Thomas Bassanetti

Stéphane Cezera

Maxime Delacroix

Ramón Escobedo

Adrien Blanchet

Clément Sire

Guy Theraulaz
email: guy.theraulaz@univ-tlse3.fr

Cooperation and deception through stigmergic interactions in human groups

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

D R A F T

Cooperation and deception through stigmergic interactions in human groups

Thomas Bassanetti a, b , Stéphane Cezera c , Maxime Delacroix a , Ramón Escobedo b , Adrien Blanchet c, d , Clément Sire a , and Guy Theraulaz b, d, 1

a Laboratoire de Physique Théorique, CNRS, Université Toulouse III -Paul Sabatier, 31062 Toulouse, France; b Centre de Recherches sur la Cognition Animale, Centre de Biologie Intégrative, CNRS, Université Toulouse III -Paul Sabatier, 31062 Toulouse, France; c Toulouse School of Economics, CNRS, 31080 Toulouse, France; d Institute for Advanced Study in Toulouse, 31080 Toulouse, France This manuscript was compiled on [START_REF] Ac Davison | Bootstrap Methods and their Application[END_REF] Stigmergy is a generic coordination mechanism widely used by animal societies, in which traces left by individuals in a medium guide and stimulate their subsequent actions. In humans, new forms of stigmergic processes have emerged through the development of online services that extensively use the digital traces left by their users. Here, we combine interactive experiments with faithful data-based modeling to investigate how groups of individuals exploit a simple rating system and the resulting traces in an information search task in competitive or non-competitive conditions. We find that stigmergic interactions can help groups to collectively find the cells with the highest values in a table of hidden numbers. We show that individuals can be classified into three behavioral profiles that differ in their degree of cooperation. Moreover, the competitive situation prompts individuals to give deceptive ratings and reinforces the weight of private information versus social information in their decisions. T he exchange of social information is the core mechanism by which groups of individuals are able to coordinate their activities and collectively solve problems [START_REF] Camazine | Self-Organization in Biological Systems[END_REF][START_REF] Garnier | The biological principles of swarm intelligence[END_REF][START_REF] Couzin | Collective minds[END_REF][START_REF] Rl Goldstone | Collective behavior[END_REF][START_REF] Moussaid | Collective information processing and pattern formation in swarms, flocks, and crowds[END_REF]. Social information allows individuals to adapt to their environment faster and/or better than through collecting personal information alone [START_REF] Tr Zentall | Social learning: Psychological and biological perspectives[END_REF][START_REF] Heyes | Social Learning in Animals: The Roots of Culture[END_REF][START_REF] Danchin | Public information: From nosy neighbors to cultural evolution[END_REF][START_REF] Kn Laland | Social learning strategies[END_REF][START_REF] Duboscq | Social information transmission in animals: Lessons from studies of diffusion[END_REF]. The use of social information thus provides evolutionary advantages to animal groups and occurs in many contexts, such as foraging, decision-making, division of labor, nest building, or colony defense [START_REF] Camazine | Self-Organization in Biological Systems[END_REF][START_REF] Garnier | The biological principles of swarm intelligence[END_REF][START_REF] Sumpter | Collective Animal Behavior[END_REF][START_REF] Td Seeley | Honeybee Democracy[END_REF]. Quite often, social information is indirectly shared between individuals: some of them leave traces of their activities in the environment and others can use this information to guide their own behavior and inform their own decisions [START_REF] Giuggioli | Stigmergy, collective actions, and animal social spacing[END_REF]. This form of indirect communication, also called stigmergy, in which the trace of an action left on a medium stimulates the performance of a subsequent action which produces another trace and so on, is widely used by animal societies and especially social insects to self-organize their collective behaviors [START_REF] Pp Grassé | La reconstruction du nid et les coordinations interindividuelles chez bellicositermes natalensis et cubitermes sp. la théorie de la stigmergie: Essai d'interprétation du comportement des termites constructeurs[END_REF][START_REF] Theraulaz | A brief history of stigmergy[END_REF][START_REF] Heylighen | Stigmergy as a universal coordination mechanism ii: Varieties and evolution[END_REF]. These stigmergic interactions that allow the emergence of coordinated activities out of local independent actions likely played a major role in the evolution of cooperativity within groups of organisms [START_REF] Chiong | The evolution of cooperation via stigmergic interactions in[END_REF][START_REF] Heylighen | Stigmergy as a universal coordination mechanism i: Definition and components[END_REF].

In humans, with the digitalization of society and economies, social information has increasingly taken the form of digital traces, which are the data individuals leave either actively or passively when using the Internet [START_REF] Mj Doyle | Stigmergy 3.0: From ants to economies[END_REF][START_REF] Cochoy | Digitalizing Consumption: How devices shape consumer culture[END_REF][START_REF] Rk Baltzersen | Cultural-Historical Perspectives on Collective Intelligence[END_REF]. New forms of stigmergic processes have been identified since these digital traces are largely exploited in social networks and in electronic commerce, in particular through the use of rating and recommendation systems that can help participants to discover new options and make better choices [START_REF] Hvd Parunak | A survey of environments and mechanisms for human-human stigmergy in Environments for Multi-Agent Systems II[END_REF][START_REF] Lü | Recommender systems[END_REF][START_REF] Hennig-Thurau | Can automated group recommender systems help consumers make better choices?[END_REF][START_REF] Sa Golder | Digital footprints: Opportunities and challenges for online social research[END_REF][START_REF] Jesse | Digital nudging with recommender systems: Survey and future directions[END_REF]. However, indi-31 viduals do not use social information in the same way. Some 32 individuals exploit it to make their choices, while others may 33 simply ignore it and only use their own private information, [START_REF] Es Gloag | Stigmergy: A key driver of self-organization in bacterial biofilms[END_REF] or can even go against the message delivered by social infor-35 mation [START_REF] Jayles | How social information can improve estimation accuracy in human groups[END_REF]. In fact, the same individual can even change the 36 way they provide and uses social information depending on 37 the context [START_REF] Steinel | The good, the bad and the ugly thing to do when sharing 1050 information: Revealing, concealing and lying depend on social motivation, distribution and 1051 importance of information[END_REF]. [START_REF] Loh | How has the internet reshaped human cognition?[END_REF] Moreover, the use of digital traces is very sensitive to noise 39 and manipulation [START_REF] Herlocker | Evaluating collaborative filtering recommender 1053 systems[END_REF][START_REF] Gunes | Shilling attacks against recommender systems: a compre-1055 hensive survey[END_REF]. Indeed, in competitive situations, 40 malicious spammers can manipulate social information by processes has been done so far [START_REF] Correia | On the role of stigmergy in cognition[END_REF][START_REF] Coucke | HuGoS: a virtual environment for studying 1059 collective human behavior from a swarm intelligence perspective[END_REF].

49

The aims of this study are twofold. First, we study through

Significance Statement

Most online services and applications on the Internet rely on digital traces resulting from choices made by their users, in particular, by means of rating-based recommendation systems. Therefore, it is crucial to understand how such traces affect individual and collective decision-making. We have conducted experiments to measure how groups of individuals interact with digital traces and determine how they could use these traces to cooperate and find the cells with the highest values in a table of hidden numbers. Our experiments and data-driven model show that digital traces spontaneously induce cooperation between individuals. However, the way individuals use the traces to deliver information to others and the reliability of that information largely depends on the degree of competition between individuals. At the start of the next round, the color of each cell in the table is updated according to the fraction of stars that have been used to rate the cell since the beginning of the experiment, that is, the number of stars in the cell divided by the total number of stars in all cells. The color scale varies between white (0 %) and black (100 %) through a gradient of shades of red (see SI-Appendix, Fig. S1C ). Thus, the cells that have received the highest fraction of stars since the beginning of the experiment will be clearly visible to all the individuals belonging to the same group. The resulting color map on the table acts like a cumulative long-term collective memory for the group, which is updated at each round. Note that the subjects cannot infer the precise value of the fraction of stars in a cell from its color, but only a rough estimate. However, they can certainly exploit the colors to compare the fraction of stars in the different cells of the table and to identify the cells with a high fraction of stars. Fig. 1A shows an example of a table displaying the participants' ratings as a color map after 10 rounds during one experiment.

D R A F T

We also investigate the impact of a competitive versus a non-competitive condition on the behavior of participants, and the individual and collective performance. In particular, we focus our analysis on the way individuals visit and rate cells and how they use the traces resulting from their ratings and those left by the other group members to guide their choices. In each experiment, we studied the individual and collective behaviors of two groups performing the same task in parallel and independently. In the non-competitive condition, hereafter called Rule 1, the actions (cell visits and ratings) of the participants do not affect the amount of reward they received at the end of an experiment that always remains constant. On the other hand, in the competitive condition, hereafter called Rule 2, the score of a participant increases at each round by the value of the cells they visit, but remains unaffected by their rating of these cells. Then, the cumulative score of participants over an entire session (12 experiments) determines their monetary reward, which depends on their ranking among the 10 participants and not just among the 5 members of their group (see Materials and Methods for the actual payment method).

This experimental design allowed us to study the impact of an intragroup competition, since each individual in a group competes with the 4 other members of their group. However, there is also an implicit intergroup competition, since each individual also competes with the 5 members of the other group for the best rank. SI-Appendix, Fig. S2 illustrates the actions performed by each participant in one group and the color maps associated with the cells in the table resulting from their ratings. SI-Appendix, Movies S1A (Rule 1) and S2A (Rule 2) show examples of the dynamics of a typical experimental run where the participants achieved a group score near the observed mean group score. In the corresponding SI-Appendix, Movies S3 and S4, we present an experimental run where the participants obtained a group score 50 % higher than the observed mean group score. SI-Appendix, Movie S5 features the same results as Movies S1-S4 but without the cell values, to better identify the different shades of red and to better reflect what the subjects actually saw during the experiment.

In the next section, we present the results of this experiment mimicking several processes at play in actual 5-star The dotted vertical lines are the mean score in the experiment, and the dashed vertical lines are the mean scores in the model. (C) Average value of the cells visited at round t, q(t) and (E) up to round t, Q(t). (D) Average value of the cells visited weighted by their ratings at round t, p(t) and (F ) up to round t, P (t). (G) and (I) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)), measuring the effective number of cells over which the visits are distributed at round t and up to round t, respectively. (H) and (J) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)), measuring the effective number of cells over which the ratings are distributed at round t and up to round t, respectively. (K ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V). rating systems: (i) the exploration by the participants of Fig. 2 A and B show respectively the probability distribu-185 tion functions (PDF) of the score S of individuals obtained after the 20 rounds and the score Ŝ of groups, i.e., the sum of the scores of the individuals belonging to the same group. In Rule 1, all scores are equal to 0. Thus, in order to compare the individual and collective performance in the two rules, each individual is assigned a virtual score computed in the same way as in Rule 2. The mean score is higher in Rule 2, showing that this competitive condition provides a stronger incentive to visit high-value cells than in Rule 1: ⟨S/Smax⟩ = 0.24 ± 0.01 in Rule 1 vs. ⟨S/Smax⟩ = 0.40 ± 0.01 in Rule 2, where Smax = 5420 is the maximum theoretical score. Fig. 2 C -F show that the average value of the visited cells increases with the number of rounds as the participants discover, visit, and rate cells with higher values. Although p(t) and P (t) are higher in Rule 1 than in Rule 2 (Fig. 2 D and F ), the average value of visited cells at round t, q(t), and up to round t, Q(t), are significantly higher in Rule 2 (Fig. 2 C andE). As we will see later, this apparent paradox is due to the fact that the strategies used by individuals to rate cells in Rule 1 and Rule 2 are very different. In particular, in the competitive Rule 2, some individuals choose to give an average or even a low rating to cells having a high value, presumably to avoid reporting these cells to the other members of their group.

Fig. 2 G and I show that individuals visit significantly more different cells in Rule 1 than in Rule 2, with IPR(Q(t)) growing up to the final round t = 20 in Rule 1, while it starts decaying after round t = 7 in Rule 2. In particular, at the final round t = 20 of the experiment, IPR(Q(t = 20)) is roughly four times larger in Rule 1 than in Rule 2. As we will see in the next section, the lower exploration observed in Rule 2 is

D R A F T

mostly due to the fact that the individuals revisit a lot more cells with high values instead of exploring new cells, in order to maximize their score. Moreover, in each round, individuals allocate more stars in Rule 1 compared to Rule 2 (see Fig. 2H ), but overall, they allocate stars to the same number of cells (see Fig. 2J ). Fig. 2 K and L show that in both conditions, the fidelity increases with the round t, suggesting that the correlations between the participants' visits/ratings and the cell values increase with time. In the final round of the experiment, the fidelity of ratings F(P(t = 20), V) is significantly higher in Rule 1 than in Rule 2. As we mentioned previously, in Rule 1, the participants explore the table a lot more and their ratings better reflect the value of the cells that they have discovered.

Individual Behaviors. In this section, we characterize the behaviors of individuals and their strategies to visit and rate cells, i.e., the way they use social information in the form of colored traces resulting from their collective past actions. Moreover, we also quantify the impact of intragroup competition on their behaviors.

Choosing the cells to be visited. The probabilities of finding the cells with the highest values are higher in Rule 1 than in Rule 2 (see Fig. 3 A-C and SI-Appendix, Fig. S3). In Rule 1, individuals find the best cells more often than would be expected if they had searched randomly, illustrating the cooperative effect induced by the use of the digital traces by individuals within groups. In Rule 2, we observe the opposite phenomenon: individuals often revisit the cells that they consider high enough to improve their score, without taking the risk of exploring new low-value cells. However, this kind of hedging also hampers their ability to discover even better cells.

We define V1(t), V2(t), and V3(t) as the average of the first-, second-, and third-best values of the cells visited by the participants at round t. Fig. 3 D-F shows that in both conditions, the average values of these 3 cells increase with round t. However, their average values are higher in Rule 2.

Note that this is not in contradiction with the results shown in Fig. 3 A-C. As a matter of fact, in Rule 1, individuals have no incentive to revisit cells with high values, so they continue exploring the table even if they have already found those cells. As already mentioned, in Rule 2, individuals have a clear incentive to revisit cells with high values that they can remember, and thus to explore the table less, so that they ultimately discover the cells with the highest values less often.

To confirm this interpretation, we quantify the way individuals revisit cells by defining, for t > 1, B1(t), B2(t), and B3(t) as the probability of revisiting at round t the cells with the first-, second-, and third-best values of the previous round (t -1). Figure 3 G-I show that individuals tend to revisit the cells with the best values, and more so as the value of the visited cells increases over time. In the final round of Rule 2, individuals revisit their first-, second-, and third-best cells of the previous round with respective probabilities 93 %, 87 %, and 66 %. In addition, these observables confirm that individuals explore the table more in Rule 1 than in Rule 2: at any round t ≥ 5, the values of B1(t), B2(t), and B3(t) in Rule 1 are typically less than one-third of the value in Rule 2.

Altogether, these results illustrate the strong impact of a competitive condition on the way individuals explore the table and select the cells they visit at each round. Rating the visited cells. SI-Appendix, Fig. S4 shows the average 278 fraction of stars ρ(v) that has been used to rate cells with 279 value v at the end of the experiment. ρ(v) increases with v, 280 showing that, on average, individuals give higher ratings to 281 cells having high values and also revisit them more often. The 282 experimental data can be fitted to the following functional 283 form:

284 ρε,α(v) = ε 1 N + (1 -ε) v α w Nww α [1] 285
where ε ∈ [0, 1] and α are two parameters, N = 225 is the 286 total number of cells in a table, and Nv is the number of cells 287 with value v, such that v Nvρε,α(v) = 1. Note that the first 288 term ε/N quantifies the fraction of stars uniformly deposited 289 in the cells, while the second term involving α accounts for 290 the fact that high-value cells should attract more stars. Thus, a cell will receive similar ratings regardless of its value between 35 and 99. This phenomenon suggests that in Rule 2, many participants adopt a non-cooperative/deceptive rating strategy, which effectively makes the information conveyed by the digital trace less discriminating. Overall, these results

show that individuals give a much fairer rating to the cells they visit in Rule 1, as the examination of the fidelity has previously revealed.

Behavioral profiles of individuals. SI-Appendix, Figs. S6 andS7 show the average number of stars used to rate a cell as a function of its value v, for each participant, in Rule 1 and Rule 2, leading to three emerging rating patterns. Some individuals rate cells somewhat proportionally to their value, some rate cells independently of their value, and some others give ratings somewhat oppositely proportional to the cell values.

To quantify and classify these three behavioral profiles, we fit the average rating of each individual with a linear function of the cell value v, u0 + u1 × 5v/99, where u0 is the intercept and u1 is the slope of the line. u0 = 0 and u1 = 1 would correspond to a strict linear rating of cells of value 0 to 99, with 0 to 5 stars. Fig. 4 shows the distribution of u0 and u1 for all individuals. We identify three classes of behavioral profiles associated with two thresholds at u def-neu = -0.5 and u neu-col = 0.5 corresponding to the two minima found in the distribution of u1. Note that the two thresholds for these three classes are close to the thresholds found using Ward's clustering method on the slope parameter u1:

• The ratings of individuals with u1 ≥ u neu-col increase with the cell values, i.e., they rate cells whose values are the lowest (resp. whose values are the highest) with a small number of stars (resp. a large number of stars; see Fig. 5A). Hereafter, we will dub these individuals as collaborators, since their rating strategy helps the other members of their group to identify the best cells (84 % in Rule 1 and 13 % in Rule 2).

• Individuals with u def-neu ≤ u1 < u neu-col rate cells with almost the same number of stars (on average, 3 stars in Rule 1, and 1.5 stars in Rule 2) regardless of their values (see Fig. 5B). Since the ratings of these individuals do not provide any distinctive information to the other group members, we will dub them as neutrals (13 % in Rule 1 and 49 % in Rule 2). Note that these neutral individuals do not form a homogeneous group. Indeed, some of them with u0 close to 0 always give 0 or a very few stars whatever the cell value, hence essentially not participating in the rating and the marking of the cells. Some other neutrals with u0 close to 5 always give a large number of stars or even 5 stars, thus marking all the cells they visit, while others do not have any consistent logic in the way they rate cells. This explains the wide range of intercepts cells, since they often revisit these cells, and hence make 354 them darker. We also address this point in the section 355 below about optimized agents and in section B. hiding them from the other members of their group. Con- Model. We now introduce a stochastic agent-based model to quantitatively identify the strategies for visiting and rating cells, and to understand their respective effects on individual and collective performance. In the model, we simulate groups of 5 agents playing a sequence of 20 consecutive rounds (3 visited and rated cells per round), exactly following the actual experimental procedure. The model, described in detail in Materials and Methods, consists of two steps that characterize the agents' visit and rating strategies.

The first step accounts for the visit strategy, i.e., which 3 cells an agent decides to visit in each round. This strategy allows for a variety of behaviors observed in the experiment:

• revisiting the first-, second-, and/or third-best cells already visited in the previous round, depending on their value (private memory; see Fig. 3 G-I );

• exploring a marked or unmarked cell (collective memory; see SI-Appendix, Fig. S4) according to its cumulative fraction of stars represented by the color of the cell in the actual experiment.

The visit strategy is the same for all agents, regardless of their behavioral profile (cooperator, neutral, or defector), as found experimentally, but is allowed to differ for the two conditions, Rule 1 and Rule 2.

The second step of the model addresses the rating strategy, i.e., the number of stars an agent uses to rate a visited cell as a function of its value. In the model, the rating strategy of agents depends on their behavioral profile (see Fig. 5 (D-I )), and is different for the two rules.

Model predictions. We consider groups of 5 agents, hereafter called MIMIC (see SI-Appendix, Movies S1B and S2B), reproducing the behaviors of human collaborators, neutrals, and defectors. Their behavioral profiles are drawn according to the corresponding fraction observed in the experiment (inset table of Fig. 4). The parameters for the rating strategies of collaborators, neutrals, and defectors have been estimated by fitting the probability to rate a cell with 0 or 5 stars (see Eqs. 5 and 6 in Materials and Methods) to the experimental data (see lines in Fig. 5 (D-I ), and SI-Appendix, Table S1). As for the parameters for the visit strategy, they have been estimated by minimizing the error between the experimental and the model results for a set of observables, using a Monte Carlo method (see SI-Appendix, Table S2). For all graphs, we ran 1,000,000 simulations, so that the error bars in our simulation results are negligible on the scale of the presented graphs.

Fig. 2 shows that simulations of the model with MIMIC agents quantitatively reproduce the performance of individuals and groups and the observables used to characterize the dynamics of collective exploration and ratings in both rules, as measured in the experiment. The model also quantitatively reproduces the dynamics of the average value of the first-best, second-best, and third-best cells visited by individuals during the different rounds (Fig. 3 D-F ), along with the probability to revisit each of these 3 best cells at the next turn (Fig. 3 G-I ). In addition, the model reproduces fairly the fraction of collaborators, neutrals, or defectors according to their rank at the end of the experiment and the negative impact of the D R A F T number of defectors on collective performance (SI-Appendix, Fig. S9). The model also predicts with great accuracy the nontrivial results of Fig. 3 (A-C ), and SI-Appendix, Fig. S3 that were commented above.

These results suggest that the behavioral mechanisms implemented in the model constitute an excellent representation of the processes by which individuals leave and use the traces to guide their choice, and how these processes are modulated in the presence of competition between individuals.

Finally, in the SI-Appendix, Supplementary Text, we also explore the model predictions for larger group sizes, larger tables, longer durations, and different types of visit and rating strategies.

Optimization of agents' performance according to specific objectives.

We have also exploited our model to find agents that are optimized in different situations. To do this, we have used a Monte Carlo method to obtain all the parameters of the model that characterize the corresponding visit and rating strategies.

We first consider a situation in which we wish to maximize the score S (as defined in Rule 2) of 5 identical agents (Opt-1 agents) in the same group and exploiting the same strategy (see SI-Appendix, Figs. S15 and S19A and SI-Appendix, Tables S1G andS2). The inspection of the Opt-1 agents' resulting parameters and SI-Appendix, Fig. S15 show that they essentially only rate cells that have very high values, which they revisit at almost every round so that there is almost no exploration. These Opt-1 agents are strong collaborators, and their average score (S/Smax = 67 %) is markedly higher than the score of the human subjects in Rule 2 (S/Smax = 40 %).

Note that, since the 5 Opt-1 agents are identical, they also maximize the total score of the group. This suggests that a situation where groups would compete (instead of individuals; intergroup instead of intragroup competition) should lead to the emergence of a collaborative behavior withing the groups, a situation that we plan to explore experimentally in a future work.

We then consider a situation in which we maximize the score of one agent competing with 4 MIMIC agents (see SI-Appendix, Figs. S16 and S19B and SI-Appendix, Tables S1H andS2).

This scenario represents a more realistic situation where an individual seeks to maximize their score while competing against four other typical individuals. In this condition, the behavior of this optimized agent (Opt-2) is markedly different from that of Opt-1 agents, since the presence of MIMIC agents behaving as neutrals and defectors forces the Opt-2 agent to adapt its visit and rating strategy to cope with indiscriminate or even false social information. Interestingly, the optimization process leads to a neutral agent assigning 0 star to every visited cell, and hence not participating at all in the rating process.

Note that, as already mentioned in the description of neutral agents above (and in section B.2 of the SI-Appendix, Supplementary Text), a neutral agent assigning a non-zero number of stars to visited cells would effectively help the other members of its group to identify the best cells, since it would often revisit these cells. The average score of the Opt-2 agents is S/Smax = 43 %, which is only slightly better than the average score of human subjects or MIMIC agents.

However, in our experiment, to obtain the maximum monetary reward, individuals were not strictly required to maximize their score but rather had to optimize their ranking among the 10 individuals in the two groups of 5 participants. In this condition, the optimized Opt-3 agent competing against 4 524 (in its group) plus 5 (in the other group) MIMIC agents be-525 haves as a defector (see SI-Appendix, Figs. S17 and S19C 526 and SI-Appendix, Tables S1I andS2). On average, the Opt-3 527 agent obtains a rank of 4.57 (compared to a mean rank of 5.5) 528 when ranked among the 10 agents of the two groups, and a 529 rank of 2.50 within its own group (mean rank equal to 3). It 530 is remarkable that the model predicts that deception is an 531 emerging behavior in the conditions of our experiment.

532

Finally, it is interesting to consider the visit and rating 533 strategies maximizing the fidelity of the distribution of rat-534 ings to the distribution of cell values in the final round, 535 F(P(t = 20), V) (see SI-Appendix, Fig. S18 and SI-Appendix, 536 Tables S1 andS2). If the number of rounds were infinite, the 537 optimal strategy for these agents (Opt-4) would be to explore 538 the table randomly and to rate cells proportionally to their 539 value on a full scale of 0 to 5 stars (corresponding to u0 = 0 and 540 u1 = 1 in Fig. 4). By using this strategy, the agents achieve a 541 fidelity of 0.76 at round 20 (compared to 0.4 in Fig. 2L), and 542 the fidelity would ultimately converge to 1 in the limit of an 543 infinite number of rounds. Clearly, these Opt-4 agents achieve 544 a very mediocre mean score of S/Smax = 11 % compared to 545 that of the previous optimized agents, and even compared to 546 MIMIC agents reproducing the experimental results, and to 547 the human participants. It is worth noting that there could 548 exist a better strategy to maximize the fidelity at round t = 20, 549 specifically tailored for the finite 20-round setting used in the 550 actual experiment.

551

Discussion

552

The ability to exploit the traces left in the environment by 553 the action of organisms is one of the simplest and oldest mech-554 anisms used to coordinate collective behaviors in biological 555 systems [START_REF] Es Gloag | Stigmergy: A key driver of self-organization in bacterial biofilms[END_REF][START_REF] Es Gloag | Bacterial stigmergy: An organising principle of multicel-1066 lular collective behaviours of bacteria[END_REF][START_REF] Khuong | Stigmergic construction and topochemical information shape ant nest 1068 architecture[END_REF]. In humans, over the past thirty years, the 556 massive development of the Internet, together with applica-557 tions that extensively use digital traces left voluntarily or 558 not by their users, have reinforced the need to understand 559 how these traces influence individual and collective behav-560 iors [START_REF] Sa Golder | Digital footprints: Opportunities and challenges for online social research[END_REF][START_REF] Karanasios | Making sense of digital traces: An activity theory driven ontological 1070 approach[END_REF][START_REF] Loh | How has the internet reshaped human cognition?[END_REF][START_REF] Firth | The "online brain": how the internet may be changing our cognition[END_REF].

561

In this work, we have measured and modeled the way groups 562 of individuals leave and use digital traces in an information 563 search task implementing a 5-star rating system similar to 564 the ones used by many online marketplaces and platforms 565 such as Amazon, TripAdvisor, or eBay, in which users can 566 evaluate products, services, or sellers. Although we certainly 567 do not claim that our experimental setup captures all the 568 processes at play in these real-life situations, it shares with 569 them an exploration of the available options (cells in our 570 experiment; products for an online store) greatly influenced 571 by their current ratings, and a rating of the selected options 572 by the participants, allowing the ratings to evolve dynamically. 573 However, real rating systems usually provide the users with 574 not only the mean rating of an available option, but also 575 the number of ratings for this option, which allows them to 576 modulate their confidence in the different ratings.

577

Our experiment considered two different rules, with Rule 2 578 implementing a monetary incentive for participants to perform 579 well, resulting in an explicit competition, absent in Rule 1. information (i.e., the individual's memory of the cells already visited) compared to social information (i.e., the collective memory of the group shown on the shared colored table) in the choice of cells that are visited.

The analysis of the model shows that a cooperative effect induced by the trace emerges as soon as there exists a minimal level of marking on cells, and that the fidelity of the ratings increases with cooperation. The model also shows that the trace induces weak cooperation even in groups of defectors, provided they rate cells with a large enough number of stars, simply because they revisit the cells whose values are the highest. In this case, individual memory plays a major role in the collective performance of these defectors. Furthermore, the model predicts that the cooperative effect induced by the traces and the average performance of individuals increases with group size. This property results from the stigmergic interactions between individuals that make it possible to amplify at the group level the information about the location of cells whose values are the highest. Similar properties are observed in many species of ants that use pheromone trail laying to coordinate collective foraging activities and to find the best food sources in their environment [START_REF] Goss | Self-organized shortcuts in the argentine ant[END_REF][START_REF] Beckers | Collective decision making through food 1083 recruitment[END_REF]. The model also allowed us to explore the dynamics of the system in different conditions (number of agents and their behavioral strategy, size of the table, number of rounds...), and to investigate the optimal agents' strategy depending on diverse specified objectives. Our analysis shows that the maximal score is obtained for collaborative agents (Opt-1), suggesting that inner-group collaboration should emerge from intergroup competition. Interestingly, the model also predicts that a defector behavior emerges for an agent (Opt-3) aiming at optimizing its rank among the 10 participants of 2 groups, in the same conditions as in our experiment.

As our model was deliberately designed to prioritize relative simplicity, it consequently presents a notable limitation by not incorporating a possible explicit time-dependence in the parameters that quantify the visit and rating strategies. Indeed, the perceived importance of a cell with a given color may vary between the beginning and the end of an experimental run. In fact, in the model, the time-dependence of a subject's actions only results from the explicit time-dependence of the cell colors and of their 3 best discovered cells. Again, we did not consider, say, time-dependent visit parameters (ε and α parameters), for the sake of simplicity of the model, but also due to the fact that identifying the possible time-dependence of these parameters with reasonable statistical accuracy would require a much larger dataset. Yet, despite the model's imperfection in reproducing certain observables, the worst agreement between experimental and model results typically remains within 2 experimental standard errors (for instance, see Fig. 2I for Rule 1). Considering the number and diversity of observables that we have considered (see figures in the main text and the SI-Appendix), this level of agreement can be regarded as very satisfactory, suggesting that the model grasps the main ingredients of the actual visit and rating dynamics.

Finally, we would like to strongly emphasize that our experimental setup coupled to our predictive model is extremely rich and versatile. Indeed, it can be straightforwardly adapted to the investigation of many other interesting aspects of stigmergic processes, as well as the respective impacts of intragroup and intergroup competition on the emergence of cooperation D R A F T in human groups. In fact, our web application also permits the inclusion of bots (for instance, MIMIC or OPT agents) trollable size, which offers the possibility to investigate the 709 behavior of a subject depending on the composition of their

Materials and Methods

Experimental procedure. We conducted two series of experiments, S1C ). participants would then move on to the next round.

773

In the non-competitive condition (Rule 1), each participant had to find the cells with the highest values in the table, but their actions (visiting and rating cells) were not translated into a score. In the competitive condition (Rule 2), the score of each participant would increase at each round by the value of the 3 cells they had visited, but it remained independent of the ratings given to these visited cells. Hence, in Rule 2, the participants' main task was to discover the cells with the highest values, while maximizing their score, and ultimately, their payment at the end of the session. Note that we ultimately introduced a notion of score in Rule 1, to compare the results in the two rules (see Fig. 2 A andB), although, again, the participants in Rule 1 experiments were never told about any notion of score.

Accordingly, all participants were paid the same 10 € at the end of a Rule 1 session. In Rule 2, the 10 participants, from the 2 groups of 5, were ultimately ranked and paid according to their cumulated score at the end of the session. The participant ranked first was paid 25 €, the second was paid 20 €, the third was paid 15 €, and the participants ranked from the 4th to the 10th place were paid 10 € each.

Observables used to quantify the collective behavior. We define pc(t) as the fraction of stars received by a cell c at round t. The set of pc(t) for all cells c forms a vector p(t) of size 225 (vectors are shown in boldface). Another vector of interest is the vector P(t) of the cumulated fraction of stars Pc(t) that have been attributed to each cell from the beginning up to round t included. Similarly, q(t) and Q(t) are vectors whose coordinates qc(t) and Qc(t) represent the fraction of visits received by each cell at round t and up to round t, respectively.

From the definition of pc(t) and Pc(t), we can define the average value of cells visited by the participants weighted by their ratings (fraction of stars) at round t, p(t) = c pc(t)Vc/vmax 1 , where vmax 1 = 99 is the highest value of a cell. In general, we have p(t) ≤ 1, and p(t) = 1 would correspond to all members of a group only giving a non-zero number of stars to the cell of value 99 at round t. Similarly, we define the cumulated quantity, P (t) = c Pc(t)Vc/vmax 1 , the average value of cells visited by the participants weighted by their ratings (fraction of stars) up to round t. Hence, p(t) and P (t) quantify the instantaneous and cumulated distribution of stars in relation to the value of the visited cells. In particular, a high value of P (t) (in particular at the final round t = 20) indicates that the participants have concentrated the allocation of stars on high-value cells. Conversely, a low value of P (t) suggests a degree of deception, with participants allocating a high fraction of stars to low-value cells, as observed for Rule 2 where many participants are defectors.

In both rules, participants were explicitly asked to discover cells having high values. However, in Rule 2, their score would increase by the value of the cells they visit, thus providing an incentive that affects the way they visit and/or revisit cells during successive rounds. To quantify this (re)visiting behavior, we consider the normalized average value of the cells visited at round t, q(t) = c qc(t)Vc × 3/(vmax 1 + vmax 2 + vmax 3 ), where V is the vector of the cell values Vc, and vmax 1 , vmax 2 , vmax 3 are respectively the first-best, second-best, and third-best values of the cells in the table. This observable is normalized so that q(t) = 1 corresponds to the best theoretical performance, i.e., when every individual would visit the three best cells of the table at round t. Similarly, we introduce Q(t) that cumulates all visits up to round t and which is defined by the same expression replacing qc(t) by Qc(t). Note that, in Rule 2, since the score of the participants is increased by the value of their visited cells, q(t) and Q(t) directly quantify the instantaneous and cumulated performance of the group. In Rule 1, the participants had no notion of score, but q(t) and Q(t) allow us to characterize the dynamics of their visits, and to compare it with that for Rule 2.

To quantify the exploration behavior of the table by the participants, we introduce the inverse participation ratio (IPR) of the probability vectors q(t), Q(t), p(t), and P(t). For a given probability distribution X = {Xc}, the IPR of X is defined as

IPR(X) = 1/ c X 2 c .

D R A F T

we have Xc = 1/n on these cells, and

IPR(X) = 1/[n × (1/n) 2 ] = n,
showing that the IPR measures the effective number of cells over which a probability distribution is spread.

We are also interested in the relationship between the hidden values of the cells in the table and the fraction of visits or ratings that these cells have received up to round t. This relation is quantified by the fidelity F, which is defined as F(X,

V) = c XcVc/ c ′ V c ′ ,
where X is Q(t) or P(t). The fidelity F takes values in the interval [0, 1] and is equal to 1 if and only if the probability vector X is proportional to the vector of cell values V, which then corresponds to a perfect fidelity. Indeed, the fidelity can be seen as the scalar product between the vector of coordinates √ Xc (of unit Euclidean norm, since

c ′ X c ′ 2 = c ′ X c ′ = 1)

Model. The agent-based stochastic model includes two components:

(i) the agents' strategy for visiting cells; (ii) their strategy for rating the visited cells.

Visit strategy. In the first round (t = 1), the agents have no information, therefore the selection of the 3 cells is fully random. For the other rounds (t > 1), the agents adopt the following strategy. For each cell i = 1, 2, 3 to visit, they either choose the ithbest cell visited in the previous round, of value V i (t -1), with probability P R i (V i (t -1)), or explore other cells with probability

1 -P R i (V i (t -1)
), with:

P R i (V i (t-1)) =    0 if V i (t -1) < a i V i (t -1) -a i 99 b i if a i ≤ V i (t -1) < a i + 99 b i 1 otherwise [2]
where a i and b i > 0 are parameters. An agent never replays a cell of value V i (t -1) < a i and always replays a cell of value V i (t -1) > a i + 99/b i (when this threshold is less than 99, the maximum value of a cell). Between these two thresholds, the probability to revisit the ith-best cell linearly interpolates between 0 and 1. The functional form in Eq. 2 is rich enough to be able to capture diverse behaviors, while only using 2 free parameters for each of the 3-best cells, and is in fact consistent with indirect measurements of these probabilities.

When an agent does not visit one of the 3 cells visited in the previous round, it explores other cells in the table. This is done by associating to each cell c a probability P E (c, t) to be selected at round t:

P E (c, t) = ε 1 N + (1 -ε) P α c (t -1) c ′ P α c ′ (t -1) [3]
where Pc(t -1) is the fraction of stars deposited in cell c up to time t -1, and ε ∈]0, 1] and α > 0 are parameters. If the selected cell is one of the 3 cells visited in the previous round, another one is selected according to Eq. 3. In Eq. 3, the parameter ε controls the amount of exploration of unmarked cells compared to the marked ones: the higher the value of ε, the more random the selection, i.e., independent of the cell color. The exponent α controls the selection of a cell among the marked ones. A high value for α would result in a preferential selection of the highly marked cells, while a small value for α would lead to a more homogeneous selection of a cell among the marked ones. The simple functional form in Eq. 3 is inspired by the experimental results of SI-Appendix, Fig. S4, which are well-fitted by the similar functional form in Eq. 1.

The values of the 8 parameters appearing in Eqs. 2 and 3 and characterizing the visit strategy of MIMIC agents in Rule 1 and Rule 2 are reported in SI-Appendix, Table S2.

Rating strategy. Looking at the probability of rating a cell with s 912 stars for each profile (SI-Appendix, Fig. S10), one notes that, except 913 for the collaborators in Rule 1, individuals mostly rate a cell with

c ′ s + d ′ s v 99 ,
for neutrals [START_REF] Moussaid | Collective information processing and pattern formation in swarms, flocks, and crowds[END_REF] 924 where cs, ds, es, fs, c ′ s , and d ′ s are parameters which must satisfy 925 the property that, for all values of v, P 0 (v) + P 5 (v) ≤ 1.

926

However, the P 1234 (v) approximation is not valid for the col-927 laborators in Rule 1, who use the whole rating scale to rate cells 928 proportionally to their values. Therefore, for these collaborators, 929 we write for s = 1, 2, 3, 4, 5, 930

Ps(v) = d ′′ s exp - v -e ′′ s 99 f ′′ s 2 , [ 6
]
931 where d ′′ s , e ′′ s , and f ′′ s are parameters which must satisfy the property 932 that, for all values of v,

5 s=1 Ps(v) ≤ 1. Finally, we set P 0 (v) = 933 1 - 5 s=1 Ps(v).

934

The functional form of Eqs. 5 and 6 are well adapted to fit the 935 corresponding probabilities observed in the experiment (see Fig. 5 936 (D-I ) and SI-Appendix, Fig. S10A), while allowing to capture very 937 diverse behaviors. SI-Appendix, Table S1 presents the values of the 938 parameters appearing in the fitting functional forms of Eqs. 5 and 939

6.

940 Determination of model parameters. For the MIMIC agents, the 8 941 parameters of the visit strategy have been determined by mini-942 mizing the error between a set of n round-dependent observables, 943 O 1 (t), . . . , On(t), measured in the experiment (by averaging them 944 over every experiment for each of the two considered rules) and 945 the corresponding set of observables, Ô1 (t), . . . , Ôn(t), obtained 946 from extensive simulations of the model (averaging over 1,000,000 947 numerical experiments for each rule). The error is hence defined by 948

∆ = n i=1 20 t=1 ( Ôi (t) -O i (t)) 2 20 t=1 O 2 i (t) [7]
949

The set of round-dependent observables considered for the compu-950 tation of this error ∆ consists in the following quantities: Finally, to obtain the parameters of the visit and rating strate-971 gies of the optimized agents (Opt-1, Opt-2, Opt-3, Opt-4), we have D R A F T exploited a similar zero-temperature Monte Carlo method as described above. However, instead of minimizing an error, we have maximized the score (Opt-1 and Opt-2) or the ranking (Opt-3) of the agent, or the fidelity F(P(t = 20), V) in the final round (Opt-4).

q(t), 951 Q(t), p(t), P (t), IPR(q(t)), IPR(Q(t)), IPR(p(t)), IPR(P(t)), 952 F(Q(t), V), F(P(t), V), V 1 (t), V 2 (t), V 3 (t), B 1 (t), B 2 (t)

Computation of the error bars.

Error bars for the experimentally measured observables correspond to a level of confidence of 68 % and were determined by exploiting the bootstrap method. Bootstrap is a particular type of Monte Carlo method that evaluates the properties of statistical parameters from an unknown probability distribution by repeated random drawings with replacement from a sample [START_REF] Ac Davison | Bootstrap Methods and their Application[END_REF]. For the numerical simulations of the model, the results correspond to an average over 1,000,000 runs, so that the error bars are negligible on the scale of the presented graphs. individual aggregated on the 10 experimental runs. The x-axis is the cell's value and goes from 0 to 100 and the y-axis is one-fifth of the number of stars used by the individual to rate a cell of a given value and goes from 0 to 1. The dots are the experimental data, and the line is a linear fit of these data with the function u0 + 5 u1v/99, where u0 is the intercept and u1 is the slope. Individuals are sorted from left to right and from top to bottom according to the value of the slope u1. The color corresponds to the behavioral profile aggregated on the 10 experimental runs: green for collaborators, brown for neutrals, and red for defectors.

Note: Although the individuals' behavior has been defined on each experimental run in Fig. 4 of the main text, we chose to represent the aggregate behavior of each individual averaged over the 10 runs they played in a session, in order to limit the number of displayed graphs (100 instead of 1000 if all runs were shown). Therefore, the proportions of each behavioral profile slightly differ from those shown in Fig. 4 (see SI-Appendix, Table S3). individual aggregated on the 10 experimental runs. The x-axis is the cell's value and goes from 0 to 100 and the y-axis is one-fifth of the number of stars used by the individual to rate a cell of a given value and goes from 0 to 1. The dots are the experimental data, and the line is a linear fit of these data with the function u0 + 5 u1v/99, where u0 is the intercept and u1 is the slope. Individuals are sorted from left to right and from top to bottom according to the value of the slope u1. The color corresponds to the behavioral profile aggregated on the 10 experimental runs: green for collaborators, brown for neutrals, and red for defectors.

Note: Although the individuals' behavior has been defined on each experimental run in Fig. 4 of the main text, we chose to represent the aggregate behavior of each individual averaged over the 10 runs they played in a session, in order to limit the number of displayed graphs (75 instead of 750 if all runs were shown). Therefore, the proportions of each behavioral profile slightly differ from those shown in Fig. 4 (see SI-Appendix, Table S3). (B) Average value of the cells visited at round t, q(t) and (C) up to round t, Q(t). (H) Average value of the cells visited weighted by their ratings at round t, p(t) and (I) up to round t, P (t). (D) and (E) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)). (J) and (K ) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)). (F ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V). (B) Average value of the cells visited at round t, q(t) and (C) up to round t, Q(t). (H) Average value of the cells visited weighted by their ratings at round t, p(t) and (I) up to round t, P (t). (D) and (E) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)). (J) and (K ) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)). (F ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V).

  social influence | stigmergy | digital traces | collective intelligence | cooperation

  41deliberately giving high (respectively, low) ratings to certain 42 low (respectively, high) quality items. Therefore, knowing 43 the way individuals share and use digital traces in different 44 contexts is a crucial step to understanding how groups of 45 individuals can cooperate through stigmergic interactions and 46 can exhibit collective intelligence. Despite their increasing 47 importance in human groups, very little research on stigmergic 48

50a

  combination of experiments and computational modeling 51 how indirect interactions between individuals in a human 52 group involving the use of traces allow them to cooperate 53 during an information search task. Secondly, we study how 54 a competitive or non-competitive context influences the way 55

Fig. 1 .

 1 Fig. 1. Experimental setup. (A) Screenshot of the table at round t = 10, as seen by a participant. In this round, the participant has already visited and rated two cells marked with black crosses. The participant just visited the third cell of value 21 and must rate that cell on a 5-star scale. The score of the participant will then depend on the considered rule: in the non-competitive Rule 1, the score will increase by 0, and by 21 in the competitive Rule 2. (B) Pictures of the experimental room and (C) of the user interface that participants used during the experiment.

Fig. 2 .

 2 Fig. 2. Collective performance and dynamics of collective exploration and ratings.For the non-competitive Rule 1 (blue) and competitive Rule 2 (orange), the symbols correspond to the experimental results, and the solid lines are the predictions of the model. (A) Probability distribution function (PDF) of the scores of individuals S, and (B) of the groups Ŝ, respectively normalized by their theoretical maxima Smax and Ŝmax = 5Smax. The dotted vertical lines are the mean score in the experiment, and the dashed vertical lines are the mean scores in the model. (C) Average value of the cells visited at round t, q(t) and (E) up to round t, Q(t). (D) Average value of the cells visited weighted by their ratings at round t, p(t) and (F ) up to round t, P (t). (G) and (I) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)), measuring the effective number of cells over which the visits are distributed at round t and up to round t, respectively. (H) and (J) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)), measuring the effective number of cells over which the ratings are distributed at round t and up to round t, respectively. (K ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V).

Fig. 3 .

 3 Fig. 3. Quantification of individual behaviors for visiting cells. For the noncompetitive Rule 1 (blue) and the competitive Rule 2 (orange), symbols correspond to the experimental results, while solid lines are the predictions of the model. (A) Probability to find the best cell, of value 99. (B) Probability to find one of the four cells whose values are 86 (× 2), 85, or 84. (C) Probability to find one of the four cells whose values are 72 (× 2) or 71 (× 2). The black dashed and dotted lines correspond to the expected probabilities of two different visiting strategies: i) cells chosen randomly (full random search, dashed lines), and ii) cells chosen randomly among those that have not been already visited (sequential random search, dotted lines). (D-F ) V1(t), V2(t), V3(t) are respectively the value of the first-best cell, second-best cell, and third-best cell visited by the participants, as a function of the round t. (G-I) Probability B1(t), B2(t), B3(t) to revisit the first-best cell, the second-best cell, and the third-best cell of the previous round, as a function of the round t > 1.

  291 SI-Appendix, Fig. S5 shows the average number of stars 292 used to rate a cell as a function of its value v. In Rule 1, the 293 D R A F T average number of stars increases almost linearly with v. On average, individuals give 1 star to the cells with low values and 4.3 stars to the ones with very high values. In Rule 2 the situation is quite different, individuals give 2.5 stars to low-value cells, and then the average rating decreases to reach a plateau at about 1.5 stars for values higher than v = 25.

Fig. 4 .

 4 Fig. 4. Behavioral profiles of individuals. (Bottom-left) Scatter plot of the values of the two parameters u0 and u1 of the linear function, u0 + u1 × 5v/99, used to fit each participant's ratings as a function of the value of the visited cells. In the non-competitive Rule 1, individuals are represented by circles, and in the competitive Rule 2, individuals are represented by squares. The color of the symbols corresponds to the behavioral profile of the individuals: collaborator (green), neutral (brown), and defector (red). The two horizontal lines at u def-neu = -0.5 and uneu-col = 0.5 are the delimitations between the profiles. (Top-left) Histogram of the values of u0. (Bottomright) Histogram of the values of u1. (Top-right) The table gives the percentage of individuals for each of the behavioral profiles. See also SI-Appendix, Fig. S8A (for Rule 1 only) and B (for Rule 2 only).

  370

Fig. 5 ADFig. 5 .

 55 Fig.5 A, D, and Gshow that collaborators mostly rate 371 cells whose values are less than 20 with 1 star, while the cells 372 whose values are greater than 80 are rated with 5 stars. By 373 contrast, Fig.5 B, E, and Hshow that for the neutral indi-374 viduals, the probability of rating a cell with a given number 375 of stars does not depend on the cell value. Finally, Fig.5376 C, F, and I show that the defectors' distribution of ratings 377 presents an inverse pattern compared to that of the collab-378 orators. Defectors poorly rate cells with high values, hence 379

580

  Our experimental results show that groups of individuals 581 can use colored traces resulting from their ratings to coordinate 582 their search and collectively find the cells with the highest 583D R A F Tvalues in a table of hidden numbers. These traces constitute 584 a form of long-term collective memory of the past actions 585 performed by the group (21, 40). Combined with the individual 586 short-term memory of the value of the cells already visited, 587 these traces determine the choice of the cells ultimately visited 588 by the participants. 589 However, our results have also revealed profound disparities 590 in the way individuals use social information resulting from 591 these colored traces to guide them in their tasks, and also 592 in the way they choose to deliver information to other group 593 members through their ratings. We have identified three be-594 havioral profiles (collaborators, defectors, and neutrals) that 595 essentially account for the way in which individuals rate cells. 596 Collaborators cooperate by leaving a trace whose intensity 597 positively correlates with the hidden value of the cells, while 598 the defectors adopt an opposite behavior. Neutral individuals 599 constitute a sizable fraction of the group members (13 % in 600 Rule 1 and 49 % in Rule 2) and their ratings are essentially 601 uncorrelated with the actual value of the cells. Yet, the marks 602 that they leave, even if they do not directly inform about 603 the value of the cells, nevertheless induce a cooperative be-604 havior, since neutrals often revisit the high-value cells in a 605 way statistically indistinguishable from the collaborators and 606 defectors. 607 The information contained in the traces can thus be manip-608 ulated by individuals depending on the context, competitive 609 or not, in which the task is performed. Therefore, one may 610 expect that when a situation becomes competitive, individuals 611 should pay less attention to the socially generated traces since 612 the reliability of the information contained in the trace de-613 creases. Previous works in social decision-making have indeed 614 shown that there exists a causal link between mistrust and 615 a decrease in information sharing, and that the fear of being 616 exploited can be a reason for group members to withhold 617 accurate information (41, 42). This clearly occurs in Rule 2, 618 where 87 % of individuals provide indiscriminate (neutrals) 619 or false (defectors) information, whereas 84 % of individuals 620 (collaborators) provide reliable information in Rule 1. 621 Despite participants achieving higher scores in the competi-622 tive Rule 2 than in Rule 1, by exploring less and revisiting the 623 best cells more, the fidelity of the cumulative trace resulting 624 from their ratings is more faithful to the actual distribution of 625 cell values in Rule 1 than in Rule 2. In other words, there is 626 a better relation (more faithful) between the final rating of a 627 cell and its true value in Rule 1 than in Rule 2, although this 628 relation that we measured remains nonlinear. 629 We used these experimental observations to build and cal-630 ibrate a model that quantitatively reproduces the dynamics 631 of collective exploration and ratings, as well as the individual 632 and collective performances observed in both experimental 633 conditions. In particular, this agreement between the model 634 and the experiment is quantified by exploiting a series of subtle 635 observables: PDF of the score, fidelity, inverse participation 636 ratio, probability of revisiting cells depending on their values... 637 Note that an important added value of our model is to offer 638 (via the analysis of its parameters) a direct and quantitative 639 interpretation of the visit and rating strategies for the three ob-640 served behavioral profiles of human participants, and also for 641 different types of optimized agents. The analysis of individual 642 behaviors combined with the simulations of the computational 643 model shows that competition reinforces the weight of private 644

  710 group. Moreover, we have also designed an identical version 711 of our interactive web application which can be deployed on 712 the Internet, and which could be used to conduct large-scale 713 experiments. We plan to explore these different avenues in 714 future works. 715 Ultimately, understanding and modeling the processes that 716 govern the influence of social information embedded in digital 717 traces on individual and collective behavior is a crucial step to 718 developing personalized decision-making algorithms as well as 719 artificial collective intelligence systems based on stigmergy (26, 720 45, 46).

  Fig.S1C). participants would then move on to the next round.

  and the normalized vector proportional to √ Vc. Hence, the fidelity measures how wellaligned these two vectors are and is in fact related to the Hellinger distance between the two distributions. In the context of a real-life 5-star rating system, a high fidelity of the cumulated ratings P(t) would indicate that the ratings provide a fair representation of the actual value of the different options. Of course, in this context, these intrinsic values of the available options are generally unknown. But our experimental setup provides a simpler context where this relation between the ratings (or the visits) of the different options (the cells, in our experiment) and their intrinsic value (the cell values) can be investigated.

914 0 or 5 5 s=0 4 ( 1 -

 5541 stars, and that the other ratings with 1, 2, 3, or 4 stars are 915 less common and have a comparable probability. Therefore, in the 916 model, the probabilities of rating a cell with 1 to 4 stars are set 917 equal and are obtained by imposing the probabilistic normalization 918 condition Ps(v) = 1, for each value of v. In other words, for 919 s = 1, 2, 3, 4, we obtain 920 Ps(v) = P 1234 (v) = 1 P 0 (v) -P 5 (v)). [4] 921 For s = 0 and s = 5, the probability Ps(v) to rate a cell of value v 922 with s stars is given by 923 Ps(v) = cs + ds tanh v -es 99 fs for collaborators/defectors

  The bootstrap method starts by creating M artificial sets of N experiments by drawing with replacement N experiments among the N original ones. This means that some actual experiments can be drawn more than once in an artificial set, while other experiments may not occur in this set. One can then compute a given observable on every artificial set and obtain its distribution, ultimately leading to confidence intervals. In our case, the independent experiments are the 10 trials played by a group of 5 individuals. Therefore, we have N = 20 experiments for Rule 1, and N = 15 experiments for Rule 2, and we used M = 10, 000 artificial sets to generate bootstrap distributions.

  Fig. S3. (A) First, (B) second, and (C) third-highest values discovered up to round t, as a function of the round t, in the non-competitive Rule 1 (blue) and the competitive Rule 2 (orange). The dots are the experimental data, and the solid lines are the predictions of the model. The highest values discovered are slightly higher in Rule 1 than in Rule 2, showing that the tendency of individuals to revisit cells (and thus to explore less) is higher in Rule 2 than in Rule 1.

2 Fig. S4 .Fig. S5 .

 2S4S5 Fig. S4. Average fraction of stars ρ(v) used to rate cells of value v at the final round t = 20 in Rule 1 (A) and Rule 2 (B). The dots are the experimental data, and the solid lines are the predictions of the model. The black dashed lines correspond to Eq. 1 used to fit the data, with ε = 0.48 and α = 2.18 in Rule 1, and ε = 0.55 and α = 1.22 in Rule 2.

Fig. S6 .

 S6 Fig. S6. Average number of stars used to rate cells as a function of the cell's value in the non-competitive Rule 1. Each of the rectangles corresponds to the behavior of a single

Fig. S7 .

 S7 Fig. S7. Average number of stars used to rate cells as a function of the cell's value in the competitive Rule 2. Each of the rectangles corresponds to the behavior of a single

Fig. S11 .Fig. S14 . 3 Fig. S15 .Fig. S16 .

 S11S143S15S16 Fig. S11. Collective performance and dynamics of collective exploration and ratings for the experiment in which individuals play alone for the non-competitive Rule 1 (blue) and the competitive Rule 2 (orange). (A) Probability distribution function (PDF) of the scores of individuals S, and (B) of the groups Ŝ, respectively normalized by their theoretical maxima Smax and Ŝmax = Smax. The dotted vertical lines are the mean score in the experiment, and the dashed vertical lines are the mean scores in the model. (C)Average value of the cells visited at round t, q(t) and (E) up to round t, Q(t). (D) Average value of the cells visited weighted by their ratings at round t, p(t) and (F ) up to round t, P (t). (G) and (I) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)). (H) and (J) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)). (K ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V). (M-O) V1(t), V2(t), V3(t) are respectively the value of the first-best cell, second-best cell, and third-best cell visited by the participants, as a function of the round t. (P-R) Probability B1(t), B2(t), B3(t) to revisit the first-best cell, the second-best cell, and the third-best cell of the previous round, as a function of the round t > 1. (S) Probability to find the best cell, of value 99. (T ) Probability to find one of the four cells whose values are 86 (× 2), 85, or 84. (U) Probability to find one of the four cells whose values are 72 (× 2) or (× 2). It is worth noting that there are two peaks in the PDF of scores in Rule 2 (A). This phenomenon results from the fact that the probability for an individual alone to find a cell with a high-value cell is very low. As a result, their final score is based solely on exploration.

Fig. S17 .

 S17 Fig. S17.Collective performance and dynamics of collective exploration and ratings in simulations with one Opt-3 agent optimizing its rank r while playing against four MIMIC agents (green solid lines) compared to the simulations results with five MIMIC agents (Rule 2, orange dashed lines) which are in good agreement with the experimental results (see Fig.2in the main text). (A) Probability distribution function (PDF) of the scores of agents S normalized by its theoretical maxima Smax. The dotted vertical lines are the mean score in the experiment and the model. (G) Probability distribution function (PDF) of the rank r of the optimized agent. The dotted vertical line corresponds to the mean rank. (B) Average value of the cells visited at round t, q(t) and (C) up to round t, Q(t). (H) Average value of the cells visited weighted by their ratings at round t, p(t) and (I) up to round t, P (t). (D) and (E) Inverse participation ratio of the visits, IPR(q(t)) and IPR(Q(t)). (J) and (K ) Inverse participation ratio of the ratings, IPR(p(t)) and IPR(P(t)). (F ) Fidelity to the cell value distribution of the distribution of visits, F(Q(t), V), and, (L) of ratings, F(P(t), V).

  values used for the rating strategy (see Eqs. 5 and 6 in the main text) for MIMIC agents (collaborator, neutral, and defector) in both rules, and for the optimized agents (Opt-1, Opt-2, Opt-3). These values result from the fitting of the probabilities of rating a cell with s stars described in the main text.

P

2 and 3

 3 in the main text) for MIMIC agents (collaborator, neutral, and defector), and optimized agents (Opt-1, Opt-2, Opt-3, and Opt-4). These values result from the optimization procedure described in the Materials and Methods section.

  We will call these individuals defectors (3 % in Rule 1 and 363 38 % in Rule 2), since we interpret that the strong traces 364 left on cells with very low values are meant to mislead 365 other group members and prevent them from finding the 366 best cells, especially in Rule 2. In addition, they also 367 decide not to share the position of the best cells they have 368 discovered, by giving them low ratings, and hence not 369 marking them on the table.

2 of the 356 SI-Appendix, Supplementary Text.

357

• Individuals with u1 < u def-neu rate the cells in the oppo-358 site way to collaborators, resulting in deceptive ratings. 359 Indeed, they attribute a small number of stars (resp. a 360 large number of stars) to the cells whose values are the 361 highest (resp. whose values are the lowest; see Fig.

5C

). 362

  , and B 3 (t).953 We checked that other sets -in particular, smaller sets -of ob-954 servables would lead to very comparable results (in particular, in If the error ∆ decreases, the new value of the parameter 962 is accepted; otherwise, the old value of the parameter is conserved. 963 The minimization procedure ends when the error stops decreas-964 ing. To account for possible multiple local minima of the error, 965 we started the Monte Carlo simulations from several initial values 966 of the parameters. We kept the final parameters, leading to the 967 smallest error. Note that the final parameters obtained in different 968 low-error Monte Carlo runs were found to result in similar functions 969 characterizing the visit strategy in Eqs. 2 and 3.

		955
	Figs. 2 and 3), fitting some observables slightly better and some	956
	others slightly worse, and leading to similar results for the functions	957
	characterizing the visit strategy in Eqs. 2 and 3.	958
	To minimize the error in Eq. 7, we have used a Monte Carlo	959
	method at zero temperature. At each Monte Carlo step, a small	960
	random change is introduced in one of the randomly selected pa-961
	rameters. 970

Table S2 . Parameters values used for the visiting strategy (see Eqs.

 S2

			E (c, t)	B1(t)	B2(t)	B3(t)
			ε	α	a1	b1	a2	b2	a3	b3
	Rule 1	MIMIC	0.78	0.89	57.6	2.19	25.0	2.29	1.4	2.64
		(col, neu, def)	0.69	1.32 -8.4	1.55 -4.1	2.11 -0.2	2.33
		Opt-1	1e-5	1.38	25.0	2.00	18.4	2.03	27.1	2.41
	Rule 2	Opt-2	0.58	2.75 -2.4	2.15	4.0	2.54	9.1	2.90
		Opt-3	0.82	4.32	22.3	4.86	13.7	3.54	8.3	3.35
		Opt-4	1	0	0	0	0	0	0	0

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ACKNOWLEDGMENTS.

This work was supported by grants from the CNRS Mission for Interdisciplinarity (project SmartCrowd, AMI S2C3) and the CNRS Project 80 | Prime ALTHEA. T.B. was supported by a doctoral fellowship from the CNRS. R.E. was supported by Marie Curie Core Grant Funding (grant no. 655235-Smart-Mass).

Supplementary Text for the analysis of these experiements). The

Supplementary Information for Cooperation and deception through stigmergic interactions in human groups

Thomas Bassanetti, Stéphane Cezera, Maxime Delacroix, Ramón Escobedo, Adrien Blanchet, Clément Sire, Guy Theraulaz Corresponding Author: Guy Theraulaz. E-mail: guy.theraulaz@univ-tlse3.fr

This PDF file includes:

Supplementary text Figs. S1 to S20 Tables S1 to S3 Legends for Movies S1 to S5 Legend for Dataset S1

Other supplementary materials for this manuscript include the following:

Supporting Information Text

A. Behavioral Profiles of Individuals Playing Alone versus in a Group. Before carrying out the experiments in groups, we studied the behavior of the participants playing alone, each individual exploring a different table during two successive rounds and seeing only their own traces (see SI-Appendix, Fig. S11). SI-Appendix, Fig. S12A shows that individuals rate the cells similarly to collaborators in groups, except that they rate a low-value cell with 1 star, presumably to remember the cells that they had already opened. Supplementary Fig. 11 B andC show that in Rule 1, the majority of individuals adopt a collaborative behavior when alone and keep this behavior when they are in a group. On the other hand, in Rule 2, many individuals who adopted a collaborative behavior when playing alone switch to a neutral or defector behavior type when they are in a group.

B. Additional Model Predictions.

B.1. Impact of the number of rounds and group size on individual performance and collective dynamics. SI-Appendix, Fig. S13 shows that after 100 rounds instead of 20 rounds, the normalized score of individuals and groups has increased by 60 % in Rule 2. Beyond round 50, the values of the observables used to quantify the dynamics of collective exploration and ratings begin to saturate. From one round to another, the MIMIC agents revisit almost exclusively the same cells whose values are very high. At the end of the 100 rounds, in Rule 2 the value of their best cell is V1(t = 100) ≃ 84, and the agents revisit their best cell with a probability B1(t = 100) ≃ 1.

SI-Appendix, Fig. S14 shows the impact of group size on the scores of individuals and groups and the dynamics of collective exploration and ratings. We compare the simulation results obtained with groups of 5 MIMIC agents exploring a table of 225 (15 × 15) cells and groups of 20 MIMIC agents exploring a table four times larger, 900 cells (30 × 30). These larger tables were obtained from the combination of four identical tables of 225 cells so that the proportion of each cell value does not change. For instance, in a table of 900 cells, there are four cells with a value of 99, but their proportion (1/225) is the same as in the smaller tables. The dynamics of the inverse participation ratio (IPR) of p(t), P(t), Q(t), and Q(t) reveal that large groups do not visit four times more cells than small groups, but instead, they concentrate their visits on a few cells with high values. Individuals also have a higher probability of finding the cells with the best values. However, despite these differences, the score remains unchanged. Finally, in Rule 1, the probability that individuals find the best cells at the end of an experiment is much larger in groups of 20 MIMIC agents. Altogether, these results suggest that cooperation induced by stigmergic interactions and the way individuals use the traces resulting from past actions increase with group size. B.2. Impact of the rating strategy on agents' performance and the fidelity of ratings. To better understand the impact of the rating strategy on individual performance, we studied the collective behaviors of groups of 5 agents having a linear rating strategy. These agents rate a cell in proportion to its value, v, with u0 + u1 × 5 v/99 stars, where u0 and u1 are respectively the intercept and the slope of the line (see Fig. 4 of the main text). When u1 > 0, the number of stars used to rate a cell increases with its value v (like for a cooperator), while when u1 < 0, the number of stars used to rate a cell decreases with its value v (like for a defector). As u0 increases, agents use a larger number of stars to rate a cell of a given value. Moreover, the combinations of parameters u0 ≤ 0 and u1 ≤ 0 correspond to a situation in which the agents rate all cells with 0 star, as some actual neutrals do in the experiment. Finally, the visit strategies of these agents are the same as those used by the MIMIC agents in each of the two conditions, Rule 1 and Rule 2. SI-Appendix, Fig. S20 presents the result of the respective impact of u0 and u1 on (i) the average performance of individuals, (ii) the average value of cells visited by the participants weighted by their ratings, and (iii) the fidelity of ratings with respect to cell values, for each condition Rule 1 and Rule 2.

We first observe that when u0 = 0, as soon as the agents start rating the cells with a non-zero number of stars, the resulting trace allows them to cooperate and significantly increase their performance, even for very low positive values of u1. The results of the simulations also show that the agents get the best scores for negative values of u0, which correspond to situations in which there exists a minimum threshold in the value of a cell that triggers the agents to rate that cell (e.g., when u0 = -0.5 and u1 = 0.5 the threshold is at v = 20). Moreover, the higher the value of u0, the worse the performance of the agents. This results from the fact that in that condition, the agents use a very high number of stars with little discrimination in the ratings for different values of v. The resulting trace left on cells then provides much less information to the agents, leading to a lower level of cooperation and lower performance. Note however that for high values of u0 (i.e., when u0 > 3) and for weakly negative values of u1 (i.e., when -1 < u1 < 0), there still exists weak cooperation between the agents. At first glance, this is rather counterintuitive, since for these parameters, agents are classified as neutrals or mild defectors. However, this phenomenon can be explained by the fact that, while the traces left by the agents in the initial rounds may not allow for the identification of cells with higher values, over time, cells with higher values will be revisited more often, resulting in a greater accumulation of marks compared to cells with lower values. Nevertheless, for values of u1 that are even more negative, indicating strong defection, the tendency of agents to revisit high-value cells is insufficient to counterbalance the negative impact of assigning high ratings to cells with low values, which ultimately leads to decreased performance.

Finally, the presence of competition between agents (Rule 2) amplifies both the positive and negative effects of the trace compared to the non-competitive situation (Rule 1). Indeed, groups of agents with cooperative behavior (u1 > 0) increase their performance in Rule 2 with respect to the reference situation (u0 = u1 = 0); conversely, groups of agents with defective behavior (u1 < 0) strongly decrease their performance with respect to the reference situation. Intercept (u 0 ) For Rule 1, we find a strong correlation between the behavioral profiles of a participant alone or in a group, in particular, for the vast majority of collaborators, and the few neutrals. For Rule 2, this correlation is mostly lost, and many collaborators while playing alone become defectors or neutrals when confronted with 4 other participants. Probability of rating a cell with 0 stars (P0(v); magenta), 1 to 4 stars (P1234(v); violet) and 5 stars (P5(v); green) as a function of its value v, for the different kinds of optimized agents. The Opt-1 agents (maximizing their score in a group of 5 identical agents) are strong collaborators, also suggesting that a competition between groups should favor intragroup collaboration. The Opt-2 agents (maximizing their score against 4 MIMIC agents) are neutrals always giving a rating of 0 start, and hence not participating at all in the coloring of the table. Finally, the Opt-3 agents (optimizing their rank against 9 MIMIC agents in 2 groups of 5) are strong defectors, illustrating that deception naturally emerges from our competitive payment structure. S3. Fractions of behavioral profiles adopted by participants, whether it is calculated on a single experimental run or over the ten experimental runs (average behavioral profile). In the table, col, neu, and def correspond respectively to collaborators, neutrals, and defectors. The lines above col, neu, and def indicate the average profiles.

Observing the table row-wise reveals that individuals

SI Dataset S1 (DATA)

All data needed to evaluate and replicate the findings of the article are present in the article, the SI-Appendix, or available at the following online repository: https://github.com/Thomas-bssnt/Stigmer-article.git. Additionally, the repository contains the movies mentioned in the article.