1 Introduction

In a wide range of economic settings such as crowdfunding projects,Footnote ¹ open-source software development, and environmental conservation initiatives, the power of sequential contributions has proven pivotal in fostering collective action and yielding positive outcomes for the common resource. In crowdfunding projects Stiver et al. (Reference Stiver, Barroca, Minocha, Richards and Roberts2015); Hudik and Chovanculiak (Reference Hudik and Chovanculiak2018); Ansink et al. (Reference Ansink, Koetse, Bouma, Hauck and van Soest2022), the act of individuals contributing money to support a cause sets off a chain reaction where others are motivated to join in, progressively building up the project's funding and increasing its chances of success. Similarly, in open-source software development Athey and Ellison (Reference Athey and Ellison2014); Xu et al. (Reference Xu, Nian and Cabral2020); Von Krogh et al. (Reference Von Krogh, Spaeth and Lakhani2003); Lerner and Tirole (Reference Lerner and Tirole2002), programmers’ contributions, whether in the form of code, bug fixes, or improved documentation, lead to an iterative improvement in the software's quality and functionality, enticing more developers to get involved in the project's advancement. Environmental conservation initiatives (see Footnote 1 for examples of environmental conservation) also benefit from sequential actions, as initial efforts by one group to preserve natural resources inspire others to participate, resulting in a cumulative and lasting improvement of the environment. In all these examples, the dynamic of sequential contributions fosters cooperation and participation, driving the collaborative spirit towards benefiting the overall well-being of the community.

In standard social dilemmas with self-interested players, co-operation is usually hindered by the players’ free-riding motives, leading often to socially sub-optimal outcomes. On the other hand, co-operation is still feasible, in the case of a repeated strategic interaction over an infinite horizon, or whenever the players’ preferences are non-standard (e.g. altruistic preferences). In a recent study, Gallice and Monzón (Reference Gallice and Monzón2019) (G &M henceforth), in the context of a discrete public goods game, show that it is possible to achieve full co-operation, even in the case where the interaction is a one-shot game, the number of players is finite, and the agents are self-interested.

In this model, individuals have to make decisions sequentially, without knowing their position in the sequence (position uncertainty), but are aware of the decisions of some of their predecessors by observing a sample of past play. In the presence of position certainty, those placed in the early positions of the sequence would want to contribute, in an effort to induce some of the other group members to co-operate Rapoport and Erev (Reference Rapoport, Erev, Schulz, Albers and Mueller1994), while late players, would want to free-ride on the contributions of the early players. Nevertheless, if the agents are unaware of their position in the sequence, they would condition their choice on the average payoff, from all potential positions, and they would be inclined to contribute so as to induce the potential successor to do so as well. G &M show that full contribution can occur in equilibrium, where given appropriate values of the parameters of the game (i.e. return from contributions), it is predicted that there exists an equilibrium where all agents contribute.

The main intuition behind the model is that an agent who observes a sample of past decisions of immediate predecessors, without defection, would decide to contribute, hoping to influence all her successors to do so. If instead, she decides to defect, then all the successors are expected to defect as well. As there is position uncertainty, the agent deals with a trade-off between inducing contributions from the remaining players in the sequence and the cost of contributing. In their main theoretical result, G &M show that incentives off the equilibrium path, largely depend on the sample size that the agents observe. When an agent observes a sample of more than one previous actions that contains defection, there is no way this agent can prevent further defection by choosing to contribute. On the other hand, when the sample size is equal to one (only the decision of the previous player is observed), the agent can induce further contributions from the remaining agents in the sequence, by deciding to contribute. The model predicts that when the sample size is equal to one there is a mixed strategy equilibrium that can lead to full contribution. Finally, when the agents are aware of their position in the sequence, the model predicts that full contribution will unravel, as late agents in the sequence will have an incentive to free-ride.

In this paper, our objective is to contribute to the understanding of agents’ decision-making process when they are presented with partial information about past contributions and face uncertainty about their position in the sequence. Motivated by the G &M theoretical contribution, the objective of our study is threefold, and in particular, we aim to identify: (1) How does the content and the size of past history, regarding contribution actions, affects contributions in the sequential public goods game when there is position uncertainty, (2) What is the effect of positional awareness on contribution decisions, when the agents receive partial information on past actions, and (3) What reasons lead to larger group contributions, when the theory predicts defection, or in other words, what is the role of agents with altruistic motives in explaining our data. To address these questions, in an incentivised economic experiment, with three different treatments, we vary the amount of information (size of sample of past actions that a subject can observe) as well as the positional awareness that subjects have.

We design a 2 $\times$ 2 fractional factorial experiment to test decision-making in sequential public goods games under varying levels of information and sequence awareness. Each session involved ten rounds of a sequential public goods game. Participants were randomly re-matched in each round into new groups of four and tasked with allocating tokens in a common project or private account. The treatments tested the impact of observing a sample of past decisions—ranging from one to two immediate predecessors—on subsequent contribution choices, without complete knowledge of one's position in the sequence, except for one treatment where participants’ sequence positions were disclosed. Using the Selten (Reference Selten1967) strategy response method, we collected comprehensive data on participant decisions across all possible information scenarios they might encounter.

We find that while participants’ behaviour is in line with the theoretical predictions, there is still a large part of behaviour that the model cannot account for. Using the Strategy Frequency Estimation Method Dal Bó and Fréchette (Reference Dal Bó and Fréchette2011); Fudenberg et al. (Reference Fudenberg, Rand and Dreber2012), we allow for the presence of various behavioural types in our subject population, and we estimate the proportion of each type in our data (see Fischbacher et al. Reference Fischbacher, Gächter and Fehr2001; Bardsley and Moffatt Reference Bardsley and Moffatt2007; Thöni and Volk Reference Thöni and Volk2018; Katuščák and Miklánek Reference Katuščák and Miklánek2023; Préget et al. Reference Préget, Nguyen-Van and Willinger2016). On top of G &M agents, we classify subjects to free-riders (who never contribute), altruists (who always contribute no matter their position), and conditional co-operators (who always contribute if they are in position 1 and contribute if at least one other person in the sample has contributed when they are in positions 2–4). Additionally, we investigate whether subjects align with the predictions of the G &M model (G &M type). We find that around 25% of the subjects behave according to the G &M model, the vast majority behaves in a conditional co-operating or altruistic way, and a non-significant proportion free rides. From a mechanism design point of view, we find that introducing uncertainty regarding the position, along with a constrained sample of previous actions (i.e. only what the immediate precedent player), maximises the public good provision.

Our work is related to and extends various strands of the literature, which we briefly summarise below. Prior to G &M's research, the timing of contributions and the level of funds raised had received considerable attention in the theoretical literature. Varian (Reference Varian1994) shows that, under appropriate assumptions, a sequential contribution mechanism elicits lower contributions than a simultaneous contribution mechanism. The crux of this result lies in the set-up of the model, where a first mover may enjoy a first-mover advantage and free-ride. On the other hand, Cartwright and Patel (Reference Cartwright and Patel2010) using a sequential public goods game with exogenous ordering, show that agents early enough in the sequence would want to contribute, if they believe that imitation from others is quite likely. In the context of fundraising, Romano and Yildirim (Reference Romano and Yildirim2001) examine the conditions under which a charity prefers to announce contributions in the form of a sequential-move game, while Vesterlund (Reference Vesterlund2003) shows that an announcement strategy of past contributions, not only helps worthwhile organisations to reveal their type, but it also helps the fundraiser reduce the free-rider problem, a result that Potters et al. (Reference Potters, Sefton and Vesterlund2005) confirm experimentally.

The early and recent experimental literature has provided substantial evidence on the superiority of a sequential contribution mechanism compared to a simultaneous one (see Erev and Rapoport Reference Erev and Rapoport1990; Rapoport and Erev Reference Rapoport, Erev, Schulz, Albers and Mueller1994; Rapoport Reference Rapoport1997, in step-level public goods games; and Andreoni et al. Reference Andreoni, Brown and Vesterlund2002; Coats et al. Reference Coats, Gronberg and Grosskopf2009; Gächter et al. Reference Gächter, Nosenzo, Renner and Sefton2010; Figuieres et al. Reference Figuieres, Masclet and Willinger2012; Teyssier Reference Teyssier2012 in public goods games without a threshold). The vast majority of the aforementioned studies conclude that the sequential protocol is significantly more effective in solving the public goods problem, compared to the simultaneous protocol, and that the effect of information on contribution is dramatic Erev and Rapoport (Reference Erev and Rapoport1990). Suleiman et al. (Reference Suleiman, Budescu, Rapoport, Schulz, Albers and Mueller1994) highlight two properties that define the sequential protocol: (1) information regarding the position in the sequence, and (2) information about the actions of preceding players. Therefore, there may be various information structures that underlie social dilemmas or the provision of a public good in real life, and relaxing or modifying these structures can lead to more realistic protocols of play. Most of the previous literature has focused on the comparison between the simultaneous and the sequential mechanism, while in terms of available information, the most commonly employed information structure is either full information on past decisions or no information at all. Both features appear to be closer to the reality that characterises the provision of public goods. In our experiment, rather than comparing the effectiveness of different contribution mechanisms, our focus is on the effect of the available information provided to the subjects, as well as the effect of positional awareness on the contribution choices. Table 1 summarises the experimental design features of the studies closest to ours.

A similar result regarding the superiority of the sequential mechanism has also been established in the literature on general public goods, particularly in the context of common pool resource games.Footnote ² This literature has identified significant ordering effects, even in the case where later subjects in a sequence could not observe past decisions (see Rapoport et al. Reference Rapoport, Budescu and Suleiman1993; Budescu et al. Reference Budescu, Suleiman and Rapoport1995; Suleiman et al. Reference Suleiman, Rapoport and Budescu1996; Rapoport Reference Rapoport1997). The model we test, makes sharp predictions regarding the behaviour of the agents for each of the potential positions in the sequence, subject to the available size and content of the sample of past decisions. Our experimental design allows us to manipulate the content of this sample, in terms of the number of past contributions, and observe the role of this information in this kind of ordering effect. While the main theoretical prediction in step-level public goods games is that the players at the early positions of the sequence will free-ride, the framework we are exploring predicts that players at the beginning of the sequence will contribute instead, to incentivise their successors. This is closely related to the leading-by-example literature, based on the linear public goods game (see Gächter et al. Reference Gächter, Nosenzo, Renner and Sefton2012; Levati et al. Reference Levati, Sutter and Van der Heijden2007; Potters et al. Reference Potters, Sefton and Vesterlund2007; Güth et al. Reference Güth, Levati, Sutter and van der Heijden2007; Figuieres et al. Reference Figuieres, Masclet and Willinger2012; Sutter and Rivas Reference Sutter, Rivas, Van Lange, Rockenbach and Yamagishi2014; Préget et al. Reference Préget, Nguyen-Van and Willinger2016, in the context of public goods games, or Moxnes and Van der Heijden Reference Moxnes and Van der Heijden2003, in a public bad experiment). They all find robust evidence of first movers contributing more than later movers, and later movers’ contributions to be positively correlated to the first movers’ contributions, indicating reciprocal motives.

Directly linked to the last observation, Figuieres et al. (Reference Figuieres, Masclet and Willinger2012) highlight that many individuals condition their decisions on the observation of others’ decisions in finitely repeated simultaneous public goods games (see Fischbacher and Gächter Reference Fischbacher and Gächter2010; Keser and van Winden Reference Keser and van Winden2000). Extensive experimental research in public goods games has shown that a large proportion of subjects’ behaviour deviates from the predictions of Nash equilibrium and they mostly behave in a conditionally co-operating manner. The latter implies that their contribution is positively correlated to their beliefs about the contributions of the other members of the group. Fischbacher et al. (Reference Fischbacher, Gächter and Fehr2001) was among the first studies to classify subjects to different types of decision makers, including the selfish type (free-rider), the reciprocator type, and the type applying a hump-shaped strategy (contributions that are first increased then decreased in others’ contributions). This methodology was later extended in Bardsley and Moffatt (Reference Bardsley and Moffatt2007), refined in Thöni and Volk (Reference Thöni and Volk2018), and recently used in Katuščák and Miklánek (Reference Katuščák and Miklánek2023), while Préget et al. (Reference Préget, Nguyen-Van and Willinger2016) use this in a sequential leader-followers public goods game. In addition to testing the predictions of G &M, we use structural econometric modelling to test the presence of alternative behavioural types of agents in our data. In particular, using the Strategy Frequency Estimation Method (SFEM), an estimation procedure introduced in Dal Bó and Fréchette (Reference Dal Bó and Fréchette2011) and Fudenberg et al. (Reference Fudenberg, Rand and Dreber2012), we explore the presence of free-riders, altruists and conditional co-operators in our data, on top of those behaving as the G &M model prescribes.

Contributing to the literature on uncertainty in social dilemmas Suleiman et al. (Reference Suleiman, Budescu and Rapoport2001); Van Dijk et al. (Reference Van Dijk, Wit, Wilke and Budescu2004); Budescu and Au (Reference Budescu and Au2002); Au and Ngai (Reference Au and Ngai2003), this paper examines how different types of uncertainty impact cooperative behaviour. Studies like Suleiman et al. (Reference Suleiman, Budescu and Rapoport2001) and Au and Ngai (Reference Au and Ngai2003) show varying effects: threshold uncertainty can increase contributions under certain conditions, while group size uncertainty paradoxically enhances cooperation. These findings, contrasting with the non-cooperative behaviour often induced by resource size uncertainty, align with our observations of increased cooperation under positional uncertainty. The nature of positional uncertainty, unlike the direct impact of resource size uncertainty on resource availability, may not readily justify non-cooperative behaviour. Our research, corroborating the significance of position in sequential decisions highlighted by Budescu et al. (Reference Budescu, Suleiman and Rapoport1995), Budescu and Au (Reference Budescu and Au2002), adds to the understanding of how context influences cooperative dynamics in social dilemmas.

In our study, the role of positional uncertainty parallels the findings of Au and Ngai (Reference Au and Ngai2003) on group size uncertainty, where certain uncertainties promote cooperative behaviour. This contrasts with Rapoport et al. (Reference Rapoport, Budescu and Suleiman1993)'s observations of non-cooperative behaviour under resource size uncertainty. Such distinctions underscore the complex impact of uncertainty types: positional and group size uncertainties might foster cooperation through a collective sense of ambiguity, while resource size uncertainty, directly affecting perceived resource availability, can lead to competitive behaviour.

The rest of the paper is organised as follows. In Sect. 2 we present the main theoretical predictions of Gallice and Monzón (Reference Gallice and Monzón2019). Section 3 presents the experimental design and the procedures, while Sect. 4 presents the descriptive statistics and results of the structural econometric analysis. We then conclude in Sect. 5.

Table 1 Summary of the experimental design of previous studies on sequential public goods games

Study	Group size	Rounds	Contribution	Threshold	Repeated	Position	Information on past contribution	Different types	No. of Subjects	Payment Method
Erev and Rapoport (Reference Erev and Rapoport1990)	5	1	Binary	Y	N	known	full	N	75	Total
Erev and Rapoport (Reference Erev and Rapoport1990)	5	1	Binary	Y	N	unknown	#past contributors	N	60	Total
Erev and Rapoport (Reference Erev and Rapoport1990)	5	1	Binary	Y	N	unknown	#past defectors	N	60	Total
Rapoport and Erev (Reference Rapoport, Erev, Schulz, Albers and Mueller1994)	5 or 7	1	Binary	Y	N	known	full	N	70	Total
Rapoport and Erev (Reference Rapoport, Erev, Schulz, Albers and Mueller1994)	6 or 7	1	Binary	Y	N	unknown	#past contributors	N	70	Total
Rapoport and Erev (Reference Rapoport, Erev, Schulz, Albers and Mueller1994)	6 or 7	1	Binary	Y	N	unknown	#past defectors	N	70	Total
Andreoni et al. (Reference Andreoni, Brown and Vesterlund2002)	2	14	Continuous	N	N	known	full	N	42	Total
Coats et al. (Reference Coats, Gronberg and Grosskopf2009)	4	20	Continuous	Y	N	known	previous	N	108	Total
Gächter et al. (Reference Gächter, Nosenzo, Renner and Sefton2010)	2	15	Continuous	N	N	known	full	N	128	Random
Teyssier (Reference Teyssier2012)	2	1	Continuous	N	N	known	full	Y	118	Random
Figuieres et al. (Reference Figuieres, Masclet and Willinger2012)	4 or 8	15	Continuous	N	Y	known	full	Y	72	Total
Figuieres et al. (Reference Figuieres, Masclet and Willinger2012)	4 or 8	15	Continuous	N	Y	known	unknown	Y	80	Total
this study	4	10	Binary	N	N	unknown	1 previous	Y	32	Random
this study	4	10	Binary	N	N	unknown	2 previous	Y	32	Random
this study	4	10	Binary	N	N	known	2 previous	Y	32	Random

Contribution: Continuous or ’all-or-nothing’. Threshold: Threshold or standard public goods game. Repeated: One-shot or repeated game (with strangers’ matching protocol). Different types: Exploration of alternative behavioural models. Payment Method: Total rounds or a randomly chosen subset. All experiments involved financially motivated participants from student subject pools

2 Theoretical framework

Gallice and Monzón (Reference Gallice and Monzón2019) propose a natural environment that sustains full-co-operation in one-shot social dilemmas among a finite number of self-interested agents, in the context of a public goods game. Here we present the main features and predictions of the model and we refer the interested reader to the original study for further details. In an effort to be as consistent as possible with the original study, we adopt the same notation as the authors. In summary, the main prediction of the model is that contributions from all individuals in a group will emerge when agents make decisions sequentially, do not know their position in the sequence, and observe the decisions of some of their predecessors.

Let $I=\{1,\dots ,n\}$ be a set of risk-neutral agents making choices in a sequence. They know the length n of the sequence but are uncertain about their position. Players are exogenously placed in the sequence that determines the order of play and they can be at any position with equal chances. Therefore, they have symmetric position beliefs regarding their position in the sequence. When the turn of play for agent $i\in I$ arrives, she can observe a sample of her predecessors’ actions. She then chooses one of two actions $a_i\in \{C,D\}$ , with $a_i=C$ being contribution of a fixed amount 1 to a common pool, while $a_i=D$ denotes defection, and therefore, investing 0. After all players choose an action, the total amount invested gets multiplied by the return from contributions parameter r, and is equally shared among all agents. Let $G_{-i}={\sum }_{j\ne i}1\left(a_j,=,C\right)$ denote the number of agents in the group who contribute, so $G_{-i}$ can take any value in the set $\{0,\dots ,n-1\}$ . The payoffs $u_i\left(a_i,,,G_{-i}\right)$ from contributing or defecting can thus be expressed as:

$\begin{array}{c}{u}_i\left(C,,,G_{-i}\right)=\frac{r}{n}\left(G_{-i},+,1\right)-1\text{and}{u}_i\left(D,,,G_{-i}\right)=\frac{r}{n}{G}_{-i}.\end{array}$

A value of r that satisfies $1<r<n$ , guarantees that although contribution by all agents is socially optimal, each agent strictly prefers to defect for any given $G_{-i}$ , as is the case in the standard public goods game.

Now we introduce the idea of sampling. Before choosing an action, each player observes how many of her $m\ge 1$ predecessors contributed. Agents in positions 1 to m observe less than m actions, given that there are less than m players before them. For instance, if $m=2$ , the player at position 1 observes a null sample, the player at position 2 observes a sample of only one previous action, while the player at position 3 observes a sample of two past actions. Each sample is denoted as a pair:

$\begin{array}{c}\xi =({\xi }^{\prime },{\xi }^{\prime \prime })\end{array}$

where the first component indicates the number of agents sampled, and the second component is the number of agents who contributed, in that sample.Footnote ³The agent has no way to identify the action of each player, but only the size of the sample and the total contribution. As a result, the first m agents can infer their position given the sample they observe. According to this set-up, players face an extensive-form game with incomplete information, and the solution concept is based on Kreps and Wilson (Reference Kreps and Wilson1982) notion of sequential equilibrium. Let $\Xi$ the set of all samples that an agent can observe. The strategy of player i is a function:

$\begin{array}{c}{\sigma }_i(C\mid \xi ):\Xi \to [0,1]\end{array}$

that specifies the probability of contributing given the sample that the agent observed.Footnote ⁴

G &M characterise an equilibrium where all the agents contribute adopting a simple profile of play, namely, agents contribute unless they observe a defection. In particular, their Proposition 1 (Gallice and Monzón Reference Gallice and Monzón2019, p. 2142) states that if:

(1) $\begin{array}{c}r\ge 2\left(1,+,\frac{m-1}{n-m+1}\right)\end{array}$

then, there exists an equilibrium in which all agents in the sequence contribute. To prove this proposition, G &M derive a series of lemmas, which our experiment aims to test. We focus on three particular cases: (1) when the sample size is greater or equal to 2, (2) when the sample size is equal to 1, and (3) when there is position certainty.

2.1 Sample $m\ge 2$

Let ${\Xi }^C$ be the set of all samples without defection. Then, Lemma 1 states that, assuming the following profile of play for all $i\in I$ :

$\begin{array}{c}{\sigma }_i^{\ast }(C\mid \xi )=\left(\begin{array}{cc}1,& \text{if}\xi \in {\Xi }^C\\ 0,& \text{otherwise.}\end{array}\right)\end{array}$

it follows that $({\sigma }^{\ast },{\mu }^{\ast })$ is a sequential equilibrium provided that the inequality in expression 1 is satisfied. In other words, what the above expression states, is that a player will always contribute, unless she observes at least one defection. A sketch of the intuition behind this follows. Consider an agent who observes a sample of full co-operation. According to the equilibrium definition, this occurs at the equilibrium path, and the agent can therefore infer that all the previous agents in the sequence contributed. By contributing herself, she knows that all subsequent players will contribute as well, and therefore, her expected payoff from contributing will be equal to $r-1$ , and this payoff is independent of the agent's beliefs regarding her position in the sequence. On the other hand, the payoff from defecting depends on these beliefs. Consider an agent who observes a sample of m. This agent can therefore infer that she is not placed in the first m positions, and there are equal chances of her being in any of the positions in $\{m+1,\cdots ,n\}$ . The expected position is $(n+m+1)/2$ , which means that she can expect that $(n+m-1)/2$ agents have already contributed. The payoff from defecting is then equal to $(r/n)(n+m-1)/2$ . Combining the two, gives the condition that r needs to satisfy in order for the agents to be willing to contribute, whenever they observe full contribution samples (condition 1).

On the other hand, the equilibrium profile requires that an agent should defect, upon observing a sample with at least one defection. G &M provide arguments why an agent who observes defection cannot affect her successors’ actions, regardless the value of r, and she is better off defecting herself as well.Footnote ⁵ This leads to our first hypothesisFootnote ⁶:

Hypothesis 1

There will be full contribution when there is position uncertainty, and the agents observe samples $m\ge 2$ of full contribution, compared to the case where agents observe at least one defection.

2.2 Sample $m=1$

A separate analysis is required for the situation in which only one previous player is sampled. In the scenario where $m=1$ , a player who spots defection has the ability to prevent additional defections by deciding to contribute. If the return from contributions is excessively high, the simple strategy of contributing, unless a defection is observed, cannot establish equilibrium. Players would deem it beneficial to contribute after witnessing a contribution but would not consider it advantageous to defect upon observing a defection. When r is high, a player who notices a defection finds the cost of contribution worthwhile to motivate all her successors to contribute. Consequently, she would choose to contribute rather than defect. However, as $r<n$ , the strategy profile in which all players contribute cannot be an equilibrium, because players who witness contribution would deviate to defecting. This means a pure strategy equilibrium with total contribution is impossible.

When the multiplication factor r is high, full contribution can emerge in a mixed strategy equilibrium. G &M construct a profile of play where players respond to contribution with contribution and permit a defection to be ’forgiven’ with a probability $\gamma \in [0,1)$ . The potential for future forgiveness makes defecting more appealing: successors could revert to contribution. This leads to Lemma 2 which states:

$\begin{array}{c}{\sigma }_i^{\ast }(C\mid \xi )=\left(\begin{array}{cc}1,& \text{if}\xi \in \left\{\right(0,0),(1,1\left)\right\}\\ \gamma ,& \text{if}\xi =(1,0).\end{array}\right)\end{array}$

for $r\in \left(3,-,3,/,(,n,+,1,),,,n\right)$ , that is agents always contribute when they observe contribution from their immediate predecessor and contribute with probability $\gamma$ if they observe defection. Our second hypothesis is summarised as follows:

Hypothesis 2

There will be full contribution when there is position uncertainty, and the agents observe sample $m=1$ of full contribution. In addition, there will be a mixed strategy equilibrium, with mixing probability $\gamma$ , when the agents observe sample $m=1$ of defection.

2.3 Position certainty

The last theoretical prediction of the model that we test, prescribes that when agents are aware of their position in the sequence, contribution unravels. Assume that player i knows her position, i.e., she knows that she is in position t. If all of her predecessors contributed, and she does not contribute, then none of her successors will contribute. This means that exactly $t-1$ players contributed. The payoff from defecting is $(r/n)(t-1)$ , while the payoff from contributing is equal to $n-1$ , as before. Figure 1 illustrates agent i's payoffs as a function of her position. The payoff from contribution, for agents early in the sequence, is larger than that of defecting. But agents placed late in the sequence would prefer defection. Consequently, if agents knew their position, contribution would unravel. Therefore, our last hypothesis states:

Hypothesis 3

Full contribution will unravel when the agents observe samples $m=2$ of full contribution and they are aware of their position in the sequence.

Fig. 1 Payoffs conditional on position and on samples without defection ( $r=3,n=4$ )

3 Experimental design and procedures

To test the predictions of the model presented in the previous section, we designed and conducted an incentivised economic experiment. The experiment took place at the Lancaster Experimental Economics Lab (LExEL) in February 2023, involving 96 subjects across three treatmentsFootnote ⁷. The subjects were primarily Lancaster University undergraduates (82.2%) from various disciplines, mainly Business and Economics (60%), Social Sciences (23%) and Science and Medicine (17%).Footnote ⁸ In terms of gender, the sample comprised 48% female and 52% male participants. The average age was 20.7 for those who provided details (4 subjects did not reply this question). Recruitment took place via the ORSEE system Greiner (Reference Greiner2015), and the experiment was computerised using the oTree platform Chen et al. (Reference Chen, Schonger and Wickens2016). None of the subjects had previously participated in a public goods experiment and every subject participated only once.

Each session consisted of 16 participants, forming groups of 4 and playing for a total of 10 rounds. In an effort to replicate the one-shot play environment, we adopted a random matching protocol where in every round, participants were randomly matched into new groups. At the beginning of all the sessions, participants were given written instructions followed by a comprehension test. Participants could retake the questionnaire until they passed. The interface would provide immediate feedback and explanation on the questions that the subjects failed to respond correctly. An experimenter would offer additional clarifications if there were further questions. Therefore, no exclusion criteria were applied. In every session, participants played a variation of a sequential, binary public goods game, as described in the previous section. At the beginning of every round, participants were endowed with 10 tokens each, with an exchange rate of £0.50 per token. In every round, subjects were randomly positioned in the sequence (with equal chances of being allocated at each position) and were sequentially asked to make a binary decision between investing all of these tokens in either a common project account or a private account. At the end of every round, feedback was provided regarding the participant's decision, the total group contribution, the individual share, and the payoff from the round. We employed the random payment mechanism, with one of the 10 rounds being randomly chosen to be played for real (different for each group).Footnote ⁹ The average payment was £17.8, including a show-up fee of £5. The experimental sessions lasted less than 45 min and payments were made via bank transfer.

The main objective of our experiment was to test the predictions of G &M when two dimensions change, namely, the size of the sample of past decisions observed by a subject, and whether subjects are aware of their position in the sequence. We, therefore, have a 2 $\times$ 2 fractional-factorial design, where the sample size can be either 1 or 2 past actions, and the position can be unknown or known.Footnote ¹⁰

There are in total three treatments. Treatment 1 ( $T_1$ ) was designed to test Hypothesis 1. In that treatment, participants received a sample of past decisions of length equal to two (they would receive information on what the combined contribution of the immediate 2 predecessors wasFootnote ¹¹) and also, participants were not informed of their position in the sequence unless they were Player 1 or Player 2.Footnote ¹² To rigorously test the predictions of the model, we need information on what subjects’ choices are, in all potential samples they can observe (i.e. the combined contribution they observe). In particular, the model predicts that if a player receives a full contribution sample (i.e. the combined contribution of the two previous players is equal to 20), she will contribute herself, otherwise, if the contribution is 0 or 10, she will defect. We elicit behaviour using the Selten (Reference Selten1967) strategy response method and collect the participants’ choices for each possible scenario. This means participants were asked to make conditional decisions for each possible information set depending on their randomly assigned position in the sequence. While this can be seen as a potential limitation of our design, this elicitation method was necessary to allow us to elicit the full strategy profile of the participants, and therefore, to be able to test the theoretical predictions of the model.Footnote ¹³ Figure 2 presents a screenshot from the experimental interface, of what a player in position 3 or 4 could see. In that case, the only information the subject has is that she is positioned in one of the two last positions of the sequence and is asked about her contribution choice, conditional on the potential total contribution of the two immediate players before her (either of players 1 and 2, if she is player 3, or players 2 and 3, if she is player 4) but she cannot distinguish her position. Therefore, the subject does not know whether there is another player following in the sequence. The available options for players at positions 1 and 2 were adapted accordingly.

Participants were incentivised to reveal their preferences truthfully, as their payment depended on what they and the other members of the group had chosen in that round. In particular, at the end of the experiment (and if this round had been selected for actual play), the computer recalled and matched all participants’ decisions in that round and calculated the payoffs based on the scenario that truly transpired. For instance, if the player at position 1 chose not to invest in the common project, the experimental software recalled the decision of the player in position 2, for that specific scenario and so forth, returning the total contribution, the total returns, and the payoffs for each participant in the group.

Fig. 2 Screen shot from Treatment 1 (players in position 3 or 4)

Treatment 2 ( $T_2$ ), was designed to test Hypothesis 2. This treatment mirrored $T_1$ , but subjects could observe a sample of past decisions of size 1 (i.e. the immediate predecessor). Participants were facing position uncertainty, unless they were assigned to be Player 1, for that round. Again, subjects were asked to indicate their contribution decision, conditional on all the potential scenarios, depending on their position in that round. Finally, Treatment 3 ( $T_3$ ), was designed to investigate the impact of position certainty on the level of contributions, thereby testing the predictions of Hypothesis 3. The features of the treatment are identical to those in $T_1$ , with the only difference being that participants were now informed about their role (i.e. position in the sequence). Subjects were asked about their contribution choice, which was conditioned on the potential total contribution of the two immediate predecessors in the sequence. Table 2 summarises the details of our treatments.

Table 2 Summary of all the Treatments

Treatment	No. of subjects	Group size	Sample Size	Position	Prediction
T1	32	4	2	Unknown	Contribute only if $r\ge 2.66$ and observe full contribution
T2	32	4	1	Unknown	Pure when $r\in [2,2.4]$ . Mixed when $r\in (2.4,4]$
T3	32	4	2	Known	Contribute only if $r\ge 4$

We conclude this section with a comment on the choice of the value of the parameter r (the return from contributions parameter). We set the value of r to be equal to 3, for three reasons: (1) this value of r satisfies the inequality in condition 1, ensuring that subjects prefer to contribute, when they observe full contribution in $T_1$ , (2) when $r\in (2.4,4]$ , the model predicts that there exists a mixed strategy equilibrium, with mixing probability $\gamma =0.528$ Footnote ¹⁴ in $T_2$ , and (3) when $r<4$ , it is predicted that full contribution will unravel in $T_3$ .