Group formation on a small-world: experiment and modelling

Abstract

As a step towards studying human-agent collectives, we conduct an online game with human participants cooperating on a network. The game is presented in the context of achieving group formation through local coordination. The players set initially to a small-world network with limited information on the location of other players, coordinate their movements to arrange themselves into groups. To understand the decision-making process, we construct a data-driven model of agents based on probability matching. The model allows us to gather insight into the nature and degree of rationality employed by the human players. By varying the parameters in agent-based simulations, we are able to benchmark the human behaviour. We observe that while the players use the neighbourhood information in limited capacity, the perception of risk is optimal. We also find that for certain parameter ranges, the agents are able to act more efficiently when compared to the human players. This approach would allow us to simulate the collective dynamics in games with agents having varying strategies playing alongside human proxies.

1. Introduction

Understanding cooperation and conflict in digitized societies is becoming increasingly important with the introduction of artificial intelligence or agents as actors in human social networks. Human–agent cooperation has been lately realized in diverse scenarios ranging from healthcare settings, to retail stores, to self-driving cars [1,2]. The task of forging cooperation or coordination between humans and agents in complex environments requires the understanding at different levels, such as the psychological underpinnings of human preferences, as well as anticipating the dynamics from human–agent collective action [3–5]. While the answers are far from simple, the use of online games to study hybrid human–agent systems seems to provide valuable insights [6,7]. For example, Shirado & Christakis [6] studied how the collective performance of humans trying to solve a coordination game on a network changes in the presence of agents (or bots) and showed the impact of the degree of randomness in agents’ behaviour on the outcomes.

In studying the dynamics of human–agent collectives in games, one needs to investigate the dynamics of groups composed solely of humans, followed by modelling human intuitions and reasoning, such that the latter model can be used for simulating human–agent hybrid systems. In this work, as a first step towards understanding a hybrid system, we develop a cooperative game that is played by human subjects on a virtual network. First, we observe how large groups can emerge, using the concept of a connected cluster of nodes. Then we construct a probabilistic model that captures the human decision-making pertinent to the game and simultaneously allows us to analyse the bounds on rationality and cognition of the players.

We choose a framework that is in the spirit of the earlier works by Kearns et al. (see [8] and references therein). In a series of experiments, they studied the effect of network structure on the efficiency of solving problems like the graph colouring and consensus by human subjects [8–10]. They also showed that the amount and quality of information in the system that is available to the subjects influences their performance, depending on the structure of the network over which the subjects are interacting. Our game also has similarities to the extensively studied matching problem that considers two distinct sets of individuals, like, men–women, producers–consumers and employers–job seekers, from which members get matched in pairs to their own mutual benefit [11,12]. The latter problem has also been studied by Coviello et al. [13] in the form of a distributed coordination game on a network where players get rewarded when all the nodes get matched in pairs. They found that human subjects participating in the experiment tend to behave prudently while averting risks, and this behavioural trait influences strongly their performance and capabilities to complete the expected goal. The complex relation between network properties, human behaviour and collective performance in problem-solving tasks has been explored in different works [14–16], and it has been shown that coordination, cooperation and other social interactions within human groups can be described and analysed through carefully designed experiments involving human subjects. In the current set-up, the players are incentivized to arrange themselves in groups rather than pairs. Closely knit groups or communities are ubiquitous in society, which act as chambers of collaboration and innovation across diverse fields of human endeavour like performing arts, science and technology development [17–20]. Therefore, the game can be also placed in the context of game-theoretic studies of social group formation, for example, games [21,22] that are based on Schelling’s segregation model [23], and more generally, hedonic coalition formation games [24,25].

During the rounds of the game, the players coordinate their actions in pairs by communicating over the network links and exchange their locations or node positions. We focus on the problem of complex decision-making by human players linked with others in a network, where the players have information at the local and global scales, i.e. colours of their neighbours and cluster sizes, respectively. The model is able to explain the decision-making in terms of the perception of risk by the players, as well as the cognition of their neighbourhood information. The model also provides insight into possible situations where excessive caution by the players would hinder their mobility and aggregation behaviour aimed towards group formation. On the other hand, lack of prudence or cautiousness would allow the groups to become fragmented. Finally, we use our model to scan the parameter space and compare other possible strategies with the strategy employed by humans.

2. Material and methods

2.1. Game rules

Human subjects playing the game are grouped into m classes having l individuals in each class, such that in sum there are n = ml individuals. A class is denoted by a colour, e.g. red, blue, green, etc. The game is played on a connected network of n nodes. Each one of the n nodes of the network is occupied by only one player, hence there are no empty nodes in the network. The goal of the experiment requires that by swapping places during successive rounds, players end positioned on the network with m clusters each of size l. A connected red cluster implies that from any red player any other red player can be reached by a connected near-neighbour path consisting of only red players (see the last network in figure 1).

Figure 1.

A game with 24 players. Starting from a graph-coloured configuration, neighbouring players exchanged places over 11 rounds, after which a solution is achieved, i.e. after the 11th round, the three largest clusters belonging to each colour have reached the maximum size of 8. Within a given cluster (for a given colour), any pair of nodes will have a path exclusively through the nodes belonging to the cluster. The colours attributed to the subjects are shown in the figure. In addition, links that connect the players in a given cluster are marked with the corresponding colour while the other links are shown in grey. Only rounds with odd numbers are shown and denoted by the label ‘rnd-’. The largest cluster size for each colour at the end of the corresponding round is indicated as a triplet (S_r, S_g, S_b) with S_r, S_g and S_b being the largest cluster size corresponding to red, green and blue, respectively. The network consisted of a regular squared mesh of 6 × 4 = 24 nodes, with periodic boundaries and three small-world links. The average of the largest cluster sizes of the three colours is scaled by the maximum value (8) and expressed as a percentage is also indicated for each round. (Online version in colour.)

The game is played in rounds, and in each round players decide to exchange their locations with other players in the network, in a requesting–accepting process. In each round, one of the m colours is chosen, and players having this colour act as requesters. Players having other colours receive the possible requests. Each round consists of two distinct stages: a first stage with requests being sent by players of the chosen colour, who want to move from their current locations. A requester can send a request to only one of the neighbouring players who has a colour different from its own (therefore, players of the same colour can not exchange places among themselves). In the second stage, acceptances occur, where those players who received requests in the previous stage from the requesters, decide on whether to accept or not, one of the received requests for exchanging places. If a request is accepted, those players involved (i.e. requester and acceptor) exchange their locations in the network. After all the possible exchanges are made, the round ends. In the next round, a different colour is chosen for the requesting activity. The colour chosen for the requesting activity on each round changes in a cyclic way, such that in m rounds, each of the m colours have been chosen once. The rounds continue until the goal is achieved or the number of rounds reaches the maximum limit, whichever happens earlier.

During each round, a player is provided the following information:

• (i)The colour of other players in the neighbourhood.
• (ii)The size of the player’s own cluster.
• (iii)The size of the cluster of each of the neighbouring players.
• (iv)The largest cluster size for each of the m colours.
• (v)The average collective progress (ACP) defined as the average of the sizes of the largest cluster in each colour normalized by the maximum possible size l. In case m largest clusters reach sizes l the ACP is 1.0 (desired outcome).

The game is terminated once the desired outcome is reached, i.e. m clusters of size l are generated, or after a fixed number of rounds T, where T is a multiple of m, which allows all the colours to have the same number of rounds to be requesters. The incentive per game is based on a score that is calculated from the ACP (see the following section).

2.2. Experimental setting

The experiment was conducted in a single session, in a computer laboratory at the Department of Computer Science of Aalto University located in Espoo, Finland, on 9 August 2017 with 30 individuals recruited from an online volunteer pool and from advertisements in social media. Before the experiment, all the subjects were provided with information sheets as well as sheets for informed consent. The informed consents, once signed by the subjects, were collected before the experiment began. No personal information on the subjects was collected other than contact emails for rewarding purposes. To help the participants better understand the dynamics of the game and its rules, a short explanatory visual presentation was given in the room to all the groups and, before the sessions started, each participant played a trial game lasting six rounds to ensure that the subjects got acquainted with the user interface of the game. During the sessions, the presentation file was accessible to the subjects on their desktop. The file is available in the electronic supplementary material. The interface and engine of this online game were implemented using the oTree framework (figure 2) [26]. Subjects’ view of other workstations was restricted and all communication during the games was forbidden.

Figure 2.

The graphical user interface for the game during requesting (a) and accepting (b) stages. The focal subject (‘YOU’) is able to view the entire network but the colour of nodes that are not immediate neighbours are masked. Links to the neighbours that have colours different from the focal subject are shown in magenta and links to subjects having the same colour are shown in dark grey. Other links are shown in grey. Buttons allowing a choice between options are displayed above the network. Adjacent to the neighbouring nodes the following two numbers were provided—(i) an identifier for the node location, and (ii) the cluster size of the neighbouring player located at that node (within brackets). (Online version in colour.)

During the 4-h session, a total of nine games were played, consisting of five games with 24 players and four games with 30 players. We ensured that during each of the first five games (24-players game), six of the 30 participants would skip exactly one game. For each game, the network consisted of a regular squared lattice (4 × 6 or 5 × 6) with periodic boundary conditions. We had three additional long-ranged links (not resulting in triangulations) to introduce a small-worldness. In every game, the positioning of these links was altered. Also, the places of subjects at workstations were shuffled between games. The networks used in the game can be considered as realizations of the Kleinberg model [27] in the limit that small-world links appear independent of distance. The time given to the players to make a decision (either requesting or accepting) was changed with rounds progressing, being 30 s during the first five rounds and 20 s during the rest.

The incentive offered for participating in the experiment were movie tickets, sent to the players by email after the whole session was completed. For a given network topology and initial positioning of subjects, we also calculated the quantity ACP_rand from simulations where agents play randomly. We obtained a normalized score for each game by scaling ACP with respect to ACP_rand and accumulated the total score (S) over nine games as

$$S=\sum _{k=1}^9\frac{\mathrm{ACP}(k)-{\mathrm{ACP}}_{\mathrm{rand}}(k)}{1-{\mathrm{ACP}}_{\mathrm{rand}}(k)}.$$

To each subject, one movie ticket was given for participation and 4 × S (nearest integer) movie tickets were given as the reward.

2.3. Model

During the game, the players take decisions on sending (or not sending) requests and accepting (or not accepting) requests based on the available information of their immediate neighbourhood, and the sizes of the largest clusters of each colour. The decision-making logic employed by the agents is expected to be heterogeneous yet far from random. It is known that exact coordination between two agents is possible in the presence of common knowledge [28]. However, in this game, the knowledge of a subject and any of its neighbours do not completely overlap. We construct a model that is based on the notion of probability matching [29–33]. This model is expected to serve both the purposes of uncovering the decision-making logic of the subjects as well as act as a template for agent-based simulations.

For an agent i having a colour c_i we assume that its neighbourhood $\mathcal{S}$ can be uniquely characterized by the following quantities—(i) the current cluster size s_i, (ii) the set of cluster sizes {s_j}, where j is a neighbour of i having a colour different from i, that is, c_j ≠ c_i. We place all the neighbourhoods $\mathcal{S}$ that have identical supersets {s_i, {s_j}} (or that can be made identical by the ordering of j’s) into a given category $\mathcal{C}$.

In a given category $\mathcal{C}$, we consider an agent i with the set of neighbours {j₁, j₂, j₃, … } that have colours different from i. In the requesting phase, it has to choose from the following set of options ω: {stay at a current location without sending a request, send a request to j₁, send a request to j₂, … }. We assume that with each option ω the player associates a probability, P_ω of it being beneficial to the progress of the game. Here beneficial may refer to an increase in the cluster size for the colour of i or increase in the cluster size for colour of j, or both. Therefore, using probability matching, the probability of choosing an option ω is given by, $p_{\omega }=P_{\omega }/\sum _{{\omega }^{\prime }}{P}_{{\omega }^{\prime }}$. Restricting ω to the set of options when a request is sent one can write,

$$p_{\omega }=\frac{P_{\omega }/P_0}{1+\sum _{{\omega }^{\prime }}{P}_{{\omega }^{\prime }}/P_0},$$

where P₀ is the estimated probability that not sending a request (that is, not moving) is a beneficial option.

Next, we focus on different categories that could be realized during the course of the game, and for each category, we identify the cases (characterized by neighbours with cluster size s_j) when one or more requests were sent from the focal players i to players j. These sets are indeed options that were actually executed. Thus in each category and for each option ω(j) we accumulate the total number of cases (N_ω) when a request was sent. Additionally, in the same category, we accumulate the total number of cases (N₀) when no request was sent, and calculate the ratio (N_ω/N₀)/(1 + N_ω/N₀), where this ratio lies in [0, 1]. Taking this ratio as the dependent variable, we perform a logistic regression on the following set of variables, (i) s_i, (ii) s_j and (iii) U_j with data from all chosen ω’s from all categories. The quantity U_j = 〈s(c_j)〉 − s_j, where $⟨s(c_j)⟩=\sum _{k=1,n_i(c_j)}{s}_k/n_i(c_j)$ is the average of the cluster sizes of the n_i(c_j) neighbours of i that have the same colour as j. We use this difference U_j as the measure of disparity that a focal player could possibly recognize in the neighbourhood in the requesting mode. A large value of this difference might encourage the focal player to send a request to a player having the colour of j, so that clusters having colour c_j could merge. There could be better measures of disparity but our choice is guided by the linearity of the model. For the actual fitting instead of U_j we use 〈s(c_j)〉 as s_j is already an independent variable. Although we base our model of decision-making on probability matching, it could as well be considered as a log-linear response model [34,35]. The fit to a logistic function allows us to use the following expression for a requester, P_ω/P₀ = exp{λ_r + α_rs_i + β_rs_j + δ_r〈s(c_j)〉} from the estimation P_ω/P₀ = p_ω/p₀ = N_ω/N₀, where, α_r, β_r, δ_r and λ_r are parameters corresponding to a requester. Once evaluated by fitting to the data from the experiment, these parameters are used in numerical simulations of agents as shown in figure 3. A similar scheme is used for data from acceptors, and the corresponding parameters α_a, β_a, δ_a and λ_a are evaluated.

Figure 3.

Example showing how the model assigns probabilities to an agent in the requesting mode in a given neighbourhood. (a) Hypothetical instance of a neighbourhood around a focal agent with label i. Agent i’s colour is blue. Agents iii, v, vi and viii are immediate neighbours of agent i. The sizes of the cluster to which each agent is connected are shown to the right of each agent. Note that agent i has two neighbours of colour red (agents iii and viii), a neighbour of colour green (agent vi), and one neighbour of the same colour (agent v). (b) Probabilities calculated using the model according to which i selects one of the four possible choices (ω), namely to send a request to one of the agents iii, vi and viii, or to not send any request and retain its place in the next round. The last option is denoted by ω(0) and represented by P₀ in the model. The parameter values α_r, β_r, δ_r and λ_r are taken from table 1. (Online version in colour.)

3. Results

3.1. Experiment

For our experiment, we chose m = 3, i.e. colours red, green and blue. We conducted five games with l = 8 (n = 24) and four games with l = 10 (n = 30) with T = 21. The network was taken as a square lattice with periodic boundary conditions and additional small-world links. For the initial condition, we tried to position the players in a graph-coloured configuration, that is, a subject with a given colour (say, red) is surrounded only by neighbours who have a different colour (i.e. can not be red). All the games with 24 players and three out of the four games with 30 players reached the desired outcome within 21 rounds. The evolution of the clusters in one of the games with 24 players is shown in figure 1. This particular game finished in 11 rounds. The games with 24 players took nine rounds to complete on average, and the completed games with 30 players took 16 rounds on average.

In general, all the games showed a fast initial growth in the sizes of the largest clusters. After which the overall activity in terms of requesting and accepting decreased. In this phase, the players appeared to become ‘conscious’ of the presence of players in the vicinity who were trapped in smaller clusters. This resulted most likely from the understanding of the collective goal of the experiment. As a result, players in larger clusters cooperated with the isolated players and with players in small clusters by exchanging locations, which sometimes caused fragmentation of the larger clusters. Such a case is shown in figure 4. The ACP and the activity for the games are shown in figure 5. It is observed that the clusters become less and less active as the game progresses in the wake of more passive requesting and accepting.

Figure 4.

A typical time evolution of cluster sizes in a game with 30 players. Cluster sizes of three randomly chosen players corresponding to the three colours (red—circles—player 21, green—squares—player 11 and blue—triangles—player 14) are shown as the game progresses from round 1 to round 21 (final round). The average collective progress (normalized value of the average of the largest cluster sizes of the three colours, ACP; see material and methods) is denoted by the dashed line. The typical dynamics leading to a solution of the problem can be observed from the evolution of the network, with an initial stage of fast progress (≈0.9 after 8 rounds), followed by a stagnation regime where only a small number of location exchanges happen in the network, eventually leading to the formation of the required three maximum-sized clusters. In the stagnation regime players that are trapped in smaller clusters are facilitated to move such that they eventually merge with larger clusters. For this particular game, we illustrate such an event occurring at round 14. The configurations during rounds 14 and 15 are shown to the right with legends similar to those used in figure 1. A player with a red colour and having cluster size of 1 (located at the bottom right corner of the mesh at round 14) exchanges place with another player with colour green. As a result of which the green cluster is fragmented. We observe this as the (chosen) green player’s cluster size decreases from the maximum possible size (10) at round 14 to the minimum possible size (1) in the next two rounds. However, the exchanges occurring after the 15th round allows the players to reorganize rather quickly to the desired configuration. (Online version in colour.)

Figure 5.

Comparison between experiments and simulations using the model for games with 24 and 30 players. (a,c) The average collective progress (normalized value of the average of largest cluster sizes corresponding to the three different colours) is plotted against the round number. The points are binned values from the five experiments in the case of 24 players and from the four experiments in the case of 30 players. The dashed lines are the result of simulations of the model with 500 runs. (b,d) The requesting and accepting activities in the games. The requesting activity is measured as the ratio between number of actual requests and the maximum possible requests per round (circles). Similarly, the accepting activity is measured as the ratio between the number of accepted requests and the total number of requests received (triangles). The dashed lines are results from simulations. The error bars in the figures indicate the standard deviations. (Online version in colour.)

3.2. Numerical results

The fitting parameters are provided in table 1. All the parameters turn out to be significantly different from zero with p < 0.01. As we have two sizes for the networks, for the purpose of fitting, we use cluster sizes that are scaled by the maximum cluster size, in such a way that the cluster size variable varies between 0 and 1. As can be observed from table 1, the corresponding parameter values for the two sets (i.e. with 24 players and 30 players) are mostly within the error bars. Therefore, using the data on all the nine games, we recalculate the coefficients. Using these coefficients, we simulate the model (see figure 3). With 24 agents and 500 simulations around 82% of the games reach a solution within 21 rounds; and the games end with a mean ACP of 0.98. In the case of 30 agents, 66% of the games reach a solution and the mean ACP is 0.95. In figure 5 we compare the ACP and the activity from the experiment with those from the simulations (averaged over runs) using the model. Note, that in addition to the probabilistic choice, we use an additional rule that prevents exchanges between agents belonging to different large clusters and increases the fraction of games completed. This rule, however, has marginal effect on the ACP and the overall activity (see the following section).

Table 1.

Values of parameters (coefficients and the intercept) derived from a logistic regression on the data.

focal	variable	coeff	24-player	30-player	joined
requester	cluster size of requester	αr	−5.33 ± 0.71	−4.06 ± 0.52	−4.68 ± 0.41
	cluster size of requested neighbour	βr	−5.36 ± 1.58	−3.80 ± 1.01	−4.52 ± 0.84
	average cluster size (neighbour’s colour)	δr	4.89 ± 1.89	2.91 ± 1.19	3.85 ± 0.99
	(intercept)	λr	3.25 ± 0.59	2.61 ± 0.43	2.96 ± 0.34
acceptor	cluster size of acceptor	αa	−5.23 ± 0.83	−3.64 ± 0.54	−4.15 ± 0.45
	cluster size of requesting neighbour	βa	−5.14 ± 2.03	−4.53 ± 2.17	−4.39 ± 1.50
	average cluster size (neighbour’s colour)	δa	5.32 ± 2.33	5.19 ± 2.30	4.76 ± 1.60
	(intercept)	λa	3.25 ± 0.62	2.31 ± 0.40	2.70 ± 0.33

Interestingly, the fact that for requesters and acceptors, the magnitudes of β and δ also overlap within the error margins, could support our initial ansatz about the inclusion of the term U_j = 〈s(c_j)〉 − s_j, as an independent variable, which would, however, diminish the importance of s_j as a separate independent variable. To investigate the importance of such a term, we slightly modify our model by considering the magnitudes of δ_r and β_r to be equal. We simulate the model by varying δ_r (keeping the relation |δ_r| = |β_r|) while taking the values for the other coefficients from table 1. The resulting plot is shown in figure 6a. The plot shows that a larger δ_r (and β_r) enhances the performance of the agents. Similarly, by varying α_r in the model, we benchmark the perception of risk in the human subjects. Agents in the model are less likely to break away from clusters when α_r is negative and large. The figure 6b shows a region near α_r = −5 where the agents perform best. Remarkably, we find that α_r obtained from the experiment coincides with the optimal value.

Figure 6.

Sensitivity of the requester model. Results from simulations of games with 30 agents demonstrating the effect of variation of the parameters in the model corresponding to the requesters on the average collective progress (red triangles) and the fraction of games completed (blue circles). (a) The parameters −β_r and δ_r are assumed to be equal and simultaneously varied. The dashed vertical line shows the location of −β_r obtained from the experiment. (b) The parameter α_r is varied. The dashed vertical line shows the location of α_r obtained from the experiment. Each of the points in the figures are averaged values from 500 runs. The shaded region in case of the ACP represents the standard deviation associated with the points. In case of the completion fraction, the region indicates the Clopper–Pearson interval. (Online version in colour.)

3.3. Model details

3.3.1. Stability rule for clusters

The expressions for probabilities in our basic model is a continuous function of the variables and is linear in terms of the argument. Actual human decision-making can be quite complex and a more accurate description might require the inclusion of nonlinearities or discontinuous dependence in terms of the variables. A lack of this in the basic model might have resulted in the excess occurrence of requesting activity between large cluster sizes. In figure 7, we show this by comparing experiment and simulations for the games with 30 players. We observed that in the model, the presence of such activity can impact the completion of the games. Requesting (and accepting) activity between players belonging to large clusters could be considered as detrimental if not useless in terms of reaching the solution. The solution might not be reached in cases in which the large clusters become unstable. Therefore, in the simulations, we prevent such requesting actions from taking place. We prohibit any requesting activity between two agents having cluster size larger than 0.6l, where l is the maximum possible value for a cluster size. The effect of such a rule can be investigated by introducing a parameter f in the model such that 1 − f is the probability of allowing such an action. With f = 0 we have the basic model and with f = 1 such requests are completely forbidden. In electronic supplementary material, figure S1, we show the effect of varying f on the ACP and the fraction of games completed. For the results of simulations reported in the main text, we take f = 1. Note, that large clusters can still fragment when requests come from smaller clusters.

Figure 7.

Requesting between large clusters in the games with 30 players—experiment (columns on the left) and simulations without the stability rule (columns on the right). Requesters having cluster sizes larger than 6 are considered (in this case the largest possible cluster size is 10). The cases when such requesters sent requests to other players having cluster size larger than 6 are counted. Also the cases when such requesters, though positioned in the neighbourhood of players with cluster sizes larger than 6, did not send any request to the latter are counted. The above two counts are normalized by the total number of cases and compared. The plots reveal that when simulating with the basic model, there is an excess of cases where players belonging to larger clusters send a request to players in larger clusters. (Online version in colour.)

3.3.2. Acceptance in the model

We observed that in the experiment the probability of a player receiving more than two requests in a given round was negligible. The dominating case was that of a single request (70%) as can be observed from electronic supplementary material, figure S2. Also, we found that when two requests were received simultaneously, the choice of acceptance by the focal player was not significantly influenced by the cluster sizes of the requesters (electronic supplementary material, figure S2, right). Therefore, we modelled acceptance by fitting our basic model (having the same structure as that of requesting) to the dominating case of single requests. In the simulations when there were cases of multiple requests, we randomly considered one of the requests to be weighed against the decision of not exchanging.

4. Discussion

Models and experiments based on game theory [36–38], especially those based on Prisoner’s Dilemma (PD) have been used extensively to study the formation of groups [39–42]. In this paradigm, the existence of cooperative ties could lead to the formation of cohesive groups, and the risk arises when players choose to defect. However, in the present framework, we interpret risk as an individual player’s decision to favour exchanging their current locations. Unlike models based on PD where the pay-off matrix would ideally determine the amount of risk, in our case, it is the propensity of an individual to relocate depending on his or her current cluster size. The dilemma arises from incomplete information, but the goal is always collective, which is how to converge to a win–win situation in a limited number of rounds.

The results from the experiment show, in general, that human players are able to process complex information about their neighbourhood as well as take into account global information about cluster sizes. It appears that the players understand well the cooperative setting of the game. In order to achieve the desired target, the players tend to coordinate their actions such that an exchange is done if it is mutually beneficial. The limited capacity of processing the neighbourhood information is apparent in figure 6a. It is found that larger magnitudes of the requester parameters of β_r and δ_r would have signified ‘wiser’ decision-making and would have led to faster progress in the game and better completion rates. However, the fact that the values of the parameters (see table 1) are significant with appropriate signs, reflects the rationality in the choice of neighbours during the action of request. The values of the acceptance parameters β_a and δ_a would also signify rationality although most of the time the acceptors receive only one request and the decision is limited to agreeing or disagreeing to exchange places.

The dynamics of fragmentation and formation of clusters crucially depends on α (the coefficient of cluster size of the focal player, the latter being a requester or an acceptor). Whereas a negative value of α would make larger clusters stable, an extremely large magnitude would make clusters inactive and render exchanges impossible. Interestingly, figure 6b shows that for requesters the value of α_r obtained from the experiment coincides with the optimal value predicted from simulations using the model. This kind of balance could result from the cooperative setting, which was well comprehended by the human subjects. This is evidenced from the cluster fragmentation illustrated in the examples shown in figure 4. Variation in α_a does not seem to have much effect. Overall, the parameter values could be reflecting as strategies of human individuals for engaging in coordination when risks are present [43]. We have also examined the aspect of learning during sessions. To do this, we partitioned the decisions into categories and checked for significant differences between their frequency distributions across the nine sessions employing chi-squared tests. The differences appeared to be non-significant (see electronic supplementary material, smallest p-value is 0.07). Similarly, we have extended the basic model to quantify the random effects resulting from the heterogeneity between the participants. By using mixed effects logistic regressions, we obtained standard deviations corresponding to the parameters of the original model (also provided in the electronic supplementary material).

The choice of the network was primarily guided by the issue of achieving convergence in a limited number of rounds while maintaining a certain level of complexity. From each player’s point of view, the accessible information comes from two sources: the local neighbourhood which describes the status of his/her neighbours, and the global information, which only shows the overall progress towards the goal, without giving any hint about the configuration of whole the network. Preliminary agent-based simulations on other topologies, for example, networks with structural communities [44], revealed extra complications and bottlenecks for the players. In general, non-regular networks with varying neighbourhoods are expected to increase the difficulty for players trying to locate themselves and their neighbours in the network. Also, frequent changes in the number of neighbours after each move would add up to the mental processing required in each round. In this regard, the chosen small-world network could have also taken different parameters (for example, using as an initial backbone a ring of size 3 × 10 instead of the chosen 6 × 5). Simulations allowed us to assess the possible layouts and possible number of long-distance links that could facilitate solving the problem in a limited amount of time. The choice of three colours was, again, with the idea of keeping the game complex enough without being extra demanding. We ran simulations of the game with more colours, and it always required considerably more rounds to be completed. Besides, increasing the number of colours would have increased the number of variables to deal with for each player, requiring more time for processing information and making decisions.

We place the game in the broad context of group formation in social networks and technological networks. The experiment illustrates how individuals located on a network and coordinating over links could achieve configurations that would in principle benefit all. In the context of real-world social networks, such dynamics would represent a mutual enhancement of individuals by the ‘bonding’ and ‘bridging’ of social capital [45–47] or the formation of social coalitions [24,25]. In a sense, the notion of a link between two nodes having the same colour is comparable to the notion of an edge in a puzzle graph (representing a compatible idea) in the recent formulation of ‘jigsaw percolation’ [48]. While an exchange of positions between two individuals is to be understood primarily as simultaneous changes in the social space of the pair, such an activity could also be considered as an exchange of physical locations of two individuals like that in a faculty exchange programme between universities. The desired nature of linking in the network with its underlying spatial structure can also be considered relevant in communication networks where the nodes are autonomous mobile agents establishing peer-to-peer radio network. The theory of cooperative games has been used to design deployment protocols of mobile agents where a coalition of agents would share a certain frequency spectrum [49].

As we have performed the experiment with a limited number of subjects, the behaviour captured via the coefficients in the model will reflect their particular characteristics and may not be universal. For a different subject pool, the individual coefficients may be different while the overall behaviour may still be close to optimal. However, we have no reason to suppose that the modelling scheme, maintaining the current level of complexity, would be entirely different and that the current formulation would exert any special influence on our inferences. Overall, we expect the broad principles revealed in this study to apply more or less universally. Our decision-making model not only serves the purpose of extracting the behavioural aspects of real human subjects, it allows us to compare other possibilities with respect to the parameter values. Although we do not perform an exhaustive search in the parameter space, we gain sufficient insight into human behaviour when we find faster convergence for some parameters. Our goal in the near future is to perform experiments with artificial agents or bots playing with humans [6,7]. Agents formulated using the model can mirror the typicalities of human behaviour who are actually willing to cooperate during tasks that require collective coordination [1]. Such agents may be used to test the collective performance in human–agent hybrid systems where humans are guided to make decisions based on less collective or selfish motivations. For instance, it would be interesting to compare the outcomes from games where the individual groups are entirely constituted by humans or bots, with games where groups are formed by mixing humans and bots.

Ethics

Approval for the experiment was obtained from the Aalto University Research Ethics Committee. The experiment was performed following the relevant guidelines and regulations of the Committee.

Data accessibility

All the relevant data used in the paper is available at https://doi.org/10.5281/zenodo.3237575. The codes for the simulation are available at https://github.com/ttakko/NetworkGameSimulations.

Competing interests

We declare we have no competing interests.

Funding

All the authors acknowledge the support from EU HORIZON 2020 FET Open RIA project (IBSEN) no. 662725. K.K. acknowledges support from Academy of Finland Research project (COSDYN) no. 276439 and from the European Community's H2020 Program under the scheme “INFRAIA-1-2014-2015: Research Infrastructures”, Grant agreement no. 654024 “SoBigData: Social Mining and Big Data Ecosystem” (http://www.sobigdata.eu). D.M. acknowledges support from CONACYT, Mexico grant 383907.