- Email: [email protected]
Awareness of wealth inequalities breeds animosity
Chaos, Solitons and Fractals 130 (2020) 109398
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Frontiers
Awareness of wealth inequalities breeds animosity Mi Zhang a, Yongjuan Ma b, Yi Tao c, Zhen Wang d, Lei Shi b,∗, Rui-Wu Wang a,∗ a
Center for Ecology and Environmental Sciences, Northwestern Polytechnical University, Xi’an 710072, China Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650221, China c Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational Biology and Evolution, Institute of Zoology, Chinese Academy of Sciences, Beijing 100101, China d Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi’an 710072, China b
a r t i c l e
i n f o
Article history: Received 15 August 2019 Accepted 20 August 2019
Keywords: Cooperation Inequality Visibility
a b s t r a c t Cooperation is a ubiquitous trait among individuals of a species which interbreeds in nature and forms part of groups [1]. Various theories and models have been developed to explain the origins, maintenance, and continuation of cooperation within a generic population. Among these theories is the prisoner’s dilemma (PD), which is a well-known framework for investigations on agents’ behavior and strategies [2]. In a traditional PD game, pairwise players are assigned the same initial endowment, and they have no access to (nor knowledge of) the opponents’ wealth throughout the course of the game. In real life, however, circumstances operate which (in most of the cases) unequally, asymmetrically or hierarchically distribute the income and resources, and the group members may be aware (completely or partially) of their opponents’ power and capacity before making their own decision to cooperate or defect. We experimentally reveal the impact of inequality and visibility by means of comparing the results of four sessions where players (1) may have equal or unequal initial endowments and (2) may be visible or invisible to opponents as far as their wealth information is concerned. Our observations disclose that wealth information makes no significant difference to the promotion of cooperation when the initial wealth distribution is fair. In the case of initially unequal endowments, hiding the payoff information enhances the frequency of cooperation while visibility of the opponent’s wealth harms group synergy. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Cooperation is found throughout nature, and its emergence in humanity is the cornerstone for the development and progress of our society. In humans, cooperation prevails at every level, embracing country to country or enterprise to enterprise relationships, as well as person to person interactions [3–6]. Therefore, the understanding of the mechanisms through which cooperation among unrelated individuals emerges (and is maintained) remains a big challenge for scientists in biology, sociology, psychology, management, and economics. In traditional pairwise settings, the power of the two sides is in general, well-balanced and the opponents’ wealth information is inaccessible, thus choices are motivated by an individual’s wealth. In more realistic conditions, however, individuals are interacting with partners, and they feature, in fact, several asymmetric properties, such as an unequal social status or social wealth and a varying
∗
Corresponding authors. E-mail addresses: [email protected] (L. Shi), [email protected] (R.-W. Wang). https://doi.org/10.1016/j.chaos.2019.109398 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
degree of cognitive abilities [7]. As a result, ties between strong and weak (or between rich and poor) individuals, are commonly present in social interactions. Therefore, the decision of cooperating (or defecting) is taken not only on the basis of the individual’s attitude, but also (and possibly prominently) on the knowledge of key information about opponents’ wealth. I.e., the question arises on whether cooperation can be promoted by combining inequality and visibility of wealth. In this paper, we will investigate whether and how individual behavior in pairwise interactions is affected by wealth status and wealth visibility. Specifically, we will consider a classic Prisoner’s Dilemma (PD) game, and experimentally study how the cooperation level of individuals is sensitive to inequalities in the initial wealth status and to visibility of opponents’ wealth information. 2. Experiment and results Similar to previous studies [8,9], we conducted an experiment based on repeated PD games with costly punishments at the Northwestern Polytechnical University, Xi’an, China. The basic game includes three options (or strategies): cooperation(C), defection (D), and costly punishment (P) [8,9]. The payoff matrix
2
M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
Fig. 1. Cooperation and punishment levels in the treatment and control groups. Panel (a): the cooperation levels (left) and the average frequencies of using P (right) in T1 and T2. Panel (b): the cooperation levels (left) and the average frequencies of using P in C1 and in C2.
associated to pair-wise interactions is
L =
4 7 4
−2 1 −2
−5 −2 . −5
(1)
In the game, when a cooperator C meets another cooperator C (or a defector D, or a punishing individual P), then it earns 4 (or −2, or −5); when D plays against C (or D, or P), it gains 7 (or 1, or −2); and finally when P is connected with C (or D, or P), it obtains 4 (or −2, or −5). A total of 158 students participated in the experiments. Specifically, 80 students were divided into two treatment groups (denoted by T1 and T2), while the remaining 78 students were divided into two control groups (denoted by C1 and C2). In the group T1, half of the participants were randomly selected and assigned an initial endowment of 50 units, while the other half were initially supplied by 100 units. Moreover, for each of the subjects in T1, the opponent’s wealth information was visible throughout the experiment. The design of the treatment group T2 was similar to T1, except that the opponent’s wealth information was hidden from the beginning to the end of the game. The groups C1 and C2 were prepared as the control group for T1 or T2, with all the subjects initially being endowed 50 units, and the opponent’s wealth information being visible (C1) or hidden (C2) throughout the game. For each pair of interactions in T1, T2, C1 and C2, agents did not know the number of rounds the experiment included, but they were told that there was 75% probability to run a next round (to avoid end effects). New interaction pairs were formed by random meetings. The experimental results show clearly that: (i) the cooperation level (defined as the average frequency of C in the game) in T1 is significantly (Wilcoxon-Mann-Whitney test, p < 0.05) lower than that in T2 (it is 0.22 in T1 and 0.49 in T2), and the average frequency of the P strategy in T1 is only slightly but significantly (Wilcoxon-Mann-Whitney test, p < 0.05) larger than that in T2 (it is 0.06 in T1 and 0.03 in T2) (Fig. 1a); (ii) the cooperation level in C1 is slightly lower than that in C2 (it is 0.50 in C1 and 0.53 in C2), and the difference is not significant (Wilcoxon-Mann-Whitney test, p = 0.53), as well as there is no significant difference between the average frequencies of P in C1 and C2 (Wilcoxon-Mann-Whitney test, p = 0.45), (Fig. 1b).
It can also be seen that the level of cooperation in the control groups are significantly higher than that in the treatment groups, that is, the average frequencies of using C in C1 and in C2 are higher than that in T1 and in T2, respectively. These results suggest that if the initial wealth is equal, then the visibility of opponent’s wealth information has not a significant impact on the average cooperation level of the system. On the contrary, if the initial wealth is unequal (the subjects have different initial wealth statuses), then the average cooperation level strongly depends on the visibility of the opponent’s wealth. Namely, in this latter case, the cooperation level will be significantly inhibited if the opponent’s wealth information is visible. The conclusion is that an increased transparency of opponents’ wealth information is not, in fact, conducive to the evolution of cooperation if the initial wealth status is unequally distributed. In order to gather a deeper understanding of the role of inequality and visibility in the development of cooperation we conducted further analysis which show that, in T1, (i) the average cooperation level of the subjects who were initially endowed 50 units (from here on called for convenience the “poor” subjects) is 0.24, while and the average cooperation level of the subjects who were initially endowed 100 units (hereinafter the “rich” subjects) is 0.2, and the difference is not significant (Wilcoxon-MannWhitney test, p = 0.17) (Fig. 2a); (ii) the frequencies of the “poor” and “rich” subjects choosing P are 0.08 and 0.05, respectively, and once again the difference is not significant (Wilcoxon-MannWhitney test, p = 0.26) (see also Fig. 2a); (iii) the average cooperation levels in the interactions between a pair of “poor” subjects, between a pair of “rich” subjects and between a “poor” and a “rich” are 0.24, 0.18 and 0.22, respectively, and the differences are again not significant (Kruskal-Walli test, p = 0.19) (Fig. 2b); and (iv) in the interactions between a “poor” and a “rich”, the average cooperation levels of the “poor” and “rich” are 0.23 and 0.19, respectively, and the difference is not significant (Wilcoxon-MannWhitney test, p = 0.19) (see also Fig. 2b). The results imply that when the opponents’ wealth information is visible, the individuals’ behavior seems to be independent on their initial wealth status. At the same time, Fig. 2c-d show results obtained in T2. To be more specific, (i) the average cooperation levels of “poor” and “rich” subjects are 0.4 and 0.58, respectively, and the difference is significant (Wilcoxon-Mann-Whitney test, p < 0.05); (ii) the frequencies of the “poor” and “rich” subjects displaying P are 0.03 and
M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
3
Fig. 2. The cooperation levels of “poor” and “rich” subjects in T1 and T2. Panel (a)& (c) shows: the average cooperation levels of “poor” (red) and “rich” (blue) subjects; and the average frequencies of the “poor” and “rich” subjects displaying P in T1 and T2, respectively. Panel (b) & (d) shows: the average cooperation levels if the interactions between a pair of “poor” subjects (50–50), between a pair of “rich” subjects (100–100) and between one “poor” subject and one “rich” subject (50–100) in T1 and T2, respectively; and the average cooperation levels of the “poor” (pink) and “rich” (green) subjects in the interactions between one “poor” subject and one “rich” subject (50–100) in T1 and T2, respectively.
0.02, respectively, with no significance in the difference (WilcoxonMann-Whitney test, p = 0.78); (iii) the average cooperation level in the interactions between pairs of “rich” subjects is significantly higher than that occurring for pairs of “poor” subjects, as well as that occurring between a “poor” subject and a “rich” subject (0.62, 0.38 and 0.49, respectively, with Kruskal-Walli test, p < 0.05); and (iv) for interactions between “poor” and “rich” subjects, the average cooperation levels of “poor” and “rich” subjects are 0.44 and 0.54, respectively, with no significance in the difference (WilcoxonMann-Whitney test, p = 0.2). In other words, if the opponents’ wealth information is hidden, not only the total cooperation level of the “rich” subjects is significantly higher than that of the “poor” subjects, but also the cooperation level in interactions between a pair of “rich” subjects is significantly higher than that pertinent to interactions between a pair of “poor” subjects. On its turn, this implies that when the opponents’ wealth information is hidden, the cooperation level of individuals is sensitive to their initial wealth status. Finally, one can see that the average final wealth level of all subjects in T1 (181.4) is significantly lower than that of subjects in T2 (262.3, T test, p < 0.05, see Fig. 3a). More specifically, (i) the average final wealth level of “poor” subjects in T1 (156.8) is significantly lower than that of “rich” subjects (203.7, T test, p < 0.05), and the normalized difference between such two wealth levels is dT1 = (203.7–156.8)/(156.8 + 203.7) = 0.13 (Fig. 3b); (ii) the average final wealth level of “poor” subjects (239.2) is also significantly lower than that of “rich” subjects (285.4) in T2 (T test, p < 0.05 with a normalized difference dT2 = (285.4–
239.2)/(239.2 + 285.4) = 0.09. (Fig. 3b); and (iii) both average final wealth levels of “poor” and “rich” subjects in T1 are significantly lower than those in T2 (T test, p < 0.05) (Fig. 3b). These results imply that the invisibility of opponents’ wealth information not only promotes the average final wealth level of single individuals, but also contributes to the elimination of initial differences in wealth levels among individuals.
3. Discussion The PD game is one of the most important theoretical tools for revealing the evolutionary mechanisms of cooperation [10–14]. We studied experimentally the role of inequalities in wealth statuses and visibility of wealth information in determining the individual behavior of players conducting repeated PD games with costly punishment. Although the role of wealth inequalities has been considered in various circumstances [7], two recent studies revealed that visibility of wealth information may be an even more important factor affecting the cooperation level of individuals [15,16]. Specifically, Nishi et al. showed that in a networked public goods game, wealth visibility facilitates the downstream consequences of initial inequalities and leads to inequality statuses more differentiated than those occurring when wealth is invisible [15]. On the other hand, Hauser et al. showed that, when wealth information is visible in a public goods game, individuals tend to punish the rich and reward the poor [16].
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M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
Fig. 3. The average final wealth levels in treatments T1 and T2. Panel (a): the average final wealth levels of all subjects in T1 (red) and T2 (blue), respectively. Panel (b): the average final wealth levels of “poor” and “rich” subjects in T1 and T2, respectively.
At variance, our experiments reveal that increased transparency of opponents’ wealth information is not conducive to the evolution of cooperation if the initial wealth status is unequal, and when the opponents’ wealth information is visible the individuals’ behavior seems to be largely independent on their initial wealth status. Conversely, when the opponents’ wealth information is hidden, individual behavior seems very sensitive to their initial wealth status. The cooperation level of the system strongly relies on the visibility of opponent’s wealth information, that is, the average frequency of displaying cooperation in the case of visible opponents’ wealth information is significantly lower than that in the case of hidden opponent’s wealth information. Finally, if the initial wealth status is unequal, the invisibility of opponents’ wealth information is not only conducive to promote the average final wealth level of each individual but also to eliminate differences in wealth levels among individuals. That is to say that the average final wealth level in the case of visible opponents’ wealth information is significantly lower than that in the case of hidden opponents’ wealth information, and the normalized difference of final wealth level in the case of visible opponents’ wealth information is larger than that in the case of hidden opponents’ wealth information. Furthermore, the effects of both inequality of initial wealth status and visibility of opponents’ wealth information on using the option of punishment are not significant in our experiment, at variance with the conclusions offered by Hauser et al. [16]. Additionally, experiments here reported correspond to selecting a dilemma strength equal to 1 [17]. We in fact conducted the same experiments with different payoff matrices (for which the dilemma strength is equal to 2) in Yunnan, China, and we obtained qualitatively similar results in both predominantly cooperative and defecting environments. In theoretical analysis, the Prisoner’s Dilemma is one extreme of a general public goods game [18]. Our finding, therefore, can make sense to prove that the Prisoner’s Dilemma and the public goods game are consistent in essence. Neurological and psychological mechanisms may shed light on the understanding of the observed behavior [19–22]. Equality allows people to act with a relatively balanced mentality, and the outcome of the last round of the game is basically agreeing with what they expect, and this leaves therefore a marginal role to hidden or visible information about the opponent’s wealth statuses. On the opposite, when the initial wealth is uneven, the visibility of wealth information re-
markably hampers cooperation. The unequal distribution of wealth has a great effect on people’s psychology [19–22]. From Fig. 2, we can see at times the inequalities are cleverly exploited by the “rich” subjects, which further leads to the rising gap between the rich and the poor. The wealth gap, social inequality and other issues have to be settled urgently, especially because the disclosure of information becomes more and more detailed and clear in our new transparent age (with the development of both science and society). Hence, it makes a difference to integrate these two factors together, which enlightens us to combine inequality with visibility to explore what drives the gap. Further research opportunities related to the inequality and visibility remain to be explored, such as the scenario under completely unequal state, the effect of varying unequal extents, and the visibility connected to more realistic situation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful to the support from the NSFC-Yunnan United fund [grant number U1302267]; the National Science Fund for Distinguished Young Scholars [grant number 31325005]; and the National Natural Science Foundation of China grants [grant number 31270433]. Author contributions R.-W.W. and L.S. designed the experiment. M.Z. conducted the experiment. M.Z and Y.J.M. performed the statistical analyses. All authors discussed the results. M.Z. wrote the manuscript. Author Information The authors declare no competing interests. References [1] Maynard-Smith J, Szathmary E. The major transition in evolution. Oxford: Univ. Press; 1998.
M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398 [2] Rapoport A, Chammah AM. Prisoner’s dilemma: a study in conflict and cooperation. The University of Michigan Press; 1965. [3] Trivers, Robert L. The evolution of reciprocal altruism. Q Rev Biol 1971;46(1):35–57. [4] Axelrod R, Hamilton WD. The evolution of cooperation. Science 1981;211(2):135–60. [5] Nowak MA. Five rules for the evolution of cooperation. Science 2006;314(5805):1560–3. [6] Fudenberg D, Maskin E. Evolution and cooperation in noisy repeated games. Am Econ Rev 1990;80(2):274–9. [7] Piketty T, Saez E. Inequality in the long run. Science 2014;344:838–43. [8] Dreber A, et al. Winners don’t punish. Nature 2008;452:348–51. [9] Wu Jia-Jia, et al. Costly punishment does not always increase cooperation. Proc Natl Acad Sci 2009;106(41):17448–51. [10] Nowak MA, et al. Tit for tat in heterogeneous populations. Nature 1992;355:250–3. [11] Gallo E, et al. The effects of reputational and social knowledge on cooperation. Proc Natl Acad Sci USA 2015;112:3647–52. [12] Sefton M, Shupp R, Walker JM. The effect of rewards and sanctions in provision of public goods. Econ Inq 2007;45:671–90.
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[13] Fehr E, Gächter S. Altruistic punishment in humans. Nature 2002;415:137–40. [14] Henrich J, et al. Costly punishment across human societies. Science 2006;312:1767–70. [15] Nishi Akihiro, et al. Inequality and visibility of wealth in experimental social networks. Nature 2015;526:426–9. [16] Hauser O, et al. Invisible inequality leads to punishing the poor and rewarding the rich. Social science electronic publishing; 2016. [17] Wang Z, et al. Universal scaling for the dilemma strength in evolutionary games. Phys Life Rev 2015;14:1–30. [18] Archatti Marco, et al. Coexistence of cooperation and defection in public goods games. Evolution 2010;65-4:1140–8. [19] Tricomi E, Rangel A, Camerer CF, O’Doherty JP. Neural evidence for inequality-averse social preferences. Nature 2010;463:1089–109. [20] Engelmann D, Strobel M. Inequality aversion, efficiency, and maximin preferences in simple distribution experiments. Am Econ Rev 2004;94:857–69. [21] Gilbert DT, Giesler RB, Morris KA. When comparisons arise. J Pers Soc Psychol 1995;69:227–36. [22] Clark AE, Oswald AJ. Satisfaction and comparison income. J Public Econ 1996;61:359–81.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Frontiers
Awareness of wealth inequalities breeds animosity Mi Zhang a, Yongjuan Ma b, Yi Tao c, Zhen Wang d, Lei Shi b,∗, Rui-Wu Wang a,∗ a
Center for Ecology and Environmental Sciences, Northwestern Polytechnical University, Xi’an 710072, China Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650221, China c Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational Biology and Evolution, Institute of Zoology, Chinese Academy of Sciences, Beijing 100101, China d Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi’an 710072, China b
a r t i c l e
i n f o
Article history: Received 15 August 2019 Accepted 20 August 2019
Keywords: Cooperation Inequality Visibility
a b s t r a c t Cooperation is a ubiquitous trait among individuals of a species which interbreeds in nature and forms part of groups [1]. Various theories and models have been developed to explain the origins, maintenance, and continuation of cooperation within a generic population. Among these theories is the prisoner’s dilemma (PD), which is a well-known framework for investigations on agents’ behavior and strategies [2]. In a traditional PD game, pairwise players are assigned the same initial endowment, and they have no access to (nor knowledge of) the opponents’ wealth throughout the course of the game. In real life, however, circumstances operate which (in most of the cases) unequally, asymmetrically or hierarchically distribute the income and resources, and the group members may be aware (completely or partially) of their opponents’ power and capacity before making their own decision to cooperate or defect. We experimentally reveal the impact of inequality and visibility by means of comparing the results of four sessions where players (1) may have equal or unequal initial endowments and (2) may be visible or invisible to opponents as far as their wealth information is concerned. Our observations disclose that wealth information makes no significant difference to the promotion of cooperation when the initial wealth distribution is fair. In the case of initially unequal endowments, hiding the payoff information enhances the frequency of cooperation while visibility of the opponent’s wealth harms group synergy. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Cooperation is found throughout nature, and its emergence in humanity is the cornerstone for the development and progress of our society. In humans, cooperation prevails at every level, embracing country to country or enterprise to enterprise relationships, as well as person to person interactions [3–6]. Therefore, the understanding of the mechanisms through which cooperation among unrelated individuals emerges (and is maintained) remains a big challenge for scientists in biology, sociology, psychology, management, and economics. In traditional pairwise settings, the power of the two sides is in general, well-balanced and the opponents’ wealth information is inaccessible, thus choices are motivated by an individual’s wealth. In more realistic conditions, however, individuals are interacting with partners, and they feature, in fact, several asymmetric properties, such as an unequal social status or social wealth and a varying
∗
Corresponding authors. E-mail addresses: [email protected] (L. Shi), [email protected] (R.-W. Wang). https://doi.org/10.1016/j.chaos.2019.109398 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
degree of cognitive abilities [7]. As a result, ties between strong and weak (or between rich and poor) individuals, are commonly present in social interactions. Therefore, the decision of cooperating (or defecting) is taken not only on the basis of the individual’s attitude, but also (and possibly prominently) on the knowledge of key information about opponents’ wealth. I.e., the question arises on whether cooperation can be promoted by combining inequality and visibility of wealth. In this paper, we will investigate whether and how individual behavior in pairwise interactions is affected by wealth status and wealth visibility. Specifically, we will consider a classic Prisoner’s Dilemma (PD) game, and experimentally study how the cooperation level of individuals is sensitive to inequalities in the initial wealth status and to visibility of opponents’ wealth information. 2. Experiment and results Similar to previous studies [8,9], we conducted an experiment based on repeated PD games with costly punishments at the Northwestern Polytechnical University, Xi’an, China. The basic game includes three options (or strategies): cooperation(C), defection (D), and costly punishment (P) [8,9]. The payoff matrix
2
M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
Fig. 1. Cooperation and punishment levels in the treatment and control groups. Panel (a): the cooperation levels (left) and the average frequencies of using P (right) in T1 and T2. Panel (b): the cooperation levels (left) and the average frequencies of using P in C1 and in C2.
associated to pair-wise interactions is
L =
4 7 4
−2 1 −2
−5 −2 . −5
(1)
In the game, when a cooperator C meets another cooperator C (or a defector D, or a punishing individual P), then it earns 4 (or −2, or −5); when D plays against C (or D, or P), it gains 7 (or 1, or −2); and finally when P is connected with C (or D, or P), it obtains 4 (or −2, or −5). A total of 158 students participated in the experiments. Specifically, 80 students were divided into two treatment groups (denoted by T1 and T2), while the remaining 78 students were divided into two control groups (denoted by C1 and C2). In the group T1, half of the participants were randomly selected and assigned an initial endowment of 50 units, while the other half were initially supplied by 100 units. Moreover, for each of the subjects in T1, the opponent’s wealth information was visible throughout the experiment. The design of the treatment group T2 was similar to T1, except that the opponent’s wealth information was hidden from the beginning to the end of the game. The groups C1 and C2 were prepared as the control group for T1 or T2, with all the subjects initially being endowed 50 units, and the opponent’s wealth information being visible (C1) or hidden (C2) throughout the game. For each pair of interactions in T1, T2, C1 and C2, agents did not know the number of rounds the experiment included, but they were told that there was 75% probability to run a next round (to avoid end effects). New interaction pairs were formed by random meetings. The experimental results show clearly that: (i) the cooperation level (defined as the average frequency of C in the game) in T1 is significantly (Wilcoxon-Mann-Whitney test, p < 0.05) lower than that in T2 (it is 0.22 in T1 and 0.49 in T2), and the average frequency of the P strategy in T1 is only slightly but significantly (Wilcoxon-Mann-Whitney test, p < 0.05) larger than that in T2 (it is 0.06 in T1 and 0.03 in T2) (Fig. 1a); (ii) the cooperation level in C1 is slightly lower than that in C2 (it is 0.50 in C1 and 0.53 in C2), and the difference is not significant (Wilcoxon-Mann-Whitney test, p = 0.53), as well as there is no significant difference between the average frequencies of P in C1 and C2 (Wilcoxon-Mann-Whitney test, p = 0.45), (Fig. 1b).
It can also be seen that the level of cooperation in the control groups are significantly higher than that in the treatment groups, that is, the average frequencies of using C in C1 and in C2 are higher than that in T1 and in T2, respectively. These results suggest that if the initial wealth is equal, then the visibility of opponent’s wealth information has not a significant impact on the average cooperation level of the system. On the contrary, if the initial wealth is unequal (the subjects have different initial wealth statuses), then the average cooperation level strongly depends on the visibility of the opponent’s wealth. Namely, in this latter case, the cooperation level will be significantly inhibited if the opponent’s wealth information is visible. The conclusion is that an increased transparency of opponents’ wealth information is not, in fact, conducive to the evolution of cooperation if the initial wealth status is unequally distributed. In order to gather a deeper understanding of the role of inequality and visibility in the development of cooperation we conducted further analysis which show that, in T1, (i) the average cooperation level of the subjects who were initially endowed 50 units (from here on called for convenience the “poor” subjects) is 0.24, while and the average cooperation level of the subjects who were initially endowed 100 units (hereinafter the “rich” subjects) is 0.2, and the difference is not significant (Wilcoxon-MannWhitney test, p = 0.17) (Fig. 2a); (ii) the frequencies of the “poor” and “rich” subjects choosing P are 0.08 and 0.05, respectively, and once again the difference is not significant (Wilcoxon-MannWhitney test, p = 0.26) (see also Fig. 2a); (iii) the average cooperation levels in the interactions between a pair of “poor” subjects, between a pair of “rich” subjects and between a “poor” and a “rich” are 0.24, 0.18 and 0.22, respectively, and the differences are again not significant (Kruskal-Walli test, p = 0.19) (Fig. 2b); and (iv) in the interactions between a “poor” and a “rich”, the average cooperation levels of the “poor” and “rich” are 0.23 and 0.19, respectively, and the difference is not significant (Wilcoxon-MannWhitney test, p = 0.19) (see also Fig. 2b). The results imply that when the opponents’ wealth information is visible, the individuals’ behavior seems to be independent on their initial wealth status. At the same time, Fig. 2c-d show results obtained in T2. To be more specific, (i) the average cooperation levels of “poor” and “rich” subjects are 0.4 and 0.58, respectively, and the difference is significant (Wilcoxon-Mann-Whitney test, p < 0.05); (ii) the frequencies of the “poor” and “rich” subjects displaying P are 0.03 and
M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
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Fig. 2. The cooperation levels of “poor” and “rich” subjects in T1 and T2. Panel (a)& (c) shows: the average cooperation levels of “poor” (red) and “rich” (blue) subjects; and the average frequencies of the “poor” and “rich” subjects displaying P in T1 and T2, respectively. Panel (b) & (d) shows: the average cooperation levels if the interactions between a pair of “poor” subjects (50–50), between a pair of “rich” subjects (100–100) and between one “poor” subject and one “rich” subject (50–100) in T1 and T2, respectively; and the average cooperation levels of the “poor” (pink) and “rich” (green) subjects in the interactions between one “poor” subject and one “rich” subject (50–100) in T1 and T2, respectively.
0.02, respectively, with no significance in the difference (WilcoxonMann-Whitney test, p = 0.78); (iii) the average cooperation level in the interactions between pairs of “rich” subjects is significantly higher than that occurring for pairs of “poor” subjects, as well as that occurring between a “poor” subject and a “rich” subject (0.62, 0.38 and 0.49, respectively, with Kruskal-Walli test, p < 0.05); and (iv) for interactions between “poor” and “rich” subjects, the average cooperation levels of “poor” and “rich” subjects are 0.44 and 0.54, respectively, with no significance in the difference (WilcoxonMann-Whitney test, p = 0.2). In other words, if the opponents’ wealth information is hidden, not only the total cooperation level of the “rich” subjects is significantly higher than that of the “poor” subjects, but also the cooperation level in interactions between a pair of “rich” subjects is significantly higher than that pertinent to interactions between a pair of “poor” subjects. On its turn, this implies that when the opponents’ wealth information is hidden, the cooperation level of individuals is sensitive to their initial wealth status. Finally, one can see that the average final wealth level of all subjects in T1 (181.4) is significantly lower than that of subjects in T2 (262.3, T test, p < 0.05, see Fig. 3a). More specifically, (i) the average final wealth level of “poor” subjects in T1 (156.8) is significantly lower than that of “rich” subjects (203.7, T test, p < 0.05), and the normalized difference between such two wealth levels is dT1 = (203.7–156.8)/(156.8 + 203.7) = 0.13 (Fig. 3b); (ii) the average final wealth level of “poor” subjects (239.2) is also significantly lower than that of “rich” subjects (285.4) in T2 (T test, p < 0.05 with a normalized difference dT2 = (285.4–
239.2)/(239.2 + 285.4) = 0.09. (Fig. 3b); and (iii) both average final wealth levels of “poor” and “rich” subjects in T1 are significantly lower than those in T2 (T test, p < 0.05) (Fig. 3b). These results imply that the invisibility of opponents’ wealth information not only promotes the average final wealth level of single individuals, but also contributes to the elimination of initial differences in wealth levels among individuals.
3. Discussion The PD game is one of the most important theoretical tools for revealing the evolutionary mechanisms of cooperation [10–14]. We studied experimentally the role of inequalities in wealth statuses and visibility of wealth information in determining the individual behavior of players conducting repeated PD games with costly punishment. Although the role of wealth inequalities has been considered in various circumstances [7], two recent studies revealed that visibility of wealth information may be an even more important factor affecting the cooperation level of individuals [15,16]. Specifically, Nishi et al. showed that in a networked public goods game, wealth visibility facilitates the downstream consequences of initial inequalities and leads to inequality statuses more differentiated than those occurring when wealth is invisible [15]. On the other hand, Hauser et al. showed that, when wealth information is visible in a public goods game, individuals tend to punish the rich and reward the poor [16].
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M. Zhang, Y. Ma and Y. Tao et al. / Chaos, Solitons and Fractals 130 (2020) 109398
Fig. 3. The average final wealth levels in treatments T1 and T2. Panel (a): the average final wealth levels of all subjects in T1 (red) and T2 (blue), respectively. Panel (b): the average final wealth levels of “poor” and “rich” subjects in T1 and T2, respectively.
At variance, our experiments reveal that increased transparency of opponents’ wealth information is not conducive to the evolution of cooperation if the initial wealth status is unequal, and when the opponents’ wealth information is visible the individuals’ behavior seems to be largely independent on their initial wealth status. Conversely, when the opponents’ wealth information is hidden, individual behavior seems very sensitive to their initial wealth status. The cooperation level of the system strongly relies on the visibility of opponent’s wealth information, that is, the average frequency of displaying cooperation in the case of visible opponents’ wealth information is significantly lower than that in the case of hidden opponent’s wealth information. Finally, if the initial wealth status is unequal, the invisibility of opponents’ wealth information is not only conducive to promote the average final wealth level of each individual but also to eliminate differences in wealth levels among individuals. That is to say that the average final wealth level in the case of visible opponents’ wealth information is significantly lower than that in the case of hidden opponents’ wealth information, and the normalized difference of final wealth level in the case of visible opponents’ wealth information is larger than that in the case of hidden opponents’ wealth information. Furthermore, the effects of both inequality of initial wealth status and visibility of opponents’ wealth information on using the option of punishment are not significant in our experiment, at variance with the conclusions offered by Hauser et al. [16]. Additionally, experiments here reported correspond to selecting a dilemma strength equal to 1 [17]. We in fact conducted the same experiments with different payoff matrices (for which the dilemma strength is equal to 2) in Yunnan, China, and we obtained qualitatively similar results in both predominantly cooperative and defecting environments. In theoretical analysis, the Prisoner’s Dilemma is one extreme of a general public goods game [18]. Our finding, therefore, can make sense to prove that the Prisoner’s Dilemma and the public goods game are consistent in essence. Neurological and psychological mechanisms may shed light on the understanding of the observed behavior [19–22]. Equality allows people to act with a relatively balanced mentality, and the outcome of the last round of the game is basically agreeing with what they expect, and this leaves therefore a marginal role to hidden or visible information about the opponent’s wealth statuses. On the opposite, when the initial wealth is uneven, the visibility of wealth information re-
markably hampers cooperation. The unequal distribution of wealth has a great effect on people’s psychology [19–22]. From Fig. 2, we can see at times the inequalities are cleverly exploited by the “rich” subjects, which further leads to the rising gap between the rich and the poor. The wealth gap, social inequality and other issues have to be settled urgently, especially because the disclosure of information becomes more and more detailed and clear in our new transparent age (with the development of both science and society). Hence, it makes a difference to integrate these two factors together, which enlightens us to combine inequality with visibility to explore what drives the gap. Further research opportunities related to the inequality and visibility remain to be explored, such as the scenario under completely unequal state, the effect of varying unequal extents, and the visibility connected to more realistic situation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful to the support from the NSFC-Yunnan United fund [grant number U1302267]; the National Science Fund for Distinguished Young Scholars [grant number 31325005]; and the National Natural Science Foundation of China grants [grant number 31270433]. Author contributions R.-W.W. and L.S. designed the experiment. M.Z. conducted the experiment. M.Z and Y.J.M. performed the statistical analyses. All authors discussed the results. M.Z. wrote the manuscript. Author Information The authors declare no competing interests. References [1] Maynard-Smith J, Szathmary E. The major transition in evolution. Oxford: Univ. Press; 1998.
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