1 Introduction

The institutions providing the services that we identify with rule of law and the regulatory framework within which modern markets operate are state sponsored and financed by coercive taxes. In the sense of imposing regulations and raising compulsory contributions even the most minimal is a welfare state of sorts. Beyond the protective as a minimum welfare state all advanced legal orders also are productive states. They also (“productively”) provide minimal guarantees of income that are financed by compulsory taxes.

According to ideals of substantively equal treatment under rule of law standards redistributive policies must be non-discriminatory (see Kliemt 1993; Buchanan and Congleton 2006; Oprea 2019 in this volume of HOEC).

Non-discrimination in redistributive policies suggests demogrant schemes of equal guarantees of basic income for all citizens, as are nowadays propagated in particular by the political left (see Van Parijs 1992), were favored originally by classical liberals [most prominently by Friedman (1962) with his proposal of what he called a “negative income tax”]. While their popularity is still on the ascent on the left of the political spectrum, unconditional basic income guarantees became increasingly unpopular among many liberal adherents of free markets (including those who otherwise think of Milton Friedman as their intellectual hero).

This paper acknowledges that, to the extent that it could be financed within financial constraints, redistribution via demogrant schemes is attractive. But it also acknowledges fundamental concerns about non-discriminatory compulsory solidarity as “embodied” in demogrant schemes. Taking sides with either the political left or the political right in this controversy is not among the aims of the paper. Rather than presenting conventional normative arguments it approaches certain aspects of the controversy experimentally.

The results of the experiments presented below speak to three central objections against implementing non-discriminatory (equal) basic income as “compulsory solidarity”:

  1. 1.

    Non-discriminatory or unconditional (not means-tested) claims to basic income would induce those who are conventionally regarded as “non-deserving poor” to opt out of the labor market. Other than the adherents of unconditional basic income, the “resenters” do not regard it as a public good to help the poor regardless of the causes of poverty. Due to such widespread resentment, the claim that providing non-discriminatory compulsory solidarity amounts to a Pareto improvement is precarious at best. Moreover, the scheme might not survive at the polls in a democracy. (In view of this it is not accidental that mandatory basic “social security” which amounts to an earmarked tax is camouflaged as “social insurance” in countries like Germany).

  2. 2.

    Many adherents of free societies not only resent the non-discriminatory but also the coercive nature of compulsory solidarity implemented as unconditional basic income. For them solidarity has moral merit only to the extent that it is voluntary. Accepting that ought presupposes can the demand for voluntariness translates into demanding that solidarity should be voluntary to the maximum viable extent.

  3. 3.

    Moreover, many adherents of rule of law and free market societies believe that relying on the fundamental coercive power of the state in exercising solidarity tends to crowd out the basic proclivity of humans to help their fellows voluntarily and/or to self-help by means of voluntary associations. They would typically add that in free market societies in particular the super-rich—the Rockefellers, Warren Buffets and Bill and Melinda Gates types—have been and still are willing to voluntarily help the deserving poor. Since voluntary donors are legally entitled to discriminate, they can select the deserving poor in targeted ways and thus exercise solidarity more efficiently than the state sponsored compulsory solidarity schemes that under the rule of law constraints rein in discrimination by state agencies and agents.

In our comparative institutional analysis we will focus on ways and means of redistribution and their more or less voluntary and involuntary character. We will do so in terms of alternative rules and how they affect experimental laboratory interactions. More specifically, alternative ways of redistribution are modeled by combining redistribution as emergent from a non-discriminatory voluntary contribution mechanism, VCM, with an outside option open to a “super-rich” participant. The super-rich may discriminate in her redistributive voluntary donations. She can do so on the basis of information accessible at a cost to her.

We distinguish two cases of “institutional settings” as represented in the laboratory. In one case to which we refer as partial compulsory solidarity, PCS, the super-rich is not a participant of the VCM but can interact with the “non-rich” participants. Her options are either granting or withholding donations which she may or may not base on information on whether the recipients are deserving or not. In the other case, to which we refer as exemplifying general compulsory solidarity, GCS, the super-rich individual participates in the VCM that distributes indiscriminately equal shares of the common pool to each participant (independent of wealth and largely independent of contributions). In GCS as in PCS, besides contributing to the common pool, the super-rich can still exercise solidarity in targeted ways.

The two precedingly sketched settings should offer some insights concerning how being involuntarily included in a general solidarity scheme, GCS, as opposed to being an outsider to it, PCS, affects voluntary contributions of the super-rich and the non-rich (in and outside the VCM). Before we turn to our experimental exploration a few remarks on related literature may be helpful and, in any event, appropriate.

Certain collective or public good experiments share certain aspects of our PCS and GCS conditions. Since Isaac and Walker (1988), wealth asymmetry via heterogeneous endowments has been frequently explored experimentally (Kachelmeier and Shehata 1997; Chan et al. 1996; Cason and Khan 1999; Buckley and Croson 2006) what allows to test for equal endowment shares of voluntary contributions.Footnote 1 The crowding out of charitable donations is explored by Blanco et al. 2012(see also Bekkers and Wiepking 2011; Wiepking and Bekkers 2012 for surveys). So far there has been no focus on retrieving costly information beforehand. Like in GCS, where we allow the super-rich to participate in collective good provision and to donate. Some experiments let participants not only contribute but also reward (or sanction) others (Fehr and Gächter 2000; 2002; Andreoni et al. 2003; Masclet et al. 2003; Sefton et al. 2007; Sutter et al. 2010). In our setup, charitable donations can be used to self-servingly influence collective good contributions of the non-wealthy.

In experimental research, generosity is typically operationalized as dictatorial reward allocation, as studied early in social psychology (Mikula 1972; Shapiro 1975) and later via impunity and dictator games (List 2007). In both our conditions the single super-rich is an allocator confronting several recipients, what renders both conditions rather unusual (see Engel 2011, for a meta study). Due to the nearly exclusive focus on dyadic interaction (Forsythe et al. 1994; Eckel and Grossman 1996; Engelmann and Strobel 2004) dictatorial reward allocation experiments with more than one recipient are not very common (Bolton et al. 1998; Selten and Ockenfels 1998; Engelmann and Strobel 2004, 2007).

In both PCS and GCS different fairness norms can apply. In the absence of a contribution norm and without punishment, low contributions have to be expected (Reuben and Riedl 2013).Footnote 2 The less discriminatory GCS should trigger stronger solidarity concerns and render it, on average, the better institutional condition without inducing a contribution norm (Keser et al. 2017). One may also compare R-individuals with privileged groups (Reuben and Riedl 2009) with no freeriding incentive in their donating activities outside VCM.

Section 2 describes the experimental protocols in more detail. Section 3 reports the main results, separately for voluntarily contributing in PCS and GCS and similarly for voluntary donations by R-participants whose choice tasks depend more crucially on the institution. Finally, the payoffs in case of PCS and GCS are compared before concluding in Sect. 4.

2 Experimental Protocols

Partial Compulsory Solidarity, PCS, excludes the super-rich, R, from the solidarity group. R is neither obliged to participate in the VCM nor entitled to an equal share of its benefits. General Compulsory Solidarity, GCS, differs from PCS by including R in the group interaction under the VCM.

Under both, PCS and GCS conditions, R has the option to improve the situation of specific other actors through “donations” to others. To target donations R can, at a private cost to her, retrieve information about individual contributions of others under the VCM. (PCS and GCS are not alternative treatments in the technical sense. In particular, including R in GCS increases the number of contributors without adjusting individual freeriding incentives).

An experimental society includes six members, one R, three P, and two M. The M- are less well-off than R- but better-off than P-types. The class specific individual initial endowments for R, M, P are, respectively, \( \rho \)= 560, \( \mu \)= 60 and \( \pi \)= 20 ECU (Experimental Currency Unit with 100 ECU = €8). This yields a total endowment of 560 + 2 · 60 + 3 · 20 = 740; that is, nearly 75% of wealth is allocated to R. In the following we also refer to super-rich R as “wealthy” and to all M and P as “non-wealthy”. The instructions refer to three different types as P- M- and R- type.

As indicated already we refer to PCS and GCS not as treatments but as institutional conditions of an exploratory exercise in comparative institutional analysis of PCS and GCS (Plott 1997; Roth and Sotomayor 1992). In PCS, but not in GCS, R can voluntarily contribute to the collective good from which in PCS only the non-wealthy, and, in GCS, all benefit equally. The non-negative integer contributions \( c_{pi} \), \( c_{mj} \) and \( c_{r} \) for \( i = 1,\;2,\;3 \) and \( j = 1, \;2 \), may not exceed 20 ECU, corresponding to the wealth constraint \( \pi \) = 20 of the poor. M-types can contribute only up to one-third of their endowment. In GCS, R is also subject to this constraint on voluntary donations in the VCM and thus can donate only up to 1/28 of \( \rho \) = 560. Payoffs of the linear collective good games of five, PCS, respectively six, GCS, contributors assume the same MPCR of 0.4.

Even when R is not engaged in collective good provision, R – as all others – is informed about its rules and about the total contribution \( C = c_{m1} + c_{m2} + c_{p1} + c_{p2} + c_{p3} \) in PCS and \( C + c_{r} \) in GCS. Before possibly donating, R can purchase information concerning individual contributions, \( c_{p1} ,\;c_{p2} ,\;c_{p3} ,\; c_{m1} \) and \( c_{m2} \) at a cost for each information retrieval (mirroring that in the field information search is in general costly). After receiving the information she has paid for, R decides on how much of the remaining endowment (560 minus the sum of information costs) she will donate to \( M_{1} \), \( M_{2} \), as well as to \( P_{1} , \;P_{2} \) and \( P_{3} \).

2.1 Stages of the Decision Process

The Decision Process Involves the Following Stages: First, in PCS the non-wealthy (in GCS) determine their contribution level; yielding the total contributions \( C = c_{p1} + c_{p2} + c_{p3} + c_{m1} + c_{m2} \) and \( C + c_{r} \) in PCS and in GCS, respectively.

Second, the collective good level C, respectively \( C + c_{r} \), becomes commonly known and the non-wealthy learn about their payoff (“wealth”) before possible donations by R:

  • \( \pi - c_{pi} + 0.4C \) for P-types in PCS; \( \pi - c_{pi} + 0.4\left( {C + c_{r} } \right) \) for P-types in GCS,

  • \( \mu - c_{mj} + 0.4 C \) for M-types in PCS; \( \mu - c_{mj} + 0.4\left( {C + c_{r} } \right) \) for M-types in GCS,

  • \( \rho \) in PCS; \( \rho - c_{r} + 0.4\left( {C + c_{r} } \right) \) in GCS.

Third, R can buy information about individual contributions at a cost of e = 10ECU each. Depending on the (costly) information R finally decides on how much of \( 560 - n \cdot e \) (with \( n \in {\mathbb{N}} \), denoting the number of information retrievals) to donate to P- and M-types.

After factoring in R’s voluntary targeted donations \( d_{pi} \) and \( d_{mj} \) for i = 1, 2, 3 and j = 1, 2 overall earnings are:

  • \( \pi - c_{pi} + 0.4C + d_{pi} \) for P-types in PCS;

  • \( \pi - c_{pi} + 0.4\left( {C + c_{r} } \right) + d_{pi} \) for P-types in GCS,

  • \( \mu - c_{mj} + 0.4C + d_{mj} \) for M-types in PCS;

  • \( \mu - c_{mj} + 0.4\left( {C + c_{r} } \right) + d_{mj} \) for M-types in GCS;

  • \( \rho - \sum\nolimits_{i = 1}^{3} {d_{pi} } - \sum\nolimits_{j = 1}^{2} {d_{mj} - n \cdot e} \) for R in PCS;

  • \( \rho - \sum\nolimits_{i = 1}^{3} {d_{pi} } - \sum\nolimits_{j = 1}^{2} {d_{mj} - n \cdot e - c_{r} + 0.4\left( {C + c_{r} } \right) } \) for R in PCS;

In the experiment all participants play 24 periods in two phases of 12 periods each. For the first phase six participants are randomly chosen to interact with the same partners for twelve periods (partner design). After the first phase again six participants are randomly chosen to interact again with the same partners for 12 periods (again partner design). Across phases the assignment of specific participants to roles R, M, or P remains invariant. Both phases are always played according to either PCS or GCS rules.

Of course, such repeated interaction in identical groups allows for both reaction to feedback information about past outcomes and anticipation of future effects. For instance, non-wealthy participants may hope that R will reward contributions via donations in PCS and possibly also by contributing more in GCS. When retrieving information repeatedly, wealthy R can identify other group members as \( P_{i} \) or \( M_{j} \). So, for instance, R could verify whether a specific \( P_{i} \) or \( M_{j} \) increases his voluntary contribution after receiving a substantial donation via purchasing the relevant information.

After having described how the two institutional alternatives, PCS and GCS, are formally defined and experimentally implemented we can compare the two institutional settings PCS and GCS, in particular, with respect to the role of R. In PCS the super-rich participant R can influence the payoffs only by redistributive donations allocated outside the VCM. In GCS both voluntary contributions and redistributive donations are viable for R. A donation costs R one additional unit and provides one unit for one of the five others (should R choose to donate to him). In PCS the marginal rate by which R’s sacrifice of substantive self-interest transforms into benefit for (any) “other” is 1. R’s contribution of one unit in GCS generates 0.4 units to each of the six members of the solidarity group or in all 6 × 0.4 = 2.4. That is 0.4 units to R and 5 × 0.4 = 2 for the five others at a “net” cost of 1–0.4 = 0.6 to self. The transformation rate of the compulsory donation amounts to 2/0.6 = 10/3. This transformation rate of contributions into collective benefits is more than three times larger than the one for the individualized voluntary donations. To put this slightly differently, by fully contributing, \( c_{r} \) = 20, R could provide 5 × (0.4 × 20) = 40 for the five others at a cost of 12 to self while R would have had to sacrifice 5 × 8 = 40 units to generate an identical positive effect on each of the other members of the community by her voluntary targeted donations outside the VCM.

Compulsory solidarity as construed here is much more effective than voluntary solidarity. Yet it is also non-discriminatory. Voluntary donations can discriminate between the members of the lower endowment classes P and M. Thereby R can redistribute in a “retributive” manner and thereby express retributive emotions that are generally essential for the workings of moral and legal institutions (Mackie 1982). She can engage in acts of targeted altruistic reward in both the PCS and the GCS setting (as opposed to acts of altruistic punishment that could only be construed as withholding gifts). In GCS as opposed to PCS super-rich R is obliged to participate in VCM. She willy-nilly participates in its benefits whether she voluntarily contributes in VCM or not. If she is concerned about fairly contributing to VCM she herself has to contribute to non-discriminatory solidarity. Her voluntary contributions to VCM may, however, implicitly reward non-deserving others. Targeted donations are more suitable to incentivize contributing behavior of the five others. Relying on a comparatively less effective process of transforming costs to self- including the necessary costly information retrieval- into benefits for others indicates a comparatively strong preference of R for retributive, targeted over non-discriminatory altruism. This part of the experimental exploration of alternative institutional rules clearly speaks to the standard conjecture that the decline of contributions in standard VCM’s is in the first place due to other-regarding resentment against free-riding and not so much a consequence of self-regarding motives to free-ride (normally rather naively discredited as greed).

Final payments are based on one randomly selected period of one randomly selected phase. We ran 13 sessions at CESARE laboratory of University Luiss with 288 students, recruited using Orsee (Greiner 2015) (11 sessions with 24 subjects and 2 sessions with 12 subjects each); 192 subjects participated in PCS and 96 in GCS. Except for the possibility to voluntarily donate, GCS is a rather usual collective good design. Compared to this, PCS additionally deviates by excluding R-members from collective good provision and deviates more fundamentally from usual collective good designs. It is because of this that we collected more PCS data. Subjects were provided with a hard copy of the instructions, which were read aloud by the experimental proctor (for an English translation of the instructions see Sect. 5.1). The experiment was fully computerized using z-Tree (Fischbacher 2007). Overall, subjects spent about 80 min in the laboratory, and earned on average €14.30. The altogether 24 participants of each session were restricted to two half groups and randomly re-matched after the first phase within the same half group: participants were not informed about this restricted re-matching.

3 Results

After analyzing contributions both in PCS and GCS, we discuss the information requests and donations by R, by distinguishing and comparing “blind” (in the sense of no information retrievals) and “non-blind” donors among R-participants, before addressing welfare implications.

3.1 Contribution Behavior

The average contribution dynamics in Fig. 1 for P, M and, in GCS, also R are consistent with the usual decline observed in repeated collective good experiments. When R is included, average contributions of P-participants (4.25 in PCS vs. 6.19 in GCS) and M-participants (4.95 in PCS vs. 6.81 in GCS) are significantly higher (p values are computed using a bootstrapped t test on group averages (GA) as the unit of observation with 1000 bootstrapped samplesFootnote 3; GA p value < 0.001).

Fig. 1
figure 1

Contributions by condition and role

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Result 1 Compared to PCS the average contributions of P- and M- participants in GCS are significantly higher.

There are two main explanationsFootnote 4:

  1. 1.

    the increased efficiency of individual contributions due to including R in GCS discourages freeriding,

  2. 2.

    the procedural fairness of GCS crowds in and strengthens solidarity concerns.

In PCS the average M-contribution significantly exceeds the P-contribution (4.95 vs. 4.25, respectively with GA p value < 0.001). In GCS this difference is also significant and positive (6.81 for M vs. 6.19 for P, GA p value = 0.016). Thus, there is solidarity across endowment classes even in PCS. R-participants contribute significantly more than the others (7.96, with GA p values ≤ 0.001 compared to both, P and M).

Result 2 M-members contribute significantly more than P-members in both PCS and GCS. Furthermore, R-participants, in GCS, contribute significantly more than both P- and M-members.

To answer whether contributions of P and M are boosted by donations and (in GCS by) contributions of R, we distinguish no and positive donations and plot individual contributions after (not) having received a donation in the previous period. Figure 2 clearly reveals that P beneficiaries contribute more, both in PCS and GCS (see top section of Table 1). The result is significant for P in both conditions (p values < 0.001 both in PCS and GCS).Footnote 5 Positive donations seem to have a positive (and significant) effect on M contributions as well (see M section of Table 1). Furthermore, contributions of P and M beneficiaries are significantly different; M-types are more cooperative which does not hold when comparing the contributions of P and M without prior donation (see bottom section of Table 1).

Fig. 2
figure 2

Contribution by donation received in previous period

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Table 1 Pairwise test on contribution by donation received in previous period
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Although non-R-participants are unaware whether being observed by R over time, P- and M-participants may infer this monitoring from changing donations. Figure 3 displays the dynamics of P- and M-contributions across time in case of uninformed vs. (more or less) informed R-participants, separately for PCS (left chart) and GCS (right chart)Footnote 6: P and M contributions are lower when R donates blindly than after investing in information in \( t - 1 \) (see bottom section of Table 1). The non-wealthy seem aware of R conditioning donations based on monitoring their individual contributions.

Fig. 3
figure 3

P and M contributions by blind and non-blind R

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Result 3 Voluntary donations of informed R-participants boost voluntary cooperation of the non-wealthy

Result 3 suggests that some non-wealthy view R-donations as a bonus system according to which R monitors individual contributions and conditions donations on individual contributions. Table 2 validates that P and M contributions are positively correlated to past contribution sum and the received donation.Footnote 7 Note also that P contributions react more strongly to donations by (partially) informed donors (see the negative and significant interaction term in Table 2Footnote 8).

Table 2 Random-effects tobit model regression on P and M contributions
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3.2 Behavior of Wealthy Participants

R-type participants donate on average less than 10 ECU in total (Table 3). While redistributive R- donations start off significantly higher in PCS, they significantly drop below those in GCS in the second phase.Footnote 9

Table 3 R-individuals’ average total donations to P only, and M only
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The decrease from phase 1 to phase 2 of R-donations applies across conditions while being weakly significant in GCS (see the p values of bootstrapped t tests in Table 3). Even though R in GCS can more efficiently display solidarity via contributing rather than donating, this does not crowd out voluntary donations. Actually, R-participants in GCS donate as much as in PCS (p value = 0.837): R-participants in GCS (PCS) donate 57.6 (39.6) percent of the time when measured as the ratio of positive donation sums of each R in each of the 24 periods in which she could donate. This implies that average donations in GCS are smaller (Table 3). This seems to suggest that at least partly R-donations have been substituted by more efficient voluntary R-contributions in GCS. But do R-participants either donate or contribute? The answer is no: R-participants, who donate, contribute on average 10.27 whereas those not donating contribute only 4.83 (t testFootnote 10 p value < 0.001).

Table 4 distinguishes three categories of R-participants, based on the average donation in each phase: those donating virtually nothing, and those who, on average, donate only small, respectively more substantial amounts. The share of R-participants not donating is 46.67% in PCS and larger than 28.13% in GCS, whose share of “Substantial Donors” is larger than in PCS (28.13% vs. 18.13%). This suggests crowding in: R-participants more frequently donate in GCS than in PCS.

Table 4 Frequency of donor types
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Figure 4 distinguishes average donations of R-participants with no and with some information retrieval (Fig. 13 in Sect. 5.2 excludes non-donors). The positive difference in information retrieval between conditions PCS and GCS in phase 1 is significant, (p value = 0.03) with GCS being higher, see Fig. 5.

Fig. 4
figure 4

Donation from blind and non-blind R

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Fig. 5
figure 5

Information purchased on P and on M

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In both conditions, average donations of R-participants with information (referred to as “info”) exceed across time those of R-types without information (referred to as “blind”) where the differences are strikingly large, see Fig. 4. Apparently (not) investing in information goes hand in hand with (not) donating: some R-participants keep their eyes closed while generous ones apparently want to learn how needy the non-wealthy are and how they could be encouraged similar to principals in principal-agent models.

Result 4 Information retrieval goes along with donating: the altogether 46 R-participants investing in information donate significantly more.

Figures 6 and 7 display how frequency shares of the different R types evolve over time. Figure 6 categorizes R-participants into four types: (1) those who donate and purchase costly information, (2) those who donate but do so blindly, (3) those who purchase information without donating, and (4) those who neither purchase information nor donate. In PCS the percentage of R-participants with possibly minimal, positive donations starts off above 60%. After the first half of the first phase the share of R-participants who donate drops by around 50%.Footnote 11

Fig. 6
figure 6

Frequency of donating (> 1 ECU) by R-participants (in PCS)

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Fig. 7
figure 7

Frequency of R-donation (> 1 ECU) and contribution to the public good in GCS

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For GCS, Fig. 7 distinguishes “blind” and “non-blind” donors, and further characterizes R-participants into four types: (1) those who both donate and contribute to the PG, (2) those who only donate, (3) those who only contribute to the PG, (4) those who neither donate nor contribute. Most non-blind R-participants engage in both, donating and contributing: 90% of the R-participants who invest in information donate as well as contribute. In comparison, blind R-participants often only contribute although blind donations are higher in GCS than in PCS.

Result 5 In GCS most R-participants (90%) reveal social concerns via costly information retrieval, voluntarily contributing and donating

Table 5 on R-donations in GCS (Model 1) rejects “crowding out” of charitable donations and confirms a positive effect of R’s own contributions, past own donations and partly of contribution sums (Models 4 for GCS, Model 2 for pooled data) on R-donations. Information retrieval enhances voluntary cooperation of the non-wealthy in GCS but not in PCS. Only few R-participants, who are substantially donating in PCS, monitor and reward higher contributions (Table 4) to inspire voluntary cooperation of the non-wealthy.

Table 5 Random-effects tobit model regression on R-donations
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Table 6, using a random-effects tobit model, reveals how R-contributions in GCS are significantly related to own previous contributions (Model 1), past donations, as well as to past contribution sums (Model 2). This suggests that R-participants, who care for the non-wealthy, do not merely substitute donations by more efficient contributions but rather employ both to improve the wellbeing of their fellow group members and their society, especially when also retrieving information about individual contributions.Footnote 12 Table 6 reveals lower R-contributions in the second phase but not across periods within a phase.

Table 6 Random-effects tobit model regression on R-contributions
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3.3 Payoff Analysis

Figure 8 compares the average payoffs for the three groups across time between GCS and PCS. Although average P-payoff (top-left graph in Fig. 8) is smoothly decreasing in both conditions, it is significantly higher in GCS than PCS. This holds also for M-participants (top-right graph in Fig. 8). Most strikingly, R-members are, on average, also better off in GCS than in PCS. On average, no endowment type, P, M and R, is worse off under GCS than under PCS institutional rules.

Fig. 8
figure 8

Payoff by role and condition. We perform a bootstrapped t test (1000 replications) on the individual payoff by role checking for significant difference between condition PCS and GCS: for all pairwise comparisons (GCS vs. PCS) p values are < 0.001

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This may be partly due to the higher efficiency of collective good provision when R is included. Let us therefore neglect R-contributions when comparing GCS with PCS and simulate efficiency without R-contributions. Figure 9 displays the average (simulated) payoffs across periods for P- (top-left graph) and M-members (top-right graph) when excluding R-contributions.Footnote 13

Fig. 9
figure 9

Payoff by role and condition, when excluding R-member from public provision in GCS. We perform a bootstrapped t test (1000 replications) on the individual payoff by role checking for significant difference between conditions PCS and GCS: for P and M pairwise comparisons (GCS vs. PCS) p values are < 0.001. The payoff for R does not significantly differ across treatments (p value = 0.602)

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Even when R-contributions are excluded average P- and M- payoffs are significantly higher in (simulated) GCS than in PCS. The bottom graph in Fig. 9 reports simulated R-payoffs for GCS when neglecting their contribution costs and collective good benefits but including donations and information costs: the average payoff is significantly lower in GCS, partly due to the significantly higher information costsFootnote 14 in GCS than in PCS (see Fig. 5).

Result 6 According to our testbed analysis in GCS all types are better off than in PCS.

This result is supported by analyzing how dispersed the payoffs are. The Gini indexFootnote 15 compares the variation from the initial dispersion of endowments (identical in PCS and GCS) with the later inequality in periodic payoffs.

Figure 10 shows that the inequality level is significantly lower across all periods in GCS: the inequality level seems to increase across time, but with considerable differences across groups.

Fig. 10
figure 10

Gini inequality index in PCS and GCS. We perform a bootstrapped t test (1000 replications) on individual observations checking for significant difference between condition PCS and GCS: p values < 0.001

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Figure 11 distinguishes the index dynamics in groups with differently generous R (Non-Donor, Small Donor, or Substantial Donor, see Table 4). Clearly, a generous donor decreases the inequality index. In particular, the inequality index is rising for groups whose R is rather opportunistic while it is stable across time when R is a substantial donor.

Fig. 11
figure 11

Gini index by donor types. We perform a bootstrapped t test (1000 replications) on individual observations (within treatment) checking for significant difference across donor types: p values < 0.001

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4 Conclusions

Our testbed exploration of potential institutional alternatives does not exclude confounding effects, but only tries to limit and control them. From the perspective of experimental methodology, PCS and GCS do not qualify as treatments since they differ in more than one aspect: The effects of the institutional variations in our setup are confounded by changing both the number of voluntary contributors as well as the efficiency of contributing.

Our results nevertheless suggest to rely on GCS rather than PCS if these are the two options of a constitutional/institutional choice: Compared to PCS the institutional setting GCS enhances voluntary cooperation letting all endowment classes gain on average. The non-wealthy contribute more in GCS and only very few R-participants react to compulsory participation in non-discriminatory solidarity by neither contributing nor donating. R-members mostly retrieve costly information about individual contributions of the others and try to enhance their contributions by conditioning own contributions and donations on behavior of others. We also find that R-participants in GCS do not merely substitute donating by more efficient contributions but maintain retrieving information about contributions by the non-wealthy and use both mechanisms to improve the wellbeing of others in the group thereby reducing there potential resentment against supporting the “non-deserving”.

We did not argue for or against compulsory solidarity nor did we discuss whether and when general compulsory solidarity is ethically, legally, or otherwise preferable. We have shown though that in our experimental setup GCS seems, economically and according to evaluative standards of customary morals, superior. We believe that the experiment raises interesting questions concerning the relation between non-discriminatory redistribution, as in unconditional basic income scheme favored under principles of equal legal treatment, and discriminatory redistribution by targeted donations through private agents.