1 Introduction

Economists have long highlighted the serious challenge that free-riding poses for the voluntary provision of public goods. In many settings, the level of the public good depends on the sum of voluntary contributions, which means that individuals may hope to free ride on the efforts of others (e.g., Zelmer, Reference Zelmer2003). Yet, many public goods do not fit this framework of a summation technology. Rather, aggregation of individual contributions into public good provision may rely on more general technologies (e.g., Hirshleifer, Reference Hirshleifer1983; Vicary & Sandler, Reference Vicary and Sandler2002) including ones where individual contributions are complements rather than substitutes. A prominent example is the weakest-link technology, where the level of the public good depends on the lowest individual contribution. Such weakest links are present, e.g., when protecting against terrorist attacks such as airplane hijacking or when fighting contagious diseases (Barrett, Reference Barrett2016; Caparròs & Finus, Reference Caparrós and Finus2020a; Vicary & Sandler, Reference Vicary and Sandler2002). In these settings, the outcome critically depends on the minimum contribution level.Footnote ¹

Moreover, there are many situations where the link between effort (contributions) and the resulting amount of the public good is stochastic: the exact precautionary effort, e.g., the vaccination rate that prevents a given mutation or the security efforts that succeed in keeping terrorists out of high-security zones, is unknown. That is, additional effort will only increase the chances of preventing an adverse event. Despite the importance of uncertain thresholds in weakest-link settings, no study has yet explored their behavioral consequences. The main research question we address in this paper is how threshold uncertainty affects the provision of a public good under a weakest-link technology, particularly in relation to the same effect under the summation technology.

We conduct a lab experiment where subjects contribute to a public good, the benefits of which (in the form of avoiding a bad event) are generated only if some contribution threshold is passed. This design allows us to explicitly compare the likelihood of reaching the threshold under certain and uncertain thresholds and for both a summation and a weakest-link aggregation of contributions. With summation, total group contributions need to exceed the threshold, while the threshold under weakest link is a minimum level of contribution that all group members must achieve. Thus, the aggregation technologies we consider represent the polar cases of perfect substitutability and perfect complementarity among different players’ contributions. We interact these investigations into the role of uncertainty and aggregation technology on contribution levels with a study of the role of (cheap talk) communication by allowing subjects to send messages on their intended contribution levels and making suggestions for the contributions by others.

Our paper contributes to diverse strands of the literature. Inspired by potential thresholds in the climate system (e.g., Steffen et al., Reference Steffen, Rockström, Richardson, Lenton, Folke, Liverman, Summerhayes, Barnosky, Cornell, Crucifix, Donges, Fetzer, Lade, Scheffer, Winkelmann and Schellnhuber2018), a prominent discussion in the theoretical and experimental literature concerns how thresholds affect the provision of public goods under a summation technology (e.g., Barrett, Reference Barrett2013; Barrett & Dannenberg, Reference Barrett and Dannenberg2012, Reference Barrett and Dannenberg2014; Croson & Marks, Reference Croson and Marks2000; Dannenberg et al., Reference Dannenberg, Löschel, Paolacci, Reif and Tavoni2015; McBride, Reference McBride2006). Thresholds provide a coordination mechanism and may increase contributions. Yet according to Barrett and Dannenberg (Reference Barrett and Dannenberg2012, Reference Barrett and Dannenberg2014), uncertainty about the threshold level substantially reduces the chances of successful coordination under summation. Intuitively, uncertainty about the threshold means that each individual is only able to impact the probability of threshold attainment at the margin. This increases the incentive to free ride compared with a known discontinuous threshold, making coordination equilibria more difficult to sustain. With this paper, we ask whether threshold uncertainty shows similar effects under a weakest-link technology.

Our experimental design is inspired by Barrett and Dannenberg (Reference Barrett and Dannenberg2012, Reference Barrett and Dannenberg2014), who explored uncertain thresholds with a summation technology. Subjects interact in groups of 10. Contributions generate payoffs that are linear in the sum of donations, and additionally, failure to reach a contribution threshold induces a discrete lump sum loss.Footnote ² Threshold attainment under the weakest link technology requires all subjects to reach the threshold. Under summation, the sum of contributions reaches the (appropriately scaled) threshold in order to avoid the adverse event.

Weakest-link structures involve an element of coordination for the voluntary provision of public goods such that it is not worthwhile for an individual to contribute more than the smallest amount provided by another group member (Barrett, Reference Barrett2016). While this often gives rise to equilibria at contribution profiles other than the free-riding outcome, successful coordination is far from guaranteed (Reference Caparrós and Finus2020b; Anderson et al., Reference Anderson, Goeree and Holt2001; Caparròs & Finus, Reference Caparrós and Finus2020a; Caparròs et al., Reference Caparrós, Blanco, Buchenauer and Finus2020; Devetag & Ortmann, Reference Devetag and Ortmann2007), not least because a single individual failing to coordinate is strongly detrimental to the outcome, thus confronting subjects with substantial strategic uncertainty.Footnote ³

Anticipating this added vulnerability in weakest-link settings, we additionally explore the effect of communication on threshold attainment. Communication is implemented as a pledge, i.e., a truthful or non-truthful statement on the player’s intended action (e.g., Cooper et al., Reference Cooper, DeJong, Forsythe and Ross1992) combined with a suggestion for the action by group members (e.g., Weber, Reference Weber2003) and thus follows closely the design in Barrett and Dannenberg (Reference Barrett and Dannenberg2014).

In previous work on cooperation games, the exact method of communication may similarly involve individual pledges by all group members or suggestions as to what other members should do. There is also variation in terms of whether all members can communicate or if only the messages of a single player are selected and broadcasted. Communication is typically efficiency enhancing (e.g. Barrett & Dannenberg, Reference Barrett and Dannenberg2016; Brosig et al., Reference Brosig and Weimann2003; Isaac & Walker, Reference Isaac and Walker1988), but may depend on the nature of communication. For example, pledges as simple numerical cheap talk had no net effect on contributions in Bochet et al. (Reference Bochet, Page and Putterman2006) and Bochet and Putterman (Reference Bochet and Putterman2009). Several extant studies consider the role of communication in weakest-link games or–more generally in coordination games. Cason et al. (Reference Cason, Sheremeta and Zhang2012a) study intra-group communication when two groups compete and thereby provide an example that communication can hurt efficiency (through the competition element).Footnote ⁴ Other studies have typically found efficiency-enhancing effects of communication, (e.g., Blume & Ortmann, Reference Blume and Ortmann2007; Charness & Grosskopf, Reference Charness and Grosskopf2004; Cooper et al., Reference Cooper, DeJong, Forsythe and Ross1992; Duffy & Feltovich, Reference Duffy and Feltovich2002, Reference Duffy and Feltovich2006; Smerdon et al., Reference Smerdon, Offerman and Gneezy2020; Van Huyck et al., Reference Van Huyck, Battalio and Beil1993).

Most of these experiments involve smaller numbers of players in a group. While larger groups are typically found to lead to less or similar coordination levels in coordination games as smaller groups (e.g., Heinemann et al., Reference Heinemann, Nagel and Ockenfels2009; Van Huyck et al., Reference Van Huyck, Battalio and Beil1990), Feltovich and Grossman (Reference Feltovich and Grossman2015) show that the beneficial effects of communication may be reduced in larger groups.

As such, it is an open question how threshold uncertainty functions in our ten-player weakest link setting and also whether or not communication can improve the coordination.

Our experiment reveals the importance of the aggregation technology. Importantly, thresholds in the weakest-link setting imply quite different behavioral patterns depending on whether the threshold is certain or uncertain. When the threshold level is certain, the threshold provides a highly successful coordination device. In contrast, groups are almost guaranteed not to reach the threshold if the level is uncertain. Thus, we conclude that the role of strategic uncertainty is further enhanced when the threshold becomes uncertain, leading to considerably lower success rates.

Under summation, we find that uncertainty regarding the threshold does not significantly affect contributions or the probability of threshold attainment. This highlights the contrast with weakest link, where threshold attainment hinges on all subjects contributing a sufficient amount, and high contributors are unable to compensate for low contributions by other group members.

The results for summation are also in stark contrast to Barrett and Dannenberg (Reference Barrett and Dannenberg2012, Reference Barrett and Dannenberg2014), who identified a strong negative effect of threshold uncertainty. One explanation for the different results is that we find subjects largely adhering to their non-binding pledges, which was not the case in the earlier studies. Another explanation might be minor differences in how the experiment was implemented in our paper compared to the Barrett and Dannenberg papers: specifically, they had subjects play five practice rounds such that subjects obtained information about the behavior of other potential group members before playing the actual experiment, while our experimental implementation does not give feedback and thus is more in line with a one-shot interaction. While we thus cannot give a definitive answer to the drivers of these differences, our experiment is designed to compare weakest links and summation technologies under threshold uncertainty.Footnote ⁵

Finally, regarding the role of pre-play communication, the chances for threshold attainment in all of our treatments turn out to heavily hinge on subjects being able to announce intended behavior (pledges) and/or make proposals for how other group members should act. A lack of communication prior to contribution decisions particularly handicaps threshold attainment under weakest link. Exploring the mechanisms through which such opportunities to communicate affect contributions, we find that the non-binding nature of the pledge induces individuals to make larger contribution announcements. These promises are more likely to be kept if other people have made similar pledges.

The paper is structured as follows: Sect. 2 discusses the experimental design, before we provide some theoretical predictions in Sect. 3. Section 4 discusses our results regarding threshold attainment, individual contribution decisions, and the role of pledges. Section 5 concludes.

2 Experimental design

The games played in our experiment are variants of the voluntary public good paradigm. In this section, we first provide general definitions and then describe our treatments and experimental protocols, which (in the summation case) are highly similar to those of Barrett and Dannenberg (Reference Barrett and Dannenberg2012, Reference Barrett and Dannenberg2014). Theoretical analysis is conducted in Sect. 3.Footnote ⁶

The games are played in groups of 10 players $i\in \left(1,\dots ,10\right)$ , following Barrett and Dannenberg (Reference Barrett and Dannenberg2012). There is mixed evidence regarding the relationship between contributions in a public good game and group size (see e.g., Zelmer, Reference Zelmer2003; Diederich et al., Reference Diederich, Goeschl and Waichman2016). For weakest-link games, coordination is clearly more difficult to achieve with larger group sizes (Riedl et al., Reference Riedl, Rohde and Strobel2016).

Each player is endowed with 20 experimental tokens. The contribution to the common pool is denoted $q_i\in \left(0,1,\dots ,20\right)$ . Each player’s opportunity cost of contributing is piecewise linear and convex, and given by

$C\left(q_i\right)=\left(\begin{array}{cc}{c}_L{q}_i,& \text{for}0\le q_i\le 10\\ 10c_L+c_H\left(q_i-10\right),& \text{for}11\le q_i\le 20\end{array}\right)$

where $c_L=€0.1$ and $c_H=€0.5$ .Footnote ⁷ One interpretation of this cost structure is that there are two technologies available to each player: one low-cost technology (L) and one high-cost technology (H). By design, and as further described below, the upper limit for the low-cost technology lies in the threshold support only if this is relatively wide.

Contributions to the common pool have two effects: (i) they generate a public-good return $r=€0.05$ to each group member per token contributed by any player, and (ii) they (weakly) increase the probability that the group reaches a threshold contribution level $\overline{Q}$ , which is uniformly distributed on integers $\left(a,a+1,\dots ,b-1,b\right)$ , with $b\ge a>0$ . Total returns from contributing (after resolving the uncertainty in $\overline{Q}$ ) are

$r\left(\sum _{i=1}^{10}{q}_i\right)-f\left(q_1,\dots ,q_{10},\overline{Q}\right)X$

where $f$ is a discontinuous function of the contributions and the threshold level, while $X=€15$ is a fixed cost of failing to reach the threshold.

Our treatments differ in the mapping of individual contributions onto threshold attainment. Under a weakest-link technology, the cost of €15 is paid unless all group members contribute at least $\overline{Q}$ . That is, the contribution by the player with the smallest contribution needs to reach the threshold, so $f=I\left(q^{\min },<,\overline{Q}\right)$ , where $q^{\min }={\text{min}}_i\left(q_i\right)$ and $I$ is the binary indicator function.Under a summation technology, the cost of €15 is paid by all players unless the group sum of contributions reaches at least $\overline{Q}$ ; formally, $f=I\left({\sum }_{i=1}^{10}{q}_i<\overline{Q}\right)$ .

Table 1 lists the six between-subject treatments included in the experiment. We use a 2-by-3 design that explores, first, the effect of technology, i.e., whether threshold attainment is based on weakest link or summation aggregation. Second, we vary whether the location of the threshold is certain (“certainty"), or alternatively exhibits uncertainty in a broad (“uncertainty”) or narrow range (“small uncertainty”).Footnote ⁸ We include the small uncertainty case as there is theoretical support that small uncertainty can actually increase contributions in a summation public good game (Barrett & Dannenberg, Reference Barrett and Dannenberg2014); we return to this point in Sect. 3. To make symmetric equilibria and per-person contributions as comparable as possible across technologies, weakest-link threshold ranges are obtained by dividing the corresponding ranges for summation by $10$ , the group size.

Table 1

Experimental treatments

Treatments	Aggregation technology	Threshold support	$N$
T1: WL, certainty	Weakest link: $q_i^{\min }\ge \overline{Q}$	$\overline{Q}=15$	100
T2: WL, uncertainty	Weakest link: $q_i^{\min }\ge \overline{Q}$	$\overline{Q}\in \left(10,,,\cdots ,,,20\right)$	100
T3: WL, small uncert	Weakest link: $q_i^{\min }\ge \overline{Q}$	$\overline{Q}\in \left(14,,,\cdots ,,,16\right)$	80
T4: Sum, certainty	Summation: ${\sum }_{i=1}^n{q}_i\ge \overline{Q}$	$\overline{Q}=150$	100
T5: Sum, uncertainty	Summation: ${\sum }_{i=1}^n{q}_i\ge \overline{Q}$	$\overline{Q}\in \left(100,,,\cdots ,,,200\right)$	100
T6: Sum, small uncert	Summation: ${\sum }_{i=1}^n{q}_i\ge \overline{Q}$	$\overline{Q}\in \left(140,,,\cdots ,,,160\right)$	100

The experiment was conducted at the laboratory of the Vienna Center for Experimental Economics, using the lab’s associated subject pool of university students. A total of 29 experimental sessions were conducted, each with 20 subjects (two groups) belonging to the same treatment.Footnote ⁹

The sessions progressed as follows. First, subjects were randomly assigned to one of the groups and acquainted with the game rules.Footnote ¹⁰ There were no practice rounds: instead, exhaustive examples and control questions were included to ensure that subjects understood the game.Footnote ¹¹ We did not allow for practice rounds since we did not want subjects to learn anything about other players’ behavior, particularly in relation to their proposed group contributions and pledges (see below).Footnote ¹² Subjects were also informed that there were two rounds, one of which would be randomly chosen for payment at the end of the session. Subjects then played a first round of the game variant corresponding to their treatment. Importantly, this initial one-shot round was not followed by feedback of any kind.

Next, group composition was re-shuffled, with subjects informed that a second round would now take place among the new groups. Our main analysis is based on this second round. Immediately prior to it, subjects were asked to make a pair of non-binding announcements to the other members of their group. Subjects submitted, first, a proposal for how many tokens the group as a whole should contribute to the joint project, and second, a pledge to personally contribute some number of tokens in the coming round. All proposals and pledges made in a group were then displayed on screen to all members immediately before the subjects made the second-round contribution decision.

To summarize, there was no interaction among group members in the first round, no feedback after round 1 (and no practice rounds), and also reshuffling between rounds. Because of this, our design arguably allows us to identify the effect of communication by comparing outcomes across the two rounds. Of course, it is difficult to fully rule out that some possibility for learning or experience (and hence confounding order effects) may remain: Weber (Reference Weber2003) found learning in a competitive guessing game over 10 rounds played with no feedback. However, our experiment only includes two rounds, and unlike the guessing game, there are multiple equilibria requiring coordinated behavior. Also, the often-observed pattern of declining contributions in public good games is not likely to hold without any information feedback between rounds (Neugebauer et al., Reference Neugebauer, Perote, Schnidt and Loos2009). Therefore, we do not believe that learning should be of any concern here, and thus we will compare across rounds in Sect. 4.3.

After the conclusion of the second round, subjects filled out an end-of-session survey with questions on risk, time, and social preferences based on Falk et al. (Reference Falk, Becker, Dohmen, Enke, Huffman and Sunde2018), as well as demographic variables.Footnote ¹³ We also asked subjects to (i) explain their reasoning behind pledging a certain contribution, (ii) rate how much they trusted the pledges of other group members, and (iii) judge whether the pledges and proposals of others made them change their own contribution level. Finally, subjects were informed of the outcome of both game rounds, including their own associated potential and actual earnings in each round.

3 Characterization of symmetric Nash equilibria

In Appendix B, we derive the best responses and characterize associated Nash equilibria within each treatment. Some simplifying assumptions are applied: first, the analysis limits attention to symmetric equilibria. Second, players are assumed to be risk neutral throughout. Finally, we do not include the proposals and pledges made prior to the second round. Table 2 summarizes the results obtained under these conditions.

Table 2

Symmetric Nash equilibria by treatment

Treatment	Symmetric Nash equilibria (all $i$ )
	Zero contribution	Coordination/cooperation
T1. Weakest link, certainty	$q_i=0$	$q_i=15$
T2. Weakest link, uncertainty	$q_i=0$	$q_i=q\in \left(10,,,\cdots ,,,20\right)$
T3. Weakest link, small uncertainty	$q_i=0$	$q_i=q\in \left(14,,,\cdots ,,,16\right)$
T4. Summation, certainty	$q_i=0$	$q_i=15$
T5. Summation, uncertainty	$q_i=0$	–
T6. Summation, small uncertainty	$q_i=0$	$q_i=16$

The symmetric equilibrium where each player has $q_i=0$ is supported in all treatments. However, options for symmetric coordination at higher contribution levels vary substantially across treatments.

Under weakest link, there is a coordination equilibrium at $q_i=15$ when the position of the threshold is known with certainty (T1). Introducing uncertainty creates a range of equilibria at every integer within the support of $\overline{Q}$ (T2 and T3). This is because the weakest-link structure leaves each player relatively pivotal to threshold attainment even in the presence of uncertainty. Thus, if every other player $j$ has chosen to contribute some $q_j=q$ in the interior of that support, then for the parameters used in our experiment, the remaining player $i$ will always prefer $q_i=q$ to $q_i<q$ (and will certainly not contribute $q_i>q_j$ , since doing so would not change the outcome).

In summation under certainty (T4), there is a single symmetric equilibrium at the threshold (Isaac et al., 1989). This payoff (Pareto) dominant equilibrium disappears under large uncertainty (T5), as stressed by Barrett and Dannenberg (Reference Barrett and Dannenberg2012) and Barrett (Reference Barrett2013). The reason is that, as uncertainty is added, each player becomes less pivotal to threshold attainment since they can now only influence the probability that ${\sum }_{i=1}^n{q}_i\ge \overline{Q}$ at the margin. Under small uncertainty, however, the threshold support is small enough, and each player correspondingly pivotal enough, that a non-zero equilibrium survives (Barrett & Dannenberg, Reference Barrett and Dannenberg2014) at the upper bound of the support of $\overline{Q}$ . Kotani et al. (Reference Kotani, Tanaka and Managi2014) obtain a similar result for a game with binary contributions.

Taken at face value, the main conclusion from Table 2 is that contributions in T5, where no “upper” equilibrium exists, might be expected to be lower than in all other treatments. Importantly, the effect of introducing large uncertainty differs substantially between the two aggregation technologies: cooperation collapses under summation but not necessarily under weakest link.