2.1 Procedures and overview

The experiment was held at the London School of Economics and Political Science's Behavioural Research Lab from 20–26 November 2015 and was advertised as a study on economic decision-making. Participation was open to registered students; subjects had no previous experience in CPR or network experiments. The experimental game was executed using z-Tree, and subjects interacted anonymously with each other by taking decisions in private (Fischbacher, Reference Fischbacher2007).

Our design consisted of a between-subjects CPR game with three monitoring and punishment network treatment groups: CN, UCN and DCN. The network treatment and the subject's node type within the network (A, B, C, or D) were held constant throughout each experimental session. An experimental session consisted of 15 decision-rounds, and this was common knowledge. A stranger-matching procedure was used, where each round started with the computer randomly forming four-person network groups by selecting one subject of each type. Thus, while the node type and the network treatment were fixed in each session, the composition of the network group in each round depended solely upon chance and was independent of group formation in any of the other rounds. We conducted seven sessions with either 16 or 20 subjects, for a total of 172 subjects, where each subject participated only in one session. Since group membership was randomized at the beginning of each round, the independent observations are at the session-level (the number of sessions was three for CN, and two for both UCN and DCN). Table 2 summarizes the experimental design and the number of observations in each network treatment.Footnote ⁵

Table 2. Experimental design

Each round in each session consisted of two stages. In stage one, subjects received an endowment of 25 tokens and decided how to divide it between a common account and a private account. As explained in more detail in the following section, a subject's decision of how much of her endowment to allocate to the common account can be interpreted as a decision about how much to appropriate from the CPR. In stage two, eligible subjects could monitor and assign punishment points to those in their network group. Alongside making an appropriation decision, subjects could also input their beliefs about the expected token placement of the other three group members in the common account in stage one in each round to estimate what their payoffs would be.

The network structure and payoff function, parameter values, punishment rules and number of rounds were common knowledge. Experimental tokens were converted to British pounds (GBP) at the end of the experiment at the rate of 25 experimental tokens = GBP 1, and average earnings were around GBP 7.5.

2.2 The common pool game with networks

Let $e$ be the total amount of tokens available to each subject (i.e., a subject's initial endowment). As previously mentioned, in the first stage subjects decide how many tokens from their initial endowment to allocate between a common account and a private account, where allocation to the private account is assumed to yield a constant marginal return $w$. We denote by ${x_i} \in [{0,\; e}]$ the amount that individual $i$ allocates to the common account. Consistent with the CPR literature, we interpret ${x_i}$ as the appropriation choice of subject $i$; the total appropriation of all subjects within the network group is given by $X = \sum\nolimits_i {{x_i}}$.

The group return from appropriation is $F (X) = aX-b{X^2}$. This nonlinear payoff structure is commonly used in the literature and aims to approximate the complexity arising from socio-ecological dynamics that agents typically face in the field (Ostrom et al., Reference Ostrom, Walker and Gardner1992). The return from appropriation to an individual agent is proportional to the ratio between her appropriation effort $({x_i})$ and the total effort of all appropriators $(X)$. The ratio ${x_i}/X$ is referred to as the ‘individual distribution factor’ (Apesteguia and Maier-Rigaud, Reference Apesteguia and Maier-Rigaud2006), and captures the negative externality accruing to the group from an individual's appropriation decision: the higher ${x_i}$ is in relation to $X$, the higher is $i$'s appropriation from the CPR, allowing an appropriator to capture a larger share of returns from the CPR $(aX - \textrm{\;} b{X^2})$. The first-stage payoff of each agent $i$ is given by:

(1)\begin{equation} \pi _i^1 = w ({e - {x_i}}) + \left( {\frac{{{x_i}}}{X}} \right) ({aX - b{X^2}}).\end{equation}

As per equation (1), subjects made their CPR appropriation decision by dividing tokens between the private and common accounts in stage one, and also stated their expectation about each of the other group members' appropriation decision (instructions and interface snapshots are provided in online appendix A and B respectively). At the end of stage one, each subject was informed only of his/her earnings from the common and private account, and of the total earnings from stage one.

In stage two, subjects simultaneously monitor and make punishment decisions regarding their neighbors, depending on the types of networks they are embedded in, as illustrated in figure 1. A tie between two subjects within a network group indicates that they are connected to each other in a monitoring and punishment relationship, and the arrowhead points to the agent whose appropriation can be monitored and punished. For each subject $i$, ${N_i}$ denotes her monitoring and punishment neighborhood, i.e., the set of subjects $j \ne i$ who can be monitored by $i$. Subject i can thus observe how many tokens subjects in ${N_i}$ placed into their private and group accounts in stage one, and can furthermore assign deduction points to target individuals in the neighborhood. An undirected tie between any two nodes implies that the pair of subjects can monitor and punish each other as in the CN and UCN (panels 1a and 1b). Conversely, a directed tie denotes an asymmetric relationship between two agents, so that only the appropriator towards whom the arrowhead points can be observed and punished but not vice versa, as in the DCN (panel 1c). Thus, if a tie runs from $i$ to $j$, then $i$ can monitor $j\; ({j\; \in \; {N_i}})$ but $j$ cannot monitor$\; i\; (i\; \notin \; {N_j})$. To summarize, in the CN all subjects can monitor and punish one another; in the UCN subjects monitor and punish those to whom they are directly connected (e.g., A can monitor and punish B and D, and A is in turn monitored by B and D, but not C); and asymmetric pairwise relationships exist in the DCN because A can only monitor and punish B, B can only monitor and punish C, and so on. In terms of network properties, the only difference between CN and UCN is that the latter has a lower network density (66 per cent for UCN versus 100 per cent for CN) and is incomplete; while the only difference between UCN and DCN is that the latter has directed rather than undirected ties.

Notes: Panels 1a, 1b and 1c illustrate three monitoring and punishment network groups with four types of nodes indexed by i = A, B, C, D. In the CN each pair of nodes is connected by an undirected tie to represent perfect monitoring, i.e., everyone can observe everyone's appropriations and can choose to punish any extractor in the network. In the UCN, the undirected tie implies connected nodes can simultaneously monitor and punish each other. In the DCN the directed tie signifies that each subject can only monitor and punish the subject in the direction it is connected to.

Figure 1. Network treatment groups: (a) Complete Network [0] (CN), (b) Undirected Circle Network [1] (UCN), (c) Directed Circle Network [2] (DCN).

Punishing others entails a personal cost which depends on the number of deduction points assigned. Each agent $i$ can punish $j$ in her monitoring and punishment neighborhood $(j \in \; {N_i})$ to reduce $j$'s payoff from the first stage by $p_j^i$ at a personal unit cost of ${c_a} = 1$ per deduction point assigned. Each punishment point received reduces $j$'s payoff by ${c_r}$ = 3. Following recent CPR experiments (table 1), the fee-to-fine ratio – which represents the cost to the punisher from levying one deduction point and the cost to the target – is 1:3, and it is held constant across all network treatments. The maximum amount of punishment points that could be assigned was bounded by the player's earnings in the first stage, thus allowing for a system of graduated sanctions for appropriators determined by the subjective choice of the agent assigning punishment (Fehr and Gächter, Reference Fehr and Gächter2000). Before proceeding to the next stage, each individual could calculate the total cost of deduction points assigned to others. Equation (2) presents the earnings to each agent from stage two, which could range between 0 and the net earnings from stage one after the cost of assigning and receiving deduction points:

(2)\begin{equation}\pi _i^2 = \max \left\{0,\; \; \pi_i^1 - {c_a}\mathop \sum \limits_{j\, \epsilon \, {N_i}} p_j^i - c_{r}\mathop \sum \limits_{i\,\epsilon \,N_{j}} p_i^{\,j}\right\}.\end{equation}

At the end of stage two, each subject was informed only of his/her total earnings from stage one, the total cost of punishment points given to and received by others, and net earnings from stage two.

Let us consider the standard prediction for this CPR game. If we assume that subjects are rational self-interested agents who choose to maximize their earnings in each round and apply the logic of backward induction, each individual will not issue punishment points in stage two since it is costly to do so ($p_i^j = 0$, for all $i$ and $j \in \; {N_i}$). Consequently, subjects will recognize that nobody will punish in this second stage and choose to appropriate at the socially-inefficient level in stage one in all types of network groups. Thus the standard prediction is that the availability of peer monitoring and punishment in stage two cannot deter free riding in stage one.

Unlike standard treatment comparisons that change only one characteristic at a time, the comparisons of this study necessitate changes in both information and punishment to reflect the underlying change in the type of network architectures studied here. However, as noted in table 1, all institutional features were kept fixed across all treatments, including the use of n = 4 individuals per network group, 15 periods of play, and a strategy space of {0, 1, …, 24, 25} tokens to be allocated to the common and/or the private account. Parameter values were chosen to ensure a good spread in values between the Nash and Pareto equilibrium allocations – 16 and 10 tokens which will yield a payoff of 189 and 225 tokens respectively – while keeping the ratio of Nash to Pareto equilibrium strategies (which determines the size of the negative externality from appropriation) comparable to recent CPR experiments in the literature (table C1 in the online appendix compares parameters to past studies).

The experimental data may show that repeated interactions can influence play across time, which can arise from subjects learning about the rules of the game, or about the preferences or ‘types' of subjects in each session, as well as from strategic play during the early rounds in the game. Since the analysis is for one-shot games, we used a stranger-matching design to balance the need for repeated experience by subjects while attempting to reduce reputation effects (e.g., as in Kreps et al., Reference Kreps, Milgrom, Roberts and Wilson1982). We followed Andreoni and Croson's (Reference Andreoni, Croson, Plott and Smitt2008) suggestion that it is more prudent to use a stranger-matching design when studying single-shot games.Footnote ⁶ While perfect stranger-matching could mitigate investments in reputations, only five rounds could be played with this protocol due to constraints like lab capacity, and this is arguably too short a period to allow for learning of the rules of the game. Moreover, past studies have not clearly established large differences in outcomes between stranger and perfect-stranger-matching (Fehr and Gächter, Reference Fehr and Gächter2000, Reference Fehr and Gächter2002). We tried to address these concerns, however, by first anonymizing an individual's identity while displaying their decisions in stage two (barring their node type) on the computer screen. Secondly, the probability of being in the same group in the following period was small (about 0.25 (0.5) per cent for 20 (16) subjects per session), also helping to mitigate strategic play in earlier periods of the session.

Finally, we paid participants for one randomly selected round of the 15 rounds played at the end of the game to mitigate strategic play in the early rounds and potential wealth effects. The round was selected at random by drawing one of 15 numbered ping pong balls at the end of the experiment. After this, subjects were paid their earnings from the CPR game, and the potential earnings from their stated expectation about the appropriation of others in the group for that round (if it matched the actual choices that the others made).Footnote ⁷ Since it was possible to receive a negative payoff in a given period, we instituted a rule that if players received negative payoffs in the round selected for payment, they would obtain no payment (we did not invoke this rule in any instance of payment).

Much care was taken to ensure that subjects understood the rules of the experiment and the incentive structure. Printed instruction booklets, with detailed explanations of the network structure and the CPR game, were placed at the computer terminals. The framing of instructions was maintained consistent across all treatments, similar to that in the literature in trying to avoid value-laden language. For example, the common pool appropriation decision was framed in terms of allocating tokens to a common account. The punishment decision was framed as the assignment of deduction points to the other people in the network group.Footnote ⁸ Apart from numerical examples to illustrate payoffs within the instruction text, an earnings table (see online appendix A) was also displayed showing potential payoffs in tokens corresponding to different token placements in the common account by the subject and total tokens placed by the three others in the group to aid comprehension, similar to the earnings table used in Apesteguia and Maier-Rigaud (Reference Apesteguia and Maier-Rigaud2006). To further familiarize subjects with the incentive structure and rules, subjects first read the instructions by themselves at the start of the session, and the instructions were then read aloud by the experimenter. After this, subjects answered trial questions at the end of the instruction booklet, which was checked by the experimenter before proceeding with the experiment.Footnote ⁹ Experimental sessions typically lasted for an hour and 15 min, with 15 min of instructional time.

2.3 Hypotheses

Building upon insights from the existing literature, we formulate hypotheses about whether and how appropriation and punishment outcomes vary across network structures. Specifically, our first hypothesis is

This hypothesis builds upon results from the role of imperfect monitoring and punishment networks in the PG game context, which suggest that average contributions tend to be similar across networks where all players are connected to one another, as is the case with the CN, UCN and DCN (Carpenter et al., Reference Carpenter, Kariv and Schotter2012; Boosey and Isaac, Reference Boosey and Isaac2016).

To better understand how the network structure impacts punishment, we consider the ‘punishment amount', which refers to the total number of punishment points received by an individual in stage two of the CPR game in a given round. In addition, we estimate the ‘punishment severity’ by weighting the average punishment received by the number of potential punishers targeting each subject (denoted by the in-degree). That is, we assign weights of 3, 2 and 1 in the CN, UCN and DCN, respectively. Lastly, we also consider the ‘punishment incidence’, which is whether the subject received any punishment in a given round. In denser networks, like the CN, each individual faces a higher number of potential punishers than in less dense networks; this, in turn, may result in a higher punishment amount being received by each individual, as well as a higher incidence of punishment. On the other hand, less dense networks like UCN, which features fewer punishment opportunities, may increase the severity of punishment allocated per opportunity. How this potential increase in punishment severity affects the amount and the willingness to punish is an open empirical question. Turning to the available experimental evidence, Carpenter et al. (Reference Carpenter, Kariv and Schotter2012) found that punishment amounts are similar between CN and UCN networks, but that the punishment amount used in the DCN is higher, possibly because the attribute of directedness specifies clearer monitoring and punishment responsibilities for each subject. Thus, we hypothesize that:

Understanding individual-level punishment behavior allows us to examine the types of punishment given in relation to the appropriation behavior of the target agent. Past studies have highlighted three broad non-standard motivations for punishing peers, namely: (i) prosocial inclinations to punish free riders, (ii) antisocial punishment to penalize cooperators, and (iii) punishment due to errors (Ostrom et al., Reference Ostrom, Walker and Gardner1992; Herrmann et al., Reference Herrmann, Thöni and Gächter2008; Casari and Luini, Reference Casari and Luini2012). When considering prosocial punishment, past studies indicate that punishing free riding is generally triggered by negative emotions that arise from a violation of either reciprocity or fairness norms, as well as from feeling exploited (being the ‘sucker’) (Ostrom et al., Reference Ostrom, Gardner and Walker1994; Fehr and Schmidt, Reference Fehr and Schmidt1999; Frey and Meier, Reference Frey and Meier2004; Fehr et al., Reference Fehr, Fischbacher and Kosfeld2005). Ostrom et al. (Reference Ostrom, Walker and Gardner1992) further differentiated between one-round prosocial punishment received because the person fined was the highest (or one of the highest) appropriator in the current round, and lagged punishment where the punished person was one of the highest appropriators in the prior round. Regarding antisocial punishment, they noted the occurrence of blind revenge, which is targeted at low appropriators by those who received punishment in a previous round. Other studies also remark on the possibility of targeted revenge aimed at an identifiable punisher when counter-punishment opportunities are available, as well as spiteful or expressive punishment by those who punish because they have a ‘hard-wired’ taste for it (Nikiforakis, Reference Nikiforakis2008; Jensen, Reference Jensen2010; Casari and Luini, Reference Casari and Luini2012). While our experiment was not designed to differentiate between these motivations, it can nonetheless shed some light on this issue.

To measure whether different network treatments affect prosocial punishment and antisocial punishment, we define them as follows. ‘Prosocial punishment’ is the punishment given to a target agent who allocated more tokens to the common account than the theoretical Pareto optimal benchmark of 10 tokens. ‘Antisocial punishment’, on the other hand, is measured as the punishment given to a target agent who appropriates at or below this Pareto optimal level. Building on Ostrom et al. (Reference Ostrom, Walker and Gardner1992), antisocial punishment is likely to stem from three sources: blind revenge by someone who was a low CPR appropriator in the previous round; free rider revenge if the punisher was a free rider who appropriated above the Pareto optimal level and was punished in the last round; or other types of antisocial punishment arising from error or a hard-wired taste for punishment. However, unlike in Ostrom et al. (Reference Ostrom, Walker and Gardner1992), we can eliminate the possibility of lagged prosocial punishment (targeting an agent who over-appropriated in the previous round) and targeted antisocial counter-punishment (as a response to being punished by the target in the previous round), since we use a stranger-matching design without a counter-punishment stage where individual players cannot be identified in the subsequent round as their identities are anonymized. Assuming that those types with a hard-wired and vengeful predisposition to punish were evenly distributed across network treatments, ex-ante, we reason as follows.

The past experimental literature on the bystander effect shows that a reduction in the number of possible ‘helpers’ increases prosocial behavior (Cryder and Loewenstein, Reference Cryder and Loewenstein2012). This implies that the burden of responsibility is more diffused in the CN due to a larger number of potential punishers who can act prosocially by sanctioning free riders in stage two. The UCN, thus, will offer less ambiguity about an individual's responsibility to punish free riders since they are less densely connected and, arguably, the DCN can offer even more clarity because it is directed. Free riders may be more likely to be punished – especially in the DCN – as it offers unambiguous responsibility to monitor and punish a single agent. Indeed, Carpenter et al. (Reference Carpenter, Kariv and Schotter2012) found that free-riders – defined as those who contribute nothing to the public good – are more likely to be punished in DCN relative to the UCN (but that the difference was not statistically significant).

By this logic, it is also possible that the clearer delineation of responsibility afforded by UCN and DCN in this CPR dilemma may also attenuate the extent of antisocial punishment. We remain agnostic about whether the pattern of types of antisocial punishment – namely blind revenge, free rider revenge and residual cases such as punishment by error – differs across network groups.