An Agent-based model for collective sanction analysis

Luca Gallo

Universita degli Studi di Torino,May 27, 2018

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Contents

1 Introduction 3

2 Methodology 4

3 The model 53.1 Variables and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 The dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Persuasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Risk and payoff formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Imitation of the others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Running experiments 104.1 Experimental setup and user interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Avoiding free-ridings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Communication vs. imitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Opposition and riot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusions 15

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1 Introduction

The collective action problem is an extensively studied situation in which a group of self-interested agentsfails to cooperate because the individual interests conflict with the benefits of the group. Following thepublication of Olson’s book The logic of collective action [1] a vast literature on the issue has arisen, rang-ing from economics [2] to policy making [3] [4], from sociology [5] [6] even to biology [7].The main issue implied by this problem is whether cooperation among the agents is possible withoutinvoking a centralized entity, as classically stated by Hobbes, and how robust to individual free-ridingthe cooperation is; in other words, the public good problem involves a question about how norms reg-ulating non-cooperative behaviors can emerge in informal groups [8] [9] [10].

A particular form of public good problem, as asserted by Heckathorn, is created when a group is sub-jected to collective sanctions or collective incentives [11] [12]; in this framework, the agents are interestedboth in “obtaining the collective awards or avoiding the collective punishments”, which represent thepublic good, and in “evading the (norm) compliance costs”, which is the individual free-riding. Further-more, collective sanctions and awards create a second collective action problem, because the sanctioning-awarding system itself is a public good, with common benefits to enjoy and individual costs to escape;thus, the agents faces a six-folded choice in a second order free rider problem [13] [14] [12]. The reason whythe agents’ choice is six-folded is that, at the first level, the group members have to decide whether ornot to comply with an external norm,1 while in the second, the choice is between defection, cooperationor opposition to the peer-sanctioning system. In this work, only collective sanctions will be considered.

Conventionally [15] [16] recognized as a form of punishment typical of primitive societies, based onclans, tribes and families, or, at least, characteristic of military boot camps [17] and prisons [18] [19],collective sanctions were supposed to “disappear from modern societies because moral responsibilityhas become individuated” [20]. Despite this prediction, collective sanctions and incentive systems or, atleast, their “functional analogues” [20], are nowadays present in both formal and informal groups, suchas corporations [21], vicarious groups of employees [22] and so on.2

The main characteristic of this framework is that collective sanctions try to exploit group members re-lationships thus the ability of agents to monitor and control their peers. However, agents threatenedby collective punishments could react opposing to the external norm thus trying to destroy the peer-sanctioning system [11].

One way to tackle the collective action problem, to analyze the norm formation and the collective sanctionregime is to formalize these issues in game-theoretic terms, typically as a single-shot or an iterated n-Prisoner’s Dilemma [8] [11] [12] [23] [24] [25]. Attempting to go beyond the limitations implied by a purelygame-theoretical description of the system, this project deals with the emergence of a normative systemimplied by a collective sanction regime using an agent-based modeling approach with a persuasion-basedmechanism; this computational paradigm is supposed to give new insights on the topic thus stimulatingthe research on the collective action problem to move also in this direction.

1 For the purpose of this project, it is not necessary to describe the nature of the external entity in detail, because only the frequencyof monitoring would be implemented in the model.

2 For a more exhaustive list of application, see Levinson’s Collective sanctions [20].

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2 Methodology

The agent-based modeling (ABM) is a computational paradigm in which autonomous entities, calledagents, make decision respecting a defined set of rules [26].Even if the agents’ decision-making process can integrate game-theoretic elements, such as a payoff-driven behavior selection, there is a significant difference between ABM and game theory. While gametheory prime interest is to describe the properties of Nash equilibria or to find under which conditionsthe systems converges towards optimal solutions i.e. Pareto efficiency, ABM mainly focuses on agent’sbehaviors and on interactions between them, capturing the emergence of a collective behavior with anhigh level of interpretability. In other words, the purpose of the agent-based modeling is not to makeprediction about the system dynamics but to point out the fundamental mechanisms underlying socialprocesses thus giving better understanding of them [27].

One major benefit of the ABM approach is the possibility to have an extensive and flexible control at amicroscopical level over both the agent’s architecture and the environment, which makes this approachremarkably suitable for managing complex interactions that might be intractable with an equation-basedmodel. From this follow two important aspects to face the norm emergence problem; first, stochasticitycan be included in the model in “the right places” [26], instead of adding thermal noise terms whichare often arbitrary or at least difficult to interpret. Secondly, a tuning of agents’ intelligence can be doneaccording to the characteristics of the experiment; this allowed, in the model, the implementation of aperception gap between the inner behaviors of a particular agent and the perception of these by its peersor the possibility to control the ability of the agents to deal with the complexity of the interactions amonggroup members.

In the context of this computational paradigm, the main objectives of the present project are a) to builda basic agent’s architecture underlying the decision-making process; b) to develop proper interactionmechanisms; c) to define quantitative measures in order to study the system dynamics and his depen-dence on some of the most significant variables and parameters.

For the code implementation, I used the multi-agent programmable environment NetLogo which al-lowed me a) to easily develop an agent-based model consistently with the characteristics of the collec-tive action problem; b) to work on an extremely user-friendly interface in order to easily manage themodel variables and parameters; c) to have a intuitive graphic representation of the model dynamics.

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3 The model

3.1 Variables and parameters

Before analyzing the model variables and parameters in detail, it is necessary to define a notation suit-able for the problem characteristics, like the presence of the perception gap; the choice of a subscript-superscript notation comes mainly from the necessity to implement the aforementioned perception gapand because of the large number of agents’ interactions. The following table presents the syntheticexplanation of the notation henceforth used:

Notation SummaryXi Indicates the real value of the i’s variable X.Y ij Indicates the value of j ’s variable Y as assumed by i.Ziji Indicates the value of j ’s variable Z about i, as assumed by i.

The fundamental mechanism underlying the model is persuasion, although other forms of intra-groupinteractions can be employed [12]. In order to reduce the risk to be sanctioned because of others’ behav-ior, the agents could try to persuade their peers, thus affecting their individual inclinations. I identifiedthree variables indicating the agents’ inclinations a) to comply with the external norm (duty); b) to sanc-tion their peers (control); c) to raise group members against the sanctioning system (riot).

One of the major feature of the present model is the implementation of a variable trust level of therelationships among group members, which regulates both the efficacy and the probability of success ofthe persuasion. It is important to stress that the trust level is asymmetric i.e. how much agent A trustsagent B differs from how much agent B trusts agent A, and unknowable by the target of trust, whichmeans that agent A can not know how much agent B trusts him, thus needing to suppose it.Variables and parameters used in the model are summarized in the following tables:

Variable Summary

dIndicates the duty i.e. the inclination of the agent to respect the

external norm.

cIndicates the control which is how much the agent wants to

increase others’ duty.

rIndicates the riot that indicates the agent’s will to reduce others’

control.

tIndicates the trust of the relationship between two agents; thisregulates the probability that the control/opposition succeeds.

e

Indicates the efficacy of an agent; this indicates the mean value oftrust, assumed by a certain agent, that the group has towardsanother member. Note that, because this variable represents asecond level of intelligence supposition, thus needing a morecomplex reasoning, the agent only considers the group mean

value of trust.Parameters Summary

mIndicates the moody i.e. how much the agent’s trust changes after

persuasion success/failure.M Indicates the frequency of monitoring.

SC , SI Indicate the extent of collective and individual sanctions.

K1, K2, K3Indicate the cost, respectively, of external norm subordination,

of peer supervision and of peer instigation to opposition.

In addition, I defined the sense of belonging as:

Bi =< tij >< tiji > (1)

This variable depends both on the mean trust towards the others and on the average supposed trustthat the group has towards the agent. Although the definition is extremely intuitive it depends only on

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the characteristics of agents’ relationships not including, for instance, parameters related to the groupnature or purpose, thus needing to be studied in deep in future researches.

The choice of global variables in order to analyze the system dynamics naturally follows from the pre-vious definitions; I identified two different groups of variables, the first being trust-related variables,namely the mean trust of a member towards other agents and the mean supposed trust of the grouptowards an agent, and the second inclination-related ones, which are the mean duty, the mean controland the mean riot. The analysis of the aforementioned global variables focuses on the evolution of themean level of duty and of the average control level, in relation both to the personal inclinations of theagents and to the relationships among them.In addition I implemented three simple dynamical counters that monitor a) the number of the first levelfree-riders; b) the number of agents trying to control their peers at the second level; c) the number ofmembers who decide to raise the group against the sanctioning system.

3.2 The dynamics

The model is based on cyclical dynamics, depicted in the figure below, composed of three bunch offunctions:

• update-team, in which each agent updates its supposition of the group inclinations and its individ-ual variables;

• act, in which agents choose and act in relation to their inner behavioral rules;

• update-globals, in which the global variables used to analyze the system are update.

Figure 1: A schematic representation of a simulation cycle. In each block of functions the agents of theteam are called in random order.

The following table summarizes the characteristics of the first two blocks, specifying the functions ofwhich they are composed:

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Bunch Function Summary

update-team

update-others The agent updates the supposed average group inclinations.

update-risk The risk of being sanctioned because of others’ behaviors isevaluated.

update-belonging

Accordingly to the trust level and to the supposed trust level,the agent updates his sense of belonging to the group.

update-payoff The payoffs of each strategy is evaluated by the agent.

actchoose The agent chooses the best fit strategy according to the

payoffs.

persuade

The agent tries to persuade the group thus affecting the innerinclinations of the others; this occurs only if the agent has

previously chosen to control the group or to raise it againstthe external norm.

mainstreamIf the sense of belonging is strong enough the agent modifies

his personal inclinations according to the supposed groupbehavior.

3.3 Persuasion

As previously mentioned, the fundamental mechanism underlying agents’ interactions is persuasioni.e. the endeavor of affecting the others’ inclination. When the costs of control or opposition do notexceed the possible benefits coming from influencing the others, agents will choose to persuade theirpeers; because the personal inclination to respect the external norm is identified with the duty, in orderto reduce the risk of being sanctioned because of others’ behavior an agent might attempt to increasethe group duty by controlling its peers. On the contrary, if the cost of complying exceeds the oppositionone, an agent might try to escape it by reducing the group inclination to control.

In any case, the group members, which are autonomous, can ignore the attempts of persuasion of thecontrolling agents. Because agents are self-interested the effort of the controlling agent to affect thegroup inclinations can be perceived as motivated by egoism and therefore the relationships between thepersuading agent and its peers could be felt by the latter as fake friendships [28].This suspicion effect is included in the model through the trust variable; thus the ratio of the trust levelof the relationship to a trust threshold, which can be set in the user interface of the model, is interpretedas the probability of success of the persuasion attempt. Furthermore, if the influence act fails i.e. thecontrolled agent mistrusts the persuading one, the trust level decreases by a definite quantity, namely,the agent’s moody; on the contrary, in case of success the trust level of the controlled agent increases bythe same quantity. Lastly, because of the perception gap, both the controlling and the controlled agentsautonomously consider the trust variation resulting from the persuasion act; this means that, whilean agent could have a certain belief about the action success the other might have a totally differentopinion. This consequently implies that the trust level update would be opposite. The following tablesummarizes the variables modification in the persuasion act:

ControlAgent Success Failure

Persuading agent i dij(n+ 1) = dij(n) + αci(n)tiji(n)

tiji(n+ 1) = tiji(n) +mij

tiji(n+ 1) = tiji(n)−mij

Persuaded agent j dj(n + 1) = dj(n) + αcji (n)tji (n)

tji (n+ 1) = tji (n) +mj

tji (n+ 1) = tji (n)−mj

OppositionAgent Success Failure

Persuading agent i cij(n + 1) = cij(n) − βri(n)tiji(n)

tiji(n+ 1) = tiji(n) +mij

tiji(n+ 1) = tiji(n)−mij

Persuaded agent j cj(n + 1) = cj(n) − βrji (n)tji (n)

tji (n+ 1) = tji (n) +mj

tji (n+ 1) = tji (n)−mj

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It is important to emphasize one aspect of this mechanism: let us suppose that more then one agentduring a cycle will try to alter the behavior of the others. If the persuasion act is accomplished privatelyi.e. the attempt to modify others’ inclinations is not communicate to the group, then it is possible thatthe perception gap between the real variables of an agent and those perceived by the group will consid-erably increase at each cycle. Because of this effect I implemented in the model the possibility to switchfrom a private persuasion act to a public one; however, also this mechanism is affected by a form ofperception gap, based on the supposed efficacy of the controlling agent, which plays the role of trust in thepersuasion act.As it will be explained in detail further on, the combination of the increase in the perception gap, impliedby a private persuasion act, and the imitation mechanism, what will be described in the next section, cansignificantly affect the model dynamics. The following table further clarifies the mechanism describedbelow; the agent targeted by the persuasion act is indicated as j, the persuading agent as k and the oneupdating its perception as i:

Act Success FailureControl dij(n+ 1) = dij(n) + αcik(n)e

ik(n)

eik(n+ 1) = eik(n) +mij

eik(n+ 1) = eik(n)−mij

Opposition cij(n + 1) = cij(n) − βrik(n)eik(n)

eik(n+ 1) = eik(n) +mij

eik(n+ 1) = eik(n)−mij

3.4 Risk and payoff formation

As the defection to the external norm of a certain agent triggers not only a sanction for itself but also forthe others, in order to choose the best strategy the agents need to evaluate the risk of being sanctionedbecause of others’ behavior. The risk of punishments depends on the group supposed duty whichcan be affected by the agent’s action; as the agent decides to persuade its peers to comply with theexternal norm thus increasing their duty, the perceived risk, supposing to have success in influencingthe others, will consequently be lowered. On the contrary, if the agent chooses to raise the others againstthe external norm, the risk of sanctions will be unchanged, because its action only affects the controllevel of the group.Considered the above, the following set of variables can be computed:

RiD = Mi

<dij>

RiC = Mi

<dij+αcitiji>

RiO = Mi

<dij>= RiD

(2)

in which α is a constant, regulated in the user-interface of the model, that can be interpreted as a quan-titative measure of difficulties and errors occurring in the process that weights the control/oppositionmechanism and D, C and O represent the corresponding second level strategy chosen.The variation entity of others’ duty depends both on the individual inclinations of the agent that per-suades its peers and on the supposed trust level of the relationship; if the agent has a significant inclina-tion to control the group and it supposes to have high-trust relationships with its peers, it will considerits ability to affect the others’ duty to be enormous thus the perceived risk of the corresponding strategywill be slight.

In a sanction-only framework, the payoff of a particular strategy consists exclusively in costs thus thebest strategy is the one that minimizes them. In the model the agents’ personal inclinations weigh thecosts related to the corresponding strategy; for instance, if an agent has a strong sense of duty the costof not complying with the external norm will be perceived as higher, whereas if its level of duty is slightthe agent would be like to free-ride.In addition, regardless of the strategy chosen the agents can be sanctioned because of others’ behaviors;however, the cost of the punishment is weighted by the perceived risk of being sanctioned so if the riskperceived is low then the charge perceived will be slight too.

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Lastly, if an agent decides to free-ride not complying with the the external norm an additional individualsanction is added to the cost of the strategy. This means that the agent’s free-riding is somehow detectedand punished; however, the nature of the individual sanction, whether it comes from an external moni-toring or from an internal formal normative system, is not specified in the model.According to these basic rules, the strategy payoffs can be computed as follows:

P iDD = −diM i(RiDSC + SI)

P iCD = −RiDSC − K1

di

P iDC = −diM i(RiCSC + SI)− K2

ci

P iCC = −RiCSC − K1

di − K2

ci

P iDO = −(di − αβriBi)M i(RiOSC + SI)− K3

ri

P iCO = −RiOSC − K1

di − K3

ri

(3)

Note that the agent’s duty, in the strategy consisting in first level defection and second level opposition,has a more elaborated expression; this comes from the fact that the opposition strategy affects the peerscontrol level and therefore the agent’s duty.3

3.5 Imitation of the others

The most original feature of the present project is the implementation of a imitation mechanism, whichcan be considered as a sort of implicit persuasion, contrary to the previously described process which canbe named explicit persuasion. It is important to underline that the characteristics of the model imitationmechanism are somehow arbitrary and need solid theoretical analysis; for reasons that will be clear fur-ther on and because of the possible fields of application, ranging from economics to biology, I believethat the combination of collective sanction systems and well-defined mimicry mechanisms is worth be-ing studied.In the present model the agents’ choice to imitate the others is only dependent on the sense of belong-ing to the group; therefore behind the agents’ will of miming the others there is no search of a best fitstrategy. Because of the sense of belonging definition, the motivation to imitate the others is duplex: inthe first place, the agents decide to mimic group behaviors because of the trust level towards it, and sec-ondly they choose to imitate their peers because of the perceived trust level that the group has towardsthem.

As the ratio of the agent’s trust to the trust threshold regulates the explicit persuasion success prob-ability, the ratio of the sense of belonging to a corresponding threshold, which can be set in the userinterface, is interpreted as the probability that the agent decides to mimic others’ behaviors. However,unlike what happens with the trust level, neither the success nor the failure of the imitation processmodify the agent’s sense of belonging.The following set of equations describes how an agent updates its personal inclinations after decidingto imitate the others:

di(n+ 1) =di(n)+<dij(n)>

2

ci(n+ 1) =ci(n)+<cij(n)>

2

ri(n+ 1) =ri(n)+<rij(n)>

2

(7)

3 As the agent considers that the group will try to persuade him to comply with the external norm, its duty will be computed asfollows:

di(n+ 1) = di(n) + α < cij(n) >< tij(n) > (4)

In order to evade the compliance cost implied in choosing a first-level-complying strategy, the agent may try to affect the groupcompliance:

diopposition(n+ 1) = di(n) + α < cij − βri(n)tiji(n) >< tij(n) > (5)

With simple steps and reminding the sense of belonging definition:

diopposition(n+ 1) = di(n+ 1)− αβri(n)Bi(n) (6)

which is the expression included in the strategy payoff.

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4 Running experiments

4.1 Experimental setup and user interface

The user interface of the present model, in which it is possible to set the initial condition of the systemand to monitor its evolution, is depicted in the next page figure.In order to define the agent’s initial personal inclinations, the user defines the mean value and thestandard deviation of a cut-tailed4 Gaussian distribution thus giving the group a certain degree of het-erogeneity. Also the trust level among the agents, the supposed frequency of monitoring and the moodyparameter can be randomly set in the same way.Furthermore, agents’ perceived values of others’ inclinations and trust levels are not totally unrelatedto the real variables; the model user can regulate the perception gap between the two by defining thestandard deviation, denoted by the word error, of a cut-tailed Gaussian distribution whose mean valueis the actual agent’s variable. Therefore, the supposed value of others’ inclinations and trust level arerandomly generated numbers accordingly with the aforementioned distribution.Lastly, it is possible to see the nudge? and communication? switches, regulating respectively the imitationmechanism and the public communicated persuasion.5

The user interface also presents a basic graphic representation of the agents and the relationships amongthem; the agents’ color depends on the strategy chosen: if the agent decides neither to control the othersnor to raise them against the sanctioning system then it is colored in gray, while it is colored in green ororange if chooses respectively to control the group or to incite them to riot.The ties among the agents are colored according to the trust level of the relationship: if the ratio of trustto the threshold is less than one third the link will be colored in red, in yellow if the ratio is less than twothirds, otherwise it will be colored in green.

Finally, both the plots of the group mean values of trust and duty-related variables and the dynamicalcounters of first level free-riders and persuading agents can be seen at the bottom of the user interface.

4.2 Avoiding free-ridings

The first case study is the searching for a parameter configuration for which the peer-controlling systemavoids or, at least, limits the number of free-riders; the free-riding prevention occurs, intuitively, whenthe controlling cost is low and the group inclination to monitor itself is high. Furthermore, the defectionavoidance depends also on the number of group members.Because the initial configuration of the system is randomized, in order to increase the statistical signifi-cance of the results I replicate the experiment fifty times.The parameter values used in the experiment can be seen in the next page figure. The following tablesummarizes the result obtained for the study of group size:

Number of free-ridersMembers Zero One Two Three Four Five Six Seven Eight Nine Tenn = 3 22% 30% 20% 28% - - - - - - -n = 4 26% 24% 18% 10% 22% - - - - - -n = 5 22% 28% 18% 14% 8% 10% - - - - -n = 6 30% 34% 8% 16% 2% 4% 6% - - - -n = 8 50% 16% 24% 4% 4% - - - 2% - -n = 10 62% 8% 12% 2% 4% 6% 4% 2% - - -

As it can be seen from the table, the more the group is populous the more the number of free-riders isreduced. This can intuitively follow from the fact that the number of controlling agent, so the average

4 This means that the negative values and the positive symmetric tail are cut off.5 A third switch, namely the all-changing? one, is present in the model, although not exploited in the present analysis; by turning

it on the persuasion mechanism is modified so that the control act increases both duty and control inclination, decreasing the riotvariable too. Moreover, he oppositional mechanism works the other way around, decreasing group duty and control variableand increasing the riot value.

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Figu

re2:

The

mod

elus

erin

terf

ace.

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increase in group duty at each cycle, raises with the group dimension.

In each case, the system reaches an equilibrium, in which agents stop their attempt to persuade theothers, thus choosing the full defection strategy (defection at both level) or the private cooperation one,consisting in first level cooperation and second level defection [12]. In the first stage of the process, acrucial role in peer-monitoring is played by hypocritical cooperators, agents who escape the external normwhile urging others to comply with it, accordingly with previous researches [11] [12]. Because the costof control does not exceeds the supposed benefit coming from lowering the risk of being sanctioned be-cause of others’ behavior and because the agents hypocritically evades the compliance cost, the strategyhas the best payoff; moreover, because all the agents autonomously choose the best fit behavior at thebeginning of each cycle, it is possible that a big number of group members decide to adopt the hypocrit-ically cooperation strategy, thus terribly affecting the others’ duty.

Another major aspect to emphasize is the role played by the imitation mechanism in affecting the systemevolution. A graphic visualization of the agent’s mimicry effects is displayed in the figure below. With

Figure 3: On the left, the evolution of duty-related global variables with the imitation mechanism turnedoff. On the right, the evolution of the same variables with the imitation mechanism turned on. Note thatthe agents’ persuasion starts after tick 50, in order to make the process more visible. In both cases thegroup consists of ten agents.

the nudge? switch turned off the duty curve resembles to a step function; when the group members startcontrolling their peers the mean value of the group duty enormously increases, immediately reachingan equilibrium state in which the agents stop affecting others’ personal inclinations and comply withthe external norm. Otherwise, when the imitation mechanism is activated the evolution of group dutyis less sharp; because of the non-rational and stochastic nature of the so-called implicit persuasion, theduty evolution curve presents small jumps, both negative and positive, which occur when an agent de-cides to mimic the others. These little leaps can significantly affect the system dynamics; for instance, asmall random increase in the agent’s inclination to control the group can result in making the strategyinvolving the peer-monitoring the best fit one, from which can follow a raise in the group mean duty.

4.3 Communication vs. imitation

Because the agents imitate theirs peers following their perception of the group mean inclinations, whenthe perception gap between supposed and actual variables is wide the imitation mechanism can dras-tically change the system evolution. Furthermore, as previously stated, when the persuasion attemptsare not communicated to the group members the perception gap may increase accordingly. The combi-nation of these two effects may result in a loss of group duty thus increasing the number of first levelfree-riders. As it can be seen in the next page figure, when no communication among the agents exists,the average duty level of the group collapses to lower values. Moreover, while in the first case free-riding is completely avoided by peer-monitoring, in the second more than an half of group membersdecide not to comply with the external norm. This result emphasizes the importance of communicationamong the agents.Let us focus on the process details: because most of the persuasion acts occurring in every cycle arenot communicated to the agents, the duty level of the group is perceived by them as lower than the ac-

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Figure 4: On the left, the evolution of duty-related global variables with public communicated persua-sion. On the right, the evolution of the same variables with private persuasion. Note that the agents’persuasion starts after tick 50, in order to make the process more visible. In both cases the group consistsof ten agents.

tual value; however, because of their high inclination to comply with the external norm and in order toavoid the cost of peer-monitoring, after a first stage of explicit persuasion, the agents choose the privatecooperation strategy, thus ending the attempts to affect others’ behaviors. Then, because the imitationmechanism is active, the agents lower their individual inclinations to the perceived level of group duty,thus starting to free-ride again.It is important to stress this result: when a mechanism based on the perception of the environment ispresent in the system, it is necessary for the success of the social action that the informations about thegroup behaviors are widespread.

4.4 Opposition and riot

After having described the possibility to create a condition in which the peer-monitoring system suc-ceeds in avoiding first level free-riding, another relevant case study is the creation of a situation in whichopposition to the sanctioning system could arise. As previously stated, when the cost of raising the oth-ers against the external norm do not exceed the complying cost or when the agents’ inclination to riot ishigh enough, it might be rational to oppose to the peer-monitoring system. The following figure depictsan example of the aforementioned situation in which riot can arise: As the agents begin to persuade

Figure 5: The parameters setting and the system evolution. Note that the agents’ persuasion starts aftertick 50, in order to make the process more visible.

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the others the group average inclination to peer-monitoring is enormously decreased thus making thestrategies involving the control of others’ behavior suboptimal; the more the system evolves the morethe agents, without any form of monitoring and because of the fluctuations induced by the imitationmechanism, start to raise against the external norm, until all of them or almost have chosen the full op-position strategy [12], consisting in first level defection and second level opposition.

The crucial point in the riot rise is represented by the trust evolution; because the best fit strategy in-volves second level opposition, the persuasion mechanism uninterruptedly goes on thus potentiallyincreasing the trust level beyond the threshold. Then, because the ratio of trust to its threshold is biggerthan one, every attempt to persuade the others will be successful, thus inexorably raising the trust levelof the relationship. Finally, because the decrease in individual duty depends on the belonging sense, soon how much trust the agent puts in the group, the more the strength of the relationships increases, themore the opposition to the sanctioning system is optimal.

The following figures give an example of how trust affects the system evolution. The first relevant

Figure 6: An example of system dynamics leading to group full opposition. Note the perception gapbetween the evolution of the level of group trust and the jumps of the control curve.

aspect to underline is the control curve evolution: the curve leaps that can be seen in the figure are as-sociated to the change in strategy by one member of the group. Furthermore, when the full oppositionstrategy becomes optimal for an agent, the slope of the trust curve increases, because of the bigger num-ber of agents’ interactions.Secondly, it is important to notice that process leading to a riot rise takes much longer than the peer-monitoring mechanism. While the latter reaches an equilibrium in few cycles, the former could lastmore than 200, as depicted in the example figure. Then, a newsworthy model modification could be theimplementation of both internal or external reaction to the riot formation, in order to balance its effects.

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5 Conclusions

Agent-based modeling can represent a versatile and functional methodology to deal with the collectiveaction problem and the studies of situation involving collective incentives and sanctions; focusing onthe analysis of the fundamental mechanisms underlying social phenomena rather than on the descrip-tion of the macroscopic evolution of the system, this computational paradigm could give new insightson issues usually tackled with game theory or other mathematical approaches.

The results presented in the present project seem to assert that it possible to find collaborative solutionsto the collective action problem i.e. situations in which all the members of a group subjected to collectivesanctions give up the free-riding. Many variables could affect the success of the peer-regulating system,including the personal inclination to control the agents, the costs coming from monitoring the othersand the group size; this latter was analyzed in detail with the result that the more the number of groupmembers is high, the more the persuasion mechanism is effective.The importance of agents’ communication in affecting the system dynamics, especially when an imi-tation mechanism is present, is one of the most relevant findings of the present study; when agents’actions are based on their suppositions about the external environment, the perception gap induced bythe lack of communication among group members may terribly change group behavior.

Otherwise, when the costs associated with the opposition to the external norm or when the group in-clination to riot is high, it is possible to observe fully non-cooperative states in which the agents do notcomply with the external norm. In these situations a major role in raising group riot is played by trust;if the oppositional strategy remains optimal for at least one member and if the mean trust level of thegroup is high enough to increase all along the system evolution, the riot will eventually spread out,involving all the agents. However, the aforementioned process could last a much longer time than theother mechanisms previously cited, so it possible to imagine the implementation of countermeasures,both external or internal, to agents’ opposition.

Although the current agent-based model, based on personal inclinations, persuasion, imitation and trustmodification, has given some relevant results about the fundamental mechanisms that could exist in acollective sanctions system, it presents some aspects that need to be studied more in detail.Firstly, the definition of a quantitative measures for human inclinations to comply or not with both exter-nal and internal norms can have some serious limitations, especially regarding a possible experimentalvalidation of the model.Secondly, as previously remarked, the imitation mechanism is somehow arbitrary but because of thefascinating results following from its exploitation, a theoretical analysis of the motivations underlyinggroup member will to imitate its peers could be extremely worth it. Similarly, it is necessary a more de-tailed characterization of trust level and its modification rules in order to corroborate the model findings.

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