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Heterogeneous resource allocation can change social hierarchy in public goods games

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ABSTRACT

Public goods games (PGGs) represent one of the most useful tools to study group interactions. However, even if they could provide an explanation for the emergence and stability of cooperation in modern societies, they are not able to reproduce some key features observed in social and economical interactions. The typical shape of wealth distribution—known as Pareto Law—and the microscopic organization of wealth production are two of them. Here, we introduce a modification to the classical formulation of PGGs that allows for the emergence of both of these features from first principles. Unlike traditional PGGs, where players contribute equally to all the games in which they participate, we allow individuals to redistribute their contribution according to what they earned in previous rounds. Results from numerical simulations show that not only a Pareto distribution for the pay-offs naturally emerges but also that if players do not invest enough in one round they can act as defectors even if they are formally cooperators. Our results not only give an explanation for wealth heterogeneity observed in real data but also point to a conceptual change on cooperation in collective dilemmas.

No MeSH data available.


Normalized pay-off Πk/(k+1) obtained in the game centred at node i as a function of the degree k for different values of the parameter α. For the static case, α=0 (a), a low dispersion around a mean pay-off value is observed for all degrees k. Increasing α leads first to an increase in the normalized pay-off and its dispersion, most notably for small degrees. A further increase of α produces a second cloud of points localized at the maximal contribution (e.g. c(r−1), in this case the maximum value is 2.5 as we have c=1 and r=3.5) and low degrees (b–d). Networks parameters are the same as in figure 3.
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RSOS170092F4: Normalized pay-off Πk/(k+1) obtained in the game centred at node i as a function of the degree k for different values of the parameter α. For the static case, α=0 (a), a low dispersion around a mean pay-off value is observed for all degrees k. Increasing α leads first to an increase in the normalized pay-off and its dispersion, most notably for small degrees. A further increase of α produces a second cloud of points localized at the maximal contribution (e.g. c(r−1), in this case the maximum value is 2.5 as we have c=1 and r=3.5) and low degrees (b–d). Networks parameters are the same as in figure 3.

Mentions: Results shown in figure 3 suggest that, once players are free to allocate their resources, a very peculiar organization emerges. The peak for large values of the investments indicates that most of the players (almost the totality for α≥3.0) allocate the majority of their resources in only one market—the most profitable one—while they distribute evenly between all the other games the remaining part of their capital creating the peak for small values of Ii,j. The previous findings could explain the observed increase in players’ cooperation [48–52], but it is not the only consequence of the observed investment distribution. Indeed, an established result in PGGs is that the most connected nodes—the hubs—are responsible for the emergence of cooperation and for the production of the majority of the pay-off. However, the results in figure 4 depict a different scenario. If we consider the total normalized pay-off produced in games centred on nodes of degree k, Πk/(k+1), we find that, in the classical case, it is distributed almost homogeneously among all the degrees, with a mean value around 0.5 (figure 4a). However, for α>0 strong differences arise. As α increases, the distribution of the pay-off for games taking place on low degree nodes starts to become more heterogenous until, finally, for α=4.0 (figure 4d) a large number of games produce very high pay-offs (note that the maximum pay-off does not depend on k, as max{Πk/(k+1)}=c(r−1) and in this case c=1 and r=3.5 leading to a maximum of 2.5). This means that all the players invested all their contributions in that game. Moreover, with the increase of α the average pay-off produced in the hubs decreases substantially.Figure 4


Heterogeneous resource allocation can change social hierarchy in public goods games
Normalized pay-off Πk/(k+1) obtained in the game centred at node i as a function of the degree k for different values of the parameter α. For the static case, α=0 (a), a low dispersion around a mean pay-off value is observed for all degrees k. Increasing α leads first to an increase in the normalized pay-off and its dispersion, most notably for small degrees. A further increase of α produces a second cloud of points localized at the maximal contribution (e.g. c(r−1), in this case the maximum value is 2.5 as we have c=1 and r=3.5) and low degrees (b–d). Networks parameters are the same as in figure 3.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5383863&req=5

RSOS170092F4: Normalized pay-off Πk/(k+1) obtained in the game centred at node i as a function of the degree k for different values of the parameter α. For the static case, α=0 (a), a low dispersion around a mean pay-off value is observed for all degrees k. Increasing α leads first to an increase in the normalized pay-off and its dispersion, most notably for small degrees. A further increase of α produces a second cloud of points localized at the maximal contribution (e.g. c(r−1), in this case the maximum value is 2.5 as we have c=1 and r=3.5) and low degrees (b–d). Networks parameters are the same as in figure 3.
Mentions: Results shown in figure 3 suggest that, once players are free to allocate their resources, a very peculiar organization emerges. The peak for large values of the investments indicates that most of the players (almost the totality for α≥3.0) allocate the majority of their resources in only one market—the most profitable one—while they distribute evenly between all the other games the remaining part of their capital creating the peak for small values of Ii,j. The previous findings could explain the observed increase in players’ cooperation [48–52], but it is not the only consequence of the observed investment distribution. Indeed, an established result in PGGs is that the most connected nodes—the hubs—are responsible for the emergence of cooperation and for the production of the majority of the pay-off. However, the results in figure 4 depict a different scenario. If we consider the total normalized pay-off produced in games centred on nodes of degree k, Πk/(k+1), we find that, in the classical case, it is distributed almost homogeneously among all the degrees, with a mean value around 0.5 (figure 4a). However, for α>0 strong differences arise. As α increases, the distribution of the pay-off for games taking place on low degree nodes starts to become more heterogenous until, finally, for α=4.0 (figure 4d) a large number of games produce very high pay-offs (note that the maximum pay-off does not depend on k, as max{Πk/(k+1)}=c(r−1) and in this case c=1 and r=3.5 leading to a maximum of 2.5). This means that all the players invested all their contributions in that game. Moreover, with the increase of α the average pay-off produced in the hubs decreases substantially.Figure 4

View Article: PubMed Central - PubMed

ABSTRACT

Public goods games (PGGs) represent one of the most useful tools to study group interactions. However, even if they could provide an explanation for the emergence and stability of cooperation in modern societies, they are not able to reproduce some key features observed in social and economical interactions. The typical shape of wealth distribution—known as Pareto Law—and the microscopic organization of wealth production are two of them. Here, we introduce a modification to the classical formulation of PGGs that allows for the emergence of both of these features from first principles. Unlike traditional PGGs, where players contribute equally to all the games in which they participate, we allow individuals to redistribute their contribution according to what they earned in previous rounds. Results from numerical simulations show that not only a Pareto distribution for the pay-offs naturally emerges but also that if players do not invest enough in one round they can act as defectors even if they are formally cooperators. Our results not only give an explanation for wealth heterogeneity observed in real data but also point to a conceptual change on cooperation in collective dilemmas.

No MeSH data available.