWoA ].

E[RWoA ] =p12 (1 − p12 )[c]+ p12 p12 (1 − p22 )[2c] + p12 p12 p22 (1 − p22 )[3c] + p12 p12 p22 p22 (1 − p32 )[4c] + · · · ∞   (3) X
1 t+1 2 t t+1 Ω Ω [(2t − 1)c] + Ω [2tc] = 2 2 2 1 − pt+1 2 t=1 While this expression for expected revenue does not reduce to a nice, compact expression, it can be expressed for specific cases. Figure 3 below plots the expected revenue of the WoA mechanism against the sunk cost sensitivity among bidders using the experimental assumptions imposed in this paper (α =

1 10 ,

γ =

1 20 ,

v = 10, c = 12 ). This figure demonstrates that as

the bidders’ sunk cost sensitivity, σ, increases from zero (fully rational) to one (fully sunk cost sensitive), bidders adjust their bidding strategies in a way that decreases the expected revenue. This result is due to the probability of bidding early in the auction decreasing with the sunk cost sensitivity. While the probability of bidding later in the auction increases with the degree of sunk cost sensitivity, this increase is too slow to overcome the overall probability of reaching these time steps, ultimately decreasing the expected revenue in this case.

Expected Revenue

Figure 3: Expected Revenue According to Bidders’ Sunk Cost Sensitivity

12 11 10 9 8

0

0.25 0.5 0.75 Sunk Cost Sensitivity (σ)

1

These theoretical results are the basis of bidding strategies and expected revenue, RWoA , in the sequential-move war of attrition for charity.

Sequential-move War of Attrition Outcomes: There exists a Markov-perfect equilibrium in

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which bidders’ strategies and expected auction revenue are described by

ptk =

E[RWoA ] =

   v−(1−α−γ)(1−σ(t−1))c v−(α−(1−α−γ)σt)c

 0

∞ X t=1

2.1.1

if k = 2 if k > 2,




1 t+1 2 Ωt2 Ωt+1 [2t] 2 [2t − 1] + Ω2 1 − pt+1 2

(4) 

Curbing Sunk Cost Sensitivities

To better understand the mechanism-specific effect the war of attrition has on bidders with sunk cost sensitivities, two variations of this auction will be studied experimentally. These variations are called Unconstrained and Self-constrained, and they are described below. The Unconstrained treatment is a standard war of attrition in which all bidding and exiting decisions are made during the auction. The Self-constrained treatment gives bidders the ability to limit their contributions by choosing, before each auction begins, a maximum number of bids they can make. These mechanism treatments may have varying effects on bidders with a sunk cost sensitivity. If this bias is present, then the Unconstrained mechanism treatment would lead sunk cost sensitive bidders to decrease the rate of exit as the auction continues. If this bias is not present, then the expectation would be to observe bidders exit at a constant rate. However, if these bidders are sufficiently sophisticated in their understanding of their bias, then the Self-constrained mechanism should give sunk cost sensitive bidders the opportunity to curb the tendency of over-bidding late in the auction. As a result, we would expect the difference in exit rates to disappear between rational and sunk cost sensitive bidders in the Self-constrained mechanism treatment. This is the basis of Hypothesis 1.

Hypothesis 1: In the Unconstrained treatment, sunk cost sensitive bidders will decrease their rate of exit as the WoA auction progresses. In contrast, bidders with no sunk cost sensitivity will exit at a constant rate as the auction progresses. In the Self-constrained treatment, sunk cost sensitive bidders will use the mechanism’s design to limit their bidding and as a result will exit the auction at the same rate as those who are not sunk cost sensitive.

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2.2 The First-price All-pay Auction The two stages of this sealed bid first-price all-pay auction are described below. Nature: the pure common value of a single auction item v, the number of potential bidders n, the constant marginal benefits to revenue α and γ, and the gain-loss utility parameters η and λ are exogenously determined. Auction Stage: bidders privately and simultaneously submit a sealed bid to the auctioneer. Each bidder is responsible for paying their own bid, and the bidder with the largest bid claims the prize. From Baye et al. (1996) it is known there is a unique symmetric equilibrium in mixed strategies for pure common value first-price all-pay auctions with complete information. Let F define the cumulative distribution over bids with density f . To model the impact loss aversion has on bidding in the FPAP mechanism, this paper follows K˝oszegi and Rabin (2006). Specifically, it is assumed the bidder’s utility from receiving some monetary outcome o from participating in an auction takes the following form. π(o|r) =

m(o) | {z }

+

consumption utility

q(o|r) | {z }

gain-loss utility

In this utility function it is assumed that the neoclassical consumption utility m and the behavioral gain-loss utility q are additively separable. For gain-loss utility, the auction outcome o is evaluated against a reference outcome, r. Thus, the gain-loss utility experienced by a bidder is q(o|r) ≡ µ(m(o) − m(r)), where µ represents a value function (Kahneman and Tversky, 1979). It is assumed there is no curvature in the value function, which gives the following linear specification. µ(x) =

  ηx

if x ≥ 0

 ηλx otherwise.

This value function indicates a bidder experiences an auction outcome o in the gain domain when o is greater than or equal to their reference outcome r. Thus, gain utility takes the form µ(m(o) − m(r)) = η(o − r), where η ≥ 0 captures the relative value of gain-loss utility to consumption utility. Conversely, a bidder experiences an auction outcome in the loss domain

if o < r. This outcome invokes loss aversion through the parameter λ > 1, which magnifies

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the (negative) change in utility to a greater extent than a similar sized outcome in the gain domain, yielding loss utility of ηλ(o − r). Taken together, the bidder’s utility for an outcome of o conditional on some reference outcome r can be expressed as π(o|r) = o + µ(o − r).

(5)

What is left to resolve is the formation of the bidder’s reference point, r. For this we again follow K˝oszegi and Rabin (2006) by endogenously determining the reference point according to the potential gains and losses within the auction mechanism. To most closely follow the experimental design that follows, it is assumed that bidders experience gain-loss utility via wide bracketing. Doing so suggests that bidders integrate the gains and losses they experience across the auctioned item and monetary outlays together, rather than separately. Eisenhuth and Grunewald (2018) offers experimental evidence that bidders use wide bracketing in a similar all-pay auction format involving cash prizes, which supports this assumption. As a result, let the gain a bidder experiences from winning the auction item, ηv, be scaled by the likelihood they do not claim the item, (1−F (x)n−1 ). Thus, as the probability they lose the auction increases the gain utility they experience from winning increases. Conversely, let the loss utility they experience, ηλv increase with the likelihood they win the auction, F (x)n−1 . Modifying Baye et al. (1996) with the addition of this gain-loss utility in charity auctions leads to the following expected utility, E[π].
E[π] = F (x)n−1 (v − (1 − α − γ)x+ 1 − F (x)n−1 ηv)

+ 1 − F (x)n−1 −(1 − α − γ)x − F (x)n−1 ηλv Z +α(n − 1) zf (z)dz

(6)

The first line in equation (6) represents the probability a bidder who submits a bid of x will win the auction, multiplied by the consumption and gain-loss utility they would receive. The second line represents the probability a bid of x loses the auction, multiplied by the consumption and gain-loss utility received. And finally, the third line in equation (6) represents the expected benefits generated from the other n − 1 bidders who also choose their bid according to the

mixed strategy, F .

To solve for the mixed strategy Nash equilibrium, the first-order condition over bids is es-

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tablished from (6) as follows. − (1 − α − γ) + (n − 1)vF (x)n−3 f (x) [2η(λ − 1)F (x)n + (1 − η(λ − 1))F (x)] = 0

(7)

The resulting differential equation in equation (7) can be solved for F (x), yielding the symmetric mixed strategy equilibrium and expected revenue, πAP , in the first-price all-pay auction for charity.

First-price All-pay Auction Outcomes: There is a symmetric mixed strategy Nash equilibrium in which bidders’ strategies and expected auction revenue are described by q

F (x) = 

E[πAP ] = n

Z

(η(λ − 1) − 1)2 +

zf (z)dz =

4η(λ−1)(1−α−γ) x v

2η(λ − 1)

+ η(λ − 1) − 1

(2n − 1) − η(λ − 1)(n − 1) v. (2n − 1)(1 − α − γ)

1  n−1



(8)

This prediction on bidding strategies suggests that the presence of gain-loss utility unambiguously decreases the expected revenue of the FPAP auction. As η → 0, indicating bid-

ders care exclusively about consumption utility, we find the expected revenue increases be
1 cause limη→0 F (x) = 1−α−γ x n−1 , which first order stochastically dominates any F where v

η(λ − 1) > 0. Since the gain-loss utility always appears in the form G = η(λ − 1) in the bidding distribution, this can be shown by evaluating the derivative

dF dG ,

which is always non-negative

over the support of the bidding distribution. The distribution of bids with and without loss aversion for n = 6 bidders is compared in Figure 4. Figure 4: First-price All-pay Bidding Distributions with and without Loss Aversion

F (x)

F =1

η = λ = 0 (No gain-loss utility) η = 12 , λ = 2 F =0

Bid (x)

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

2.2.1

16

Curbing Loss Aversion

To better understand the mechanism-specific effect the first-price all-pay auction has on bidders with loss aversion, two variations of this mechanism will be explored experimentally. These variations are called Sealed-bid and Incremental Revelation, and they are described below. The Sealed-bid treatment is a standard auction format in which bids are privately and simultaneously submitted, and the winner is revealed immediately. The Incremental Revelation treatment is a modification of the Sealed-bid treatment in which bidders submit their bids privately and simultaneously, but then reveal their bids incrementally à la the war of attrition. The design of these mechanisms may motivate bidders differently, depending on the relative importance of gain-loss utility (i.e. the magnitude of η and λ). If bidders are relatively more loss averse, then the Sealed-bid mechanism may suppress bid levels since it can produce perception of loss that is relatively large and instantaneous.4 However, the Incremental Revelation mechanism may limit this effect by way of presenting losses incrementally, despite bids being sunk before the bidding process begins. When a bidder considers the reaction they might have in response to winning or losing an auction with a particular bid, it is reasonable to imagine the Incremental Revelation mechanism focuses their attention on many smaller losses rather than on one large loss. If this is true, then studying the Incremental Revelation mechanism may reduce the differences observed between bidders who are loss averse and bidders who are not. Furthermore, this auction can provide insights into behavior in the war of attrition, as it isolates any effect the bidding procedure itself might have on bids. This is the basis of Hypothesis 2. Hypothesis 2: In the Sealed-bid mechanism treatment, the distribution of bids from loss averse bidders will be first order stochastically dominated by the distribution of bids from those who are not loss averse, leading to an average bid that is less among loss averse bidders. In the Incremental Revelation mechanism treatment, the distribution of bids will not vary based on the bidders’ loss aversion, leading to no difference in their average bids. The following section outlines the experimental design utilized to compare the stated hypotheses against empirical tests. 4 See Chen et al. (2017) and Ernst and Thöni (2013) for theoretical and experimental evidence, respectively, in similar auction settings.

17

3

FOSTER, JOSHUA

Experiment Design In this study, a total of 192 subjects were recruited to participate in one of sixteen sessions.

Using a between-subjects design, sessions were equally split to test four charity auction settings: two involving a first-price all-pay auction, and two involving a sequential-move war of attrition. The variations explored within each mechanism are outlined in Table 2. Both mechanisms included a standard protocol, called the Benchmark, and a modified protocol, called the Variant. In the first-price all-pay auction, the Benchmark was a sealed-bid protocol where bidders simultaneously submitted contributions and winners and losers were simultaneously revealed. In the Variant, bidders again chose sealed-bid contributions simultaneously, however they revealed their bids sequentially in fifty cent increments. The sequence order of revelation was randomly determined. In the war of attrition, bidders were also randomly assigned to a sequence order for bidding, and each bid cost fifty cents. In the Benchmark mechanism, each bidder chose to bid or exit without constraint. However, in the Variant mechanism each bidder was given the opportunity to set a maximum contribution for themselves before the auction began. Sessions took place in behavioral research laboratories at two large American universities in groups of twelve subjects. Subjects were seated at partitioned computer workstations to ensure private decision-making and given computerized instructions about the experiment, at which point they were allowed to ask procedural and clarifying questions.5 The entire experiment lasted approximately seventy minutes, and subjects earned $21.27 on average. Table 2: Session Variations by Mechanism Mechanism

Benchmark

Variant

First-price All-pay War of Attrition

Sealed-bid Unconstrained

Incremental Revelation Self-constrained

The instructions explained the experiment consisted of two phases. In Phase 1, subjects were given multiple opportunities to complete a task and earn cash. This task served as a method of making the resources bidders used in the auction phase of the experiment feel like earned resources, thus it consisted of a real-effort task. Following Foster (2014), subjects were responsible for counting how many numbers (0-9) there were in a series of 120 random characters that 5 Instructions for a treatment of the war of attrition can be found in the appendix. All sessions were computerized and programmed using z-Tree (Fischbacher, 2007)

18

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

included both digits (0-9) and letters (A-Z) within 60 seconds. Each of the 120 characters was equally likely to be one of the 36 alphanumeric characters. If the subject reported the correct number or was off by at most 3 in absolute value then they received $2, otherwise they received nothing. Table 3 is an example of one such task.6 To avoid income effects or the creation of disparities in bidding ability, a relatively large margin of error was allowed in this task, as its purpose was principally to avoid risk seeking behavior that may occur with unearned endowments. Subjects were given eight opportunities to complete this task for a maximum possible endowment of $16, though they were not informed on the number of opportunities they would have. This task seemed to serve its purpose well, as 187 of the 192 subjects earned either $14 or $16 in Phase 1 while the remaining five subjects earned either $10 or $12. Average endowments across sessions varied from $15.58 to $15.67. Table 3: Effort Task Example 8 6 R 7 F V

3 E 9 Q N R

P Y Y 7 N H

2 R Y M L R

M 4 P I Y D

9 D 2 2 J 8

L S 4 D N R

K S A Z N 0

V 9 V V G 7

5 L 0 6 D C

W V D W X G

9 X R Z J P

G 3 L U L P

5 Q Z P Q X

2 S X L 7 3

X M 8 2 5 L

M O 3 G 4 C

E H A P T X

G B L Q F 9

P A J J Q 0

In Phase 2, subjects were given multiple opportunities to participate in auctions using the money they earned in Phase 1. Though the auction mechanism varied by session, the prize for winning the auction was always a fictitious item with a value of $10. Thus, if at the end of the auction a bidder was awarded for winning an auction, then they knew they would receive a payout of $10. There were twenty periods of Phase 2 in which bidders were randomly matched in auctions. Ten of these auctions were two bidder auctions while the other ten were six bidder auctions, giving a total of 80 per session. In an attempt to control for order effects, half of the sessions began with two bidder auctions while the other half began with six bidder auctions. One limitation of this study is the lack of randomization of auction order across universities. All sessions that began Phase 2 with two bidder auctions were run at one university, while all session that began Phase 2 with six bidder auctions were run at another university. Should an order or session effect exist, which the experimental results suggest one does, then it is not clear whether it is due to the ordering of the auctions or differences in the subject pools. However, as per the study’s motivation, any differences across the subject pools on sunk cost sensitivity and loss aversion will be controlled for directly. 6 This

example has 32 numbers in it. If the subject reported 29, 30, 31, 32, 33, 34 or 35 then they earned $2.

19

FOSTER, JOSHUA

The computer placed the twelve subjects for a given session into groups of four during the two bidder auctions and then randomly rematched within that fixed group. For the six bidder auctions, randomized two groups of six across the twelve subjects each period. This is summarized in Table 4. Table 4: Session Groups for Each Auction Treatment Rematching by Auction Size 12 Subjects per Session Bidders per Auction Groups of No. of Auctions 2 4 4 4 10 6 12 10 Notes on Table: two of the four sessions for each treatment began with two bidder auctions while the other two sessions began with six bidder auctions. To maintain a fixed financial position across auctions, bidders’ endowments were set to the amount they earned in Phase 1 regardless of previous auction outcomes. To further aid in the simulation of the one-shot nature of the theory, subjects were paid according to a randomly selected auction from Phase 2. As a result, a subject’s final earnings consisted of their cumulative earnings from Phase 1, one randomly selected auction in Phase 2, and a show up payment of $5. In accordance with the theoretical assumptions, two constant marginal benefits were induced in the auction to simulate a charity setting. The particular parameters for these marginal benefits were chosen to be the same as Carpenter et al. (2014) to aid in the comparison to previous studies. The first marginal benefit α = 0.1 is the indirect benefit associated to the revenue generated by the auction. In the experiment, subjects were informed that for every dollar of revenue generated, each of them would receive $0.10. The second marginal benefit γ = 0.05 is the “warm glow” from personally contributing to the auction. As a result, for each dollar a bidder personally contributed, they received $0.05. After each auction, bidders were shown a break down of their earnings, which consisted of 1) their auction contribution, 2) the outcome of the auction (claimed the item or not), 3) the total revenue generated by the auction and their indirect benefit earnings from revenue (α), and 4) the money returned from their own contributions (γ). Table 5 outlines the theoretical predictions over revenue for the parameters of this experiment. Following Phase 2, subjects were asked to answer survey questions regarding their expe-

20

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

Table 5: Contribution Predictions with Rational Bidders by Mechanism (in $) Experimental Parameters v = 10

α = 0.1

γ = 0.05

Predictions of Average Contributions by Treatment First-price All-pay Sealed-bid Incremental Revelation

War of Attrition

2 Bidders 5.88

6 Bidders 1.96

Unconstrained

2 Bidders 6.38

6 Bidders 2.12

5.88

1.96

Self-constrained

6.38

2.12

rience in the auction, as well as questions regarding hypothetical scenarios that could address their susceptibility to the sunk cost fallacy and loss aversion. Many of the survey questions came from Carpenter et al. (2010), some of which were slightly modified for regional context purposes. These questions are included in the instructions found in the appendix.

4 4.1

Experimental Results Bidder Characteristics Analysis begins with the summary of bidder characteristics, which were measured through

survey responses collected at the end of each session. Four questions in this survey provide the basis of subjects’ loss aversion and sunk cost sensitivity, which are summarized in Table 6. To identify sunk cost sensitivity, subjects were presented with two hypothetical scenarios involving sunk costs. In the first scenario, the subject had lost a movie ticket and asked if they’d be willing to buy another. In the second, the subject was asked if they’d forego the deposit on a weekend trip after learning there was a preferred alternative trip at the same time. If both answers indicated a sunk cost sensitivity, then that subject was coded as sunk cost sensitive. In Table 6 we see there is a consistent estimate of 25% to 33% of bidders as being sunk cost sensitive across mechanisms. Two questions regarding loss aversion were asked, one on whether they tend to worry about losses in financial situations and another on their tendency to worry about losses in general. Their responses were coded from 1 to 5, with 1 being “Strongly Disagree” and 5 being “Strongly Agree.” The average responses across mechanisms are relatively similar, with values ranging from 4.21 to 4.35 for financial losses and 3.77 to 4.02 for general losses. These two measures

21

FOSTER, JOSHUA

Table 6: Summary of Bidder Characteristics Mean Scale Response First-price All-pay Incremental Sealed-bid Revelation

War of Attrition SelfUnconstrained constrained

0.29

0.25

0.25

0.33

Financial Loss (1-5)

4.21

4.21

4.27

4.35

General Loss (1-5)

3.77

3.83

4.02

3.85

Combined (2-10)

7.98

8.04

8.29

8.21

48

48

48

48

Sunk Cost Sensitive (0-1) Loss Sensitivity

Subjects

of loss aversion were combined for each subject to create an aggregate measure allowing for values from 2 to 10. Standard deviations for the combined measure ranged from 1.14 to 1.75 across mechanisms. All analysis with respect to loss aversion utilizes this combined measure.

4.2 Bidder Contributions in the War of Attrition A summary of bidder contributions in each of the war of attrition mechanisms is reported across two and six bidder auctions in Figure 5. The box plots therein combine information on the distribution of the bidders’ contribution with the theoretical (T) and experimental (E) means (dashed lines). Included with the experimental means are 95% confidence intervals calculated using robust standard errors clustered at the subject level.7 Dark gray box plots represent all two bidders auctions, while the light gray box plots represent all six bidder auctions. Across these treatments, we see that bidders tended to under-bid in the two bidder auctions and over-bid in the six bidder auctions, as compared to their theoretical predictions. Each of the treatments had average contributions that were statistically significantly different at the 1% level from their theoretically predicted averages. The average contributions in the two bidder Unconstrained and Self-constrained treatments were 27.3% and 31.8% below their theoretical predictions, respectively. Average contributions in their six bidder counterparts were 69.4% and 7 Clustering standard errors at the group level, as described in Table 4, would have been the preferred method of accounting for the repeated interactions in the experiment. However, with only sixteen groups per treatment, clustering at this level would provide a poor approximation of the critical values needed to reject a null hypotheses. Clustering at the subject level accounts for the correlation of errors while providing 48 clusters per treatment. See Cameron and Miller (2015).

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

22

91.8% above their respective predictions. This pattern of results corroborates previous findings regarding wars of attrition. Namely, the mechanism greatly out-paces theoretical revenue predictions with several bidders present, but falls short with only two bidders. Figure 5: Bidder Contribution Box Plots with Theoretical (T) and Experimental (E) Means

Notes on Figure: Dark gray represents two bidder auctions, light gray represents six bidder auctions. Confidence intervals around experimental means were calculated with robust standard errors clustered at the subject level.

Without controlling for covariates, the lack of adjustment to the number of bidders in the auction led contributions to be statistically insignificantly different in the Self-constrained treatment (p = 0.499). However, there is a statistically significant difference in average contributions between two and six bidders auctions in the Unconstrained treatment (p = 0.020). Additionally, there is very little difference in the distributions of contributions across treatments by number of bidders. The difference in average contributions in two bidder auctions across treatments was statistically insignificantly different (p = 0.595), as it was in the difference in six bidder auctions (p = 0.418). Since the winner’s willingness to contribute is censored by the losing bidder’s contribution in the war of attrition, analysis turns to bidder survival analyses via nonparametric cumulative hazard rates using a Nelson–Aalen estimator. Recall Hypothesis 1 predicts that sunk cost sensitive bidders in the Unconstrained treatment will reduce their rate of exit as the auction progresses, while those who are not sunk cost sensitive and exit at a constant rate. In addition, it is predicted that there will be no difference between these bidder types in the Self-constrained treatment. To begin testing this hypothesis, Figure 6 shows the difference in the cumulative

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FOSTER, JOSHUA

Figure 6: Cumulative Hazard Rates for War of Attrition Treatments Unconstrained War of Attrition

3

3

Cumulative Hazard Rate

Cumulative Hazard Rate

Self-constrained War of Attrition

2

1

0

2

1

0 0

5 10 Contribution

15

Sunk Cost Sensitive Bidders with 95% CI

0

5 10 Contribution

15

Non Sunk Cost Sensitive Bidders with 95% CI

rates of exit across sunk cost sensitive (dark gray) and non sunk cost sensitive (light gray) bidders with 95% confidence intervals across the war of attrition treatments. Two and six bidder auctions are pooled in these plots, as their hazard rates are identical for all contribution levels, aside from at zero where instant exit for all but two bidders is theorized to occur. In this model, a linear hazard estimate represents a constant rate of exit through time, while a concave hazard estimate represents a rate of exit that reduces through time. In the Selfconstrained treatment, where bidders can voluntarily choose to limit their ability to bid in the auction, we find there is no meaningful difference in the exit rates between sunk cost sensitive and non sunk cost sensitive bidders. A log relative-hazard exponential regression of the data suggest there is no meaningful difference in the hazard rates (p = 0.977). In the Unconstrained treatment, however, we find that the cumulative hazard rate of those who are sunk cost sensitive falls meaningfully below that of their non sunk cost sensitive counterparts at high contribution levels. According to this parametric estimate, sunk cost sensitive bidder is only 80% as likely to lose an auction, as compared to a non sunk cost sensitive bidder. This result is statistically significant at the 5% level. The combination of these results suggests that sunk cost sensitive bidders are using the opportunity to limit their bids in the Self-constrained treatment in a ‘sophisticated’ way, so as to not over-bid beyond what they would prefer. However, in the absence of a commitment device, the

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

24

Unconstrained treatment demonstrates these sunk cost sensitive bidders are susceptible to overbidding. Figure 6 helps show this effect is particularly pronounced over large contributions, where sunk costs are likely to weigh most heavily on the bidders. To further capture the behavioral effects in these war of attrition treatments, Table 7 presents a set of Tobit estimates that address the censored nature of winning bidders’ contribution data. Two Tobit estimates are reported for each treatment. Tobit I is an estimate that utilized all nominal contribution observations from all bidders for its respective treatment, while Tobit II used above average contribution observations (according to auction size) only. The data in these estimates were coded as right censored if the bidder won the auction with their contribution. Specifically, the inherent variability in when the winning bidder was censored, as determined by the last exiting bidder, was addressed directly for each auction. The independent variables controlled for in these estimates include the behavioral measures for loss aversion and sunk cost sensitivity, as measured by the survey responses presented in Table 6. They also control for period of the auction (1-20), the number of bidders in the auction with a dummy variable (1=two bidders), and the order in which they experienced the number of bidders in the auction, again with a dummy variable (1=two bidder auctions were before the six bidder auctions). The results in Table 7 demonstrate that sunk cost sensitive bidders are statistically significantly more likely to contribute greater amounts in the Unconstrained treatment at the 1% level, as compared to non sunk cost sensitive bidders. However, this effect only holds if the bidder has already contributed an above-average amount to the auction. This suggests that bidders’ sunk cost sensitivity increases with the size of the sunk cost. The resulting statistic is also practically significant, as it suggests sunk cost sensitive bidders are willing to contribute 2.54 more dollars on average to auction, conditional on it having above-average revenues, which is approximately 25% of the value of the item. Moreover, this effect in sunk cost sensitive bidders disappears in the Self-constrained treatment where bidders can impose limits on their own bidding. These results lead to Experimental Result 1.

Experimental Result 1: Sunk cost sensitive bidders are willing to contribute more in the war of attrition, and thus win high-cost auctions more frequently. However, if given a commitment device that constrains their ability to bid, large contributions and winning rates are no longer significantly higher.

Among the other covariates in the full Tobit I estimates, we find that bidders in the sessions

25

FOSTER, JOSHUA

Table 7: Tobit Estimate on Contributions in the War of Attrition Treatments DV: Bidder Contribution Mechanism Self-constrained I II Loss Aversion (2-10) Sunk Cost Sensitive (0-1) Period (1-20) 2 Bidder Auctions (0-1) 2 Bidder Auctions First (0-1) Constant Obs. Uncensored Obs. Censored Obs.

Unconstrained I II

0.423 (0.275) -0.628 (1.078) -0.074 (0.048) 2.750∗∗∗ (0.585) 2.168∗∗ (0.948) 1.346 (2.653)

0.135 (0.322) -1.207 (1.125) 0.159∗∗ (0.072) -0.747 (0.812) -0.095 (0.858) 9.238∗∗∗ (3.042)

-0.343 (0.277) 1.168 (0.856) -0.116∗∗ (0.051) 3.569∗∗∗ (0.625) 2.282∗∗∗ (0.747) 7.403∗∗∗ (2.148)

-0.297 (0.417) 2.542∗∗∗ (0.925) 0.016 (0.072) 0.501 (0.776) 1.224 (0.881) 13.570∗∗∗ 3.657

960 644 316

325 177 148

960 644 316

297 158 139

Notes on Table: Robust standard errors are reported in parentheses. ∗∗∗ ∗∗ , , and ∗ indicate significance at the 1%, 5% and 10% level, respectively. I: All contribution observations included. II: Only above average contribution observations (by auction size) included.

starting with two bidder auctions were willing to contribute more on average at the 5% and 1% levels. One limitation of this study is its ability to fully identify the source of the order/session effect, as previously mentioned, due to how the sessions were run. All sessions that began with two bidder auctions were run at one university, while all sessions that began with six auctions were run at another university. Therefore, it is not at all clear whether this effect, and the differences in average bids across auction sizes, are due to the order of the sessions or the subject pools at these universities. While we can rely on the survey responses to shed light on the similarities of these subject pools, the concern exists nonetheless. Despite this order/session effect, the primary objective of this paper is not to directly compare two bidder and six bidder auctions. Rather, the objective is to compare auction mechanism variants to understand the role they play in eliciting behavioral responses. Finally, since this order/session effect adds noise to the experiment it introduces a bias against finding differences across auction treatments. In this

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

26

sense, the results that are reported can be viewed as stronger.

4.3 Bidder Contributions in the First-price All-pay Auction As presented for the war of attrition treatments, a summary of bidder contributions in each all-pay treatment is reported in Figure 7 by auction size. A similar pattern to what was seen in the war of attrition auction contributions is present in the all-pay auctions as well. Namely, bidders tended to under-contribute in two bidder auctions and over-contribute in six bidder auctions. Both treatments had statistically significantly greater than predicted average contributions at the 1% level in the six bidder auctions. However, the deviations from theoretical predictions are not as prominent in the two bidder auctions, as the average contribution in the Incremental Revelation treatment was only slightly less and not significantly different (p = 0.9025), and the average contribution in the Sealed-bid treatment was marginally significantly less (p = 0.06). Figure 7: Bidder Contribution Box Plots with Theoretical (T) and Experimental (E) Means

Notes on Figure: Dark gray represents two bidder auctions, light gray represents six bidder auctions. Confidence intervals around experimental means were calculated with robust standard errors clustered at the subject level.

Analysis of bidders’ behavioral tendencies in the all-pay treatments begins with a comparison of the cumulative distribution functions (CDFs) of bidder contributions in Figure 8. Hypothesis 2 predicted that the CDF of bids from loss averse bidders would be first order stochastically dominated by the CDF from bidders who were not loss averse. To highlight the differences in bidders’ loss aversion, each plot includes two experimental CDFs. The dashed

27

FOSTER, JOSHUA

CDFs represent those bidders who were measured to be highly loss averse (75th percentile or higher), while the dotted CDFs represent contributions from bidders who were measured to be relatively neutral toward losses (25th percentile or lower). The theoretical predictions for rational bidders is included as solid line CDFs. Figure 8: CDFs of Bidder Contributions in First-price All-pay Auctions Sealed-bid Treatment with Two Bidders

Incremental Revelation Treatment with Two Bidders

0.9

0.9

0.8

0.8 Pr(X ≤ Contribution)

1

Pr(X ≤ Contribution)

1

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

Theory High Loss Aversion

0.2 0.1 0

Low Loss Aversion 0

5

10

Theory High Loss Aversion

0.2 0.1 0

15

Low Loss Aversion 0

Contribution in $

10

15

Contribution in $

Sealed-bid Treatment with Six Bidders

Incremental Revelation Treatment with Six Bidders 1

0.9

0.9

0.8

0.8 Pr(X ≤ Contribution)

Pr(X ≤ Contribution)

1

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

Theory High Loss Aversion

0.2 0.1 0

5

Low Loss Aversion 0

5

10

Contribution in $

15

Theory High Loss Aversion

0.2 0.1 0

Low Loss Aversion 0

5

10

Contribution in $

15

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

28

The leftmost panels of Figure 8 show that the contribution distributions for those with a low loss aversion response first order stochastically dominate the contribution distribution for those with a high loss aversion response in the Sealed-bid treatment. Comparing these empirical distributions with a two-sample Kolmogorov-Smirnov test suggests these distributions are not the same (p ≤ 0.001 for two bidders; p ≤ 0.001 for six bidders). Additionally, these panels illus-

trate that highly loss averse bidders are particularly unwilling to make smaller contributions, as compared to those with low loss aversion.

The right panels of Figure 8 shows that contributions more closely follow the theoretical prediction for both low and high loss aversion bidders in the Incremental Revelation mechanism. A two-sample Kolmogorov-Smirnov test finds no statistical difference between these empirical distributions (p = 0.371 for two bidders; p = 0.055 for six bidders, which is marginally significant). Low loss aversion bidders’ contribution distribution did not differ from the theoretically predicted distribution for two bidder auctions (p = 0.275), however the high loss aversion bidders’ distribution did (p ≤ 0.001). Both groups of loss averse bidders had contribution distributions that were statistically significantly different from the theoretical distribution (p ≤ 0.001 for each). Table 8 reports a broader investigation of the bidders’ characteristics on their contributions in the first-price all-pay auctions through OLS estimates with robust standard errors clustering observations at the subject level. The dependent variable and covariates in these estimates are the same as those found in Table 7, with the variables controlling for bidders’ behavioral tendencies again being measured by the survey responses presented in Table 6. In the Sealed-bid treatment, the results show there is a statistically significant effect from bidders’ loss aversion that reduces their willingness to contribute. This effect is also practically significant. Replacing a bidder whose loss aversion is in the 75th percentile with one in the 25th percentile is associated with an increase in the average bid that is approximately 15% of the auctioned item’s value. In the Incremental Revelation mechanism, the OLS estimate indicates that loss aversion does not impact a bidder’s willingness to contribute. This statistical comparison suggests that bidders evaluate losses differently across these treatments, which in turn influences their willingness to contribute. As described in Behavioral Hypothesis 2, the Sealed-bid mechanism reveals payouts nearly instantly and bidders experience gains and losses simultaneously. However, in the Incremental Revelation mechanism, bidders are put through a process of discovery in which they experience their losses while evaluating their chances of winning as other bidders exit. The first experience leads one to imagine that an aversion to losses weighs heavily on bidders,

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FOSTER, JOSHUA

Table 8: OLS Estimates on Contributions in the All-pay Auction Treatments DV: Bidder Contribution Mechanism Loss Aversion (2-10) Sunk Cost Sensitive (0-1) Period (1-20) 2 Bidder Auctions (0-1) 2 Bidder Auctions First (0-1) Constant

Sealed-bid

Incremental Revelation

-0.768∗∗∗ (0.255) 0.798 (0.834) -0.084∗∗∗ (0.031) 1.805∗∗∗ (0.367) 1.367∗∗ (0.674) 10.042∗∗∗ (2.047)

-0.297 (0.234) -0.544 (0.814) -0.070∗ (0.040) 1.633∗∗∗ (0.472) 0.974 (0.700) 7.530∗∗∗ (2.188)

960

960

Obs.

Notes on Figure: ∗∗∗ , ∗∗ , and ∗ indicate significance at the 1%, 5% and 10% level, respectively. Robust standard errors are reported in parentheses.

while the second experience may be attenuated by the mechanism design. These findings lead to Experimental Result 2.

Experimental Result 2: Loss averse bidders submitted smaller contributions on average in the Sealed-bid mechanism, as compared to those who were less loss averse. When contributions were revealed in a manner that mimicked the war of attrition via the Incremental Revelation mechanism, the influence from a bidder’s loss response was attenuated and not statistically significant. These results support Hypothesis 2.

The results in Table 8 point to a potential order/session effect in the Sealed-bid treatment. As mentioned in the analysis of the war of attrition treatments, one limitation of this study is that an order effect cannot be disentangled from the subject pools used for this treatment. However, the purpose of the study is not to understand the differences in auction size, per se, but instead to compare various auction mechanisms. And, since the mechanism effect is found despite the noise introduced by the order/session effect, the support for Experimental Result 2 is that

LOSS AVERSION AND SUNK COSTS IN ALL-PAY AUCTIONS FOR CHARITY

30

much stronger.

5

Conclusion This paper adds to a growing literature on all-pay auctions and their ability to raise money

for charity. The theory developed to date on charity auction design uniformly touts the superior fund-raising capacity of these mechanisms, as they reduce the free-riding problem that is associated with generating revenue from their winner-pay counterparts. However, the experimental literature to date demonstrates that the revenue from all-pay auctions often fails to meet its theoretical expectations. Into this discussion, the present study offers both theoretical insights and experimental evidence of all-pay auctions’ limitations in generating charitable contributions in the presence of specific behavioral biases among bidders. By studying the first-price all-pay auction and the war of attrition in parallel, this paper finds a distinct behavioral bias for each mechanism that is responsible for its under-performance. In the first-price all-pay auction, the auction design moderates bidders’ loss aversion by making it overwhelmingly likely that all but one bidder (the winner) will end up worse off than they were before the auction began. As a result, the contributions bidders are willing to make with their bids is reduced by the presence of this psychological cost. In the case of the war of attrition, bidders are able to make many decisions over relatively small contributions to revenue, and this process can produce a mental account among sunk cost sensitive bidders. The tendency to create a mental account deficit motivates many sunk cost sensitive bidders to exit the war of attrition faster than if the bias were absent, however it can also be the reason why some choose to make very large contributions as they become increasingly reluctant to realize their deficit. While this leads sunk cost sensitive bidders to occasionally contribute large amounts to revenue, the net effect this bias has on bidder strategies leads to relatively lower revenues in expectation. Moreover, when bidders are provided with a commitment device that allows them to limit over-bidding, the proportion of large contributions received in the war of attrition is reduced. Finally, this study helps guide organizations in their use of all-pay mechanisms by providing a ranking of their performances according to the distribution of sunk cost sensitive and loss averse bidders. In particular, the war of attrition may be a powerful revenue raiser when there are several bidders who are susceptible to sunk costs. However, in the absence of these behavioral responses, a first-price all-pay auction with an incremental revelation of one’s bid may

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provide a more stable source of revenue that is close to theoretical predictions. Future studies can further aid organizations in this area by identifying relevant characteristics correlated with such behavioral tendencies.

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6

Appendix

6.1 Subject Directions for the War of Attrition Welcome! You have already earned $5 for showing up on time. It is important that you read these instructions carefully and understand them, so that you can make good decisions, and potentially make a considerable amount of money today. If you have any questions during the experiment, please raise your hand, and an experimenter will come to assist you. Otherwise, please do not talk or communicate with anyone.

Overview: This experiment has two phases, each of which involves a repeated activity. Phase 1: In this phase all participants here today will have multiple opportunities to complete a task. Each time you complete the task successfully you will earn $2. If you do not complete the task successfully you will earn $0. Phase 2: In this phase all participants here today will have multiple opportunities to participate in an auction. The item being auctioned is a fictitious item worth $10. At first, each of you will be randomly and anonymously matched with one other bidder (participant in this experiment) for each auction. Later, each of you will be randomly and anonymously matched with five other bidders for each auction. However, only one of you may claim the $10 item in a given auction.

Phase 1: The Task Instructions for the task are as follows: You will be shown a sequence of 120 random characters consisting of numbers (0-9) and letters (A-Z). The task is to count how many numbers (0-9) there are in the 120 characters. You will have 60 seconds to complete this task and report the number to the computer. If you count how many numbers are in the sequence correctly within 3 then you will receive $2. In other words, you can be off by at most 3 (over or under) and still receive $2. To the left is an example of the task in Phase 1 [shown on screen]. Take a moment to see what this task is like. (The "REPORT" button does not work right now because this is just an example.)

Hint: in this example there are [actual] numbers. If you had reported [actual-3], [actual-2], [actual1], [actual], [actual+1], [actual+2] or [actual+3] you would have received $2. Note that zeros (0) and Os look very similar. Because the example here was randomly generated, it may or may not have zeros or Os.

Phase 2: The Auction In Phase 2 you will have the opportunity to participate in auctions using the money you earned in Phase 1. To participate in an auction you must have Tokens. Tokens can be purchased, using the money you earned in Phase 1, for $0.50 each before the auction begins. You cannot buy more tokens than you can afford, and you cannot buy tokens once the auction begins. Bidders will take turns Bidding (i.e. contributing) their Tokens, one Token at a time. The last bidder to Bid a Token will claim the auctioned item, which is a fictitious item worth $10. If at any point you prefer to not Bid, you may choose to Exit, and forfeit the auction. If you choose to Exit you will not claim the item. A bidder can claim the item only if they are the last to Bid (at least one) Token. If, when the auction ends, you have Tokens left over, these Tokens will be redeemed for $0.25 each.

Important Aspects of this Auction: 1. Each auction you will be randomly and anonymously matched with 1 or 5 other participants for the auction. So, you will never know who the other bidders are. 2. No one will ever know how many Tokens you buy, and you will not know how many Tokens other bidders buy. 3. Tokens are purchased at the beginning of each auction only, and cannot be transferred from one auction to the next. 4. One of you will be randomly chosen to start bidding. This will be revealed after all bidders have bought their Tokens and has nothing to do with how many Tokens were purchased. 5. You may choose to buy 0 Tokens at no cost, if you like. 6. If no one chooses to Bid a Token, then no one will claim the item.

7. If only one bidder chooses to buy Tokens, then that bidder will claim the item by bidding their first Token. Bidders’ endowments will be reset to the amount they earned in Phase 1 at the beginning of each new auction. As a result, the decision to participate in an auction does not impact a bidder’s ability to participate in future auctions.

Important Aspects of this Auction: Furthermore, in these auctions we will be simulating charity auctions. Charity auctions are different in two ways, both of which will impact how much money you earn. These two ways are outlined here: 1. In charity auctions, bidders often receive benefits from how much revenue is raised (i.e. how many Tokens are contributed by all bidders combined). 2. Bidders often feel good for contributing to a charity. The auction raises revenue through the total value of Tokens that were Bid by everyone in the auction and the unredeemable value of unused Tokens. Because bidders often receive benefits from the total revenue raised, for every $1 the auction raises in revenue the bidders in that auction will receive $0.10 each. Example: if all bidders combined contribute $15 to the auction, then each of them would receive $1.50 at the end of the auction. Because individual bidders feel good for contributing to charity, for every $1 you personally contribute to the auction, you will receive $0.05. Example: if you contribute $6 to the auction (i.e. 12 Tokens), then you would receive $0.30 at the end of the auction on top of any money received because of the total amount raised.

Summary of the Auction To recap, in each auction of Phase 2 you will be placed in an auction where, 1. Each period you will be randomly and anonymously paired with one or five other participants(s) for the auction. So, you will never know the other bidders. 2. All bidders will privately buy Tokens for the auction using the money earned in Phase 1. 3. One bidder will be randomly chosen to start bidding.

4. The last person to contribute a Token will claim the fictitious item worth $10. 5. Each of you, regardless of winning or losing the auction, will earn an additional $0.10 for each $1 of revenue generated by all bidders. 6. Each of you, regardless of winning or losing the auction, will earn an additional $0.05 for each $1 personally contributed to the auction. As mentioned earlier, there will be many opportunities for you to participate in an auction. At the end of the experiment, the computer will choose one of these auctions randomly. Your payment for Phase 2 will be based on the outcome of this one randomly selected auction.

Almost Ready to Begin Your total earnings for today’s experiment will consist of your cumulative earnings from Phase 1, your earnings from one randomly selected auction in Phase 2, and your $5 show up payment for coming on time. So: Total Earnings = Phase 1 Earnings + Phase 2 Earnings + Show Up Payment ($5) If you have any questions you would like to ask at this point, please raise your hand. Otherwise, you will now go through a series of comprehension questions to make sure you understand the experiment. After everyone has correctly answered these questions, there will be one practice auction, then Phase 1 will begin.

Comprehension Questions 1. True or False: The other bidders in my auction are the same every period. (a) True. (b) False. [A: Bidders are randomly assigned to auctions each time.] 2. True or False: You must buy Tokens for every auction. (a) True.

(b) False. [A: You can choose to not participate in an auction by not buying any Tokens.] 3. True or False: You won’t know who will bid first when you buy your Tokens. (a) True. [A: Bidders will buy Tokens, then one bidder will be randomly selected to start bidding.] (b) False. 4. True or False: If you choose not to buy any Tokens you cannot win the auction, but you will not use any money you earned in Phase 1. (a) True. [A: If you don’t buy any Tokens, you cannot win the auction and you will not use any money you earned in Phase 1.] (b) False. 5. True or False: Your earnings for Phase 2 will be based on one randomly selected auction. (a) True. [A: One auction will be randomly selected, and the outcome of this auction will determine your earnings for Phase 2.] (b) False. [At the beginning of Period 11] The number of bidders in each auction has increased to 6! Bidders are still randomly assigned to auctions. (Everything else about the auction is the same.)

6.2 Exit Survey The experiment is now over. Thank you for participating! Below, there are a few questions that will help us understand the decisions you’ve made in this experiment. Please give them your full consideration. How difficult did you find this auction? Very Easy 

Easy 

Moderate 

Hard 

Very Hard 

Briefly, explain what your strategy was for bidding. [open response] How many auctions have you participated in over the last 2 years (including online auctions like eBay)? [numerical response] How fair is this auction to bidders? Very Unfair 

Unfair 

Neither Fair Nor Unfair 

Fair 

Very Fair 

If present at a charity auction like this in the future, how likely would you be to participate? Very Unlikely 

Unlikely 

Neither Likely Nor Unlikely 

Likely 

Very Likely 

We would like to ask you a few questions about your preferences and attitudes. Please try to answer these as accurately as possible. In general, do you see yourself as someone who is willing to take risks? Strongly Disagree 

Disagree 

Neither Agree Nor Disagree 

Agree 

Strongly Agree 

Financially, do you see yourself as someone who is willing to take risks? Strongly Disagree 

Disagree 

Neither Agree Nor Disagree 

Agree 

Strongly Agree 

In general, when you are faced with an uncertain situation, do you worry a lot about possible losses?

Strongly Disagree 

Disagree 

Neither Agree Nor Disagree 

Agree 

Strongly Agree 

In financial situations, when you are faced with an uncertain situation, do you worry a lot about possible losses? Strongly Disagree 

Disagree 

Neither Agree Nor Disagree 

Agree 

Strongly Agree 

In general, how competitive are you? Not at All 

Usually I Am Not 

Undecided 

Usually I Am 

Very Competitive 

We would like to ask you a few hypothetical questions. Please try to answer these as accurately as possible. Imagine that you’ve decided to see a movie in town and have purchased a $10 ticket. As you’re waiting outside the theater for a friend to join you, you discover that you’ve lost the ticket. The seats are not marked and the ticket cannot be recovered because the person who sold it doesn’t remember you. Would you buy another $10 ticket? [Response: Yes or No] Imagine that a month ago, you and a friend made a nonrefundable $100 deposit on a hotel room in New Orleans for the coming weekend. Since the reservation was made, however, the two of you have been invited to spend the same weekend at another friend’s cabin in Colorado. You’d both prefer to spend the weekend at the cabin but if you don’t go to New Orleans, the $100 deposit will be lost. Would you still go to New Orleans? [Response: Yes or No]