- Email: [email protected]
The evolution of fairness in the coevolutionary ultimatum games
Chaos, Solitons & Fractals 56 (2013) 13–18
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
The evolution of fairness in the coevolutionary ultimatum games Kohei Miyaji a, Zhen Wang b,c,⇑, Jun Tanimoto a, Aya Hagishima a, Satoshi Kokubo a a
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka, Japan Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong c Center for Nonlinear Studies, The Beijing–Hong Kong–Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong b
a r t i c l e
i n f o
Article history: Available online 4 July 2013
a b s t r a c t In the ultimatum games, two players are required to split a certain sum of money. Through the responder accepting the offer of proposer, the money will be shared and the fairness is built finally. Here, we figure out five coevolutionary protocols, where strategy (offering rate p and threshold for accepting an offer q) and underlying network topology can coevolve, to demonstrate how the link severing scenario affects the evolution of fairness. We show that the equilibrium of the games is significantly influenced by these coevolutionary protocols. The deterministic rules lead to overly lavish or overly generous result that is inconsistent with the outcome of human behavior experiment. However, the probabilistic rules produce fair division, similar to the realistic case. Moreover, we also introduce an amplitude parameter b to verify the plausibility of assumed link severing protocols. By means of enhancing b we analytically exhibit that preferable performance can be obtained in the game, since the total amount of agents increases as well. Last, we further support our conclusion by showing the so-called unrealistic severing events under these coevolution scenarios. We thus present a viable way of understanding the ubiquitous fairness in nature and hope that it will inspire further studies to resolve social division. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Since the seminal introduction by Güth et al. [1], the ultimatum game has become a powerful tool for studying the social questions and human behaviors [2–6]. In the basic version of ultimatum game, two players are required to split a sum of money. One of them is designated to the role of proposer, the other acts as the responder. The proposer suggests how to split the resource, and then the responder can accept or reject the offer. No haggling is allowed. If the proposer’s offer p reaches the responder’s acceptance threshold q, the agreement goes ahead, otherwise both re⇑ Corresponding author at: Hong Kong Baptist University, Department of Physics, Kowloon Tong, Hong Kong. Tel.: +852 34115688. E-mail addresses: [email protected] (Z. Wang), tanimoto@cm. kyushu-u.ac.jp (J. Tanimoto). 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.007
main empty handed. The most important problem in their deal is the emergence of fairness, which has fascinated extensive interest from both experimental and theoretical studies [7–12]. In the case of well-mixed population, a proposer always claims that a majority of the sum, and the responser should accept the smallest offer as otherwise he would get nothing. This scenario is the Nash Equilibrium (NE) of the game, which can be predicted based on the theoretical assumption that each player is fully rational and focused only on maximizing his own utility. Interestingly, however, a large number of observations and experiment proofs, ranging from different countries to various age structures, report that people do not behave in such a rational way. It is found that a majority of the proposers offer 40%–50% of the total sum, and about half of all responders reject offers below 30% [1,6,5,7–9]. It thus turns out that fair division is
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ubiquitous in real human society. Now, the questions become why fairness can appear and how fairness evolves in a population of selfish players. Similarly to earlier investigations of social dilemmas [13–26], several mechanisms have been identified which can provide a reasonable explanation for fair share and lead to the evolution of fairness. Examples include the role of reputation [3], effective links to costly punishment [11] or empathy [27,28], the framework of adaptive dynamics [29], heterogeneous distribution [30,31], and of course spatially structured populations [2,32–36]. Most notably, if players are arranged on the vertices of a network and interact only with their neighbors, then much fairer outcomes evolve [2]. Along this pioneering line of research studies on the evolution of fairness have received a substantial boost. For instance, the further analysis has validated that fair share is robust against different types of players, spatial topologies and updating rules [33]. The increase of neighborhood also allows for individuals to pursue fair behavior even if degree of disorder is large [34]. Moreover, Szolnoki et al. show that the defense mechanisms of empathetic agents can provide beneficial conditions for the formation of fairness [37]. However, apart from the exploration of single mechanism, the coevolution scenario of ultimatum game is receiving more and more considerable attention recently [38–41], since the conjunct adaptation of both strategy and spatial topology is much closer to the genuine situations in society. We can look at some examples more specifically, in a recent work [40], where players are allowed to either adjust their strategy or switch adverse partnership, a suitable partner rewiring frequency leading to fairer offers is reported. In [38], it is shown that the difference between the ratio of structure updating and that of strategy updating can affect the emergence of fair behavior. Inspired by these successful research efforts, an interesting question poses itself, which we aim to address in what follows. If we consider more coevolution scenarios where network adaptation is fully dominated by parameters p and q, are all these always beneficial for the evolution of fairness? Here we construct five coevolution rules to explore how the evolution of fairness is affected. Under these rules, players will rewire their neighbors or update their strategies. By means of systematic simulations we demonstrate, compared with the case of alone network reciprocity, that the final equilibrium changes significantly. We provide an interpretation of the observed phenomena via examining some special evolution events. In the remainder of this paper we will first detail the coevolution models, subsequently we will present our main results, and finally summarize our conclusion. 2. The coevolution model We consider the evolutionary ultimatum game with N players located on the vertices of a network. The game is iterated forward in accordance with the following procedures. A randomly chosen player i obtains its accumulated payoffs by playing the game with all its ki neighbors, where ki is the degree of player i. Subsequently, we evaluate in the same way the payoffs of all the neighbors of player i. Last,
player i decides whether to adapt its network or to update its strategy: player i rewires the link with a randomly selected neighbor j according to a probability w, otherwise he will update his strategy via the comparison of their payoffs (for details see the following protocols). Evidently, for w = 0 the classical spatial reciprocity is recovered where only strategy updating is considered. For w = 1, the strategy adaption is eliminated, and the strategy distribution will be frozen. In this work, we focus on w to the interval [0,1). Irrespective of the value of w one full simulation step involves all players having a chance either to rewire its neighbor or to adopt a strategy from one of their neighbors once on average. 2.1. Extended ultimatum game For simplicity, but without loss of generality, we set the sum which is divided by the two players to be one, and consider a strategy denoted by two parameters p and q in the unit interval. Before the game each agent i is initially assigned a strategy Si = (pi ; qi ). If the player i acts as the proposer, the parameter pi denotes the amount offered to other player. While he is in the role of the responder, qi denotes the minimum acceptance level (or aspiration level). Then player i will have an interaction with his neighbor j (whose strategy is Sj = (pj ; qj )) and can obtain the expected payoff as follows,
pij
8 1 pi þ bpj ; > > > <1 p ; i ¼ > bpj ; > > : 0;
if pi P qj and pj P qi ; if pi P qj and pj < qi ; if pi < qj and pj P qi ; if pi < qj and pj < qi :
ð1Þ
where the parameter b is a multiplied amplification factor. Evidently, for b ¼ 1 the classical situation is recovered where player i can get the formal share from his neighbor j. For b > 1, however, the offer from his neighbor can be enhanced. In all our following studies, we simply fix the value of b to be b = 1 or b = 2. Along the same way, when player i carries out the interaction with all his ki neighbors, he is P able to obtain the overall payoff Pi ¼ ki pij . 2.2. Network adaptation During the process of network adaptation, an edge, i–j link, which connects the focal player i and his neighbor j is randomly chosen. Then, either player i or his neighbor j can sever the link according to one of the following proposals, which depend on both players’ performance. When an agent severs the link, he must re-construct a new connection with a randomly selected player i (the duplicate links and self-loops are not allowed). Specially, the case that agents are isolated is inevitable, but we confirm that this occurs rarely and does not affect the final results. Now we define the five protocols as follows; (The first two rules are deterministic, the others are probabilistic.) I. Responder rule: responder determines whether to sever or maintain a link, which is similar with the proposal [40].
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– If pi P qj and pj P qi , the i–j link is maintained; – If pi P qj and pj < qi , player i severs the link with j and build a new connection with a randomly chosen individual i; – If pj P qi and pi < qj , individual j dismiss the partnership with i and then randomly picks up an agent i from the rest of population as its new partner; – If pi < pj and pj < qi , the i–j link is broken and either of them has the chance to connects to a new player i. II. Proposer rule: proposer determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If pi P qj and pj < qi , individual j severs the link with i and build a new connection with a randomly chosen individual i; – If pj P qi and pi < qj , player i dismiss the partnership with j and then randomly picks up an agent i from the rest of population as its new partner; – If pi < pj and pj < qi , the i–j link is broken and either of them has the chance to connects to a new player i. III. Offer rule: offer proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, pi =ðpi þ pj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner. IV. Generous rule: generous proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, ðpi qi Þ=ðpi qi þ pj qj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner.
to update probability,
wðSj !Si Þ ¼
his
strategy
Pj Pi ; ð1 þ bÞmaxðki ; kj Þ
in
accordance
with
the
ð2Þ
where ki and kj are the degrees of i and j respectively. However, if Pi > Pj , player i keeps his strategy constant. Moreover, when the strategy updating event succeeds, some small noise dp and dq which are regarded as the mutation of reproduction are introduced into the new strategy of agent i. It becomes Si = (pi þ dp ; qi þ dq ), where dp and dq are two independent random numbers from the interval [d; þd]. 2.4. Simulation setting In our model, we consider N = 1600 individuals associated with the nodes of a graph. The initial graph topology starts from an Erdös–Rényi (ER) network with the average degree hki = 12, whose degree distribution decays exponentially for large k [42]. At the beginning of each simulation episode, individual strategy S = (p; q) denoted by two random numbers is uniformly distributed in the interval [0, 1]. Last, we fix the mutation noise d = 0.005. Results of computer simulations presented below were determined within 106 full time steps after sufficiently long transients were discarded. Moreover, since the adaption of network topology may introduce additional disturbances, final results were averaged over up to 100 independent runs for each set of parameter values in order to assure suitable accuracy. 3. Results First we investigate when taking a coevolution protocol into account, is it beneficial for the evolution of fairness or not? Fig. 1 illustrates a typical evolution course of average offer and acceptance threshold of Responder rule for a low value of w. It shows clearly that in the early stages of the evolution process, average values of both offer and acceptance threshold undergo a fast decrease. Then, with the game proceeding forward, offer still needs sufficient long time to reach the stationary status. In fact, the sufficient
V. Incentive rule: incentive proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, ð2pi qi pj Þ=ðpi þ pj qi qj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner.
2.3. Strategy adaptation When the strategy adaptation occurs, player i can randomly choose one neighbor j and decide whether
Fig. 1. Time course depicting the evolution of the average offer p and acceptance threshold q for the Responder rule. Note compared with [40], we need sufficient long time to reach the equilibrium state. (The evolution has reached steady state, but sufficient long relaxation time makes it not enough explicit.) Depicted results are obtained for w = 0.1 and b ¼ 1:0.
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long relaxing time is caused by the utilized rule, where the connection tone between edges is mainly controlled by the proposer. Interestingly, though the final equilibrium status is somewhat inferior to the statistical average of economical experiments [1,6,5,7–9], the fair division is remarkably promoted, which outperforms the rational action and is beneficial for the emergence of bargain behavior in society. We argue that such a particular coevolution proposal plays a significant role in the emergence of fair traits and their sustenance, which eventually halts the performance of no division. Next, we will systematically study how the fairness evolves under other protocols, especially compared with the effect of simple spatial population. Fig. 2 illustrates average p and q of equilibrium status obtained on five coevolution rules for different values of w and b. Evidently, it can be observed, except for the case of Proposer rule, that other coevolution protocols obviously result in more divisions with the increment of w, which goes beyond what can be warranted by the spatial reciprocity alone [2,32– 36]. Because more frequent adaptation of network under our scenarios allows players to avoid unfair treatment through richly making use of the game exit option [43], which is similar to the observation in the study of various dilemma games [44,45]. Moreover, we can also see in despite of the fact that enhanced b enables the observation of a little flourishing fairness, these scenarios are universally effective in promoting the evolution of fairness for different values of b. In what follows we will detailedly examine the intrinsic reasons for these phenomena. It is visible, compared with other scenarios, that the Responder rule shows the larger values of p and q at equilibrium. When w approaches to 1.0, the monotonously increasing values of p and q even exceed 0.9 and 0.4, which is remarkably different from the view of an optimal p [40].
In the Responder rule, a player who does not accept the offer always possess the right of severing the link. Along with this way, when a proposer provides a reasonable offer, he is still dismissed by the responder whose aspiration level is higher than the offer. And this interaction occurs deterministically. Consequently, each proposer tends to offer more against losing his neighbor (i.e., decreasing his degree). Only when his neighbor is maintained, he has the possibility of retaining higher payoff. Naturally, it is simple to understand why agents become more generous with enhancing w. On the contrary, the Proposer rule leads to the smallest p and q values. In particular, q ! 0 for the case of w P 0:1. In this scenario, the proposer holds the right of determining whether to adjust the initial connection. When a generous responder who requires a small aspiration level meets a stingy proposer, he is confronted with the risk of being dismissed. Because the proposer only provides a tiny offer that is even smaller than the acceptance threshold of the proposer. This interaction is deterministic as well. In order to avoid losing his neighbor the responder is inclined to reduce his own acceptance threshold. Hence the fairness disappears, which is caused by the excessive generosity of proposer. And this effect can be further consolidated with increasing w. Due to the fact that there is no need for offering too much, the offer level of responder for the next round spontaneously decreases. We have inspected the evolution case of two deterministic rules, both of which product unrealistic outcomes when referring to experimental results [2,32–36], namely, either overly lavish (large p) or overly generous (small q). In the following, we will focus on the other rules based on probabilistic rather than deterministic concepts, which seem more likely to imitate the behavior in real human society.
Fig. 2. Average p and q in dependence on w for different coevolution rules, where the bars denote the standard deviation. The top panel is the result for b = 1.0, while the bottom panel is the result for b = 2.0.
K. Miyaji et al. / Chaos, Solitons & Fractals 56 (2013) 13–18
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Fig. 3. Average p versus average q at equilibrium for different w. For each coevolution rule, note that the larger symbol corresponds to larger value of w. The background color denotes the average payoff of per link of one node at equilibrium, which is calculated through applying mean-field approximation.
Fig. 4. The number of unrealistic severing link (i.e., a proposer offering p P 0:5 is dismissed) for each coevolution rule. Depicted results are obtained for b ¼ 1:0. Moreover, open circles and small bars mean average p at equilibrium in case of w ¼ 0:95 for b ¼ 1 and 2.
From Fig. 2, we can clearly see, compared with the case of deterministic rules, that the average offer p undergoes a non-monotonous change under such three coevolution protocols. This observation, to large extent, is caused by the intrinsic uncertainty of these coevolution rules. For the Offer rule, the right of breaking a connection to proposer or responder fully depends on the ratio of their offers. Hence larger p enables the proposer more likely to obtain the right of severing a link. This further decreases the likelihood of loosing his neighbor and allow him to retain higher payoff. While for the Generous rule the situation becomes somewhat complicated: the right of severing a link given to proposer or responder entirely relies on the generosity defined by p q. In this sense, a generous agent who offers more as well as accepts less profit is unlikely to be severed, which eventually results in its aspiration level lower than that of Offer rule. Now we focus on Incentive rule, where the generosity difference is considered. As mentioned above individual generosity is measured by pi qi , and his opponent’s generosity can be evaluated by pj pi . Combining two terms, we get the generosity difference 2pi qi pj . Obviously, increasing pi is beneficial for protecting individual connection, which causes the final offer larger than that of Offer rule. It is also instructive to analyze the observation that larger b is beneficial for the enhancement of p and q, as shown
in Fig. 2 for b = 2. In fact, only when the social total payoff is profitable, players are likely to choose larger p and q. In order to approve our argument, we apply mean-field approximation approach to calculate the social payoff Ptotal . When b = 1 (i.e., the traditional case), we can obtain the following result,
Ptotal ¼ hkiNð1 p þ bpÞ ¼ hkiN ¼ Const:
ð3Þ
where hki denotes the average degree of the network, N is the overall number of nodes on the graph, and p represents the average offer of all agents at equilibrium. It is notable that the social payoff becomes irrelevant to the average offer, thus p ¼ 0 is the Nash Equilibrium (NE) [46]. However, when b > 1 is taken into account, the social payoff Ptotal starts to depend on this value, namely,
Ptotal ¼ hkiNð1 p þ bpÞ ¼ hkiN½1 ð1 bÞp:
ð4Þ
In such a situation, when each agent offers more, the social payoff can be maximized [47]. Quite interestingly, based on these mathematical analysis, we validate our argument that more profit (here it corresponds to higher b) stimulates larger p and q. Moreover, to confirm our conclusion, we also calculate the average payoff of the agents at equilibrium. Results presented in Fig. 3 clearly attest to the fact that larger value of b
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promotes the enhancement of individual profit, which is further beneficial for increment of p and q. An important remaining question concerns how the case of severing link develops under these coevolution protocols. In order to quantify the observation, we define an unrealistic event in which a proposer offering p P 0:5 is dismissed. Fig. 4 features the number of unrealistic severing link for five scenarios. It is obvious that two deterministic rules (Responder rule and Proposer rule) lead to higher number of unrealistic events, implying that individual offer is dominated by the unrealistic events. However, with the consideration of probabilistic factor, the situation changes: the number of unrealistic severing link decreases notably. This is because under these rules the probability of severing connection is jointly determined by both players, which is not like the unilateral case in deterministic rules. Naturally, the values of p are not overly high or excessively low, which is closer to the result of human behavior experiment [1,6,5,7–9]. 4. Summary To sum, we have studied the evolution of fairness in the extended ultimatum games, where the coevolution of strategy and the underlying network is considered. Under five proposed rules, we have found that the equilibrium of the games changes notably. Two deterministic rules where either the proposer or the responder has the certain right of severing the connection lead to overly high or low benefit, which is not consistent with the result of human behavior experiment [1,6,5,7–9]. However, when the probabilistic rules are introduced, the quantitatively similar description with realistic survey can be guaranteed, which, to large extent, attributes to the uncertainty within network adaptation. Interestingly, the effect of these coevolution scenarios on fairness is further supported by exploring the case of unrealistic severing link. In addition, our conclusion is robust with different values of parameter b, and larger b enables agents to get more benefit. Evidently, these coevolution scenarios seem widely applicable as well justifiable with realistic examples. For example, when people deal with contract in economic society, they are more likely to choose either different strategy or potential opponent according to specifically rules. Under some deterministic cases, they can obtain extremely high or low gain. However, with the consideration of uncertain factor, they can in general get the fair division. Since this work appears very reasonable, we hope that it will inspire further studies, especially in terms of the combination with human behavior. Acknowledgments This study was partially supported by a Grant-in-Aid for Scientific Research by JSPS awarded to Prof. Tanimoto
(#23651156), and by the Kurata–Hitachi Foundation and the Hayao Nakayama Foundation. We would like to express our gratitude to these funding sources. References [1] Güth W, Schmittberger R, Schware B. J Econ Behav Oragniz 1982;3:367. [2] Page KM, Nowak MA, Sigmund K. Proc R Soc Lond B 2000;267:2177. [3] Nowak MA, Page KM, Sigmund K. Science 2000;289:1773. [4] Roth A, Prasnikar V, Okuno-Fujiwara M, Zamir S. Am Econ Rev 1991;81:1068. [5] Roth A. Bargaining experiments in handbook of experimental economics. Princeton University Press; 1995. [6] Bolton GE, Zwick R. Games Econ Behav 1995;10:95. [7] Güth W, Tietze R. J Econ Psychol 1990;11:417. [8] Thaler RH. J Econ Perspect 1988;2:195. [9] Fehr E, Schmidt K. Q J Econ 1999;114:817. [10] Sigmund K, Nowak MA. Sci Am 2002;286:82. [11] Li X, Cao L. Phys Rev E 2009;80:066101. [12] Iranzo J, Floı´a LM, Moreno Y, Sánchez A. PLoS ONE 2012;7:e43781. [13] Perc M, Szolnoki A. BioSystems 2010;99:109. [14] Szolnoki A, Wang Z, Perc M. Sci Rep 2012;2:576. [15] Perc M, Szolnoki A. Phys Rev E 2008;77:011904. [16] Vainstein MH, Arenzon JJ. Phys Rev E 2001;64:051905. [17] Wang L, Wang Z, Zhang Y, Xiang L. Sci Rep 2013;3:1468. [18] Szolnoki A, Perc M. New J Phys 2012;14:093016. [19] Wang Z, Szolnoki A, Perc M. Sci Rep 2013;3:1183. [20] Xia CY, Wang J, Wang L, Sun SW, Sun JQ, Wang JS. Chaos Solitons Fract 2012;45:1239. [21] Wang Z, Szolnoki A, Perc M. Sci Rep 2012;2:369. [22] Xia CY, Ma ZQ, Wang Z, Wang J. Physica A 2012;391:6440. [23] Xia CY, Zhao J, Wang J, Wang YL, Zhang H. Phys Scr 2011;84:025802. [24] Wang Z, Wang L, Yin ZY, Xia CY. PLoS ONE 2012;7:e40218. [25] Perc M, Gómez-Garden´es J, Szolnoki A, Florı´a LM, Moreno Y, Soc JR. Interface 2013;10:20120997. [26] Jin Q, Wang Z, Wang Z, Wang YL. Chaos Solitons Fract 2012;45:395. [27] Page KM, Nowak MA. Bull Math Biol 2002;64:1101. [28] Sánchez A, Cuesta JA. J Theor Biol 2005;235:133. [29] Duan W-Q, Stanley HE. Phys Rev E 2010;81:026104. [30] Da Silva R, Kellerman GA. Braz J Phys 2007;37:1206. [31] Da Silva R, Kellerman GA, Lamb LC. J Theor Biol 2009;258:208. [32] Killingback T, Studer E. Proc R Soc London B 2001;268:1797. [33] Iranzo J, Román J, Sánchez A. J Theor Biol 2011;278:1. [34] Sinatra R, Iranzo J, Gómez-Gardeñes J, Florı´a LM, Latora V, Moreno Y. J Stat Mech 2009;9:P09012. [35] Szolnoki A, Perc M, Szabó G. EPL 2012;100:28005. [36] Lieberman E, Hauert C, Nowak MA. Nature 2005;433:312. [37] Szolnoki A, Perc M, Szabó G. Phys Rev Lett 2012;109:078701. [38] Deng L-L, Tang W-S, Zhang J-X. Chin Phys Lett 2011;28:080204. [39] Deng L-L, Tang W-S, Zhang J-X. Physica A 2011;390:4227. [40] Gao J, Li Z, Wu T, Wang L L. EPL 2011;93:48002. [41] de Jong S, Uyttendaele S, Tuyls K. J Artif Intell Res 2008;33:551. [42] Ballobas B. Random graphs. London: Academic Press; 1985. [43] Schuessler R. J Conf Resolut 1989;33:728. [44] Tanimoto J. Int J Mod Phys C 2011;22:401. [45] Santos FC, Pacheco JM, Lenaerts T. PLoS Comput Biol 2006;2:1284. [46] We have known that the Nash Equilibrium (NE) of the traditional ultimatum games is p ¼ q ¼ 0 (i.e., S ¼ ð0; 0Þ). In economics, any arbitral strategy with p P q can be the Equal Pareto Optimum (EPO), which contains S ¼ ð0; 0Þ. Thus, the Nash Equilibrium (NE) is comprised by the Equal Pareto Optimum (EPO) and the game has no dilemma. [47] When b > 1, the Nash Equilibrium (NE) is consistent with the Equal Pareto Optimum (EPO) that becomes p ¼ 1. Hence, the dilemma appears.
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
The evolution of fairness in the coevolutionary ultimatum games Kohei Miyaji a, Zhen Wang b,c,⇑, Jun Tanimoto a, Aya Hagishima a, Satoshi Kokubo a a
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka, Japan Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong c Center for Nonlinear Studies, The Beijing–Hong Kong–Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong b
a r t i c l e
i n f o
Article history: Available online 4 July 2013
a b s t r a c t In the ultimatum games, two players are required to split a certain sum of money. Through the responder accepting the offer of proposer, the money will be shared and the fairness is built finally. Here, we figure out five coevolutionary protocols, where strategy (offering rate p and threshold for accepting an offer q) and underlying network topology can coevolve, to demonstrate how the link severing scenario affects the evolution of fairness. We show that the equilibrium of the games is significantly influenced by these coevolutionary protocols. The deterministic rules lead to overly lavish or overly generous result that is inconsistent with the outcome of human behavior experiment. However, the probabilistic rules produce fair division, similar to the realistic case. Moreover, we also introduce an amplitude parameter b to verify the plausibility of assumed link severing protocols. By means of enhancing b we analytically exhibit that preferable performance can be obtained in the game, since the total amount of agents increases as well. Last, we further support our conclusion by showing the so-called unrealistic severing events under these coevolution scenarios. We thus present a viable way of understanding the ubiquitous fairness in nature and hope that it will inspire further studies to resolve social division. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Since the seminal introduction by Güth et al. [1], the ultimatum game has become a powerful tool for studying the social questions and human behaviors [2–6]. In the basic version of ultimatum game, two players are required to split a sum of money. One of them is designated to the role of proposer, the other acts as the responder. The proposer suggests how to split the resource, and then the responder can accept or reject the offer. No haggling is allowed. If the proposer’s offer p reaches the responder’s acceptance threshold q, the agreement goes ahead, otherwise both re⇑ Corresponding author at: Hong Kong Baptist University, Department of Physics, Kowloon Tong, Hong Kong. Tel.: +852 34115688. E-mail addresses: [email protected] (Z. Wang), tanimoto@cm. kyushu-u.ac.jp (J. Tanimoto). 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.007
main empty handed. The most important problem in their deal is the emergence of fairness, which has fascinated extensive interest from both experimental and theoretical studies [7–12]. In the case of well-mixed population, a proposer always claims that a majority of the sum, and the responser should accept the smallest offer as otherwise he would get nothing. This scenario is the Nash Equilibrium (NE) of the game, which can be predicted based on the theoretical assumption that each player is fully rational and focused only on maximizing his own utility. Interestingly, however, a large number of observations and experiment proofs, ranging from different countries to various age structures, report that people do not behave in such a rational way. It is found that a majority of the proposers offer 40%–50% of the total sum, and about half of all responders reject offers below 30% [1,6,5,7–9]. It thus turns out that fair division is
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ubiquitous in real human society. Now, the questions become why fairness can appear and how fairness evolves in a population of selfish players. Similarly to earlier investigations of social dilemmas [13–26], several mechanisms have been identified which can provide a reasonable explanation for fair share and lead to the evolution of fairness. Examples include the role of reputation [3], effective links to costly punishment [11] or empathy [27,28], the framework of adaptive dynamics [29], heterogeneous distribution [30,31], and of course spatially structured populations [2,32–36]. Most notably, if players are arranged on the vertices of a network and interact only with their neighbors, then much fairer outcomes evolve [2]. Along this pioneering line of research studies on the evolution of fairness have received a substantial boost. For instance, the further analysis has validated that fair share is robust against different types of players, spatial topologies and updating rules [33]. The increase of neighborhood also allows for individuals to pursue fair behavior even if degree of disorder is large [34]. Moreover, Szolnoki et al. show that the defense mechanisms of empathetic agents can provide beneficial conditions for the formation of fairness [37]. However, apart from the exploration of single mechanism, the coevolution scenario of ultimatum game is receiving more and more considerable attention recently [38–41], since the conjunct adaptation of both strategy and spatial topology is much closer to the genuine situations in society. We can look at some examples more specifically, in a recent work [40], where players are allowed to either adjust their strategy or switch adverse partnership, a suitable partner rewiring frequency leading to fairer offers is reported. In [38], it is shown that the difference between the ratio of structure updating and that of strategy updating can affect the emergence of fair behavior. Inspired by these successful research efforts, an interesting question poses itself, which we aim to address in what follows. If we consider more coevolution scenarios where network adaptation is fully dominated by parameters p and q, are all these always beneficial for the evolution of fairness? Here we construct five coevolution rules to explore how the evolution of fairness is affected. Under these rules, players will rewire their neighbors or update their strategies. By means of systematic simulations we demonstrate, compared with the case of alone network reciprocity, that the final equilibrium changes significantly. We provide an interpretation of the observed phenomena via examining some special evolution events. In the remainder of this paper we will first detail the coevolution models, subsequently we will present our main results, and finally summarize our conclusion. 2. The coevolution model We consider the evolutionary ultimatum game with N players located on the vertices of a network. The game is iterated forward in accordance with the following procedures. A randomly chosen player i obtains its accumulated payoffs by playing the game with all its ki neighbors, where ki is the degree of player i. Subsequently, we evaluate in the same way the payoffs of all the neighbors of player i. Last,
player i decides whether to adapt its network or to update its strategy: player i rewires the link with a randomly selected neighbor j according to a probability w, otherwise he will update his strategy via the comparison of their payoffs (for details see the following protocols). Evidently, for w = 0 the classical spatial reciprocity is recovered where only strategy updating is considered. For w = 1, the strategy adaption is eliminated, and the strategy distribution will be frozen. In this work, we focus on w to the interval [0,1). Irrespective of the value of w one full simulation step involves all players having a chance either to rewire its neighbor or to adopt a strategy from one of their neighbors once on average. 2.1. Extended ultimatum game For simplicity, but without loss of generality, we set the sum which is divided by the two players to be one, and consider a strategy denoted by two parameters p and q in the unit interval. Before the game each agent i is initially assigned a strategy Si = (pi ; qi ). If the player i acts as the proposer, the parameter pi denotes the amount offered to other player. While he is in the role of the responder, qi denotes the minimum acceptance level (or aspiration level). Then player i will have an interaction with his neighbor j (whose strategy is Sj = (pj ; qj )) and can obtain the expected payoff as follows,
pij
8 1 pi þ bpj ; > > > <1 p ; i ¼ > bpj ; > > : 0;
if pi P qj and pj P qi ; if pi P qj and pj < qi ; if pi < qj and pj P qi ; if pi < qj and pj < qi :
ð1Þ
where the parameter b is a multiplied amplification factor. Evidently, for b ¼ 1 the classical situation is recovered where player i can get the formal share from his neighbor j. For b > 1, however, the offer from his neighbor can be enhanced. In all our following studies, we simply fix the value of b to be b = 1 or b = 2. Along the same way, when player i carries out the interaction with all his ki neighbors, he is P able to obtain the overall payoff Pi ¼ ki pij . 2.2. Network adaptation During the process of network adaptation, an edge, i–j link, which connects the focal player i and his neighbor j is randomly chosen. Then, either player i or his neighbor j can sever the link according to one of the following proposals, which depend on both players’ performance. When an agent severs the link, he must re-construct a new connection with a randomly selected player i (the duplicate links and self-loops are not allowed). Specially, the case that agents are isolated is inevitable, but we confirm that this occurs rarely and does not affect the final results. Now we define the five protocols as follows; (The first two rules are deterministic, the others are probabilistic.) I. Responder rule: responder determines whether to sever or maintain a link, which is similar with the proposal [40].
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– If pi P qj and pj P qi , the i–j link is maintained; – If pi P qj and pj < qi , player i severs the link with j and build a new connection with a randomly chosen individual i; – If pj P qi and pi < qj , individual j dismiss the partnership with i and then randomly picks up an agent i from the rest of population as its new partner; – If pi < pj and pj < qi , the i–j link is broken and either of them has the chance to connects to a new player i. II. Proposer rule: proposer determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If pi P qj and pj < qi , individual j severs the link with i and build a new connection with a randomly chosen individual i; – If pj P qi and pi < qj , player i dismiss the partnership with j and then randomly picks up an agent i from the rest of population as its new partner; – If pi < pj and pj < qi , the i–j link is broken and either of them has the chance to connects to a new player i. III. Offer rule: offer proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, pi =ðpi þ pj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner. IV. Generous rule: generous proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, ðpi qi Þ=ðpi qi þ pj qj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner.
to update probability,
wðSj !Si Þ ¼
his
strategy
Pj Pi ; ð1 þ bÞmaxðki ; kj Þ
in
accordance
with
the
ð2Þ
where ki and kj are the degrees of i and j respectively. However, if Pi > Pj , player i keeps his strategy constant. Moreover, when the strategy updating event succeeds, some small noise dp and dq which are regarded as the mutation of reproduction are introduced into the new strategy of agent i. It becomes Si = (pi þ dp ; qi þ dq ), where dp and dq are two independent random numbers from the interval [d; þd]. 2.4. Simulation setting In our model, we consider N = 1600 individuals associated with the nodes of a graph. The initial graph topology starts from an Erdös–Rényi (ER) network with the average degree hki = 12, whose degree distribution decays exponentially for large k [42]. At the beginning of each simulation episode, individual strategy S = (p; q) denoted by two random numbers is uniformly distributed in the interval [0, 1]. Last, we fix the mutation noise d = 0.005. Results of computer simulations presented below were determined within 106 full time steps after sufficiently long transients were discarded. Moreover, since the adaption of network topology may introduce additional disturbances, final results were averaged over up to 100 independent runs for each set of parameter values in order to assure suitable accuracy. 3. Results First we investigate when taking a coevolution protocol into account, is it beneficial for the evolution of fairness or not? Fig. 1 illustrates a typical evolution course of average offer and acceptance threshold of Responder rule for a low value of w. It shows clearly that in the early stages of the evolution process, average values of both offer and acceptance threshold undergo a fast decrease. Then, with the game proceeding forward, offer still needs sufficient long time to reach the stationary status. In fact, the sufficient
V. Incentive rule: incentive proportion determines whether to sever or maintain a link. – If pi P qj and pj P qi , the i–j link is maintained; – If the case is other than pi P qj and pj P qi , player i severs the link with j according to the probability, ð2pi qi pj Þ=ðpi þ pj qi qj Þ, otherwise individual j severs the link with i. The agent who dismiss the partnership will connect with an agent i from the rest of population as its new partner.
2.3. Strategy adaptation When the strategy adaptation occurs, player i can randomly choose one neighbor j and decide whether
Fig. 1. Time course depicting the evolution of the average offer p and acceptance threshold q for the Responder rule. Note compared with [40], we need sufficient long time to reach the equilibrium state. (The evolution has reached steady state, but sufficient long relaxation time makes it not enough explicit.) Depicted results are obtained for w = 0.1 and b ¼ 1:0.
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long relaxing time is caused by the utilized rule, where the connection tone between edges is mainly controlled by the proposer. Interestingly, though the final equilibrium status is somewhat inferior to the statistical average of economical experiments [1,6,5,7–9], the fair division is remarkably promoted, which outperforms the rational action and is beneficial for the emergence of bargain behavior in society. We argue that such a particular coevolution proposal plays a significant role in the emergence of fair traits and their sustenance, which eventually halts the performance of no division. Next, we will systematically study how the fairness evolves under other protocols, especially compared with the effect of simple spatial population. Fig. 2 illustrates average p and q of equilibrium status obtained on five coevolution rules for different values of w and b. Evidently, it can be observed, except for the case of Proposer rule, that other coevolution protocols obviously result in more divisions with the increment of w, which goes beyond what can be warranted by the spatial reciprocity alone [2,32– 36]. Because more frequent adaptation of network under our scenarios allows players to avoid unfair treatment through richly making use of the game exit option [43], which is similar to the observation in the study of various dilemma games [44,45]. Moreover, we can also see in despite of the fact that enhanced b enables the observation of a little flourishing fairness, these scenarios are universally effective in promoting the evolution of fairness for different values of b. In what follows we will detailedly examine the intrinsic reasons for these phenomena. It is visible, compared with other scenarios, that the Responder rule shows the larger values of p and q at equilibrium. When w approaches to 1.0, the monotonously increasing values of p and q even exceed 0.9 and 0.4, which is remarkably different from the view of an optimal p [40].
In the Responder rule, a player who does not accept the offer always possess the right of severing the link. Along with this way, when a proposer provides a reasonable offer, he is still dismissed by the responder whose aspiration level is higher than the offer. And this interaction occurs deterministically. Consequently, each proposer tends to offer more against losing his neighbor (i.e., decreasing his degree). Only when his neighbor is maintained, he has the possibility of retaining higher payoff. Naturally, it is simple to understand why agents become more generous with enhancing w. On the contrary, the Proposer rule leads to the smallest p and q values. In particular, q ! 0 for the case of w P 0:1. In this scenario, the proposer holds the right of determining whether to adjust the initial connection. When a generous responder who requires a small aspiration level meets a stingy proposer, he is confronted with the risk of being dismissed. Because the proposer only provides a tiny offer that is even smaller than the acceptance threshold of the proposer. This interaction is deterministic as well. In order to avoid losing his neighbor the responder is inclined to reduce his own acceptance threshold. Hence the fairness disappears, which is caused by the excessive generosity of proposer. And this effect can be further consolidated with increasing w. Due to the fact that there is no need for offering too much, the offer level of responder for the next round spontaneously decreases. We have inspected the evolution case of two deterministic rules, both of which product unrealistic outcomes when referring to experimental results [2,32–36], namely, either overly lavish (large p) or overly generous (small q). In the following, we will focus on the other rules based on probabilistic rather than deterministic concepts, which seem more likely to imitate the behavior in real human society.
Fig. 2. Average p and q in dependence on w for different coevolution rules, where the bars denote the standard deviation. The top panel is the result for b = 1.0, while the bottom panel is the result for b = 2.0.
K. Miyaji et al. / Chaos, Solitons & Fractals 56 (2013) 13–18
17
Fig. 3. Average p versus average q at equilibrium for different w. For each coevolution rule, note that the larger symbol corresponds to larger value of w. The background color denotes the average payoff of per link of one node at equilibrium, which is calculated through applying mean-field approximation.
Fig. 4. The number of unrealistic severing link (i.e., a proposer offering p P 0:5 is dismissed) for each coevolution rule. Depicted results are obtained for b ¼ 1:0. Moreover, open circles and small bars mean average p at equilibrium in case of w ¼ 0:95 for b ¼ 1 and 2.
From Fig. 2, we can clearly see, compared with the case of deterministic rules, that the average offer p undergoes a non-monotonous change under such three coevolution protocols. This observation, to large extent, is caused by the intrinsic uncertainty of these coevolution rules. For the Offer rule, the right of breaking a connection to proposer or responder fully depends on the ratio of their offers. Hence larger p enables the proposer more likely to obtain the right of severing a link. This further decreases the likelihood of loosing his neighbor and allow him to retain higher payoff. While for the Generous rule the situation becomes somewhat complicated: the right of severing a link given to proposer or responder entirely relies on the generosity defined by p q. In this sense, a generous agent who offers more as well as accepts less profit is unlikely to be severed, which eventually results in its aspiration level lower than that of Offer rule. Now we focus on Incentive rule, where the generosity difference is considered. As mentioned above individual generosity is measured by pi qi , and his opponent’s generosity can be evaluated by pj pi . Combining two terms, we get the generosity difference 2pi qi pj . Obviously, increasing pi is beneficial for protecting individual connection, which causes the final offer larger than that of Offer rule. It is also instructive to analyze the observation that larger b is beneficial for the enhancement of p and q, as shown
in Fig. 2 for b = 2. In fact, only when the social total payoff is profitable, players are likely to choose larger p and q. In order to approve our argument, we apply mean-field approximation approach to calculate the social payoff Ptotal . When b = 1 (i.e., the traditional case), we can obtain the following result,
Ptotal ¼ hkiNð1 p þ bpÞ ¼ hkiN ¼ Const:
ð3Þ
where hki denotes the average degree of the network, N is the overall number of nodes on the graph, and p represents the average offer of all agents at equilibrium. It is notable that the social payoff becomes irrelevant to the average offer, thus p ¼ 0 is the Nash Equilibrium (NE) [46]. However, when b > 1 is taken into account, the social payoff Ptotal starts to depend on this value, namely,
Ptotal ¼ hkiNð1 p þ bpÞ ¼ hkiN½1 ð1 bÞp:
ð4Þ
In such a situation, when each agent offers more, the social payoff can be maximized [47]. Quite interestingly, based on these mathematical analysis, we validate our argument that more profit (here it corresponds to higher b) stimulates larger p and q. Moreover, to confirm our conclusion, we also calculate the average payoff of the agents at equilibrium. Results presented in Fig. 3 clearly attest to the fact that larger value of b
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promotes the enhancement of individual profit, which is further beneficial for increment of p and q. An important remaining question concerns how the case of severing link develops under these coevolution protocols. In order to quantify the observation, we define an unrealistic event in which a proposer offering p P 0:5 is dismissed. Fig. 4 features the number of unrealistic severing link for five scenarios. It is obvious that two deterministic rules (Responder rule and Proposer rule) lead to higher number of unrealistic events, implying that individual offer is dominated by the unrealistic events. However, with the consideration of probabilistic factor, the situation changes: the number of unrealistic severing link decreases notably. This is because under these rules the probability of severing connection is jointly determined by both players, which is not like the unilateral case in deterministic rules. Naturally, the values of p are not overly high or excessively low, which is closer to the result of human behavior experiment [1,6,5,7–9]. 4. Summary To sum, we have studied the evolution of fairness in the extended ultimatum games, where the coevolution of strategy and the underlying network is considered. Under five proposed rules, we have found that the equilibrium of the games changes notably. Two deterministic rules where either the proposer or the responder has the certain right of severing the connection lead to overly high or low benefit, which is not consistent with the result of human behavior experiment [1,6,5,7–9]. However, when the probabilistic rules are introduced, the quantitatively similar description with realistic survey can be guaranteed, which, to large extent, attributes to the uncertainty within network adaptation. Interestingly, the effect of these coevolution scenarios on fairness is further supported by exploring the case of unrealistic severing link. In addition, our conclusion is robust with different values of parameter b, and larger b enables agents to get more benefit. Evidently, these coevolution scenarios seem widely applicable as well justifiable with realistic examples. For example, when people deal with contract in economic society, they are more likely to choose either different strategy or potential opponent according to specifically rules. Under some deterministic cases, they can obtain extremely high or low gain. However, with the consideration of uncertain factor, they can in general get the fair division. Since this work appears very reasonable, we hope that it will inspire further studies, especially in terms of the combination with human behavior. Acknowledgments This study was partially supported by a Grant-in-Aid for Scientific Research by JSPS awarded to Prof. Tanimoto
(#23651156), and by the Kurata–Hitachi Foundation and the Hayao Nakayama Foundation. We would like to express our gratitude to these funding sources. References [1] Güth W, Schmittberger R, Schware B. J Econ Behav Oragniz 1982;3:367. [2] Page KM, Nowak MA, Sigmund K. Proc R Soc Lond B 2000;267:2177. [3] Nowak MA, Page KM, Sigmund K. Science 2000;289:1773. [4] Roth A, Prasnikar V, Okuno-Fujiwara M, Zamir S. Am Econ Rev 1991;81:1068. [5] Roth A. Bargaining experiments in handbook of experimental economics. Princeton University Press; 1995. [6] Bolton GE, Zwick R. Games Econ Behav 1995;10:95. [7] Güth W, Tietze R. J Econ Psychol 1990;11:417. [8] Thaler RH. J Econ Perspect 1988;2:195. [9] Fehr E, Schmidt K. Q J Econ 1999;114:817. [10] Sigmund K, Nowak MA. Sci Am 2002;286:82. [11] Li X, Cao L. Phys Rev E 2009;80:066101. [12] Iranzo J, Floı´a LM, Moreno Y, Sánchez A. PLoS ONE 2012;7:e43781. [13] Perc M, Szolnoki A. BioSystems 2010;99:109. [14] Szolnoki A, Wang Z, Perc M. Sci Rep 2012;2:576. [15] Perc M, Szolnoki A. Phys Rev E 2008;77:011904. [16] Vainstein MH, Arenzon JJ. Phys Rev E 2001;64:051905. [17] Wang L, Wang Z, Zhang Y, Xiang L. Sci Rep 2013;3:1468. [18] Szolnoki A, Perc M. New J Phys 2012;14:093016. [19] Wang Z, Szolnoki A, Perc M. Sci Rep 2013;3:1183. [20] Xia CY, Wang J, Wang L, Sun SW, Sun JQ, Wang JS. Chaos Solitons Fract 2012;45:1239. [21] Wang Z, Szolnoki A, Perc M. Sci Rep 2012;2:369. [22] Xia CY, Ma ZQ, Wang Z, Wang J. Physica A 2012;391:6440. [23] Xia CY, Zhao J, Wang J, Wang YL, Zhang H. Phys Scr 2011;84:025802. [24] Wang Z, Wang L, Yin ZY, Xia CY. PLoS ONE 2012;7:e40218. [25] Perc M, Gómez-Garden´es J, Szolnoki A, Florı´a LM, Moreno Y, Soc JR. Interface 2013;10:20120997. [26] Jin Q, Wang Z, Wang Z, Wang YL. Chaos Solitons Fract 2012;45:395. [27] Page KM, Nowak MA. Bull Math Biol 2002;64:1101. [28] Sánchez A, Cuesta JA. J Theor Biol 2005;235:133. [29] Duan W-Q, Stanley HE. Phys Rev E 2010;81:026104. [30] Da Silva R, Kellerman GA. Braz J Phys 2007;37:1206. [31] Da Silva R, Kellerman GA, Lamb LC. J Theor Biol 2009;258:208. [32] Killingback T, Studer E. Proc R Soc London B 2001;268:1797. [33] Iranzo J, Román J, Sánchez A. J Theor Biol 2011;278:1. [34] Sinatra R, Iranzo J, Gómez-Gardeñes J, Florı´a LM, Latora V, Moreno Y. J Stat Mech 2009;9:P09012. [35] Szolnoki A, Perc M, Szabó G. EPL 2012;100:28005. [36] Lieberman E, Hauert C, Nowak MA. Nature 2005;433:312. [37] Szolnoki A, Perc M, Szabó G. Phys Rev Lett 2012;109:078701. [38] Deng L-L, Tang W-S, Zhang J-X. Chin Phys Lett 2011;28:080204. [39] Deng L-L, Tang W-S, Zhang J-X. Physica A 2011;390:4227. [40] Gao J, Li Z, Wu T, Wang L L. EPL 2011;93:48002. [41] de Jong S, Uyttendaele S, Tuyls K. J Artif Intell Res 2008;33:551. [42] Ballobas B. Random graphs. London: Academic Press; 1985. [43] Schuessler R. J Conf Resolut 1989;33:728. [44] Tanimoto J. Int J Mod Phys C 2011;22:401. [45] Santos FC, Pacheco JM, Lenaerts T. PLoS Comput Biol 2006;2:1284. [46] We have known that the Nash Equilibrium (NE) of the traditional ultimatum games is p ¼ q ¼ 0 (i.e., S ¼ ð0; 0Þ). In economics, any arbitral strategy with p P q can be the Equal Pareto Optimum (EPO), which contains S ¼ ð0; 0Þ. Thus, the Nash Equilibrium (NE) is comprised by the Equal Pareto Optimum (EPO) and the game has no dilemma. [47] When b > 1, the Nash Equilibrium (NE) is consistent with the Equal Pareto Optimum (EPO) that becomes p ¼ 1. Hence, the dilemma appears.