I. INTRODUCTION

Lotteries, or raffles, are widely used as a voluntary fund-raising mechanism for a variety of causes. Lottery pot sizes range from hundreds of dollars at small-scale church and community organized events to hundreds of millions of dollars in state lotteries. (1) Small-scale raffles are often conducted for charitable causes, such as disaster relief efforts, with proceeds transferred to charitable organizations.

The implementation of fixed-prize raffles in the field may be challenging because the organizers can end up losing money if the total proceeds from the raffle are smaller than the value of the fixed-prize. Self-financing raffles in which the prize is generated through donations themselves are, therefore, an attractive alternative. Very popular in this class are simple 50-50 raffles, in which one half of the ticket proceeds goes to the winner and the other half is used for the raffle's charitable cause.

The goal of a 50-50 charity raffle is simply to sell as many tickets as possible in order to maximize revenue for the charity. One way to sell more tickets may be to advertise the size of the pot while raffle tickets are for sale. One common example is to announce the pot size of a 50-50 raffle at halftime of a sporting event. (2) If a larger pot size makes raffle tickets more attractive, then revealing the intermediate pot size increases the sales to second-half buyers, especially if the pot size is larger than what they expected it to be. Another possibility is that the halftime announcement actually changes the incentives that participants face when buying tickets. It may be possible to observe strategic contributions by the more charitable participants in the first half to make less charitable participants contribute more in the second half as the pot size becomes more attractive. We refer to the effect, be it strategic or not, of a larger intermediate pot size inducing higher contributions in the second half as priming the pump of a raffle.

The goal of this paper is to better understand bidder behavior in these sorts of raffles where the intermediate pot size is revealed, so that organizers can earn more money for their charity. By comparing different raffle mechanisms in a controlled laboratory environment, we can isolate those aspects of bidder behavior that an organizer may capitalize on. More specifically, we study whether priming the pump occurs with modifications of a simple 50-50 raffle. In our two-stage raffles participants can buy raffle tickets twice. The proceeds from ticket sales in the first stage are added to the pot of the second stage. We consider two types of two-stage raffles. In the complete draw down two-stage raffle, all the tickets purchased at the first stage are destroyed, that is, the proceeds of the first stage are used as seed money for the second stage. In the no draw down two-stage raffle, the tickets purchased at the first stage continue to participate in the second-stage draw, that is, the difference between this raffle and the standard (one-stage) raffle is purely informational: participants observe the first-stage donations and can donate more if they like.

Our study can be classified as a mixture of a laboratory and framed field experiment (Harrison and List 2004; List 2008). Although the experiment is conducted in standard laboratory conditions with students as subjects, we do not use induced valuations of the public good, but instead allow subjects to contribute money to an actual charity. (3) This reduces the amount of control we have over subjects' incentives but provides a more immediate external validity to our results.

Using a model of a charity raffle where bidders have a heterogeneous warm glow of donating to the charity, we show that under certain conditions on the participants' preferences priming the pump of the two-stage raffle occurs in equilibrium. However, priming the pump increases revenues only if there is a drawing down of tickets after the first stage. Therefore, the rational warm glow model predicts that the complete draw down raffle outperforms the others. We also show that in the class of two-stage raffles with a draw down, the complete draw down raffle should produce the strongest priming effect.

In the experiment, we find that initially both two-stage raffles yield significantly higher contributions than the one-stage raffle. Over time, contributions in the complete draw down raffle decline, similar to the often observed pattern of declining public good contributions in voluntary contribution mechanisms. Surprisingly, contributions remain high in the no draw down raffle.

People often purchase lottery tickets when no charity is involved. However, a rational self-interested bidder would not choose to participate in a 50-50 raffle regardless of pot size, unless they were non-trivially risk seeking. Still, a large pot is attractive in a pecuniary sense, and one can easily imagine that some participants mistakenly believe that purchasing tickets when the pot size is large is a good bet for their pocket books. Beliefs like that may arise in the presence of boundedly rational cognitive biases. For example, a subject who ignores the strategic element of the raffle and thinks that her probability of winning depends only on her own number of tickets purchased would hold such beliefs. Unlike in the rational warm glow model, this effect would work whether or not the tickets in the first stage are drawn down. Therefore, this type of bounded rationality can explain why the no draw down raffle outperforms the others.

The rest of the paper is structured as follows. In Section II, we provide a brief review of the related literature. In Section III, we set up a rational warm glow-based model of two-stage raffles and derive its testable predictions for our treatments. Thereafter, we introduce the experimental design and procedures (Section IV) and analyze our experimental data (Section V). Section VI provides a discussion of the results and Section VII concludes.

II. RELATED LITERATURE

The literature on the efficiency of public good provision through raffles goes back to Morgan (2000) who showed that a raffle can generate more money than a standard voluntary contribution mechanism (VCM). (4) The reason is that the free-riding problem of public good provision is partially mitigated by the negative externalities imposed on one another by the raffle participants. This finding is supported by a number of experimental studies. (5) While in some environments fixed-prize raffles can produce more money than 50-50 self-financing raffles (Dale 2004), they also create a possibility for organizers to lose money (e.g., Landry et al. 2006).

Several authors explored multistage fundraising mechanisms where the proceeds from earlier stages are revealed. Carpenter, Holmes, and Matthews (2014) proposed a "bucket" all-pay auction in which bidders sequentially add money to the pot until all but one bidder drop out and find that it generates more money than the standard all-pay auction in a laboratory experiment. Lange (2006) proposes a two-stage modification of the public good lottery mechanism of Morgan (2000): the second stage lottery prize is comprised of voluntary public good contributions of the first stage. (6) It is shown that for a wide range of conditions the resulting equilibrium donations exceed those of the standard VCM under the same preferences. Although our complete draw down two-stage raffle is similar to the mechanism proposed by Lange (2006), the important differences in our setting are in that the second-stage proceeds are also added to the prize, and the prize is then divided equally between the charity and the winner. Thus, our second-stage raffle has both the fixed prize and the variable prize elements to it.

III. MODEL

We model a two-stage charity raffle where there is a potential drawing down of tickets from the first stage to the second. There are n [greater than or equal to] 2 risk- neutral bidders indexed by i = 1, ..., n, whose utility functions are given by

[u.sub.i] = [w.sub.i] + [g.sub.i] ([d.sub.i]) + [[beta].sub.i] [n.summation over (j=1)] [d.sub.j], i = 1, ..., n.

Here, [w.sub.i] is the amount of a numeraire good of bidder i and [d.sub.i] is bidder i's donation to the charity. The functions [g.sub.i](*) are common knowledge and [g'.sub.i] > 0, [g".sub.i] < 0 for all bidders. Following Andreoni (1990), we model charitable preferences with a concern for the total group contribution, parameterized by commonly known coefficients [[beta].sub.i] [member of] [0,1), as well as a concern for one's personal level of contribution, measured by [g.sub.i](*). Each bidder has an initial endowment e of the numeraire. Tickets are sold at a unit cost. In Stage 1, bidder i chooses to buy [x.sub.i] tickets, with the constraint [x.sub.i] [less than or equal to] e.

Let X = [[summation].sub.i] [x.sub.i] denotes the total number of tickets sold in the first stage. After Stage 1, all but S(X) tickets are removed from the active pool of tickets. In Stage 2, bidder i chooses to buy [y.sub.i] tickets, with the constraint [x.sub.i] + [y.sub.i] [less than or equal to] e. Let Y = [[summation].sub.i] [y.sub.i] denotes the total number of tickets sold in the second stage, and these are added to the S(X) tickets that remain after the first stage. One ticket out of the S(X) + Y active tickets is randomly selected as the winner, and the owner of the ticket receives a fraction [alpha] [member of] (0,1) of the total amount collected while the charity receives a fraction 1 - [alpha] of the proceeds.

In the following two sections we consider two special cases of the general two-stage raffle described above corresponding to our experimental treatments. First is the benchmark case of no draw down, S(X) = X, which leads to no priming of the pump and the same aggregate donations in equilibrium as the one-stage raffle. Second, at the other extreme, is the case of complete draw down, S(X) = 0, which, in equilibrium, produces the strongest priming effect in the class of two-stage raffles with partial draw down (for a proof of this statement, see Appendix S1, Supporting Information).

A. No Draw Down

Consider the two-stage charity raffle with no drawing down of tickets, or S(X) = X. In the second stage, bidder i chooses [y.sub.i] [member of] [0, e - [x.sub.i]]. Given the first-stage outcomes of [x.sub.i] and X, bidder i's expected utility is equal to

[EU.sub.i] ([y.sub.i]|X) = e - (1 - [alpha]) ([x.sub.i] + [y.sub.i]) + [g.sub.i] ((1 - [alpha]) ([x.sub.i] + [y.sub.i])) + [[beta].sub.i](1 - [alpha]) (X + Y).

The first-order condition for maximization of expected utility over [y.sub.i] is

(1) 1 - [[beta].sub.i] [greater than or equal to] [g'.sub.i] ((1 - [alpha]) ([x.sub.i] - [y.sub.i])),

with equality for [y.sub.i] [member of] (0, e - [x.sub.i]).

Equation (1) shows that each bidder's total optimal contribution, [x.sub.i] + [y.sub.i] is independent of other bidders' actions and constitutes a dominant strategy of buying a total number of tickets equal to [(1 - [alpha]).sup.-1] [g'.sup.-1.sub.i] (1 - [[beta].sub.i]), or e if their endowment is reached. It does not matter to the bidders at which stage they purchase the tickets. Let [P.sup.ND] = X + Y be the pot size when each bidder chooses their optimal number of tickets in the two-stage charity raffle with no draw down. Note that PND is the same as the pot size of a standard one-stage raffle. This is because tickets purchased in the first and second stage are perfect substitutes, and having multiple opportunities to buy them does not change the aggregate equilibrium outcome. Let [[??].sub.i] = min {[(1 - [alpha]).sup.-1] [g'.sup.-1.sub.i] (1 - [[beta].sub.i]), e} be bidder i's optimal number of tickets purchased in the second stage if X = 0, and [??] = [[summation].sub.i] [[??].sub.i]. Then, because bidders are indifferent to purchasing all their tickets in the second stage, [P.sup.ND] = [??].

B. Complete Draw Down

Consider the two-stage charity raffle with a complete draw down where every ticket from the first stage is eliminated from the active pool of tickets, that is, S(X) = 0. In the second stage, bidder i chooses [y.sub.i] [member of] [0, e - [x.sub.i]]. Given the first-stage outcomes of [x.sub.i] and X, bidder i's expected utility is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition for maximization of expected utility over [y.sub.i] is

(2) 1 - [[beta].sub.i] - {[[alpha]X (Y - [y.sub.i])]/[(1 - [alpha])[Y.sup.2]]} [greater than or equal to] [g'.sub.i] (1 - [alpha]) ([x.sub.i] + [y.sub.i])),

with equality for [y.sub.i] [member of] (0, e - [x.sub.i]). Let [P.sup.CD] = X + Y denotes the resulting pot size under complete draw down. By comparing the first-order conditions (1) and (2), we arrive at the following proposition.

PROPOSITION 1. Provided that there is at least one active bidder in the first stage (X > 0), there are at least two active bidders in the second stage (Y > [y.sub.i]), and for at least one active bidder the budget constraint does not bind under no draw down ([[??].sub.i] < e), the equilibrium pot size under complete draw down is greater than under no draw down, [P.sup.CD] > [P.sup.ND].

The intuition behind this result is that the marginal cost of purchasing a ticket under complete draw down is offset by the chance that the bidder could win the raffle, when X > 0, and it is this term (which appears in (2) but not in (1)) which implies [P.sup.CD] > [P.sup.ND] because [g".sub.i] < 0.

Equation (2) also shows that every bidder's total contribution, [x.sub.i] + [y.sub.i] is an increasing function of the aggregate first-stage contribution, X. Thus, if bidder i did not contribute in the first stage ([x.sub.i] = 0), her second-stage contribution is an increasing function of the first-stage contributions of other bidders. (7) We refer to this effect as "priming the charitable pump" because it is through this mechanism that more charitable bidders, by bidding in the first stage, can induce less charitable bidders to bid more in the second stage.

PROPOSITION 2. (Priming the charitable pump) Suppose bidder i is not active in the first stage, [x.sub.i] = 0. Then, provided [[??].sub.i] < e and [[??].sub.i] < [??], bidder i's optimal second-stage bid, [y.sub.i], is increasing in the first-stage donations X.

We now turn to the analysis of equilibrium bidding in the first stage and derive conditions under which X can be larger than zero in equilibrium.

PROPOSITION 3. There exists a subgame perfect equilibrium of the two-stage raffle with complete draw down with at least one bidder priming the charitable pump by purchasing tickets in the first round (X > 0) if for some bidder i,

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In Appendix S1, we prove Proposition 3 by showing that when X = 0, the marginal utility of buying a ticket in the first stage is positive for some bidder given the assumptions of the model and condition (3). Proposition 3 establishes a sufficient condition for priming the charitable pump in equilibrium. In Appendix SI, we demonstrate that the condition can hold for a variety of parameterizations.

C. Hypotheses

Given the theoretical findings of this section, we establish the following experimental hypotheses.

HYPOTHESIS 1: Two-stage raffles with no draw down generate the same aggregate contributions as one-stage raffles.

HYPOTHESIS 2: Two-stage raffles with complete draw down generate larger aggregate contributions than two-stage raffles with no draw down (and one-stage raffles).

HYPOTHESIS 3: In a two-stage raffle with complete draw down, an increase in group donations in the first stage leads to an increase in individual donations in the second stage.

IV. EXPERIMENTAL DESIGN

The previous section demonstrated that in theory two-stage raffles might have an advantage over traditional one-stage raffles. To provide empirical support, we conducted an experiment with three different raffle types: a classic one-stage raffle (1STAGE), a two-stage raffle with complete draw down (2STAGECOMPDRAW), and a two-stage raffle with no draw down (2STAGENODRAW). In all three treatments, a real charity organization (United Way of the Big Bend, a local chapter of United Way) was the beneficiary of half of the proceeds.

All sessions were conducted at the XS/FS laboratory at Florida State University between November 2012 and January 2013. The experiment was programmed in z-Tree (Fischbacher 2007), with 216 subjects recruited through ORSEE (Greiner 2004) from a pool of roughly 3,000 pre-registered FSU students. We conducted three sessions per treatment with 24 subjects in each session divided into three groups of eight. Table 1 gives an overview of observations and some of the subjects' characteristics.

Upon arrival, subjects were seated in cubicles, the printed instructions were distributed and read to the subjects. Thereafter, a printout of the mission statement of the charity (United Way of the Big Bend) was distributed and read to the subjects. (8) Providing the subjects with the mission statement ensured that all subjects were informed about the main goals of the charity.

After the instructions had been provided and all questions posed by the subjects were answered, the corresponding charity raffle was played. Subjects received an endowment of $8.00 and could buy raffle tickets at a cost of 10 cents per ticket. In the two treatments with two-stage raffles, total proceeds after the first stage were shown before subjects decided how many tickets to buy at the second stage.

After the first raffle had been played, it was announced that it would be followed by an additional 10 periods of the same raffle in the same fixed groups of eight subjects. In each of the 10 additional periods, subjects started with an initial endowment of $8.00, but only one of the 10 periods was randomly selected at the end for actual payments, in addition to the first raffle.

At the end of the experiment, risk attitudes were elicited with the Holt and Laury (2002) instrument, followed by a set of post-experimental questionnaires. First, subjects self-assessed their risk attitudes in different domains (Dohmen et al. 2011). Second, subjects were asked whether they donated any amount during the last year. Third, subjects reported their gender and age. For summary statistics of the Holt and Laury (2002) task and the other post-experimental questionnaires refer to Table S1 in Appendix S1. (9) After the experiment, all participants were paid and one random participant monitored the transfer of half of the total donations to the charity. Each session lasted between 60 and 90 minutes with average earnings per subject of around $22.45, including a show-up fee of $7.00.

V. RESULTS

In this section we present the results of our experiment. We first focus on the aggregate treatment effects, that is, identify the ranking of the three raffles in terms of the revenue they generated. We then take a closer look at the two two-stage raffles and investigate whether priming of the charitable pump actually occurred in our experiment.

A. Overall Comparison of Raffles

Table 2 gives the mean, median, and standard deviation of the number of raffle tickets bought in each treatment for the duration of the whole experiment, as well as separately for the first and second part of the experiment. Overall, both two-stage raffles lead to higher mean/median donations than the one-stage raffle. However, only the overall comparison of 2StageNoDraw with ISTAGE is statistically significant (p=.0035, two-sided Mann-Whitney U test),10 while the comparison of 2StageCompDraw with ISTAGE remains insignificant (p=.4735). Overall, 2STAGENODRAW generates significantly higher ticket sales than the other two-stage raffle, 2StageCompDraw (p=.0751).

The majority of the subjects participated actively in each period: 61% of subjects in 2STAGENODRAW buy raffle tickets in all 11 periods, 54% and 64% do so in 2StageCompDraw and 1STAGE, respectively. As a result, no significant differences in the average number of periods in which subjects participate in the raffle through the purchase of tickets are observed (two-sided Kruskal-Wallis rank test p=.83).

Thus, the significant differences in overall contributions are driven by an increased amount of tickets bought per subject and not by a change in the inclination to buy raffle tickets at all. This is shown in Figure 1 which gives the distribution of the fraction of the endowment spent on raffle tickets. This distribution is skewed to the right in ISTAGE with the mass of the distribution on low amounts of endowment spent. This skewness is reduced in 2StageCompDraw and even more so in 2STAGENODRAW.

We now turn to the change in donations over time: Figure 2 shows average total donations by period. Already in the first period are donations significantly larger in 2STAGENODRAW compared to those of 1Stage (p=.0210, two-sided Mann-Whitney U test). However, the difference between 2STAGECOMPDRAW and 2StageNoDraw develops only over time. In the first period it is not significant (p=.5652), but the difference increases over time as median ticket sales decrease in 2STAGECOMPDRAW. Ultimately, it becomes significant for the second part of the experiment (p=.0455). The comparison between2STAGECOMPDRAW and 1STAGE reveals a significant difference for the first period (p=.0069), but over the course of the second part of the experiment this difference vanishes (p=.6546). Comparing the median ticket sales in the first part of the experiment with the ones in the second part demonstrates rather stable behavior in 2StageNoDraw (p=.5533, two-sided Wilcoxon signed-rank test), while raffle sales drop significantly in 2StageCompDraw (p=.0283) and 1Stage (p=.0074).

Before taking a closer look at the two-stage raffles, we complement the analysis presented above with a series of tobit regressions given in Table 3. Models (1) and (2) replicate our previous results: significantly higher donations in 2STAGENODRAW than in 1Stage and donations that significantly decrease over time in 2STAGECOMPDRAW. Donations increase with the number of raffle wins in the previous periods included in Model (3). (11) In Model (4) we include additional variables to control for subjects' individual characteristics. Age and gender do not have a significant impact on the number of tickets bought, but a higher willingness to take risks (12) and previous donations to charities are associated with an increase in the number of tickets bought. Our main treatment effects remain significant; we thus arrive at the first result, which stands in sharp contrast to our Hypotheses 1 and 2.

RESULT 1: (a) Overall, significantly more tickets are purchased in the two-stage raffle with no draw down than in the one-stage raffle and the two-stage raffle with complete draw down.

(b) In the two-stage raffle with complete draw down, the number of tickets purchased decreases over time until they reach the level of the one-stage raffle.

B. Comparing the Two-Stage Raffles

For comparison of the two-stage raffles, we analyze first-stage and second-stage donations separately. Figure 3 gives the absolute number of tickets bought as well as the fraction of possible tickets bought in the first and second stages. (13)

Clearly, first stage donations are significantly higher in 2StageNoDraw compared to 2StageCompDraw (p = .0380, two-sided Mann-Whitney U test). In 2StageCompDraw, donations in the first stage are only driven by the concern for the charity as they do not increase the likelihood of winning the raffle. However, tickets bought in the second stage increase the chance to win the raffle in both treatments and thus similar fractions of tickets are bought (p = .1711).

The tobit regressions presented in Table 4 explain the number of tickets bought at each stage in the two-stage raffles. The dependent variable in these regressions is the fraction of available tickets, which is x/80 at stage 1 and y/(80 - x) at stage 2. Models (1) and (2) replicate our nonparametric results and show that in the first stage significantly less tickets were bought in 2StageCompDraw than in 2StageNoDraw. This difference increases even more over time as the number of tickets bought decreases significantly in 2StageCompDraw. Winning the raffle in previous rounds, being female, and/or having a higher willingness to take risks leads to an increase in the number of tickets bought in the first stage, while the main treatment and time effects remain significant.

RESULT 2: Significantly more tickets are purchased in the first stage of the two-stage raffle with no draw down than in the first stage of the two-stage raffle with complete draw down. This difference increases even further over time.

As Models (3) and (4) demonstrate, no significant treatment effect (neither overall nor as a treatment-specific time trend) is observed in the second stage. As in the first stage, lower risk-aversion leads to more purchased tickets in the second stage. In addition, subjects who won the raffle in previous rounds are more willing to buy new raffle tickets. The most important result, however, emerges when we take a closer look at the influence of the first-stage donations on second-stage donations: an increase in the aggregate donations of the other group members in the first stage ([X.sup.Others.sub.Stage1]) leads to an increase in the number of tickets bought at Stage 2. Variable [X.sup.Others.sub.Stage1] is defined as the sum of tickets bought in a group minus the tickets bought by oneself and, thus, its impact on tickets bought in the second stage can be substantial. The average total number of tickets bought by others in a group in the first stage ([X.sup.Others.sub.Stage1] = 113) would increase the average fraction of income spent on tickets in the second stage by roughly 13.5%. (14) This effect was predicted by Hypothesis 3 for the treatment 2StageCompDraw. However, it turns out that the effect is also present in 2StageNoDraw and that it is equally strong in both two-stage raffles. (15)

RESULT 3: There is evidence of priming the charitable pump in that the number of tickets purchased at the second stage increases in the aggregate first-stage contributions of other group members.

VI. DISCUSSION

Our analysis reveals two important results. First, we identified a two-stage raffle mechanism that performs significantly better than the standard one-stage 50-50 raffle. Interestingly, and in stark contrast with the theoretical predictions, the mechanism that leads to the highest donation levels is the no draw down raffle. In what follows we discuss possible alternate explanations for this result. Second, in agreement with the theory, priming the charitable pump does occur. Higher group donations in the first stage lead to higher donations of individuals in the second stage. So, what is driving these results?

We can dismiss explanations based on the learning of charitable preferences of the other group members. In each treatment the subjects played 11 rounds, which should have given them enough time to learn those preferences by observing past group donations even in the one-stage raffle. Still we find that the difference between treatments remains strong or even increases over time. Neither can the results be explained by joy of winning as it should increase donations in all treatments equally. Rather, we believe the observed results are amenable to an explanation involving bounded rationality and heterogeneous charitable preferences.

Recall that priming the pump works in both two-stage raffles and now consider subjects' behavior in Stage 2 of the no draw down raffle. If a subject has charitable preferences, it is rational for her to purchase tickets in one of the two stages, and it might as well be the second stage. This incentive to give does not depend on the pot size. Furthermore, it is never rational to buy tickets for purely pecuniary concerns regardless of the pot size unless the subject is very risk seeking. (16) One plausible explanation for why many subjects purchase more tickets in Stage 2 when the pot size is larger is a version of bounded rationality termed solipsism bias', that is, some bidders fail to account for the negative effect of pot size on their probability of winning, or equivalently, they simply do not consider the expected value of a raffle ticket when deciding how many tickets to buy. The notion of solipsism bias was first introduced by Guarino, Huck, and Jeitschko (2006) as a behavioral explanation for the surprisingly high degree of coordination stability they observed in networks. For raffles, the presence of solipsism bias is suggested by the results of Lim, Matros, and Turocy (2009) who found that in a one-stage raffle experiment subjects do not change the spending as the group size increases from two to nine participants. A bidder with a solipsistic bias will mistakenly believe that buying raffle tickets is more attractive when the pot size is large. Thus, priming the charitable pump can occur in an environment where theoretically it should not.

We also believe that subjects have heterogeneous preferences over the total amount donated to the charity. (17) In the no draw down raffles, the most charitable subjects have an incentive to buy more tickets than they otherwise would in the first stage, in order to induce noncharitable subjects to buy more tickets in the second stage. In our experiment, the first-stage contributions in no draw down were larger than the first-stage contributions in complete draw down. This is exactly what we would expect to happen given the bounded rationality and heterogeneous charitable preferences described above because buying a first-stage ticket in the no draw down treatment has approximately the same power to prime the charitable pump with the added bonus that the ticket can win a substantial amount of money.

VII. CONCLUSIONS

In this paper, we experimentally investigated two-stage charity raffles in which participants can buy tickets in two stages. The proceeds of the first stage are used as the seed money for the second stage. In the complete draw down two-stage raffle, the first stage tickets are eliminated from the active pool of tickets, while in the no draw down raffle they remain in the active pool. In addition, we conducted a standard one-stage 50-50 raffle as a benchmark.

We find that both two-stage raffles initially perform significantly better than the standard one-stage 50-50 raffle. Over time, the aggregate contribution level in the complete draw down raffle declines and approaches that of the one-stage raffle. In contrast, contributions in the no draw down raffle are stable and remain significantly higher than in the other two mechanisms. In agreement with the theory, priming the charitable pump does occur. Higher group donations in the first stage lead to higher donations of individuals in the second stage, and this effect is especially strong in the two stage raffle with complete draw down.

The charity raffles we studied have the advantage of being self-financed, that is, the participants in the raffle generate the prize. Thus, the organizers bear no risk while fixed-prize auctions or raffles might lead to donations below the value of the prize and to losses for the charity (e.g., Landry et al. 2006). Our paper demonstrates that two-stage raffles can significantly increase the proceeds of those risk-free raffles.

Our results cannot be explained by a standard warm glow model of charitable giving or a combination of this model with joy of winning and learning about others' preferences. One possible explanation is a boundedly rational behavior which involves "solipsistic" bidders who mistakenly believe that their probability of winning the raffle does not depend on the pot size. The presence of such bidders is consistent with the results of prior experiments on raffles (e.g., Lim, Matros, and Turocy 2009). Of course, our experiment was not designed to test for the presence of solipsism bias; therefore, we only propose it as one possible explanation for our results and cannot exclude other versions of bounded rationality as explanations.

The superiority of the no draw down two-stage raffle and the fact that priming the charitable pump occurs has implications for a large class of mechanisms designed to generate monetary contributions. If a charity must use a self-financed raffle mechanism to generate revenue, the organizers would benefit by making the number of tickets sold public information. Having a first stage may be a useful way to coordinate the efforts of charitable participants who will collectively create a pot size large enough to attract less charitable individuals to participate. Our experiment shows that it is not necessary for these first-stage participants to be purely altruistic (as in the case of complete draw down) and the optimal mechanism should exploit both the desire to win and the incentive to prime the charitable pump as motivations for contribution.

doi: 10.1111/ecin.12245

ABBREVIATION

VCM: Voluntary Contribution Mechanism

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SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Appendix SI. (A) Proofs and Examples, (B) Instructions, (C) Additional Tables and Figures.

(1.) For example, the Florida Lottery transferred more than $1.31 billion to the Educational Enhancement Trust Fund in fiscal year 2011-2012; see http://flalottery.com/education .do.

(2.) Similarly, state lotteries regularly advertise the size of potential winnings.

(3.) Isaac, Pevnitskaya, and Salmon (2010) used a similar approach in one of their treatments.

(4.) The total amount of money raised would still be inefficient. Franke and Leininger (2013) propose an asymmetric mechanism similar to Morgan (2000) that leads to efficient levels of public good provision.

(5.) See, for example, Morgan and Sefton (2000), Dale (2004), Davis et al. (2006), Lange et al. (2006), Orzen (2008), and Corazzini, Faravelli, and Stanca (2010).

(6.) See also Varian (1994) and Vesterlund (2003), among others, who find that sequential VCMs can generate more revenue than the standard VCM.

(7.) This would not be the case only for bidders who are giving their full endowment already, or for those who would have been the only one buying tickets in the second stage.

(8.) The instructions are provided in Appendix S1. The mission statement was taken from the charity's website http:// uwbb.org/.

(9.) We did not identify any significant differences in subjects' responses to the Holt and Laury (2002) measure and other questionnaire items across the treatments (age: two-sided Kruskal-Wallis rank test p=.87; gender: two-sided Fisher's exact test p=.50; Holt and Laury (2002): two-sided Kruskal-Wallis rank test p=.77 financial risk: two-sided Kruskal-Wallis rank test p=31). We conclude that the randomization of the subjects into the treatments was successful.

(10.) All nonparametric tests are based on group medians, as subjects can observe their group's behavior between periods/stages. Thus, our nonparametric test results are rather conservative.

(11.) The result stays qualitatively the same if we replace the cumulative number of wins with a dummy for a raffle win in the previous period or for winning ever in the previous periods.

(12.) Risk attitudes elicited with the Holt and Laury(2002) measure are significantly correlated with the self-assessed risk attitudes in the domain of financial risks (Spearman's rank correlation [rho] = .2464 withpc.001). In the following we base our analysis on the Holt and Laury (2002) measure, but the results remain robust if the self-assessed risk attitudes (given on a Likert scale) are used instead.

(13.) In the first stage a maximum of 80 tickets could be bought; in the second stage the maximal possible number of tickets, 80 - x, depends on the tickets already bought in the first stage.

(14.) For the highest sum of tickets bought by others in the first stage ([X.sup.Others.sub.Stage1] = 337) the estimated effect would be 40%.

(15.) The coefficient on interaction 2StageCompDraw x [X.sup.Others.sub.Stage1], if it is added, would be insignificant, while the main effect would remain significant.

(16.) This is because the probability of one ticket winning the raffle is 1/[P.sup.ND], while the revenue from winning is only [P.sup.ND]/2, where [P.sup.ND] is the pot size of the no draw down raffle.

(17.) We clearly have evidence that subjects differ in terms of their charitable donations outside of the experiment. Overall, 76% reported that they donated to charities at least once during the previous year, while 24% did not.

SEBASTIAN J. GOERG, JOHN P. LIGHTLE and DMITRY RYVKIN *

* We are grateful to two anonymous referees and the audiences at the 2012 ESA meeting in Tucson and the 2013 SEA meetings in Tampa for very helpful comments. Special thanks to Philip Brookins for programming and help with running the experiment.

Goerg: Department of Economics, Florida State University, Tallahassee, FL 32311-2ISO. Max Planck Institute for Research in Collective Goods, Bonn D-53113, Germany. Phone (850) 544-7950, Fax (850) 644-4535, E-mail sgoerg@fsu.edu

Lightle: Department of Economics, Virginia Commonwealth University, Richmond, VA 23284-4000. Phone (804) 828-1717, Fax (804) 828-9103, E-mail johnlightle@gmail.com

Ryvkin: Department of Economics, Florida State University, Tallahassee, FL 32311-2180. Phone (850) 644-7209, Fax (850) 644-4535, E-mail dryvkin@fsu.edu

TABLE 1
Subjects' Characteristics and Observations by Treatment

                                     Number    Number
                 Average     %         of        of      Number of
Treatment          Age     Female   Subjects   Groups   Observations

1Stage            19.9       53        72         9          792
2StageCompDraw    19.9       60        72         9          792
2StageNoDraw      20.2       58        72         9          792
Overall           20         57       216        27        2,376

TABLE 2
Summary Statistics for Raffle Tickets Bought

                      Periods 1-11           Part 1 (Period 1)

Treatment        Mean    Median    SD     Mean    Median    SD

1Stage           21.95     10     26.91   23.99    12.5    25.70
2StageCompDraw   25.88     16     27.60   29.59     25     23.83
2StageNoDraw     32.91     25     30.20   35.36     30     26.44

                  Part 2 (Periods 2-11)

Treatment        Mean    Median    SD

1Stage           21.75     10     27.04
2StageCompDraw   25.50     15     27.94
2StageNoDraw     32.66     25     30.55

TABLE 3
Tobit Regression--Tickets Bought per Subject

Total Tickets Bought             (1)            (2)

2StageCompDraw                 -9.2846        -3.3653
                               (5.731)        (5.621)
1Stage                       -12.7211 **    -12.7163 **
                               (5.687)        (5.684)
Period                        -0.6060 **      -0.2758
                               (0.273)        (0.346)
2StageCompDraw*Period                        -0.9954 *
                                              (0.548)
Number of raffle wins

Age

Female

Holt and Laury

Donated within a year

Constant                     34.7797 ***    32.8126 ***
                               (4.001)        (3.967)
Observations                    2,376          2,376
Number of subjects               216            216
Probability > [chi square]      p <.01       p < .005

Total Tickets Bought             (3)            (4)

2StageCompDraw                 -3.0854        -3.1659
                               (5.198)        (5.002)
1Stage                       -12.3652 ***   -10.9904 **
                               (4.535)        (4.310)
Period                       -2.6001 ***    -2.4927 ***
                               (0.413)        (0.413)
2StageCompDraw*Period         -1.0105 *      -1.0181 *
                               (0.577)        (0.569)
Number of raffle wins        18.3409 ***    17.5149 ***
                               (1.853)        (1.836)
Age                                           -0.8278
                                              (1.040)
Female                                        3.4178
                                              (3.753)
Holt and Laury                               1.6184 **
                                              (0.743)
Donated within a year                       10.5861 **
                                              (4.171)
Constant                     32.9018 ***      30.8921
                               (3.596)       (22.745)
Observations                    2,376          2,376
Number of subjects               216            216
Probability > [chi square]     p<.0001       p < .0001

Notes: 2StageNoDraw is the reference group, 522 of 2,376 observations
were left-censored (i.e., equal to zero) and 312 were right-censored
(i.e., equal to 80). Subject-level clustered robust standard errors
are in parentheses.

*** p < .01, ** p < .05, * p < .1.

TABLE 4
Tobit Regression--Fraction of Possible Tickets Bought

                                              First Stage

Fraction of possible tickets bought        (1)           (2)

2StageCompDraw                         -0.0850 *     -0.0822 *
                                       (0.049)       (0.049)
Period                                 -0.0084 *     -0.0201 ***
                                       (0.005)       (0.006)
2StageCompDraw*period                  -0.0185 ***   -0.0185 ***
                                       (0.007)       (0.007)
Number of raffle wins                                 0.0857 ***
                                                     (0.020)
Holt and Laury                                        0.0242 ***
                                                     (0.008)
Female                                                0.1028 **
                                                     (0.044)
Age                                                  -0.0158
                                                     (0.014)
Donated within a year                                 0.0237
                                                     (0.048)
[X.sup.Others.sub.Stage 1]

Constant                                0.2059 ***    0.3228
                                       (0.036)       (0.297)
Observations                              1,584         1,584
Number of subjects                         144           144
Probability > [chi square]              p < .0001     p < .0001

                                             Second Stage

Fraction of possible tickets bought        (3)           (4)

2StageCompDraw                          0.0420        0.1014
                                       (0.069)       (0.067)
Period                                  0.0004       -0.0240 ***
                                       (0.006)       (0.007)
2StageCompDraw*period                  -0.0090       -0.0065
                                       (0.008)       (0.009)
Number of raffle wins                                 0.2264 ***
                                                     (0.032)
Holt and Laury                                        0.0314 **
                                                     (0.012)
Female                                                0.0042
                                                     (0.057)
Age                                                   0.0072
                                                     (0.013)
Donated within a year                                 0.0765
                                                     (0.063)
[X.sup.Others.sub.Stage 1]                            0.0012 ***
                                                     (0.000)
Constant                                0.2002 ***   -0.3337
                                       (0.048)       (0.314)
Observations                              1,532         1,532
Number of subjects                         144           144
Probability > [chi square]              p = .0635     p < .0001

Notes: 2StageNoDraw is the reference group. Subject-level clustered
robust standard errors are in parentheses.

* p < .1; ** p < .05; *** p < .01.