1 Introduction

The voluntary contributions mechanism (VCM) has become a widely used experimental framework for studying cooperation. In typical experiments participants are assigned to n-player groups. Each group member is endowed with tokens and chooses how many to place in a private account, from which that person earns α money units per token, and how many to place in a group account, from which every person in the group earns β money units per token. Parameters are chosen so that private and collective interests are in conflict: the group maximizes earnings by contributing all tokens to the group account (nβ>α), but each group member has a private incentive to place her tokens in her private account (α>β). This task is then repeated over a number of periods.Footnote ¹

One factor that might be expected to affect contributions to the group account is the size of the group. Previous VCM studies have found that, if there is an effect of group size, it is in the direction of higher contributions in larger groups (see Sect. 2 for a review). An explanation for this finding can be based on the private and social costs and benefits of a contribution. For each token contributed to the group account a contributor incurs a cost of α and a benefit of β money units regardless of group size, whereas the social benefits of the contribution, n times β money units, increase with group size. If individuals care about more than just own earnings, and instead internalize some of the social benefits of contributing, they may be willing to contribute more in larger groups where the social benefits are larger.

Interestingly, findings from other experimental settings where there is a tension between private and collective interests, such as n-person prisoner’s dilemma or oligopoly experiments, suggest that increasing the size of the group may have a negative impact on cooperation, as subjects find it more difficult to attain collectively optimal outcomes in larger groups. For example, Marwell and Schmitt (Reference Marwell and Schmitt1972) and Bonacich et al. (Reference Bonacich, Shure, Kahan and Meeker1976) study n-person iterated prisoner’s dilemmas where they vary n while keeping constant the private and social costs and benefits of cooperating.Footnote ² They find that cooperation rates are lower in larger groups. Numerous subsequent studies report further evidence that cooperation is inversely related to group size, usually using groups of size between two and seven, although, as Kollock (Reference Kollock1998) notes in a review of this literature, in some studies the decrease in cooperation as group size increases tapers off quickly.Footnote ³

Similarly, a large literature on experimental oligopolies finds that cooperation (i.e. collusion) is more difficult in larger groups (i.e. when there are more competitors). For example Fouraker and Siegel (Reference Fouraker and Siegel1963) examine the textbook Bertrand model under duopoly and triopoly treatments. Although equilibrium predictions are the same for both treatments, prices are higher among duopolies. Similarly, Dolbear et al. (Reference Dolbear, Lave, Bowman, Lieberman, Prescott, Rueter and Sherman1968) study a model of price competition with differentiated goods where equilibrium predictions are independent of the number of competitors, and they too find prices to be higher in smaller markets. Relatedly, Isaac and Reynolds (Reference Isaac and Reynolds2002) design two- and four-firm posted-offer markets with capacity constraints so that benchmark predictions (competitive, collusive, Cournot-Nash) are comparable, and find that prices are higher in two-firm markets. Other studies examine models where equilibrium predicts a numbers effect. Even here, there is evidence of a negative effect of group size on collusion beyond that predicted. Huck et al. (Reference Huck, Normann and Oechssler2004), for instance, study the textbook homogeneous goods Cournot model with two, three, four or five firms per market, where the equilibrium prediction is that quantities are higher in larger markets. In the experimental two-firm markets there is evidence of collusion and firms produce less than the Nash level. As the number of firms increases markets become more competitive and firms tend to produce even more than the Nash level. Orzen (Reference Orzen2008) studies a price-competition setting where standard theory predicts that expected prices increase with the number of firms in the market. In the experiment he finds the opposite effect: outcomes are more collusive in two- than four-firm markets.

There are several possible explanations for these negative effects of group size on cooperation. One is based on the idea that a number of factors conducive to cooperation, such as social pressure and social incentives, may be more effective in small than larger groups (Olson Reference Olson1965). Other authors (Marwell and Schmitt Reference Marwell and Schmitt1972) have suggested the ‘bad apple’ hypothesis. Many individuals are willing to cooperate, but only as long as others do so as well. Thus, if a group contains one non-cooperator (a ‘bad apple’) cooperation will unravel. In a population containing a fixed proportion of non-cooperators larger groups are more likely to unravel. Another possibility is that, as discussed by Kim and Walker (Reference Kim and Walker1984), individuals in small groups have a greater perception that their free-riding may have an impact on others’ willingness to cooperate in the future.

These considerations suggest that the observed group size effects in VCM experiments reflect a combination of factors that may operate in opposite directions. A positive effect of group size stems from the increased social benefits from a contribution, while a negative effect stems from the difficulty of sustaining cooperative outcomes in larger groups. This led us to conjecture that the moderately positive effects observed in previous VCM experiments may reflect the focus of the literature on groups of four or more players, which in the standard VCM setup is already sufficiently large to make it difficult to sustain cooperation, and where the positive effects may dominate. We conjectured that the negative effect of group size may be more evident in smaller groups, while the positive effect is more evident in larger groups.

To test these conjectures we compare cooperation rates in VCM experiments varying the number of players matched into a group. As in the previous VCM literature on group size effects, we also vary the marginal per capita return from contributions to the group account (MPCR=β/α): either low (0.3) or high (0.75). For both low and high MPCR we compare four- and eight-person groups, as in previous studies. For the high MPCR setting we extend the analysis to smaller groups of two and three players.Footnote ⁴

As in previous studies, in our low-MPCR treatments we observe a significantly positive effect of group size on contributions. This positive effect is already evident in the first period of the experiment where contributions are 12 % higher in eight- than four-person groups. In both treatments contributions steadily decline across periods, but we do not observe a faster unraveling of cooperation in larger groups. The picture is different in the high-MPCR treatments. Here initial contributions are around 75 % of endowments in all treatments. However, treatments differ in how cooperation unravels across periods, with contributions declining faster in larger groups. The overall effect of group size on cooperation is negative: average contributions are highest in two-person groups and lowest in eight-person groups, with contributions in three- and four-person groups taking intermediate values.

Thus, in our VCM experiments we observe both a positive and negative effect of group size on cooperation. However, contrary to our initial conjecture, whether the positive or negative effects dominate does not seem to depend on the size of the group: in our high-MPCR treatments we do observe a negative effect of group size on cooperation also for groups of four or more players. Rather, our findings suggest that which effect dominates depends on how conducive the VCM environment is to cooperation. In settings that are particularly conducive to cooperation (like our high-MPCR treatments) contributions are already high, and there is limited scope for improving on such high levels of cooperation. Here the negative effects of group size are more evident as the initially high cooperation levels deteriorate faster in larger groups. In contrast, in settings that are unfavorable to cooperation (like our low-MPCR treatments) there is more scope for group size to have a positive effect on initial cooperation, and less scope for it to affect the decline of contributions over time.

The remainder of the paper is organized as follows. In the next section we review previous evidence on group-size effects in VCM experiments. In Sect. 3 we describe our experimental design and procedures. In Sect. 4 we present the results, and Sect. 5 concludes.

2 Group size effects in previous VCM experiments

We are aware of six previous studies that have systematically examined the effects of group size in VCM games holding other game parameters constant. These studies are listed in Table 1, along with the MPCR and group sizes used in each treatment. We also report the average contributions as a percentage of endowments and levels of statistical significance.Footnote ⁵

Table 1 Group size effects in previous VCM experiments

Study	MPCR	Group size (average contribution as % of endowment)	Statistical significance
Isaac et al. (Reference Isaac, Walker and Thomas1984) (inexperienced subjects)	0.3	4 (27 %); 10 (33 %)	n.a.
	0.75	4 (65 %); 10 (65 %)	n.a.
Isaac et al. (Reference Isaac, Walker and Thomas1984) (experienced subjects)	0.3	4 (12 %); 10 (34 %)	n.a.
	0.75	4 (50 %); 10 (54 %)	n.a.
Isaac and Walker (Reference Isaac and Walker1988)	0.3	4 (13 %); 10 (29 %)	∗∗
	0.75	4 (50 %); 10 (46 %)	n.s.
Isaac et al. (Reference Isaac, Walker and Williams1994)	0.3	4 (18 %); 10 (26 %);	4 vs. 10 n.s.; 4 vs. 40∗∗; 4 vs. 100∗∗ 10 vs. 40∗∗; 10 vs. 100∗ 40 vs.100 n.s.
		40 (44 %); 100 (40 %)
	0.75	4 (43 %); 10 (44 %);	any comparison: n.s.
		40 (39 %); 100 (38 %)
Goeree et al. (Reference Goeree, Holt and Laury2002) (UVA subject pool)	0.8	2 (48 %); 4 (39 %)	n.s.
Goeree et al. (Reference Goeree, Holt and Laury2002) (USC subject pool)	0.8	2 (50 %); 4 (45 %)	n.s.
Carpenter (Reference Carpenter2007)	0.375	5 (37 %); 10 (54 %)	n.s.
	0.75	5 (50 %); 10 (70 %)	n.s.
Weimann et al. (Reference Weimann, Brosig-Koch, Hennig-Schmidt, Keser and Stahr2012)	0.02	60 (11 %); 100 (13 %)	∗∗
	0.04	60 (20 %); 100 (23 %)	∗

Levels of statistical significance are based on two-sided Fisher’s randomization tests (see footnote Footnote ⁵ for details). ^∗∗ = significant at the 5 % level; ^∗ = significant at the 10 % level; n.s. = not significant; n.a. = insufficient number of independent observations for a meaningful test

With the exception of Goeree et al. (Reference Goeree, Holt and Laury2002) who use one-shot games, all studies use repeated VCM games. In most studies group composition does not change across periods (partners matching protocol). In Carpenter (Reference Carpenter2007) groups are randomly reformed at the beginning of each new period (strangers matching protocol). Group size effects are studied by varying the number of players matched in a group. In all studies group sizes are varied in a between-subject design, except in Goeree et al. (Reference Goeree, Holt and Laury2002) who use a within-subject design. Most studies have investigated group size effects using groups of 4 or more players. An exception is Goeree et al. (Reference Goeree, Holt and Laury2002) who compare groups with 2 and 4 players.Footnote ⁶

Most studies report a positive effect of group size on cooperation. The effect seems stronger in settings where the MPCR is low. The reported effect is not always statistically significant, although in some cases this may reflect our conservative testing procedure. In the studies where subjects interact repeatedly we treat groups in which subjects interact as a single observation, resulting in a small effective sample size. For example, the averages reported for Carpenter (Reference Carpenter2007) are based on 2100 choices (210 subjects × 10 periods), but only 13 independent groups. It is therefore not surprising that the large observed effect is insignificant.Footnote ⁷ In some cases, increasing group size is found to reduce average contributions (e.g., in Goeree et al. Reference Goeree, Holt and Laury2002, or when comparing 40- and 100-person groups in Isaac et al. Reference Isaac, Walker and Williams1994), although these negative effects are always statistically insignificant.

Overall, the picture that emerges from previous studies is that group size has a moderate, positive effect on cooperation in VCM games. This conclusion is reinforced by the meta-analysis results reported by Zelmer (Reference Zelmer2003). She uses data from 27 VCM experiments conducted using different parameterizations and procedures, and finds a positive and significant (at the 10 % level) effect of group size on contributions. In the next section we describe a new experiment comparing four- and eight-person VCMs across low and high MPCR conditions, as in several previous studies, and extend the design to study group size effects in smaller two- and three-person groups.

3 Experimental design and procedures

The experiment was conducted at the University of Nottingham using the software z-Tree (Fischbacher Reference Fischbacher2007) and 364 student subjects from a wide range of disciplines, recruited through the online recruiting system ORSEE (Greiner Reference Greiner, Kremer and Macho2004). Multiple sessions were conducted and no participant took part in more than one session.

At the beginning of each session participants were randomly matched into groups that remained the same for the whole experiment. Participants did not know the identities of the other subjects in the room with whom they were grouped. They were given instructions for the experiment (reproduced in the Electronic Supplementary Material) and these were read aloud by the experimenter. Any questions were answered by the experimenter in private, and no communication between participants was allowed. No information passed across groups during the entire session.

All groups played a ten-period VCM game. In each period, players received an endowment of 20 tokens and had to choose how many to allocate to a public account and how many to keep in a private account. A player earned α points for each token she kept in her private account, and β points from each token allocated to the public account (regardless of which group member had contributed it). At the end of the period players were informed of the decisions and earnings of each group member.

Across sessions we varied the MPCR from contributions to the group account and the number of players matched into a group. In our LOW_4 and LOW_8 treatments we used a low MPCR (β/α=3/10) and subjects played the VCM game in four- and eight-person groups, respectively. In four other treatments we used a high MPCR (β/α=3/4) and subject played the VCM game either in two-person groups (HIGH_2 treatment), three-person groups (HIGH_3), four-person groups (HIGH_4), or eight-person groups (HIGH_8). Table 2 summarizes the experiment design and provides details on the number of sessions, subjects and independent observations for each treatment.

Table 2 Experiment design

Treatment	MPCR	Group size	Number of sessions	Number of subjects	Number of independent observations
LOW_4	0.3	4	3	36	9
LOW_8	0.3	8	5	80	10
HIGH_2	0.75	2	3	36	18
HIGH_3	0.75	3	5	60	20
HIGH_4	0.75	4	6	72	18
HIGH_8	0.75	8	5	80	10

At the end of a session the accumulated point earnings from all periods were converted into cash at a rate of £0.003 (£0.0075) per point in the low (high) MPCR treatments.Footnote ⁸ Participants’ earnings ranged from £4.60 to £30.25, averaging £12.09, for sessions lasting between 30 and 60 minutes.