1 Introduction

A vast number of laboratory and field studies have shown that many people contribute voluntarily to the provision of public goods, even when it is not in their monetary interest (e.g., Chaudhuri, Reference Chaudhuri2011; Fehr & Fischbacher, Reference Fehr and Fischbacher2003; Zelmer, Reference Zelmer2003). The declining pattern of contributions over time in these social dilemmas is consistent with two behavioral explanations. First, conditional cooperation (or reciprocity) has been identified as being important for voluntary contributions to public goods; i.e., many decision makers contribute to public goods when others also contribute or are expected to do so (Fischbacher et al., Reference Fischbacher, Gächter and Fehr2001; Fischbacher & Gächter, Reference Fischbacher and Gachter2010; Thöni & Volk, Reference Thöni and Volk2018). Second, confusion (or decision errors) have been invoked as an alternative explanation for the prevalence of voluntary contributions; i.e., many decision makers, particularly in laboratory experiments, are supposed to contribute to a public good not out of a preference motive or out of a reciprocity norm, but instead because they misunderstand the incentives of the game and therefore are unaware how to correctly pursue their self-interest (e.g., Andreoni, Reference Andreoni1995; Burton-Chellew & West, Reference Burton-Chellew and West2013; Ferraro & Vossler, Reference Ferraro and Vossler2010; Houser & Kurzban, Reference Houser and Kurzban2002). Confusion might be especially relevant in one-shot interactions or at the start of repeated interactions, and the observed decay in contribution levels, if it is due to learning, is consistent with this explanation. For example, Burton-Chellew et al. (Reference Burton-Chellew, El Mouden and West2016), using the strategy method elicitation developed by Fischbacher et al. (Reference Fischbacher, Gächter and Fehr2001), report that decision makers exhibit the same conditional contribution pattern, irrespectively of whether they interact with humans or computers, which seems to corroborate the second explanation based on confusion.

In this paper, we report on a novel way of testing whether confusion about optimal strategies, i.e., optimally implementing one’s preference, could be an important driver of voluntary contributions in a laboratory public goods game. Given the importance of public goods games in analyzing social dilemmas and developing policy-relevant designs and incentives for problems outside the laboratory (see, for instance, Schmidt & Ockenfels, Reference Schmidt and Ockenfels2021, for an application in climate policy), it seems relevant to assess the internal validity of the main paradigm used in experimental research. To this end, we experimentally vary in a linear public goods game whether decision makers receive information about the individually optimal strategy and the socially optimal strategy or not, and we analyze how this information affects elicited contribution preferences and cooperation in a public goods game. We believe that an assessment of whether experimental participants fully understand their strategic options and the relevant incentives provides us with the most direct test of the confusion hypothesis.

2 Experimental design and procedures

Our experimental design builds on the standard voluntary contribution mechanism with the following linear payoff function:

(1) ${\pi }_i=20-g_i+0.5\sum _{j=1}^3{g}_i,$

where $g_i$ denotes the contribution of participant i to the public good. Each group consists of n = 3 randomly assigned participants, and each participant receives an endowment of 20 points. The marginal per capita return (MPCR) from investing in the public good is 0.5 and the social return is 1.5. The parameters are all known by participants. Assuming that participants are rational and selfish payoff maximizers, these parameters guarantee that it is individually optimal to contribute zero. From a social or efficiency perspective, they guarantee that it is collectively optimal to contribute the entire endowment. Hence, the setup and the parameters imply a social dilemma.

Participants were randomly assigned to one of four treatments in a 2 (information: standard vs. optimal strategies) × 2 (control questions: incentivized vs. not incentivized) between-subjects design (see Table 1). In the standard information treatment, participants received the standard instructions that explained the public goods problem (see Online Appendix A; online link is in the acknowledgments). In the optimal strategies information treatment, participants received the same instructions, but with an additional paragraph that was explicit about which strategies maximize their individual income and their group’s income. Specifically, participants received the information that their individual income is maximized by contributing zero to the public good, regardless of the behavior of the other group members, and why this is the case (see Online Appendix A); i.e., we explained strategic dominance. Participants additionally received the information that their group’s income is maximized by contributing one’s entire endowment to the public good if everybody does so, and why this is the case (see Online Appendix A). Participants were also explicitly informed that in case they contribute more to the public good than their group members, the other group members benefit more from their contributions and end up earning more, i.e., we explained the sucker’s payoff.

Table 1

Summary statistics for the four treatments

Treatment	Part 1: control questions	Part 2: strategy method		Part 3: repeated public goods game		N
	Av. corr. contr. questions	Av. uncond. contrib.	Av. slopes	Av. contrib.	Av. beliefs
Control questions incentivized and optimal strategies information	0.80 (0.30)	7.79 (6.15)	0.60 (0.58)	7.28 (5.50)	7.73 (4.85)	24
Control questions not incentivized and optimal strategies information	0.82 (0.21)	8.29 (6.89)	0.49 (0.46)	7.70 (7.01)	9.00 (6.47)	21
Control questions incentivized and standard information	0.73 (0.31)	10.08 (7.04)	0.47 (0.54)	10.08 (4.91)	11.08 (3.27)	24
Control questions not incentivized and standard information	0.76 (0.28)	8.38 (6.32)	0.54 (0.51)	8.19 (6.03)	10.28 (5.16)	24
Treatments with optimal strategies information (pooled)	0.81 (0.26)	8.02 (6.44)	0.55 (0.52)	7.47 (6.18)	8.32 (5.63)	45
Treatments with standard information (pooled)	0.75 (0.30)	9.23 (6.68)	0.50 (0.52)	9.14 (5.53)	10.68 (4.28)	48

Av. corr. contr. questions = average correct control questions; Av. uncond. contrib. = average unconditional contributions; Av. contr. = average contribution; N = number of participants. Standard deviations in parentheses

Slopes are calculated as the slope coefficient of an individual regression with own contribution as the dependent variable and average others’ contributions as the independent variable

At the beginning, participants learned that the experiment consists of four parts.Footnote ¹

Part 1 Participants were asked to answer 16 standard control questions in four separate blocks (see Online Appendix B). We often use these or similar questions in related public goods experiments to ensure a basic understanding. In the control questions incentivized treatment, participants could earn a bonus. They were told that, after completing all questions, one question would be randomly chosen and, if answered correctly, would result in a bonus of 12 experimental points. In the control questions not incentivized treatment, participants could not earn a bonus. After completing all questions, all participants received information for every question about the correct answer, their actual answer, and if their answer was correct or wrong.

Part 2 We then elicited contribution preferences using the strategy method of Fischbacher et al. (Reference Fischbacher, Gächter and Fehr2001), validated for repeated interactions in Fischbacher and Gächter (Reference Fischbacher and Gachter2010). Group members first make an unconditional contribution to the public good, which is a single integer number that satisfies 0 ≤ $g_i$ ≤ 20. Thereafter, group members make a conditional contribution for each of the 21 possible rounded averages from 0 to 20 (i.e., they submit a contribution schedule). Both the unconditional as well as the conditional contribution are potentially payoff relevant (for the way of how to incentivize both, see Fischbacher et al., Reference Fischbacher, Gächter and Fehr2001). Participants did not receive any information about other participants’ decisions at the end of Part 2.

Part 3 Finally, participants played ten periods of a repeated public goods game with partner matching (i.e., with constant group composition). We emphasized that the group composition was determined randomly and thus most likely different from the previous strategy method decisions in Part 2. In each period, we elicited beliefs about the other group members’ average contributions. At the end of each period, group members were informed about their group members` average contributions and their own payoffs. To avoid hedging, at the end of the session two periods were randomly selected for payment: In one period, the outcome of the public goods game was payoff relevant, and in another one, beliefs were incentivized.

We ran the experiment in the Munich Experimental Laboratory for Economic and Social Sciences (MELESSA), using the software z-Tree (Fischbacher, Reference Fischbacher2007) and the organizational software ORSEE (Greiner, Reference Greiner2015). In total, 93 undergraduates took part in the experiment with average earnings of €19 (including a show-up fee of €4).

3 Experimental results

Table 1 provides summary statistics for the three parts of the four treatments and shows the number of independent observations per treatment. Our analysis reveals that incentivizing control questions does not significantly affect participants’ behavior in any of the three parts (see Online Appendix C for a detailed discussion of the results). Since there is no effect of the incentivizing control questions, we pool the data and compare results for the two information treatments in the remainder of the paper.

In this context, it is also important to see whether different experimental instructions influence the number of correct answers. As a matter of fact, information about optimal strategies has neither a significant effect on how many correct answers participants give [p = 0.313; two-sided Mann–Whitney-U (MWU) test; all p values $\ge$ 0.084 (MWU) on the level of the 16 individual control questions], nor on how many participants answer all control questions correctly (p = 0.804; two-sided Chi-square test). Hence, information about optimal strategies does not affect how well participants do in answering control questions.

3.1 Treatment information: standard instructions vs. optimal strategies instructions

We observe that information about the optimal strategies does not affect the distribution of contribution preferences elicited by the strategy method (p = 0.366; two-sided Chi-square test; definitions of types according to Fischbacher et al., Reference Fischbacher, Gächter and Fehr2001; see Table 2). Especially, the relative frequency of conditional cooperators is almost identical. Further, Mann–Whitney-U tests do not reveal any significant differences in slopes of the conditional cooperation schedulesFootnote ² (p = 0.561) or mean unconditional contributions in Part 2 (p = 0.363) for the two treatments. Thus, we conclude that decision makers exhibit the same elicited preferences for cooperation, regardless of whether they receive standard instructions or instructions that explain the payoff consequences of all strategies in detail. Confusion does not seem to play a major role.

Table 2

Distribution of player types for the two information treatments

Treatment	Free rider	Conditional cooperators	Hump-shaped/others
Optimal strategies information	17.8% (N = 8)	66.7% (N = 30)	15.5% (N = 7)
Standard information	12.5% (N = 6)	60.4% (N = 29)	27.1% (N = 13)
Chi2 test	p = 0.366

There are N = 2 humped-shaped contributors in the optimal strategy information treatment and N = 4 in the standard information treatment. To avoid that a cell has fewer than five entries, we pool hump-shaped and others types (see Kocher et al., Reference Kocher, Martinsson, Matzat and Wollbrant2015)

In Fig. 1, we show the dynamics of contributions over the ten periods of Part 3 for the two information treatments. We find that information about optimal strategies does not affect mean contributions (p = 0.304; MWU; group means as independent observations) and mean beliefs (p = 0.149; MWU; group means as independent observations) in the repeated public goods game (Part 3). To investigate this relationship, we run a regression of participants’ contributions in the public goods game in period 1, using OLS regressions (see Models 1–4; Table 3) and averaged over periods 1–10 using GLS regressions (see Models 5–8; Table 4). The models include different predictors. Models 1 and 5 include an information treatment dummy; Models 2 and 6 additionally include beliefs on others’ contributions; Models 3 and 7 add predicted contributions (i.e., contributions based on elicited preferences from Part 2 and the belief); and Models 4 and 8 further add interaction terms of the information treatment dummy with beliefs and predicted contributions as well as period (only Model 8). The analysis reveals that information about the optimal strategies has no significant main effect on average contributions, neither in period 1 (see Models 1–4) nor overall (see Models 5–8). Moreover, information about optimal strategies does not significantly interact with beliefs and predicted contributions, neither in period 1 (see Models 1 and 4) nor overall (see Models 5 and 8), nor with period (see Model 8). Based on these results, we conclude that participants’ contributions in a repeated public goods game, and their beliefs about others’ contributions, do not change when they receive extended information about which strategies maximize individual and group payoffs.

Fig. 1

Average contributions over ten periods in Part 3

Table 3

Regression models of contributions in the repeated public goods game in period 1

Model	Dependent variable: contribution
	1	2	3	4
Optimal strategies information	–0.651 (1.460)	0.596 (0.800)	0.580 (0.801)	–0.003 (1.036)
Belief		1.011*** (0.036)	0.930*** (0.080)	0.980*** (0.096)
Predicted contribution			0.100 (0.080)	–0.004 (0.088)
Optimal strategies information × belief				–0.069 (0.154)
Optimal strategies information × predicted contribution				0.183 (0.154)
Constant	9.563*** (0.925)	–0.354 (0.658)	–0.272 (0.686)	–0.024 (0.847)
Observations	93	93	93	93
R-squared	0.002	0.719	0.723	0.728

Coefficients from OLS regressions with robust standard errors reported in parentheses

***p < 0.01, **p < 0.05, *p < 0.1

Table 4

Regression models of contributions in the repeated public goods game in periods 1–10

Model	Dependent variable: contribution
	5	6	7	8
Optimal strategies information	–1.662 (1.976)	–0.129 (0.979)	–0.051 (0.884)	–1.511 (1.158)
Belief		0.651*** (0.068)	0.521*** (0.094)	0.429*** (0.128)
Predicted contribution			0.236*** (0.089)	0.290** (0.143)
Optimal strategies information × belief				0.182 (0.170)
Optimal strategies information × predicted contribution				–0.103 (0.168)
Optimal strategies information × period				0.073 (0.119)
Period	–0.349*** (0.096)	–0.202*** (0.057)	–0.201*** (0.060)	–0.229*** (0.075)
Constant	11.052*** (1.147)	3.293*** (0.818)	2.956*** (0.786)	3.689*** (0.854)
Observations	930	930	930	930
Number of individuals	93	93	93	93

Coefficients from GLS regressions with standard errors in parentheses, random effects on the participant level and standard errors clustered at the matching group level

***p < 0.01, **p < 0.05, *p < 0.1