Evolution of collective fairness in complex networks through degree-based role assignment
Andreia Sofia Teixeira, Francisco C. Santos, Alexandre P. Francisco, Fernando P. Santos
EEvolution of collective fairness in complex networks through degree-based role assignment
Andreia Sofia Teixeira , Francisco C. Santos , Alexandre P. Francisco , andFernando P. Santos INESC-ID and Instituto Superior Técnico, Universidade de Lisboa, R. Alves Redol 9, 1000-029Lisboa, Portugal Indiana Network Science Institute, Indiana University, 1001 IN-45 Bloomington IN, USA Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544,United States
March 1, 2021 a r X i v : . [ phy s i c s . s o c - ph ] F e b bstract From social contracts to climate agreements, individuals engage in groups that must col-lectively reach decisions with varying levels of equality and fairness. These dilemmas alsopervade Distributed Artificial Intelligence, in domains such as automated negotiation, conflictresolution or resource allocation. As evidenced by the well-known Ultimatum Game — wherea Proposer has to divide a resource with a Responder — payoff-maximizing outcomes arefrequently at odds with fairness. Eliciting equality in populations of self-regarding agentsrequires judicious interventions. Here we use knowledge about agents’ social networks toimplement fairness mechanisms, in the context of Multiplayer Ultimatum Games. We focuson network-based role assignment and show that preferentially attributing the role of Pro-poser to low-connected nodes increases the fairness levels in a population. We evaluate theeffectiveness of low-degree Proposer assignment considering networks with different averageconnectivity, group sizes, and group voting rules when accepting proposals (e.g. majorityor unanimity). We further show that low-degree Proposer assignment is efficient, not onlyoptimizing fairness, but also the average payoff level in the population. Finally, we show thatstricter voting rules (i.e., imposing an accepting consensus as requirement for collectives toaccept a proposal) attenuates the unfairness that results from situations where high-degreenodes (hubs) are the natural candidates to play as Proposers. Our results suggest new routesto use role assignment and voting mechanisms to prevent unfair behaviors from spreading oncomplex networks.
Keywords—
Modeling and Simulation; Social networks; Evolutionary Game Theory; Populationdynamics; Complex Systems; Fairness; Wealth inequality
Introduction
Fairness has a profound impact on human decision-making and individuals often prefer fairoutcomes over payoff maximizing ones [6]. This evidence is pointed through behavioralexperiments, frequently employing to the celebrated Ultimatum Game (UG) [10]. In theUG, one Proposer decides how to divide a given resource with a Responder. The game onlyyields payoff to the participants if the Responder accepts the proposal. Human Proposerstend to sacrifice their share by offering high proposals and Responders often prefer to earnnothing rather than accepting unfair divisions. These counter-intuitive results motivatedseveral theoretical models that aimed at justifying, mathematically, the evolution of fairintentions in human behavior [15, 14, 3, 20].In Distributed Artificial Intelligence and Multiagent Systems, fairness concerns are importantin domains that go beyond pairwise interactions. Autonomous agents have to take partin group interactions that must decide upon outcomes possibly favoring different partsunequally. Examples of such domains are automated bargaining [11], conflict resolution [17]or multiplayer resource allocation [2]. To capture some of the dilemmas associated withfairness versus payoff maximization in these interactions, we use a multiplayer extensionof the Ultimatum Game [23] (MUG). Here, a proposal is made by a Proposer to a groupof N − Responders that, collectively, decide to accept or reject it. As in the pairwiseUG, the strategy of a Proposer, p , is the fraction of resource offered to the Responder; thestrategy of the Responder, q , is the personal threshold used to decide about acceptance orrejection [14, 15]. Groups decide to accept or reject a proposal through functions of theindividual acceptance thresholds, q . Group acceptance depends on a decision rule: if thefraction of acceptances equals or exceeds a minimum fraction of accepting Responders, M ,the proposal is accepted by the group. In that case, the Proposer keeps what she did notoffer ( − p ) and the offer is divided by the Responders — each receiving p/ ( N − . If thefraction of acceptances remains below M , the proposal is rejected by the group and no oneearns anything. As in the UG, the sub-game perfect equilibrium of MUG consists in a verylow value of proposal p and very low values of threshold q [24].Previous studies with the UG [15, 14, 3, 20] and the MUG [23, 26], assume that the rolesof Proposer and Responder are attributed following uniform probability distributions: eachagent has the same probability of being selected to play as Proposer. These assumptionsare naturally at odds with reality. In real-life Ultimatum Games, being the Proposer or1he Responder depends on particular agents’ characteristics. Proposers, such as employers,investors, auction first-movers or rich countries, are in the privileged position of having thematerial resources to decide upon which divisions to offer. This advantageous role is notoriousif, again, one considers the theoretical prediction of payoff division in the UG ( sub-gameperfect equilibrium ) posing that Proposers will keep the largest share of the resource beingdivided. The benefits of Proposers are more evident when proposals are made to groups, asResponders need to divide the offers — thus increasing the gap in gains between the singleProposer and the Responders. In this multiplayer context, punishing Proposers becomesharder: any attempt to punish unfair offers is only effective if there is a successful collectiveagreement — on Responders — to sacrifice individual gains and reject an offer. Assertingthat these two roles are asymmetric, so should be the criteria to assign them, leading us totwo main questions:• How should a Proposer be selected within a group, in multiplayer ultimatum games toguarantee efficiency and fairness?• In networked games, which network-based criteria can be used to maximize long-termefficiency and fairness?Here we introduce a model, based on evolutionary game theory (EGT) [32, 31] and complexnetworks, to approach the previous questions. We analyze multiplayer ultimatum gamesin heterogeneous complex networks through network centrality-based role assignment. Thefact that networks are heterogeneous allows us to test several node properties and centralitymeasures as base criteria for defining how to select Proposers in a group. We focus on degreecentrality. We find that selecting low-degree Proposers elicits fairer offers and increases theoverall fitness (average payoff) in a population. The questions we address in this work — and the model proposed to tackle them — lie onthe interface between mechanism for fairness elicitation in multiagent systems, multilayerbargaining interactions, dynamics on complex networks and network interventions to sustainsocially desirable outcomes. 2ome of the most challenging contexts to elicit fairness involve the tradeoff between payoff-maximizing and fair outcomes. As stated, the UG [10] has been a fundamental interactionparadigm to study such dilemmas. In this context, reputations [14] and stochastic effects [20]were pointed as mechanisms that justify fair behaviors. Page et al. found that, in a spatialsetting, fairer proposals emerge as clusters of individuals proposing high offers are able togrow [15]. Also in the realm of interaction networks, De Jong et al. concluded that scale-freenetworks allow agents to achieve fairer agreements; rewiring links also enhances the agents’ability to achieve fair outcomes [3]. A game similar to the UG assumes that Responders areunable to reject any proposal and Proposers unilaterally decide about a resource division.This leads to the so-called Dictator Game. In this context, reputations and mechanismsbased on partner choice were also pointed as drivers of fair proposals [34].The previous works assume that all agents have the same probability of playing in the roleof Proposer or Responder. Going from well-mixed (i.e., all individuals are free to interactwith everyone else) to complex networks, however, provides the opportunity to implementnetwork-based role assignment that considers network measures. In this context, Wu et al.studied the pairwise UG in scale-free networks, with roles being attributed based on networkdegree. The authors show that attributing the role of Proposer to high-degree nodes leadsto unfair scenarios [33]. Likewise, Deng et al. studied role assignment based on degree,concluding that the effect of degree-based role assignment depends on the mechanism ofstrategy update [4]. When considering a pairwise comparison based on accumulated payoffsand social learning (as we do in the present work), the levels of contribution in the populationincrease if lower-degree individuals have a higher probability of being the Dictators. Bothworks consider the pairwise Ultimatum Game.In this work we use a multiplayer version of the UG (MUG) proposed in [23]. Other forms ofmultiplayer ultimatum games can be found in [7, 9, 29]. Santos et al. studied this game in thecontext of complex networks, showing that fairness is augmented whenever networks (wherethe game is played) allow agents to exert a sufficient level of influence over each other, byrepeatedly participating in each others’ interaction groups. The authors also find that strictergroup decision rules allow for fairer strategies to evolve under MUG. Here we use networks todefine group formation as suggested in the previously mentioned work (and originally in [27]).Departing from previous works that study degree-based role assignment in pairwise UltimatumGames [33, 4], we focus on a multiplayer game. As mentioned, this version highlights theasymmetries in the Proposer and Responder role: here, Proposers are likely to receive a3ven higher share of payoffs than each Responder – as the latter need to divide acceptedoffers – and punishing unfair Proposers depends on a group decision by the Responders –that naturally may call for extra coordination mechanisms. Also, differently than in [33, 4],here we show that, whenever highly connected nodes are the natural candidates to play in therole of Proposer, stricter voting rules (i.e., imposing an accepting consensus as requirementfor collectives to accept a proposal) attenuates reduces the emergent level of inequality.Finally, the approach we follow in this work is akin to testing network interventions for socialgood. In this realm, we shall underline a recent work that employs EGT — as we do in thepresent paper — to study interventions to sustain cooperation in complex networks [13]. Theauthors conclude that local interventions — i.e., based on information about the neighborhoodof the affected node — outperform global ones. A similar conclusion is presented in [16].Several works study social dilemmas on top of complex networks and stress the conditionsleading, in this context, to socially desirable outcomes [21, 18, 22, 16].
Here we detail the proposed evolutionary game theoretical model to evaluate the effect ofdegree-based role assignment on fairness under MUG. We start by providing details on thepayoff calculation under MUG.
In the 2-player UG, a Proposer has a resource and is required to propose a division with aResponder. The game only yields payoff to the participants if the Responder accepts theproposal [10]. Given a Proposer with strategy p ∈ [0 , and a Responder with strategy q ∈ [0 , , the payoff for the Proposer yields Π P ( p, q ) = − p, p ≥ q , p < q , (1)4 A - � - � � � ���������������� A12 α , role-degree dependence p r ob s e l ec t a s P r opo s e r Figure 1: Example of group formation and Proposer selection based on degree. Each nodeand its neighborhood define an interaction group. In the figure, node A plays in 5 groupsand its fitness results from the payoff sum after playing in all those groups. In general, anode plays in a number of groups equal to its degree plus one. For each group, the payoff iscalculated after one individual is selected to be the Proposer. Proposer selection dependson the degree of each individual, and a parameter α controls this dependence (see Methodssection). To exemplify this process, we represent the probability of each individual — A (high-degree), (medium-degree) and (low-degree) — to be selected as a Proposer whenplaying in the group centered in A , as a function of α .and for the Responder Π R ( p, q ) = p, p ≥ q , p < q . (2)In the MUG, proposals are made by one Proposer to the remaining N − Responders, whomust individually reject or accept it [23, 26]. Since individuals may act both as Proposersand Responders (with a probability that will depend on node characteristics), we assumethat each individual adopts a strategy ( p , q ). When playing as Proposer, individuals offer p to the Responders. Responders will individually accept or reject the offer having their q asa baseline: if the share of an offer p is equal or larger than q (i.e., pN − ≥ q ) the individualaccepts the proposal. Otherwise, the Responder rejects that proposal. We can regard q asthe minimum fraction that an individual is willing to accept, relatively to the maximumto be earned as a Responder in a group of a certain size. Alternatively, we could assumethat individuals ignore the group size and as such, when faced with a proposal, they must5udge the absolute value of that proposal (an interpretation that also holds if we assume thatindividuals care about the whole group payoff).Overall group acceptance will depend upon M , the minimum fraction of Responders thatmust accept the offer before it is valid. Consequently, if the fraction of individual acceptancesstands below M , the offer will be rejected. Otherwise, the offer will be accepted. In this case,the Proposer will keep − p to himself and the group will share the remainder, that is, eachResponder gets p/ ( N − . If the proposal is rejected, no one earns anything. All together, ina group with size N composed of Proposer with strategy p ∈ [0 , and N − Responderswith strategies ( q , ..., q N − ) ∈ [0 , N − the payoff of the Proposer is given by Π P ( p, q , ..., q N − ) = − p, (cid:80) N − i =1 Θ( pN − − q i ) / ( N − ≥ M , otherwise , (3)where Θ( x ) is the Heaviside step function, that evaluates to when x ≥ and evaluates to when x < . The payoff of any Responder in the group yields, Π R ( p, q , ..., q N − ) = pN − , (cid:80) N − i =1 Θ( pN − − q i ) / ( N − ≥ M , otherwise . (4)We assume that MUG is played on a complex network, in which individuals are assignednodes and links define who can interact with whom. Following [27, 25], every neighborhoodcharacterizes a N-person game, such that the individual fitness (or success) of an individual isdetermined by the payoffs resulting from the game centered on herself plus the games centeredon her direct neighbors. We provide a visual representation of such group formation in Figure1. Degree heterogeneity will create several forms of diversity, as individuals face a differentnumber of collective dilemmas depending on their degree (and social position); groups wheregames are played may also have different sizes. Such diversity is introduced by consideringtwo types scale-free networks. One is generated with the Barabási-Albert algorithm (BA)of growth and preferential attachment [1] leading to a power-law degree distribution, anda low clustering coefficient. The clustering coefficient offers a measure of the likelihood offinding triangular motifs or, in a social setting, how likely two friends of a given node arealso friends of each other, a topological property of relevance in the context of fairness and6-person games [25]. In the second case, we consider the Dorogotsev-Mendes-Samukhin(DMS) duplication model [5], exhibiting the same power-law degree distributions, yet withlarge values of the clustering coefficient. In the BA model [1], at each time step, the network grows by adding a new node andconnecting it to m other nodes already in the network. These connections are probabilistic,depending on the degree of the nodes to be connected with: having a higher degree increasesthe probability of having new connection. This process result in heterogeneous degreedistributions, in which older nodes become highly connected (creating so-called hubs ). This isthe combination of two processes – growth and preferential attachment . In the DMS model [5],at each time step, a node is added; instead of choosing other nodes to connect with, itchooses one edge randomly and connects to both ends of the edge. The networks generatedby the DMS model have higher cluster coefficient than those with BA model, combining thehigh-clustering and high-heterogeneity that characterizes real-world social networks. Previous works show that anchoring the probability of nodes being selected for the role ofProposer or Responder on their degree has a sizable and non-trivial effect on the evolvinglevels of proposal in traditional two-person Ultimatum Games [33, 4]. Considering multiplayerultimatum games, however, opens space to study the interplay between group characteristics(such as group sizes) and network-based criteria to select Proposers in completely unexploreddirections. So far, we assume that nodes are selected to be Proposers based on their degree.As such, in a group with N individuals, where each individual i has degree k i , the probabilitythat j is selected as Proposer is given by p j = e αkj (cid:80) i e αki , where α controls the dependence ofdegree on role selection. One node is selected as Proposer and the remaining N − play asResponders. 7 lgorithm 1: Pseudo-code of the main cycle of our simulations. We perform 100runs over 10 different networks of each type (BA and DMS) with × generationsper run. Initialize all p i , q i ∈ Z = X ∼ U (0 , for t ← to Gens do Main cycle of interaction and strategy update: for j ← to Z do Select agent to update: /* Sample two neighbors the population */ A ← X ∼ U (1 , Z P ) (agent to update) B ← Y ∈ neighbours ( A ) if X ∼ U (0 , < µ then Mutation: p A ← X ∼ U (0 , q A ← X ∼ U (0 , else Imitation: f A ← fitness( A ) f B ← fitness( B ) prob ← / ( e − β ( fB − fA ) ) if X ∼ U (0 , < prob then p A ← p B + imitation error ∼ U ( − ε, ε ) q A ← q B + imitation error ∼ U ( − ε, ε ) We simulate the evolution of p and q in a population of size Z , much larger than the groupsize N . Initially, each individual has values of p and q drawn from a discretized uniformprobability distribution in the interval [0 , . The fitness f i of an individual i of degree k isdetermined by the payoffs resulting from the game instances occurring in k + 1 groups: onecentered on her neighborhood plus k others centered on each of her k neighbors (see Figure1). Values of p and q evolve as individuals tend to imitate (i.e., copy p and q ) the neighborsthat obtain higher fitness values.The numerical results presented below were obtained for structured populations of size Z = 1000 . Similar results were obtained for Z = 10000 . As already mentioned, we considernetworks generated with both BA and DMS algorithms, with average degree (cid:104) k (cid:105) = { , , } .Simulations take place for × generations, considering that, in each generation, all theindividuals have (on average) the opportunity to revise their strategy through imitation.At every (discrete and asynchronous) time step, two individuals A and B (neighbors) areselected from the population. Given the group setting of the MUG, B is chosen from one ofthe neighbours of A . Their individual fitness is computed as the accumulated payoff in all8ossible groups for each one, provided by the underlying structure (in each group the roleof Proposer or Responder is selected following the section below); subsequently, A copiesthe strategy of B with a probability χ that is a monotonic increasing function of the fitnessdifference f B − f A , following the pairwise comparison update rule: χ = e − β ( fB − fA ) [30]. Theparameter β specifies the selection pressure ( β = 0 represents neutral drift and β representsa purely deterministic imitation dynamics). Imitation is myopic: the value of p and q copiedwill suffer a perturbation due to errors in perception, such that the new parameters will begiven by p (cid:48) = p + ζ p,ε and q (cid:48) = q + ζ q,ε , where ζ p,ε and ζ q,ε are uniformly distributed randomvariables drawn from the interval [ − ε, ε ] . This feature not only i) models a slight blur inperception but also ii) helps to avoid the random extinction of strategies, and iii) ensuresa complete exploration of the strategy spectrum. To guarantee that new p and q are notlower than 0 or higher than 1, we implement reflecting boundaries at 0 and 1. Furthermore,with probability µ , imitation will not occur and the individual will adopt random values of p and q , proceeding through a random exploration of behaviors. We use µ = 1 /Z , β = 10 and ε = 0 . throughout this work. The effect of varying µ is similar to the one verifiedwhen changing ε : an overall increase of randomness leads to higher chances of fairer offers(as in [20, 23]). For each combination of parameters, the simulations are repeated times(using different networks from each class studied), whereas each simulation starts from apopulation where individuals are assigned random values of p and q drawn uniformly from [0 , . We provide a summary of the algorithm used to revise agents’ strategies in Algorithm1. The average values of p , q and f (denoted by (cid:104) p (cid:105) , (cid:104) q (cid:105) and (cid:104) f (cid:105) ) are obtained as a time andensemble average, taken over all the runs (considering the last generations, disregardingan initial transient period). We run the proposed model and record the average strategies played by the agents over timeand over different runs (starting from different initial conditions, see Methods). We find thatattributing the role of Proposer to low-degree nodes (or low-degree Proposer assignment )increases the average level of proposal, p , adopted in the population of adaptive agents. Thismeans that the payoff gap between Proposers and Responders is alleviated. Figure 2 showsthat, for low α ( α < ), we obtain higher levels of average proposal when considering BA(low clustering coefficient) and DMS (high clustering coefficient) networks. We observe asteep decline in average proposals when the role of Proposer and Responder is attributed9igure 2: The average proposal played by agents in a population, (cid:104) p (cid:105) , decreases with α . Thismeans that attributing the role of Proposer to high-degree nodes reduces the overall fairnesslevel in a population. We present results for BA and DMS networks with average degree (cid:104) k (cid:105) = 4 . We verify that low-degree Proposer assignment maximizes (cid:104) p (cid:105) for different groupdecision rules, M = { . , . , . } , i.e., the fraction of Responders that needs to accept aproposal for it to be accepted by the group.regardless the degree of individuals ( α = 0 ). The low-proposal tendency is maintained if therole of Proposer is assigned to high-degree nodes ( α > ).We also confirm that high-degree Proposer assignment leads to unequal (unfair) results withina population. Figure 3 depicts the average payoff gains for individuals with a certain degree.We can observe that, for α = 2 , high-degree nodes obtain much higher values of payoff thanlow-degree nodes. This situation is ameliorated if individuals with lower degree are given ahigher chance of becoming Proposers (lower α ) and, to a lower extent, if more Respondersare required to accept a proposal in order for it to be accepted (higher M , bottom panels inFigure 3).We can further verify the effect of α on fairness through the so-called Lorenz curves [12],often used to compute the
Gini coefficients [8] that quantify income inequality. In Figure 4we represent the Lorenz curves associated with different role-assignment rules ( α ) and votingrules, M . Each curve is generated by ordering individuals by increasing value of incomeplotting the corresponding cumulative distribution. A curve closer to the perfect equality line(45 degree) represents a most egalitarian distribution of resources and a lower Gini coefficient.As we verify in Figure 4, the most unequal outcomes (higher Gini) are obtained for higher α .10igure 3: On top of decreasing the average level of proposal in the population, (cid:104) p (cid:105) , we foundthat attributing the role of Proposer to highly connected nodes decreases the level of fairnessand equality within the population. Here we use scatter plots to observe the average payoffobtained per game, (cid:104) Π (cid:105) , for individuals with a certain degree (horizontal axis). The leftpanels represent a low-degree Proposer assignment scenario ( α = − ); the center panelsrepresent random — and degree-independent — role attribution ( α = 0 ); the right panelsrepresent a high-degree Proposer assignment scenario ( α = − ). Each gray cross represents anode in a degree- (cid:104) Π (cid:105) space; the orange line represents the mean taken over all nodes with acertain degree. Top panels represent M = 0 . and bottom panels M = 0 . . High α — i.e.,high-degree Proposer assignment — implies that highly connected nodes earn (approximately)five times more payoff per game than low-connected nodes (top-right panel). This effect isalleviated for higher M ; for M = 0 . , highly connected nodes earn (approximately) threetimes more payoff per game than low-connected nodes (bottom-right panel).We further verify that, when fixing α = 2 , having stricter voting rules (high M , in this case M = 0 . ) attenuates the unfairness associated with having hubs being the Proposers.Not only does low-degree Proposer assignment reduce unfairness, as it also sustains moreefficient outcomes — taken as higher values of average fitness observed in the population.As stated previously, fairness represents here the payoff that a node obtains after playing ineach possible group (see Figure 1). In Figure 5 we confirm that low values of α maximize theaverage fitness of populations. This occurs when considering heterogeneous networks withdifferent average degrees ( (cid:104) k (cid:105) ) and group decision rules ( M ). This effect is more evident whenconsidering less strict group decision rules (that is, lower M , meaning that less Respondersare required to accept a proposal for the group to accept it) and networks with higher (cid:104) k (cid:105) .11igure 4: Selecting high-degree nodes as Proposers increases unfairness. Here we representthe so-called Lorenz curves, often used to compute the Gini coefficients – a typical measureof income inequality. Each curve is generated by ordering individuals by increasing value ofincome and plotting the corresponding cumulative distribution. Curves closer to the perfectequality line (45 degree line) represent more egalitarian outcomes. Here we observe, yetagain, that assigning the role of Proposer to high-connected nodes ( alpha = 2 ) yields unfairoutcomes (orange line). While this is evident for soft (panel a, M = 0 . ), medium (panel b, M = 0 . ) and strict decision rules (panel c, M = 0 . ), we also verify that whenever hubs arethe Proposers ( α = 2 ), having strict decision rules (high M ) reduces unfairness.Figure 5: Low-degree Proposer assignment maximizes the average fitness (i.e., sum of payoffstaken over all games, see Figure 1) in a population. Here we observe that the average fitness, (cid:104) f (cid:105) , increases as α decreases. We show results for BA networks with different values of (cid:104) k (cid:105) and M . A similar conclusion is obtained when considering DMS networks with the sameparameters as Figure 2 and 5. Note that, increasing (cid:104) k (cid:105) implies that the average groupsize to play MUG also increases, which leads offers to be divided by larger groups (hencecontributing to lower values of average payoff per game). On the other hand, increasing (cid:104) k (cid:105) means that more games are played, thus contributing to an increase in accumulated fitness(taken as the sum of payoffs in all games played).Finally, we confirm that low-degree Proposer assignment maximizes the average proposalplayed in the population (and thus fairness) when considering networks with higher (cid:104) k (cid:105) and,as a result, larger average group sizes. As Figure 6 conveys, the higher values of averageproposal, (cid:104) p (cid:105) are obtained for α < . Notwithstanding, we are able to find parameter spaces12igure 6: We confirm that low-degree Proposer assignment maximizes populations’ averagelevel of proposal, (cid:104) p (cid:105) , for both BA and DMS networks with higher average degree ( (cid:104) k (cid:105) = 8 ,top, and (cid:104) k (cid:105) = 16 , bottom). For networks with higher (cid:104) k (cid:105) — leading to MUGs played inlarger groups — and low M , random role attribution ( α = 0 ) configures the worst scenario interms of fair proposals.where the dependence of (cid:104) p (cid:105) on α is seemingly affected by i ) the average connectivity of thenetwork — and thus on the average size of the groups in which MUG is played — and ii )particular values of M . Also, we confirm that increasing M increases (cid:104) p (cid:105) for all values of α . Our results suggest that offering the first move to low-degree nodes balances the naturalpower of highly connected nodes in scale-free networks, leading to a significant increase in theglobal levels of fairness. Interestingly, we also find that particular voting rules ( M ) are ableto attenuate the negative effect of high α ( i.e. privileged high-degree nodes being selected tobe Proposers) on fairness. In this paper we address the general problem of 1) deciding how to attribute bargaining rolesin a social network and, in particular, 2) understanding the impact of different criteria on theemerging levels of fairness in Multiplayer Ultimatum Games. We verified that attributing therole of proposer to low degree nodes boost both fairness and overall fitness. This conclusionremains valid for different network structures (BA and DMS networks with average degreesranging from 4 to 16) and interaction scenarios (in terms of group sizes and group decisionrules) . 13ne possible intuition for this result can be understood as follows: consider the situationwhere two hubs ( H and H ) are connected to each other and each is linked to k − nodeswith degree (i.e., leafs); given that imitation is based on the accumulated payoff, these hubshave, potentially, a higher fitness than the leafs. When highly connected nodes are selectedto be the Proposers, H and H always play in that role. If this is the case, the hubs alwaysuse the values of p they adopt and the only way that leafs earn some payoff is to lower theiracceptance thresholds and always accept hubs’ offers. When low connected nodes are selectedto be the Proposers, however, the hubs will always play in the role of Responder. Assumingthat all leafs adopt strategy p , the fitness of H and H will be given by pk + p ( k − —the first term corresponding to the payoff earned when playing in a group centered in H or H and the second term corresponds to the payoff earned when playing with each of theProposer-leafs. Notice that the fitness of the hubs (and hence the probability of gettingimitated) increases with p , the offers made by the leafs. As leafs are only connected with H or H , the only way they have to spread their strategies — which occurs when the hubsimitate each other — is by increasing their offered values, p . Fairer strategies thus spreadunder low-degree Proposer assignment.We also find that the perils of having high-degree Proposers can be softened with strictgroup decision rules. This means that, whenever role selection is constrained and Proposersare necessarily the better connected nodes (by having the needed resources to propose andbe the first movers in a bargaining situation) unfairness can be reduced by imposing thatproposals need to be validated by a large fraction of Responders. The effect of M on elicitingfairer offers is similar to that found in recent literature [23, 26]. Also, our results are in linewith works showing that selecting low-degree Proposers maximizes fairness in the context ofpairwise Ultimatum Games [33] and Dictator Games [4].This work can underlie several extensions of interest for social and engineering sciences. Herewe consider that role assignment is endogenously imposed. In reality, these rules are likelyto evolve side-by-side with individual strategies, being another self-organized property ofthe system, as fairness and wealth distributions. Moreover, the fact that network-basedrole assignment elicits fairness in rather complicated scenarios — as multiplayer bargaininggames — suggests that such approach could also be used within the broader context ofactive interventions aiming at fostering fairness in hybrid populations comprising humans andmachines [26, 28, 19]. In this context, it would be relevant to assess — both experimentallyand through numerical simulations — the impact on human decision-making of having virtualregulators dynamically deciding the role to adopt by their group peers, depending on their14osition in the interaction structure. 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