1. Introduction
Organizations generate coherent, intricate patterns of coordinated activity. This is no mean feat. Van Huyck et al. (Reference Van Huyck, Battalio and Beil1990) famously documented a consistent pattern of coordination failure in large groups playing the minimum-effort game, where the group members stand to benefit if they successfully coordinate on high actions, but each player is incentivized to match the lowest action in the group.
The classic minimum-effort game captures a stylized organization where incentives are coarse and untargeted: each individual is rewarded based on the entire group’s performance, and thus is held responsible for coordinating with everyone else. In practice, the scope and complexity of task interdependencies within most organizations are limited – by design. Production may be organized so that workers on an assembly line coordinate amongst themselves but operate almost independently of the rest of the organization. Incentives may be tailored so that a team member is responsible only for completing his own assignments while the manager is responsible for the team’s overall performance.
A particularly salient feature of many real organizations is that task and incentive interdependencies are predominantly acyclic. Work often flows along a chain of responsibility, from upstream tasks to downstream ones, without substantial feedback in the opposite direction. A production line typically imposes a strict order of operations: the quality of a downstream worker’s output depends on the care taken upstream, but upstream workers are not directly penalized for mistakes made further down the line.
Hierarchical reporting structures provide another example of acyclic interdependencies. An entry-level worker may be tasked with mechanically following procedures and instructions, so that his payoff is independent of others’ actions. In contrast, a senior manager is often held accountable for coordination failures even when her subordinates are at fault. In such environments, responsibility and exposure to coordination failures flow upward through the hierarchy, forming directed chains rather than reciprocal loops.
Suppose we represent the set of interdependencies in an organization’s task and incentive design as a network across agents. Put loosely, if agent 
$2$’s payoff depends on agent 
$1$’s actions, we draw a link from 
$1 \rightarrow 2$. Given a network representation of organizational structure, we seek to understand how organization-wide coordination emerges from the ensemble of network interactions.
To do so, we study a generalization of the minimum-effort game where players are linked by a directed network, and each player is incentivized to match the lowest action amongst their direct links. We refer to this setting as the network minimum game.
The classic minimum-effort game corresponds to the special case of a complete network. In contrast, our analysis allows for incomplete networks, thus capturing the notion of limited within-organization interdependencies. Furthermore, by considering directed networks, we allow for asymmetric interdependencies between players: player 
$i$’s payoff may depend on player 
$j$’s actions, but not vice versa. Such asymmetries allow us to capture acyclic task dependencies within organizations, as discussed above.
Our experimental setting represents repeated interactions within long-lived organizations with persistent structure: participants play in fixed groups, with fixed network structure, for ten rounds. The treatments vary two aspects of network structure: network cyclicity (the existence of cycles of dependencies in the network; see Figure 1) and network density (the number of links per player).
Cyclic vs. acyclic networks (examples)
We find that network cycles inhibit coordination. Table 1 summarizes our key findings. On acyclic networks of all densities, players coordinate on almost-maximal actions (the maximum action being 
$7$). Cyclic networks perform worse than acyclic networks. Dense cyclic networks perform worst: they generate almost-minimal actions (the minimum action being 
$1$). That is, cycles matter, especially for dense networks.
Mean final-round action, by network structure
Table 1 Long description
The table reports average final-round action values for groups of six, comparing sparse versus dense networks and acyclic versus cyclic structures. In sparse networks, the mean action is 6.52 for acyclic and 4.90 for cyclic. In dense networks, the mean action is 6.72 for acyclic and 2.08 for cyclic. Acyclic networks are consistently high and similar across density, staying around the mid sixes. Cyclic networks drop as density increases, from just under five in sparse to just above two in dense. The variability is smaller for acyclic conditions than for cyclic conditions, with the largest spread in sparse cyclic and the smallest in dense acyclic. These values are averages, so individual group outcomes may differ from the means.
Mean [st. dev.] action for each network structure, averaged over groups of size 
$n=6$. Action set 
$=\{1,2,\dots,7\}$.
Our experiment’s acyclic networks are hierarchical: they take the form of a chain of command where higher-indexed participants depend on lower-indexed participants. This construction captures the essence of an organizational hierarchy, where supervisors are held responsible for their subordinates’ activities, and each worker has a unique chain of command leading to the CEO. Some papers study the efficiency of hierarchical structures from the perspective of organizational design (e.g.,Harris & Raviv, Reference Harris and Raviv2014; Radner, Reference Radner1993; Sah & Stiglitz, Reference Sah and Stiglitz1986). The same network structure can also be interpreted as a stylized sequential production technology: directed paths represent production lines or process flows in which intermediate outputs move from upstream to downstream stages, so that a downstream stage’s payoff depends on the care taken upstream but not on actions further down the line. Our results highlight that acyclic structures – including hierarchies and sequential production lines – are particularly effective at fostering coordination.Footnote 1
What mechanisms underlie our results? We have in mind that dependency cycles create feedback loops that allow ‘seeds’ of strategic uncertainty to circulate and amplify, potentially leading to coordination failure: eventually, each player selects a low action simply because he anticipates that the next player in the cycle may do the same, ad infinitum. In contrast, destructive feedback loops do not arise in acyclic networks, thus enabling successful coordination.
This logic may be starkly interpreted in terms of the Nash equilibrium. For acyclic networks, the unique Nash equilibrium is for all players to take a maximum action. Whereas, for our cyclic networks, every common action level is a Nash equilibrium – that is, the strategic uncertainty associated with network cycles translates to equilibrium multiplicity. Such multiplicity, however, also means that the Nash equilibrium is silent about why sparse cyclic networks coordinate more successfully than dense cyclic networks. (Indeed, the set of pure-strategy Nash equilibria for cyclic networks is independent of network density.) Nor does it speak to some of our other experimental findings. We find, for instance, that in cyclic networks, the level of coordination is independent of cycle length. We also find that in acyclic networks, participants who are higher in the pecking order – that is, participants whose payoffs depend more on others’ actions – take lower actions.
To address these experimental findings and to enrich our intuitions about how network structure amplifies or dampens strategic uncertainty, we analyse logit (quantal-response) equilibria of the network minimum game (in Appendix A). In logit equilibrium, seeds of strategic uncertainty are introduced by assuming that each player inevitably makes small mistakes when choosing actions (Anderson et al., Reference Anderson, Goeree and Holt2001; Goeree et al., Reference Goeree, Holt and Palfrey2016; McKelvey & Palfrey, Reference McKelvey and Palfrey1995). This modelling device allows us to tractably capture the feedback-loop mechanisms discussed above. Furthermore, logit equilibrium produces sharp comparative static predictions that match our experimental findings.
We also examine how participants learn to coordinate over time. We find that coordination is more resilient in acyclic networks. In initial rounds, play is noisy, and average actions are intermediate for both cyclic and acyclic networks. Participants in acyclic networks tend to overcome such initial miscoordination: average actions increase towards the maximum level over time. In contrast, in cyclic networks, initial miscoordination is exacerbated: average actions decrease over time. We interpret our results as being largely consistent with a notion of ‘learning to coordinate’ where participants gradually improve at predicting others’ actions and at making optimal decisions. Viewed in this light, acyclic networks empower players to develop and maintain coordinated outcomes.
1.1. Related literature
Our paper builds on a large experimental literature on weakest-link/minimum-effort games and on coordination games played on networks.
1.1.1. Minimum-Effort/Weakest-Link Games
The classic minimum-effort game of Van Huyck et al. (Reference Van Huyck, Battalio and Beil1990) has been used extensively to study coordination with Pareto-ranked equilibria and strategic uncertainty. Early experiments by Van Huyck et al. (Reference Van Huyck, Battalio and Beil1990), Knez and Camerer (Reference Knez and Camerer1994), and Camerer and Knez (Reference Camerer and Knez2000) documented robust coordination failure in baseline designs: in large fixed groups without communication, play typically converges to the lowest-effort equilibrium, whereas small fixed pairs often reach the efficient equilibrium.Footnote 2 These findings have motivated a vast literature that examines how structural, cognitive, and behavioural factors influence coordination outcomes.Footnote 3
A first strand asks when and why coordination failure can be overcome. Weber (Reference Weber2005); Weber (Reference Weber2006) show that efficient coordination in large weakest-link groups can be achieved by gradual growth: start with a small, efficiently coordinated group, then add new members who observe the group’s history. Brandts and Cooper (Reference Brandts and Cooper2006) demonstrate that temporarily increasing the payoff from high-effort coordination can move groups to efficient outcomes that persist even after incentives are scaled back. Weber et al. (Reference Weber, Camerer and Knez2004) study a ‘virtual observability’ treatment in which players move sequentially in the minimum-effort game but cannot observe predecessors; although the information and payoff structure is unchanged, such timing reduces uncertainty and modestly improves coordination. Chen and Chen (Reference Chen and Chen2011) show that inducing a salient group identity among participants increases effort in weakest-link games, even in relatively large groups.
More recent work continues to refine our understanding of how design and environment shape coordination in weakest-link games. Leng et al. (Reference Leng, Friesen, Kalayci and Man2018) and Gärtner et al. (Reference Gärtner, Östling and Tebbe2023) study long-horizon and continuous-time versions of the weak-link game and find that longer horizons or continuous time do not by themselves resolve coordination failure. Feri et al. (Reference Feri, Gantner, Moffatt and Erharter2022) use belief elicitation and structural modelling in a repeated minimum-effort game to show that some agents systematically play above their stated beliefs about others’ effort, effectively acting as ‘leaders’ trying to pull the group towards more efficient outcomes. Szkup and Trevino (Reference Szkup and Trevino2025) study costly information acquisition in coordination games and find that higher-precision information choices lead to more coordination attempts and higher success rates, pointing to information design as another lever for mitigating strategic uncertainty. Caparrós Gass et al. (Reference Caparrós Gass, Blanco and Finus2025) investigate endogenously formed institutions in weakest-link games with fixed neighbourhoods and show that even weak unanimous institutions can substantially raise effort, while also providing a quantal-response-equilibrium interpretation of their data.
A second, closely related strand examines dynamic and intergenerational aspects of coordination in minimum-effort settings. Cooper et al. (Reference Cooper, DeJong, Forsythe and Ross1992), Charness (Reference Charness2000), and Blume and Ortmann (Reference Blume and Ortmann2007) show that pre-play cheap-talk communication – including intergenerational advice – can greatly improve coordination in weakest-link games, although not all groups manage to sustain high effort. Van Huyck et al. (Reference Van Huyck, Gillette and Battalio1992), Brandts and MacLeod (Reference Brandts and MacLeod1995), and Weber et al. (Reference Weber, Camerer, Rottenstreich and Knez2001) model leadership by allowing one participant to send a pre-play message; such leadership can help, but is no panacea. Weber (Reference Weber2006) demonstrate that efficient coordination can be ‘built’ through gradual organizational growth, while Brandts and Cooper (Reference Brandts and Cooper2006) show that once high-effort profiles are reached under favourable incentives, they may persist even when incentives are later weakened. Avoyan and Ramos (Reference Avoyan and Ramos2023) show that partially binding communication – which falls between cheap talk and full commitment – can further improve coordination. Relative to this work, our contribution is to show that even in the absence of communication, leadership, or institutional change, appropriately designed acyclic network structures can deliver coordination levels in large groups of size 
$6$ or 
$12$ that are comparable to those achieved with these richer institutions.Footnote 4
1.1.2. Coordination on Networks
A growing experimental literature studies coordination games played on fixed networks. Keser et al. (Reference Keser, Ehrhart and Berninghaus1998) and Berninghaus et al. (Reference Berninghaus, Ehrhart and Keser2002) show that ‘circle’ and ‘lattice’ networks can sustain higher coordination than small complete networks in certain threshold and coordination games. Cassar (Reference Cassar2007) compares local, random, and small-world networks and finds that small-world networks – which combine short path length with some clustering – achieve better coordination than purely local or purely random networks. Charness et al. (Reference Charness, Feri, Meléndez-Jiménez and Sutter2014) study a class of network games with strategic complementarities and show that networks with greater clustering tend to coordinate more successfully. Gallo and Yan (Reference Gallo and Yan2015) implement a network game with a unique but inefficient Nash equilibrium and find that simple, symmetric networks facilitate informal cooperation to improve on the equilibrium outcome. Relatedly, Choi et al. (Reference Choi, Gallo, Kariv, Bramoullé, Galeotti and Rogers2016) and Calford and Chakraborty (Reference Calford and Chakraborty2020); Calford and Chakraborty (Reference Calford and Chakraborty2021) study network dilemmas with public goods and higher-order beliefs embedded in network formation.
Our paper is particularly close to work that combines weakest-link technologies with endogenous or rich network structure. Riedl et al. (Reference Riedl, Rohde and Strobel2016) introduce endogenous (undirected) network formation in a weakest-link game and show that efficient coordination can emerge because subjects are motivated to take high actions by the threat of exclusion by others. Riyanto and Teh (Reference Riyanto and Teh2020) study weak-link games on networks with adjustable neighbourhoods and find that greater flexibility in choosing whom to interact with substantially improves coordination efficiency. Caparrós Gass et al. (Reference Caparrós Gass, Blanco and Finus2025) consider weakest-link games with fixed neighbourhoods but allow subjects to form ‘institutions’ endogenously to coordinate their choices. Relative to this literature, we work with exogenously fixed networks but vary the presence of directed cycles and the density of dependencies, focusing on how the architecture of a given network – rather than the ability to choose partners or institutions – affects the emergence and robustness of efficient coordination. A key difference is that the networks studied in the papers just cited are undirected, so any link creates a two-player dependency cycle. By contrast, we work with directed networks and can cleanly separate acyclic (hierarchical or sequential) structures from those containing cycles.
1.1.3. Network Design as an Organizational Tool
Finally, from the perspective of organizational design, our results complement theoretical work on hierarchies, polyarchies, and the processing of information in organizations.Footnote 5 The network minimum game highlights a downside of dense interdependencies: additional links strictly reduce performance in logit equilibrium, and closing a single directed cycle in an otherwise hierarchical structure can dramatically undermine coordination. At the same time, our experiment shows that acyclic structures – including hierarchies, sequential production lines, and other directed acyclic graphs – are particularly effective at fostering and sustaining high-effort coordination, even without communication or explicit leadership. Relative to the many other devices that have been shown to improve coordination in weakest-link games (communication, leadership, identity, dynamic growth, institutional design), network design is simple and robust: merely avoiding cycles of responsibility can substantially reduce the scope for destructive feedback loops of strategic uncertainty.
2. Framework
2.1. Background: the (classic) minimum-effort game
The classic Minimum-Effort Game models coordination amongst an 
$n$-player group under a ‘weakest-link’ production technology where output is determined by the group’s worst performer. Each player 
$P_i$ in the group 
$\{P_1,...,P_n\}$ simultaneously chooses an action 
$x_{i}$ from a compact action set 
$X \subset \mathbb{R}$, and receives the minimum group action less a private action cost:
\begin{equation*}
\pi_i \left( x_{1},...,x_{n} \right) = \min \left\{x_{1},...,x_{n} \right\} - c x_{i} \text{with }c \lt 1.
\end{equation*} In most existing experimental implementations, the action set 
$X$ is discrete: each participant chooses an integer action between 
$1$ and 
$7$. We follow this convention in the laboratory, but impose a continuous action set in our theoretical analysis (Appendix A).
With at least two (self-interested) players, any symmetric action profile 
$\left(x,\dots ,x \right)$ is a Nash equilibrium. These equilibria are Pareto-ranked, with higher-action equilibria being more efficient. The natural interpretation is that low-action equilibria represent coordination failure.
Table 2 reports existing experimental results from papers to highlight one of the most consistent findings about the classic minimum-effort game (absent communication or network structure): coordination deteriorates dramatically with group size, and failure (in the form of almost-minimal actions) is almost inevitable in groups with four or more players.
Average final-round actions in classic minimum-effort games, by group size 
$n$
Table 2 Long description
The table reports average actions chosen in the final round of classic minimum-effort games, summarized by study and group size. In pairs, averages are high: 6.39 in Van Huyck et al. (1990) and 6.85 in Camerer and Knez (2000). For groups of 3, averages are lower at 4.05 in Knez and Camerer (1994) and 3.48 in Camerer and Knez (2000). For groups of 6, Knez and Camerer (1994) reports a much lower average of 1.52. For groups of 9, Cachon and Camerer (1996) reports 2.06, and for groups of 14 to 16, Van Huyck et al. (1990) reports 1.74. Overall, the available entries suggest a decline in final-round actions as group size increases, but not every study reports every group size, so comparisons are based on a sparse set of cells.
Mean action out of action set 
$\{1,2,...,7\}$, averaged over groups. List of papers is from Table 2 of Weber (Reference Weber2006), excl. Chaudhuri et al. (Reference Chaudhuri, Schotter and Sopher2009) where data for final-round average actions was unavailable.
2.2. The network minimum game
Let’s augment the classic minimum game with a (connected) directed network 
$\textbf{g}$ over the group 
$\{P_1,...,P_n\}$. If there is a link 
$P_j \rightarrow P_i$, then we say that 
$P_i$ depends on 
$P_j$. We require that players always depend on themselves: 
$P_i \rightarrow P_i$ for all 
$P_i$. Each player 
$P_i$’s neighbourhood 
$S_{i}$ is the subset of players that 
$P_i$ depends on.Footnote 6
The network minimum game differs from the classic minimum game in just one respect: each player 
$P_i$’s payoff depends only on actions of those in 
$P_i$’s neighbourhood,
\begin{equation}
\pi_i \left( x_{1},...,x_{n} \right) = \min \left\{x_{j}: {j} \in S_{i} \right\} - c x_{i} \text{with} c \lt 1.
\end{equation} Notice that the classic minimum-effort game is a special case of the network minimum game where 
$\textbf{g}$ is the complete network: 
$P_i \rightarrow P_j$ for all 
$i,j \in \{1,...,n\}$.
Some terminology: a network path is a sequence of players where each player depends on, and is distinct from, his predecessor. A network cycle is a network path that starts and ends with the same player. A network without network cycles is acyclic.
In any acyclic network, the unique Nash equilibrium is for everyone to play the maximum action, 
${x}_{i} \equiv \max X$. Network cycles introduce equilibrium multiplicity: for any network cycle, any common action 
$x_i \equiv x \in X$ is a Nash equilibrium. Taken together, these observations hint at the central point of our paper: network cycles introduce strategic uncertainty, potentially leading to coordination failure. However, this multiplicity also implies that the Nash equilibrium is silent about how coordination varies across different cyclic networks. Instead, we appeal to an alternative solution concept.
In Appendix A, we analyse logit equilibria of the (one-shot) network minimum game. Logit equilibrium introduces seeds of strategic uncertainty into the interactions between players. Each player best-responds ‘noisily’ – by playing a distribution over actions where higher-payoff actions are chosen (exponentially) more frequently – to the similarly noisy play of others. Such noise captures the strategic uncertainty inherent in our coordination-game setting. Furthermore, as in Anderson et al. (Reference Anderson, Goeree and Holt2001)’s logit-equilibrium analysis of the classic minimum-effort game, the introduction of noise shrinks the set of equilibria relative to the Nash case, and thus serves as an effective equilibrium-selection device. Consequently, logit equilibria produce intuitive predictions about the circumstances under which coordination failure occurs. When discussing the intuitions underlying our experimental results in Section 4, we will often refer the interested reader to theoretical results in Appendix A.
3. Experimental design
3.1. Network treatments
In the experiment, groups of participants played a version of the network minimum game where actions are integers between one and seven, i.e., 
$X = \{1,2,...,7\}$.
The experiment adopted a between-subject design: each participant in each session was assigned to a specific network position within a group of fixed size and network structure, and each group played ten rounds of the network minimum game. Groups differed in size: 
$n \in \left\{2,3,4,6,12 \right\}$. Modulo size, network structure took one of four (
$=2 \times 2$) forms, which differed along two dimensions: density and cyclicity. Each treatment thus corresponded to some combination of group size 
$\times$ network density 
$\times$ network cyclicity. This taxonomy is illustrated in Figure 2 for groups of size 
$n=6$.
Six-player networks
Fig. 2 Long description
The image consists of four circular diagrams, each representing a different network structure. The first diagram, labeled Acyclic Sparse, shows six nodes labeled P subscript 1 through P subscript 6 arranged in a circle with directed arrows connecting each node to the next in a clockwise manner, forming a simple path. The second diagram, labeled Acyclic Dense, also has six nodes labeled P subscript 1 through P subscript 6, but each node is connected to every other node with directed arrows, creating a fully connected network. The third diagram, labeled Cyclic Sparse, is similar to the Acyclic Sparse diagram but includes an additional arrow from P subscript 6 to P subscript 1, forming a complete cycle. The fourth diagram, labeled Cyclic Dense, mirrors the Acyclic Dense structure but also includes an additional arrow from P subscript 6 to P subscript 1, completing a cycle within the dense network. Each diagram is enclosed in a circle with its respective label below it.
Consider our acyclic networks. Both the sparse and dense acyclic networks are hierarchical: participant 
$P_i$ depends on participant 
$P_j$ only if 
$i \geq j$, so higher-indexed participants depend (directly or indirectly) on lower-indexed participants. The sparse acyclic network is minimally connected, in the sense that removing any link will partition the group into two distinct components. The dense acyclic network is maximally connected, in the sense that adding any link will introduce a network cycle. So, our sparse and dense networks represent extremes in density amongst the set of connected acyclic networks.
Our cyclic networks are defined in relation to our acyclic networks. To create the sparse/dense cyclic network of size 
$n$, we add a single link – chosen judiciously to complete a network cycle – to the sparse/dense acyclic network of size 
$n$. Each cyclic network is thus ostensibly identical to its acyclic counterpart, except for the single additional link.
Some combinations of group size 
$\times$ network density 
$\times$ network cyclicity were not implemented. Table 3 summarizes which, and how many, network treatments were run; Appendix E illustrates each such treatment.Footnote 7
Experimental design
Table 3 Long description
The table indicates which experimental conditions were run across sample sizes 2, 3, 4, 6, and 12, split by structure type (acyclic or cyclic) and connectivity (sparse or dense). For sample size 2, only the cyclic dense condition was run, with ten groups. For sample sizes 3 and 4, three conditions were run: acyclic sparse (nine groups at 3; ten groups at 4) and cyclic sparse (ten groups for both), while the dense conditions were not run. Sample size 6 is the most complete: all four conditions were run, each with ten groups. For sample size 12, only the acyclic dense condition was run, with two groups. Blank cells mean that condition was not run, and group counts vary by condition, so comparisons across sample sizes should consider the unequal number of groups.
$\checkmark_{[m]}$ = treatment was run; 
$[m]=\text{number of groups}$.
3.2. Procedural details
Experimental sessions ran from April to August 2017 at UNSW Sydney’s BizLab. Participants were recruited from the university’s subject pool and administered via ORSEE (Greiner, Reference Greiner2015); the experiment was programmed in zTree (Fischbacher, Reference Fischbacher2007). Overall, 421 participants participated in 33 sessions plus a pilot study, with 12 to 30 participants per session. Each treatment was played by ten groups of participants.Footnote 8 Groups were fixed throughout.
Participants faced a version of the payoff function from Equation (1). Specifically,
\begin{align*}
\pi_i(x_1,...,x_n) = 6 + 3 \min \{x_j:j\in S_i \} - 2 x_i,
\end{align*}where payoffs were denominated in AUD. Payoff information was presented to participants in the form of Table 4. Participants were paid their experimental earnings from one randomly selected round plus a show-up fee of AUD 5. No participant was allowed to participate in more than one session. On average, each session lasted about 50 minutes, and each participant earned AUD 16.27.
Network minimum game payoffs
Table 4 Long description
The table lists minimum game payoffs for combinations of your chosen action and the minimum action observed in your neighbourhood. Rows are your action from one to seven, and columns are the neighbourhood minimum from one to seven. Values are provided only when your action is at least as large as the neighbourhood minimum; other cells are not applicable and shown as dashes. The highest payoff shown is thirteen when both your action and the neighbourhood minimum are seven. The lowest payoff shown is minus five when your action is seven and the neighbourhood minimum is one. For any fixed neighbourhood minimum, payoffs generally increase as your action increases; for example, when the neighbourhood minimum is four, payoffs rise from ten at action four to four at action seven in the opposite direction is not shown because lower actions are not applicable. For any fixed action, payoffs decrease as the neighbourhood minimum drops; for example, at action seven the payoff falls from thirteen at minimum seven to minus five at minimum one. Negative payoffs appear only when your action is high while the neighbourhood minimum is very low, such as action seven with minimum two or one. Interpret comparisons only within the filled triangular portion of the table, since missing cells indicate combinations that are not defined in this setup.
The experimental design corresponds to a complete-information setting. At the start of the experiment and at the start of each round, each participant was reminded about the network structure and their position within the network. At the end of each round, each participant was informed about every participant’s action within their group and every participant’s neighbourhood’s minimum action in that round. Each participant only participated in one session and was only exposed to one network structure and one position within that network.
Participants received written instructions and were then shown a 
$5$-minute video which explained each step of the experiment.Footnote 9 Before the start of the experiment, participants had to pass two on-screen comprehension tests, after which all participants in the session started the experiment simultaneously.
4. Experimental findings
This section studies how network density and network cyclicity affect coordination. Here, the analysis is static: we report mean final-round actions for each treatment, averaged over groups. In contrast, Section 5 will study dynamics: that is, how mean actions evolve over the ten rounds. Throughout our analysis, the mean action for one group is treated as a single observation. The approach controls for potential within-group correlations in a conservative fashion.
4.1. Acyclic Networks
Our first experimental finding is that groups coordinate well in acyclic networks regardless of group size or network density.
Table 5 lists average final-round actions for sparse and dense acyclic networks of various group sizes. Participants achieved close to the maximum action in the final round of each treatment. Across the various acyclic networks, the final-round average action ranged from 
$6.52$ to 
$6.72$ out of 
$7$.Footnote 10
Mean actions by network structure (final round)
Table 5 Long description
The table reports average action in the final round for groups under different treatments defined by group size and network density, comparing acyclic versus cyclic network structures. For acyclic networks, mean actions are consistently high where reported: 6.52 for size 3 sparse, 6.70 for size 4 sparse, 6.52 for size 6 sparse, 6.72 for size 6 dense, and 7.00 for size 12 dense, with small standard deviations shown in brackets. For cyclic networks, mean actions are lower and more variable: 4.50 for size 2 dense, 4.13 for size 3 sparse, 4.13 for size 4 sparse, 4.90 for size 6 sparse, and 2.08 for size 6 dense, with larger standard deviations. Comparing acyclic to cyclic within the same treatment, acyclic exceeds cyclic by about 2.4 points for size 3 sparse, about 2.6 for size 4 sparse, about 1.6 for size 6 sparse, and about 4.6 for size 6 dense. Two-sided tests indicate these differences are statistically significant for size 3 sparse, size 4 sparse, and size 6 dense under both the rank-sum and clustered regression approaches, while size 6 sparse is weaker but still significant under the clustered regression test. Some cells are missing, including acyclic for size 2 dense and cyclic for size 12 dense, so not all treatments allow a direct comparison.
Notes: Mean action in the final round, averaged across groups. Standard deviations in brackets below means. Each column corresponds to a distinct treatment (group size and network density). 
$p$-values from two-sided tests comparing mean actions between acyclic and cyclic networks: (i) Wilcoxon rank-sum test (non-parametric), and (ii) OLS regression with robust standard errors clustered at the group level. There were 10 groups per treatment, with the exception of sparse acyclic 
$n=3$, and acyclic dense 
$n=12$.
To highlight the point that high density does not hinder coordination in acyclic networks, we collected data for two ‘super-dense’ acyclic networks with 
$n=12$ (Figure 3). These networks coordinated remarkably successfully. In the final round, all participants played the maximum action 
$7$.
Large dense acyclic network
Fig. 3 Long description
A circular boundary encloses a directed network drawn with 12 labeled nodes placed around the perimeter. The labels read P1 at the top, then P2, P3, P4, P5, P6, P7 at the bottom, then P8, P9, P10, P11 and P12 continuing around back toward the top. Numerous straight line segments connect pairs of nodes across the interior and many of these segments end with arrowheads at one end, indicating direction along that connection. The connections form a dense web of intersecting lines in the center, with additional arrowed connections drawn close to the perimeter between neighboring labels.
To sum up, neither group size nor network density made a substantial dent on coordination in acyclic networks. Such insensitivity is in stark contrast with the fact that coordination deteriorates rapidly with group size (and thus with network density) in the classic minimum-effort game (see Table 2). But the same insensitivity is consistent with the Nash equilibrium prediction that participants in any acyclic network will coordinate well. Pushing this point further, we find in Proposition A.1 of Appendix A that under logit equilibrium, participants in any acyclic network will choose close-to-maximal actions as long as players’ mistakes are not too large. We will elaborate on this point shortly, when we compare our experimental results for cyclic versus acyclic networks.
4.2. Cyclic (vs. Acyclic) Networks
Our second experimental finding is that coordination weakens when cycles are introduced into a network, especially when the existing network is dense. Recall that in our experimental design, the only difference between each acyclic network and its cyclic counterpart is the addition of a single link.
Table 5 also shows the mean final-round action in sparse and dense cyclic networks of various group sizes. Sparse cyclic networks produce intermediate levels of coordination, but perform substantially worse than sparse acyclic networks. Average final-round actions are significantly lower in each size-
$n$ sparse cyclic network than in the corresponding size-
$n$ sparse acyclic network: Wilcoxon rank-sum tests produce p-values of 
$0.0208$, 
$0.0033$, and 
$0.0571$ for 
$n=3,4,$ and 
$6$, respectively.
Comparing the reported results in Table 5, the difference in final-round actions between the dense acyclic treatment and the dense cyclic treatment is particularly dramatic. With 
$n=6$, the mean final-round action in the dense cyclic network is 
$2.08$ [s.d. 
$1.33$], compared to 
$6.72$ [s.d. 
$.63$] in the dense acyclic network (
$p \lt 0.0001$). That is, the addition of a single link – to a dense network – that completes network cycles has devastating effects on coordination.Footnote 11
Relatedly, actions decrease with network density in cyclic networks. With 
$n=6$, the mean final-round action in the sparse cyclic network is 
$4.90$ [s.d. 
$2.02$], versus 
$2.08$ [s.d. 
$1.33$] in the dense cyclic network (
$p=0.0035$).
Moving beyond point estimates, Figure 4 shows the empirical distribution of action distributions by network treatment for 
$n=6$. Action distributions are significantly higher in stochastic dominance in each acyclic treatment than in its cyclic counterpart: two-sample Kolmogorov-Smirnov (KS) tests and two-sample Epps-Singleton (ES) tests both produce 
$p \lt 0.001$ for final-round actions.Footnote 12
Empirical distributions of acyclic vs. cyclic networks with 
$n=6$
Fig. 4 Long description
The image A showing a step line graph titled n equals 6 Sparse: Round 1. The horizontal axis label is Action with tick labels 1, 2, 3, 4, 5, 6, 7. The vertical axis label is Cumulative Probability with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Two step lines are shown with a legend: n equals 6 acyclic sparse and n equals 6 cyclic sparse. The n equals 6 acyclic sparse line stays at 0 from action 1 through action 3, rises to about 0.5 at action 4, rises to about 0.75 at action 5, rises to about 0.8 at action 6 and reaches 1 at action 7. The n equals 6 cyclic sparse line stays at 0 from action 1 through action 2, rises to about 0.1 at action 3, rises to about 0.5 at action 4, rises to about 0.8 at action 5, rises to about 0.9 at action 6 and reaches 1 at action 7. The image B showing a step line graph titled n equals 6 Sparse: Round 10. The horizontal axis label is Action with tick labels 1, 2, 3, 4, 5, 6, 7. The vertical axis label is Cumulative Probability with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Two step lines are shown with a legend: n equals 6 acyclic sparse and n equals 6 cyclic sparse. The n equals 6 acyclic sparse line stays at 0 from action 1 through action 4, rises to about 0.1 at action 5, rises to about 0.2 at action 6 and reaches 1 at action 7. The n equals 6 cyclic sparse line rises to about 0.2 at action 1, rises to about 0.3 at action 2, rises to about 0.4 at action 3, rises to about 0.5 at action 4, rises to about 0.6 at action 5, rises to about 0.7 at action 6 and reaches 1 at action 7. The image C showing a step line graph titled n equals 6 Dense: Round 1. The horizontal axis label is Action with tick labels 1, 2, 3, 4, 5, 6, 7. The vertical axis label is Cumulative Probability with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Two step lines are shown with a legend: n equals 6 acyclic dense and n equals 6 cyclic dense. The n equals 6 acyclic dense line rises to about 0.2 at action 1, rises to about 0.3 at action 2, rises to about 0.5 at action 4, rises to about 0.7 at action 5, rises to about 0.8 at action 6 and reaches 1 at action 7. The n equals 6 cyclic dense line stays at 0 from action 1 through action 2, rises to about 0.1 at action 3, rises to about 0.3 at action 4, rises to about 0.5 at action 5, rises to about 0.6 at action 6 and reaches 1 at action 7. The image D showing a step line graph titled n equals 6 Dense: Round 10. The horizontal axis label is Action with tick labels 1, 2, 3, 4, 5, 6, 7. The vertical axis label is Cumulative Probability with tick labels 0, 0.2, 0.4, 0.6, 0.8, 1. Two step lines are shown with a legend: n equals 6 acyclic dense and n equals 6 cyclic dense. The n equals 6 acyclic dense line rises to about 0.6 at action 1, rises to about 0.7 at action 2, rises to about 0.8 at action 3, rises to about 0.9 at action 4 and reaches 1 at action 5. The n equals 6 cyclic dense line stays at 0 from action 1 through action 6 and reaches 1 at action 7. Across the four graphs, each panel shows two step lines that reach 1 by action 7. In the sparse graphs, the cyclic sparse line increases earlier than the acyclic sparse line in round 10, while both lines show larger steps around actions 4 and 5 in round 1. In the dense graphs, the acyclic dense line increases earlier than the cyclic dense line in round 10, reaching 1 by action 5, while the cyclic dense line remains at 0 until action 7.
What forces underlie coordination failure in cyclic networks, and why are these forces muted in acyclic networks? We have in mind that small seeds of strategic uncertainty are amplified by strategic interactions in cyclic networks, but not in acyclic networks. We formalize this intuition in Propositions A.1 and A.2 of Appendix A. We have in mind that as best-response dynamics play out over time, doubt unfolds – and potentially reverberates – across the network. Suppose players on a network cycle all (initially) play the maximum action. Inject a small ‘seed’ of noise by adding noise to perturbing some player 
$P_i$’s action downward. In the subsequent best-response dynamic, each player who depends on 
$P_i$ best-responds by lowering his action; in this way, the negative shock propagates along network paths from 
$P_i$. In fact, this negative shock will circle back to 
$i$ along the network cycle, potentially inducing 
$P_i$ to further lower their action. In other words, the network cycle serves as a feedback loop for the initial seed of strategic uncertainty. Indeed, players may be embedded in multiple network cycles, in which case the feedback effect is multiplied. In dense cyclic networks where many players are embedded in many cycles, the feedback effect is sufficiently strong that any small shock becomes self-reinforcing and eventually leads to coordination failure, where everyone involved plays almost-minimal actions. In contrast, seeds of strategic uncertainty may propagate across acyclic networks but are eventually damped due to the absence of feedback loops, and thus do not substantially damage coordination.Footnote 13
4.3. Cycle Length
Our third experimental finding is that cycle length has little, if any, effect on coordination. Consider the sparse cyclic networks. In a sparse cyclic network of size 
$n$ (Figure 5), there is a single cycle of length 
$n$ and every neighbourhood has size 
$|S_i| \equiv 2$. To wit: we can study how cycle length affects coordination, keeping density fixed at 
$\ell=2$, by comparing the sparse cyclic networks with different group sizes.
Sparse cyclic networks
Table 5 also shows average final-round actions in each of the sparse cyclic treatments. Average final-round actions in the various sparse cyclic networks are not significantly different: testing for differences between pairs of treatments, we get p-values of 
$p_{23}=0.7596$, 
$p_{24}=0.6212$, 
$p_{26}=0.7010$, 
$p_{34}=0.9095$, 
$p_{36}=0.4452$, and 
$p_{46}=0.3620$ (where 
$p_{n_1 n_2}$ is the p-value from comparing group sizes 
$n_1$ vs. 
$n_2$).Footnote 14
This experimental result is consistent with Proposition A.2 in Appendix A, which implies that logit equilibrium action distributions are independent of cycle length for sparse cyclic networks.Footnote 15
These findings may shed some light on the mechanisms leading to coordination failure in large groups in the classic minimum-effort game, which (as we recall) corresponds to the complete network. Given a complete network, an increase in group size corresponds to both (i) an increase in network density and (ii) the introduction of longer cycles into the network. What role does each of these factors play in inducing coordination failure in the classic minimum-effort game? Our finding that cycle length has little effect on coordination suggests that (ii) is not a major factor; and thus that coordination failure in large groups in the classic minimum-effort game is due to high network density.
4.4. Pecking Order
We also find tentative evidence for a fourth regularity: within each group with an acyclic network structure, participants in ‘higher’ positions (higher index 
$i$) choose lower actions. We refer to this as the pecking-order hypothesis. Our acyclic network structures are hierarchical in the sense that there is a strict ordering of players, with higher-indexed participants depending on lower-indexed participants but not vice versa (see Appendix E). Thus, the pecking-order hypothesis states that higher-indexed players in the hierarchy, who bear greater responsibility for coordination, take lower actions than lower-indexed players.
Table 6 reports mean final-round actions by participant position for each treatment. In every acyclic treatment, mean actions generally decline with position. The differences across positions are modest: in each acyclic treatment, the highest-indexed participants still choose higher actions, on average, than the average participant (at any position) in the corresponding cyclic treatment.
Mean actions by network position (final round)
Table 6 Long description
The table reports mean final-round actions for players P1 through P6 by network structure and group size, with standard deviations in brackets, plus two tests of whether actions fall as network position increases. In acyclic networks, actions are high and generally decrease with position: for example, in 6-player sparse groups the mean moves from 6.80 at P1 to 6.00 at P6, and the trend tests indicate strong evidence of a decreasing pattern. Acyclic 3-player sparse and 4-player sparse groups also show modest declines from P1 to the last listed position, with trend evidence ranging from weak to moderate. Acyclic 6-player dense groups are less monotonic, with P1 and P4 at 7.00 and mid positions lower, and the trend evidence is not strong. In cyclic networks, mean actions are lower overall and patterns are mixed: 3-player sparse slightly increases from P1 to P3, and 6-player sparse varies without a clear ordering, both with little evidence of a decreasing trend. The clearest cyclic decline appears in 6-player dense groups, where means fall from 2.80 at P2 to 1.60 at P6 and the trend tests support a decreasing pattern. Aggregated results indicate strong overall evidence of decreasing actions with higher position in acyclic networks, but not in cyclic networks. Because the regression results are based on ranked values and clustered standard errors, and the trend tests are nonparametric, the reported p-values reflect evidence for monotonic ordering rather than the size of differences in raw means.
Notes: Mean [st. dev.] final-round action. The 
$\beta$ column reports the slope estimate (
$p$-value) from a linear regression of ranked actions on ranked network position, with standard errors clustered by group; one-sided 
$p$-values in parentheses correspond to the hypothesis that higher positions take lower actions. The J-T column reports 
$p$-values from Jonckheere–Terpstra (J-T) tests of the hypothesis that, within each group, final-round actions are weakly decreasing in network position. Group-level J-T 
$p$-values are aggregated by network structure using Fisher’s combined probability test. Network positions for each 
$P_i$ are given in Appendix E.
We test the pecking-order hypothesis in two ways. First, for each network structure, we regress participants’ final-round action ranks on network position, clustering standard errors by group. Second, for each group, we run a J-T test of the hypothesis that higher-indexed participants take lower actions, then use Fisher’s method to combine the group 
$p$-values within each network structure.Footnote 16 The 
$\beta$ and J-T columns of Table 6 report these statistics.
The acyclic-network results are broadly consistent with the pecking-order hypothesis. The slope estimates 
$\beta$ are negative for every network structure, with significance at the 5% level for the 
$n=4$ sparse and 
$n=6$ sparse networks. Similarly, the Fisher J-T 
$p$-values point in the same direction, with significance at the 5% level for the 
$n=4$ sparse and 
$n=6$ sparse networks. Combined across all acyclic-network groups, the Fisher J-T 
$p$-value is below 
$0.001$.
The pecking-order hypothesis relates only to acyclic networks; for comparison, we report the same tests for the cyclic treatments. The cyclic-network slope estimates 
$\beta$ are mixed in sign and generally smaller in magnitude, with only one slope estimate (for the 
$n=6$ dense cyclic network) being significant at the 
$5\%$ level. The J-T results for cyclic networks are similarly mixed: only the 
$n=6$ dense cyclic network is individually significant (
$p=0.017$), and the Fisher J-T 
$p$-value combined across the four cyclic networks is not significant (
$p=0.148$).
Intuitively, in acyclic networks, higher-indexed players find high actions more costly. Their neighbours are themselves relatively high-indexed and therefore choose lower actions than the neighbours of lower-indexed players. In dense acyclic networks, higher-indexed players also have larger neighbourhoods and thus face a greater risk of at least one low action in their neighbourhood. More generally, within a hierarchy higher-ups bear more of the coordination risk, which induces them to choose lower actions.Footnote 17 We formalize these points in Proposition A.3 of Appendix A.
4.5. Additional Tests
We conducted three robustness tests on this section’s results. Our first test excludes participant 
$P_1$ from the analysis of acyclic networks. Unlike the other participants, in acyclic networks, 
$P_1$’s neighbourhood contains nobody else, so 
$P_1$ faces no strategic uncertainty – thus their dominant strategy is always to play the maximum action, 
$x_{i,t}=7$. Excluding 
$P_1$ from the analysis of acyclic networks thus serves, in a sense, to ‘level the playing field’ between cyclic and acyclic networks. Table B.5 reports mean actions for acyclic networks without 
$P_1$, and Table B.6 tests for differences between cyclic and acyclic treatments after excluding 
$P_1$. Our results remain qualitatively unchanged.
Our second test is to consider, instead of the mean action, the mean neighbourhood-minimum action; i.e., the minimum action in each participant’s neighbourhood, averaged over participants. Each participant’s neighbourhood-minimum action captures the extent to which coordination failure affects the participant. Table C.4 compares cyclic versus acyclic treatments and shows that the differences in such mean neighbourhood-minimum actions are, if anything, even more pronounced than the differences in mean actions.
Our third robustness test examines an alternative ‘anchor’ hypothesis. Notice that in our acyclic networks, player 
$P_1$’s neighbourhood consists only of themself, and thus has a strictly dominant strategy: play 
$7$. Indeed, we observe in the laboratory that 
$P_1$ consistently and quickly arrive at a stable choice of 
$7$ in almost all sessions. The anchor hypothesis is that acyclic networks coordinate successfully because the anchor player serves as a focal point for coordination on the highest action 
$7$ by other players. On the other hand, cyclic networks have no anchor position, so successful coordination on high actions is less likely.
We test this hypothesis by introducing anchored cyclic networks. Each anchored cyclic network is constructed by adding an additional anchor position 
$P_0$ to either the six-player sparse cyclic or the six-player dense cyclic network. Figure 6 illustrates. For each of the two anchored cyclic networks, we consider two variants. In the first variant, the anchor is isolated, i.e., does not belong to anyone else’s neighbourhood. In the second variant, the anchor is connected in the sense that they are in 
$P_1$’s neighbourhood (and nobody else’s).Footnote 18
Anchored cyclic networks
Fig. 6 Long description
The image shows four diagrams of cyclic networks. The first diagram is labeled ′Cyclic Sparse (connected anchor)′ and features nodes P subscript 0 to P subscript 6 connected in a sparse cyclic pattern with P subscript 0 connected to P subscript 1. The second diagram, ′Cyclic Dense (connected anchor),′ shows a dense network with all nodes from P subscript 0 to P subscript 6 interconnected. The third diagram, ′Cyclic Sparse (isolated anchor),′ is similar to the first but with P subscript 0 isolated. The fourth diagram, ′Cyclic Dense (isolated anchor),′ mirrors the second but with P subscript 0 isolated from the network.
Table 7 shows average final-round actions from unanchored cyclic treatments (reproduced from Table 5) as well as average final-round actions from their anchored counterparts.
Mean actions in cyclic networks with additional anchor player (final round)
Table 7 Long description
The table reports the mean final-round action for six-player cyclic networks, with values shown for sparse versus dense networks under three setups: no anchor, a connected anchor, and an isolated anchor. In every setup, sparse networks have higher mean actions than dense networks. Unanchored networks show 4.90 in sparse versus 2.08 in dense. With a connected anchor, non-anchor players average 3.86 in sparse and 1.69 in dense. With an isolated anchor, the mean is 4.54 in sparse and 1.08 in dense, the lowest value in the table. Standard deviations indicate more variability in sparse conditions, while dense conditions are less variable, especially with an isolated anchor where the spread is very small. These are group averages from the final round, so they summarize outcomes but do not by themselves establish causation.
Mean [st. dev.] action, averaged across groups. The first two columns (
$n=6$ unanchored) reproduce the mean final-round actions from our original 
$n=6$ sparse and dense cyclic networks (Table 5). The next two columns (
$n=6$ connected) report, for the sparse and dense cyclic networks with a connected anchor player, the mean final-round actions of the non-anchor players (
$P_1$ to 
$P_6$) in each group. The last two columns report the analogous figures for the sparse and dense cyclic networks with an isolated anchor player.
We find that the addition of anchor players to our cyclic networks does not significantly improve coordination amongst the remaining players, and may even worsen coordination. In one case, the anchor player may have significantly worsened coordination: the cyclic dense network with an additional isolated anchor player produced average final-round actions (
$1.08$) that were significantly worse compared to those from the anchor-less dense cyclic network (
$2.08$). Overall, we have neither a sufficiently strong signal nor a plausible mechanism to infer with confidence that anchor players worsen coordination in our cyclic networks.Footnote 19 We view our findings as relatively strong evidence against the hypothesis that ‘anchor’ players improve coordination.
5. Experimental findings: dynamics
Section 4’s analysis was static: it focused on final-round actions. Complementarily, Appendix A studies logit equilibria of the one-shot minimum effort game. In this section, we step away from the static setting and discuss how coordination evolved over the ten rounds of play in our experiment.
5.1. Coordination Levels
Table 8 shows average action choice by treatment, for round 1; rounds 1–5; rounds 6–10; and round 10.Footnote 20 A clear pattern emerges: actions increase over time (culminating in relatively high actions) in acyclic networks, but decrease over time (culminating in relatively low actions) in cyclic networks.Footnote 21
Time trends in mean actions – acyclic vs. cyclic networks
Table 8 Long description
The table reports mean action levels (with standard deviations) for acyclic versus cyclic networks across four treatments: sparse networks with 3, 4, or 6 participants, and a dense network with 6 participants. In every treatment, average actions are higher in acyclic than cyclic networks; for example, at round 10 the acyclic means range from 6.52 to 6.72, while cyclic means range from 2.08 to 4.90. Early-round averages show the same pattern: for rounds 1 to 5, acyclic means are about 5.70 to 6.18 versus 3.32 to 5.11 for cyclic, with the largest gap in dense n equals 6 (5.70 vs 3.32). Later rounds (6 to 10) also favor acyclic networks, including dense n equals 6 where acyclic averages 6.52 and cyclic averages 2.29. Time-trend estimates indicate actions rise over rounds in all acyclic treatments (about 0.07 to 0.16 per round, statistically significant), while cyclic trends are negative or near zero, with the steepest decline in dense n equals 6 (about minus 0.21 per round, statistically significant). Sample sizes are 270 to 600 observations per treatment, with 9 to 10 clusters, so comparisons should consider that one treatment has slightly fewer clusters.
Mean [st. dev.] action, averaged across groups. Each cell reports the estimated coefficient on Period from an OLS regression of 
$x$ on Round for the specified treatment (network structure and size). Standard errors (clustered by group) in parentheses. *** 
$p \lt 0.01$, ** 
$p \lt 0.05$, * 
$p \lt 0.1$. There were 10 groups per treatment with the exception of sparse acyclic 
$n=3$.
Figure 7 paints a richer picture – showing means and 25th to 75th percentiles of action distributions round-by-round.Footnote 22 In the acyclic networks, actions generally increased over time. In contrast, in cyclic networks, action distributions decrease over time. This is especially so in dense networks.Footnote 23
Evolution of mean actions in acyclic & cyclic networks
Fig. 7 Long description
The composite figure contains eight scatter plots arranged in two groups of four. The first group covers acyclic network conditions and the second group covers cyclic network conditions. Each plot has the x-axis labeled Round and the y-axis labeled Mean Action. Axis ranges are not explicitly marked with numeric increments in the image. Each plot displays two series of data points per panel: one series shown as filled markers representing mean actions and a second series shown as distinct markers representing a separate condition or baseline. Error bars on each point indicate the 25th to 75th percentile range of the action distribution at each round. For acyclic conditions, the four panels correspond to n equals 3 sparse acyclic, n equals 4 sparse acyclic, n equals 5 sparse acyclic and n equals 6 dense acyclic. Across these panels, mean action values start at lower levels in early rounds and rise toward higher levels in later rounds. The increase is visible across both series in each panel. Error bars remain relatively consistent across rounds. For cyclic conditions, the four panels correspond to n equals 3 sparse cyclic, n equals 4 sparse cyclic, n equals 5 sparse cyclic and n equals 6 dense cyclic. Across these panels, mean action values start at higher levels in early rounds and decline toward lower levels in later rounds. The downward trend is present in both series per panel. The dense cyclic panel shows a more pronounced decline compared to the sparse cyclic panels. Error bars in cyclic panels are visible at each round data point. The legend shown at the bottom of the figure identifies the two plotted series, labeled as Main series and 100 periods, though exact numeric starting and ending values are not clearly legible at the available image resolution.
Overall, these time trends suggest that participants may be playing adaptively, adjusting their actions over time in response to recent play. Next, we consider whether such adaptive play reduces the uncertainty that participants face.
5.2. Learning
We provide suggestive evidence that participants in both cyclic and acyclic networks learn to coordinate over time, in the sense that they improve their predictions of others’ actions. Note that a participant 
$P_i$ who perfectly anticipates the play of other group members should optimally match the minimum action elsewhere in his neighbourhood by choosing period-
$t$ action
\begin{align*}
x^*_{i,t} =
\begin{cases}
\min \{x_{j,t}: P_j \in S_i \setminus P_i \}&\text{if } |S_i \setminus P_i| \geq 1,\\
7 &\text{if } |S_i \setminus P_i| = 0.
\end{cases}
\end{align*} Accordingly, call the difference between the participant 
$P_i$’s optimal action and realized action, 
$| x_{i,t}-x^*_{i,t} |$, the participant’s prediction error.
Table 9 shows time trends in average prediction errors by treatment.Footnote 24 Two patterns emerge.
Time trends in prediction errors – acyclic vs. cyclic networks
Table 9 Long description
The table reports average prediction error levels (with standard deviations) and the estimated per-round change in error for eight treatments defined by network size, density, and whether links are acyclic or cyclic. In every treatment, the per-round change is negative, indicating errors decrease over time; the largest decline occurs in the dense acyclic network (about minus 0.17 per round) and the dense cyclic network (about minus 0.21 per round). Early errors are higher in cyclic than acyclic networks within each size and density: for example, in sparse networks with size 6, round 1 error is 1.77 in cyclic versus 1.28 in acyclic, and in dense networks with size 6 it is 2.70 in cyclic versus 1.90 in acyclic. Mid-to-late errors are lower than early errors across treatments; for instance, in sparse size 6 acyclic, the average error drops from 0.96 in rounds 1 to 5 to 0.41 in rounds 6 to 10, while sparse size 6 cyclic drops from 1.53 to 1.09. By round 10, errors are generally below 1.0, with sparse size 6 acyclic at 0.23 and sparse size 6 cyclic at 0.83; dense size 6 ends at 0.52 for acyclic and 0.43 for cyclic. Standard deviations are sizable in many cells, so differences in means should be interpreted with caution. Sample sizes are similar across treatments (typically 10 clusters and 600 observations), except sparse acyclic size 3, which has 9 clusters and 270 observations.
Mean [st. dev.] prediction error, averaged across groups. In the 
$n=6$ dense acyclic network, eliminating a single outlier reduces final-round mean prediction error to 0.13 [0.39]. Each cell reports the estimated coefficient on Period from an OLS regression of 
$x$ on Round for the specified treatment (network structure and size). Standard errors (clustered by group) in parentheses. *** 
$p \lt 0.01$, ** 
$p \lt 0.05$, * 
$p \lt 0.1$. There were 10 groups per treatment with the exception of sparse acyclic 
$n=3$.
First, prediction errors are generally substantially larger in cyclic treatments than in the corresponding acyclic treatments.Footnote 25 Using two-sided Wilcoxon rank-sum tests to compare prediction errors between acyclic and cyclic treatments – where each observation is the average prediction error in one treatment over a specific time range – we find that, for most network structures and most time ranges, prediction error is significantly higher in a given cyclic treatment than the corresponding acyclic treatment.Footnote 26
Second, prediction error decreases over time in both cyclic and acyclic networks: the average prediction error over the last five rounds is lower than over the first five rounds in every treatment.Footnote 27 For acyclic networks, such improvements in prediction go hand-in-hand with improvements in actions, in the sense that actions increase over time (Table 6) – but not for cyclic networks, where actions decrease over time.
We prefer to interpret prediction error as arising from a combination of strategic uncertainty and noisy decision-making: participants make smaller prediction errors when they face less strategic uncertainty about actions in their neighbourhood, and when they understand better how to make optimal choices given others’ actions. That is, our experimental findings from Table 9 suggest that over repeated play, participants improve their decision-making and strategic uncertainty diminishes. In this context, it is not surprising that our results from Section 4 – which focus on final-round play – are consistent with the predictions of low-
$\mu$ logit equilibrium (Appendix A), which posits that agents optimize with a small amount of noise.
6. Conclusion
This paper argues that introducing cycles of interdependencies into an organization’s design may trigger coordination failure. In this sense, the network minimum game provides a perspective that highlights the downsides of interdependencies within organizations; indeed, Lemma A.2 states that performance decreases monotonically whenever interdependencies are introduced. This stark result arises because the network minimum game framework essentially assumes that interdependencies produce no benefits for organizations. While this assumption is clearly unrealistic, we prefer not to take a stand on how to model such benefits (see, e.g., Becker & Murphy, Reference Becker and Murphy1992; Dessein & Santos, Reference Dessein and Santos2006 for some options). Instead, we show that the disadvantages of additional interdependencies may be mitigated by avoiding network cycles. In other words, if interdependencies are necessary, an acyclic structure is preferable to a cyclic structure. Where cycles are technologically unavoidable, our results point to the value of complementary mechanisms studied in the minimum-effort literature – such as communication, endogenous neighbourhood choice, and targeted group incentives – that can help organizations sustain high-effort outcomes even in the presence of feedback loops.
Acknowledgements
We thank the editor (Ragan Petrie), the consulting co-editor, three anonymous referees, Evan Calford, Anujit Chakraborty, Rahul Deb, David Freeman, Ben Greiner, Yoram Halevy, Axel Ockenfels, Marco Ottaviani, Dotan Persitz, Colin Stewart, and Lanny Zrill for comments and suggestions. This research has been approved by the Human Research Ethics Committee of UNSW Sydney under approval number HC16664.
Funding statement
This work was supported by the UNSW Bizlab, UNSW Business School, and Australian Research Council Discovery Project DP240103257.
Replication material
The replication material for the study is available at https://doi.org/10.17605/OSF.IO/Z35YS.
Open access
