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Cluster evolution in public goods game with fairness mechanism
Physica A 532 (2019) 121796
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Physica A journal homepage: www.elsevier.com/locate/physa
Cluster evolution in public goods game with fairness mechanism ∗
Baojian Zhang a , Zeguang Cui a , , Xiaohang Yue b a b
School of Management Science and Engineering, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee 53211, WI, USA
highlights • • • •
The problem in public goods game is solved, and the process of cluster evolution is studied. Same-strategy group is utilized to obtain full cooperation in small-scale cluster. Punishment is the emphasis for the formation of cooperation, punishment cost is paid by cooperators evenly. Fairness mechanism that employs punishment is established.
article
info
Article history: Received 29 October 2018 Received in revised form 27 May 2019 Available online 18 June 2019 Keywords: Public goods game Fairness mechanism Punishment Cluster evolution
a b s t r a c t Defector contributes nothing but gains benefit in public goods game. This phenomenon creates an incentive for free riding and causes inefficient equilibrium in which everyone is defector. However, coexistence of cooperation and defection is pervasive for a large well-mixed population in nature society. Inspired by the fact that humans have strong but also emotional tendencies for fair play, we employ a fairness mechanism with peer punishment to solve the public goods problem of a well-mixed cluster and aim to analyze cluster evolutionary process. The process is divided into three stages. In the first stage called cluster formation, this problem can be effectively solved by same-strategy groups. The second stage is cluster extension, we utilize multiplayer game and prove that increasing fairness factor and decreasing punishment cost are two effective ways to improve cooperation. Finally, we obtain the coexistence of cooperation and defection in the third stage called boom and bust. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Each player contributes an amount to the common pool in public goods game (PGG), the total contributions will be multiplied by a factor, and then equally allocated by all players. PGG plays an important role in a wide range of clusters, including the management of rural public goods [1,2], the voluntary contribution of social public goods [3], and the supply of international public goods [4–6]. Obviously, coexistence of cooperation and defection can be often found for a large well-mixed population in human society [7]. It has aroused extensive attention of social scientists, economists, and biologists [8–13]. For the presence of public goods, in theory everyone will favor defective behavior because defector contributes nothing but gains benefit, which leads to social inefficient equilibrium. Individual self-interest is at odds with group interest. ∗ Corresponding author. E-mail address: [email protected] (Z. Cui). https://doi.org/10.1016/j.physa.2019.121796 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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However, the vast majority of studies show that total contributions will lie between 40% to 60% of group optimum in one-off experiments [14], while the number of experiments is not one, the total contributions are also not zero obviously. It is true that there is a free rider problem, but the strong free rider prediction is clearly wrong, a range of behavior exists from fully selfish to fully altruistic. It is same as business incubators, and participants in incubator share information. There must be information sharing to maintain operation of the incubator, but each participant will not fully share their core information to completely lose their competitiveness. Therefore, the coexistence of cooperation and defection is a logical outcome. In order to solve the free-rider problem, we utilize commitment and form same-strategy group. Commitment has an immediate positive effect [15–17] because it makes the interests of players more aligned when they commit to making a similar decision [18]. For instance, a player commits to contributing 0.5a when another contributes a. Then if certain players commit to contributing the same amount in PGG, such action represents the formation of same-strategy group, which is utilized to form full cooperation in our study. But we also prove that this way can be useful only when the cluster scale is small because of the limitation and difficulty of forming same-strategy groups. On the other hand, punishment has been verified to promote the evolution of cooperative behavior [19–33], and punishment entails paying a cost. Obviously, punishment is harmful to defectors because punishment compels all defectors to pay a fine. Therefore, cooperators paying the cost of punishment is logical [34–40]. Then we utilize the assumption that cooperators equally pay the cost together [34,41,42], which is an effective means without a second-order free-rider problem. However, the emergence of punishment remains a challenge. Punishment brings a loss of profit to cooperators while they pay the cost, and puts them at the risk of being retaliated by defectors [43]. So, why do cooperators choose to punish defectors? Inspired by the fact that humans have a strong but also emotional sense of fairness [44–49], we employ a fairness mechanism to solve the above problem. With the emergence of fairness mechanism, the individual utility will be divided into two parts, which are profit utility and fairness utility. We know that punishment brings a decrease of profit utility to cooperators, but it also causes an increase of fairness utility, then cooperator’s strategy depends on the change of total utility in the pursuit of maximizing utility level. For example, if the increase of fairness utility is higher than the decrease of profit utility, cooperators’ total utility increase and punishment is a dominant strategy. So, punishment can appear because of the emergence of fairness. After the above consideration, in this work, we propose same-strategy group and fairness mechanism with peer punishment to solve the public goods problem in a well-mixed cluster, and aim to study the process of cluster evolution. In particular, we will focus on the phenomenon of coexistence of cooperation and defection [50–56]. We divide player’s utility into profit utility and fairness utility, and set the numbers of cooperators and players to change based on that everyone pursues maximum utility. Then we find same-strategy groups can solve this public goods problem effectively and equilibrium is full cooperation, we also prove that this result is useful when the cluster scale is small. In addition, we find that the coexistence of cooperators and defectors is the dominant outcome after cluster scale grows, and increasing fairness factor and decreasing punishment cost are two effective ways. Our paper except Introduction is organized as follows. We compare the utilities of cooperator and defector with a different choice about punishment in Section 2, it is the reason why the number of cooperators changes. In Section 3, we obtain equilibrium after the change of cooperators’ number by two-player and multiplayer game, and analyze stabilization based on the positive and negative of utilities of all players at equilibrium. In Section 4, cluster evolutionary process is simulated, the process is divided into three stages. We conclude the paper in Section 5. 2. The simple model We analyze PGG in a well-mixed population, n individuals of the population are randomly chosen and formed a cluster. The notations related to the model are shown in Table 1. 2.1. Fairness mechanism when cooperators do not punishment In the model, each player in the cluster has two strategies, namely, cooperation and defection, so we utilize the initials c and d to distinguish everything of cooperator and defector at the subscript. For example, we describe the contributions of cooperator and defector as qc and qd respectively, and set qc is 1 and qd is 0. Thus, the quantity of total contributions is equal to the number of cooperators nc . The total contributions will be multiplied by an income factor k1 (1 < k1 < n), and then split evenly among all players. Therefore, the player’s profit utility is Up =
k 1 nc n
− qi (i = c , d) .
(1)
Huaizhou et al. [57] have described a Fairness Optimization Model and used two parameters to define unfairness, one parameter is the difference between the mean of total utility and individual utility, another is the variance of total utility. However, in our model, player’s fairness utility Uf is reflected by the comparison of contributions rather than utility, thus we set mean and variance of contributions as µ and σ 2 , and get fairness utility by using a combination of µ − qi and σ 2 . In human society, it is logical that Uf and µ − qi are positively related when σ 2 is fixed, but the correlation of Uf and σ 2
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Table 1 Notations list including all key symbols. Symbol
Description
Symbol
Description
n qc k1 C
The number of players The contributions of cooperator The income factor The punishment cost paid by all cooperators The mean of contributions of all players The total utility of cooperator The profit utility of player The threshold of cooperators’ number for cooperators to get more An intermediate variable for analyzing F The difference of utilities of cooperator and defector without punishment The difference of utilities of cooperator and defector after the choice about punishment when the number of cooperators is nc A symbol representing the existence of punishment, which could appear in the top right of other symbols
nc qd k2
The The The The The The The The
µ Uc Up A g f Fn∗c ′
δ σ2 Ud Uf B
k3 F hn c
∗
number of cooperators contributions of defector fairness factor fine paid by defector variance of contributions of all players total utility of defector fairness utility of player number of cooperators at equilibrium
The cost factor The difference of utilities of cooperator and defector with punishment The difference of utilities of cooperators with different choice about punishment when the number of cooperators is nc A symbol representing the finish of choice about punishment, which could appear in the top right of other symbols
is negative when µ − qi is constant. Then we combine the principle of ‘‘Efficiency is priority and fairness should also be taken into consideration’’ and set Uf = k2 (µ − qi − 1) σ 2 (i = c , d) .
(2)
where k2 is the fairness factor in a range of (0, 1). The principle can be embodied from two aspects. First, the player’s fairness utility is obviously not positive because k2 > 0, µ − qi − 1 < 0 and σ 2 ≥ 0. Accordingly, the total utility must be negative if one’s profit utility is negative. So, efficiency, represented by profit, is priority. Second, fairness is an influential factor but not decisive when k2 reflects the importance of fairness and 0 < k2 < 1. After the proof in Appendices, we can know players’ total utilities Ui of two strategies (i = c , d) as
( Uc =
k 1 nc n
k 1 nc
) nc (n − nc ) (2n − nc ) − 1 − k2 . 3 n
(3)
nc (n − nc )2
. (4) n n3 Then we set f = Uc − Ud , and determine whose total utility is bigger through the positive and negative of f . However, Ud =
− k2
it must be that f = −1 − k2 punishment.
nnc −n2c n2
< 0 in any condition. Therefore, the social dilemma is the inevitable outcome without
2.2. Fairness mechanism when cooperators do punishment If cooperators punish defectors by paying a cost C evenly, defectors all should pay a fine δ . After punishment, the new contribution of cooperator is q′c = 1 + nC , and defector’s is q′d = δ . Of course, punishment cannot appear if no cooperator or c defector exists, so the range of nc is (0, n) when punishment is implemented. In order to simplify the following formulas, we firstly set g = (1 − δ) nc + C .
(5)
After the analysis of new utilities of cooperator and defector in Appendices, we can work out n − nc (n − nc )2 − 1+ − k2 g 3 − k2 g 2 2 . n nc n3 n2c n nc ( ) k n n − n n − n 1 c c c Ud′ = − δ + k2 g 3 3 − k2 g 2 2 . ′
Uc =
[
k1 nc
(
C
)]
n
n nc
n nc
(6) (7)
Our paper sets an equation F = Uc′ − Ud′ . When cooperators implement punishment, we can determine the better strategy between cooperation and defection through the analysis of F , which is same to f . F = Uc′ − Ud′ = −g
[
1 nc
+
k2 ( nc
σ′
)2
]
.
(8)
( )2 1 + nk2c σ ′ > 0 by nc > 0, k2 > 0 nc ( ′ )2 and σ > 0, and Eq. (5) has given that g is dependent on C , δ and nc , so we can confirm that F is positive or negative Obviously, F and g have opposite positive and negative values when we know
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B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796 Table 2 Game model of the two groups.
through the analysis about these three factors. We firstly analyze C and δ .
∂F 1 n − nc = − − 3k2 g 2 2 2 < 0. ∂C nc n nc n − n ∂F c = 1 + 3k2 g 2 2 > 0. ∂δ n nc
(9) (10)
It is clear that F and C are negatively related, and the correlation between F and δ is positive. Hence, decreasing C and increasing δ are two effective strategies to increase F when cooperator implement punishment, in other words, these two strategies are beneficial to cooperators with punishment. However, F and nc do not have such obvious relationship, we will utilize C and δ to get a new factor A for the next analysis about nc . g We assume that Cn + 1 ≥ δ , then it must be nC + 1 > δ because nc < n, and it is certain that nC + 1 − δ = n > 0. c c c Subsequently, we can know F < 0 because of the relationship between F and g. Therefore, the total utility of cooperator cannot be bigger than defector’s even punishment is implemented. The assumption Cn + 1 ≥ δ must be illogical, and it should be Cn + 1 < δ , so C + n (1 − δ) < 0. Also, it is obvious that C + 0 (1 − δ) > 0, we combine them and form C + n (1 − δ) < 0 < C + 0 (1 − δ) .
(11)
Then we can determine the rationality of existence of C + A (1 − δ) = 0, in which A can change from 0 to n, and it can C be transformed into A = δ− . There we must emphasize that 1 − δ < 0, if not, C + A (1 − δ) = 0 cannot be logical because 1 C > 0 and A > 0. Based on these, the positive and negative of F can be obtained by comparing nc and A. If nc < A, it must be g = (1 − δ) nc + C > (1 − δ) A + C = 0 because 1 − δ is negative. Then, F is negative because of the correlation of g and F . If nc ≥ A, then g ≤ 0 and F ≥ 0 can be proved in the same way. We get Proposition 1. Proposition 1. If the number of cooperators nc is less than A when cooperators punish defectors, g > 0 and F < 0 is certain, defection is the superior strategy. If nc ≥ A, we can obtain g ≤ 0 and F ≥ 0, then cooperation is better. 3. Game 3.1. Two-player game Cooperators can choose punishment or not, so each player in the well-mixed cluster has three strategies, which are defection, cooperation with punishment and cooperation without punishment. However, all cooperators equally pay the cost together when punishment is implemented, accordingly the three strategies cannot coexist, and two strategies can coexist at the most. Based on this argument, we divide all players into two groups when the size of cluster n is fixed, namely, Groups I and II, and their quantities are represented as n1 and n2 respectively. Players choose a same strategy in each group, it is the formation of same-strategy groups. Each group could choose a strategy in its own strategy space, then we obtain the following game model. To review our settings in this paper, punishment does not exist if there is no defector, and it means everyone’s strategy is cooperation without punishment. So, there are three blanks in this table, which is reasonable. We can work out Uc , Ud , Uc′ , and Ud′ when all the factors are certainly based on Eqs. (3) (4) (6) and (7), and utilize 1 them to analyze each utility level in Table 2. For example, U11 is the utility of player in Group I when all groups 1 choose defection together. Also, U11 is the utility of defector when no cooperator and punishment. So, it is obvious that 1 U11 = Ud (nc = 0) = 0 because of Eq. (4). Subsequently, we analyze equilibrium. First, if the equilibrium of this model is that all players are defectors, which becomes a social dilemma, everyone cannot obtain an increment of the total utility and thus tends to leave the cluster. This equilibrium should be avoided. Second, the coexistence of cooperator and defector can be equilibrium as well. The third equilibrium is that everyone chooses to
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cooperate and it is called full cooperation. Everyone’s total utility is Uc = k1 − 1 > 0 because of Eq. (3) when nc = n. This scenario is actually the ideal state that the two-player game hopes to achieve. Cluster scale is initially in a small state. It is easy to form same-strategy groups because the correlation between formation difficulty and cluster scale are positive, and it will be proved in Section 4.1. After the two-player game, equilibrium will be obtained as the basis to analyze large-scale cluster by a multiplayer game. 3.2. Multiplayer game In a large-scale cluster, it is difficult to form same-strategy groups, and each player should decide individually to be a cooperator or defector. Player may change strategy when other player’s utility is bigger, which will lead to a change in the number of cooperators nc . After the change of nc , all cooperators should reconfirm their strategies on whether to punish or not, then everyone will get a new total utility. However, the phenomenon that some players’ utility is bigger may also exist here, the above process will be repeated. The number of players n can also change when a player enters or leaves, which will be analyzed in Section 3.2.2. We study the cluster when n is constant. 3.2.1. Game with fixed scale The number of players n represents the scale of cluster, so n is constant when the scale is fixed. Then players can transform their strategies between cooperation and defection in the pursuit of maximizing utility level. If the number of cooperators nc does not change or fluctuates in a small range, it means the formation of equilibrium. Before the analysis of equilibrium, we set that hnc = Uc′ − Uc and Fn∗c = Uc∗ − Ud∗ , where Uc∗ is defined as the final utility of cooperator after the choice about punishment and Ud∗ is that of defector. We use hnc to judge whether to punish or not, and use Fn∗c to judge whose final utility is bigger. For example, hnc > 0 represents that cooperators can obtain additional utility by implementing punishment, punishment is the better strategy and Uc∗ = Uc , also, the final utility of defector Ud∗ equals to U ′ d , Fn∗c = F = Uc′ − Ud′ . When hnc ≤ 0, we can utilize the same way to confirm that cooperators will not punish defectors, it will be Uc∗ = Uc , Ud∗ = Ud , and Fn∗c = f = Uc − Ud . Based on these setting, we obtain these following equilibria, and set equilibrium quantity of cooperator as B (0 ≤ B ≤ n). (1) Stable equilibrium If all players will not transform their strategies in a state, the state is called stable equilibrium. Three stable equilibria exist. First, if defector’s final utility Ud∗ is bigger than cooperator’s when only one cooperator exists, the latter will become a defector. So, this scenario is a social dilemma when F1∗ < 0, and the equilibrium quantity of cooperator B is 0. Second, if Fn∗−1 > 0, it means that the quantity of cooperators is n − 1 and the only defector will definitely choose to be a cooperator. This scenario reaches an equilibrium called full cooperation, in which everyone chooses cooperation. We also know B = n. Third, the equilibrium conditions are FB∗ = 0, FB∗+1 < 0 and FB∗−1 > 0. When nc = B, it lets all players’ final utilities be same. In addition, defector’s final utility Ud∗ is better with the increase in B by 1, and cooperator’s final utility Uc∗ is larger if B decreases by 1. The equilibrium value of cooperator’s number must be B and is stable. (2) Vibratory equilibrium If FB∗ < 0 and FB∗−1 > 0, it will form the vibratory equilibrium. Defector is more satisfied with his/her final utility when nc = B, so nc will change from B to B − 1. Then it will let cooperator’s final utility Uc∗ be higher because FB∗−1 > 0, and cause that nc changes from B − 1 to B again. Therefore, we can determine nc will change continuously between B − 1 and B. It is the formation of vibratory equilibrium. We see B as the equilibrium quantity. If FB∗+1 < 0 and FB∗ > 0, it is the same as the above analysis, so we can use FB∗ < 0 and FB∗−1 > 0 as the only premise of vibratory equilibrium. (3) Unstable equilibrium This equilibrium is similar to the third stable equilibrium. When nc = B, all players’ final utilities are also same. But cooperator obtains more when nc = B + 1, it will promote nc to increase because FB∗+1 ≥ 0, or nc will decrease when nc = B − 1 because FB∗−1 ≤ 0. This state is an equilibrium, which will be broken with the change of nc , so it is unstable. We have known all equilibria and the formation conditions of them by now. However, several of them are illogical in our paper. Starting the analysis on feasibilities of all equilibria is the basis for next study. Based on the results about f and F in Section 2 and the setting of hnc , we can acquire the following proposition. Proposition 2. It must be Fn∗c = f < 0 when hnc < 0, because cooperators will not choose punishment. But if hnc ≥ 0, the positive and negative of Fn∗c is undetermined, it is related to the difference between nc and A. Then we must let hnc ≥ 0 and nc > A for obtaining Fn∗c = F > 0. We use the Table 3 to present the proposition. So, there should be enough cooperators to choose punishment and spare the cost, it is the sufficient and necessary condition of that cooperators can get more than defectors. Then we will prove that the second stable equilibrium, called full cooperation, is a part of vibratory equilibrium and the third stable equilibrium cannot exist. For full cooperation, its formation condition is Fn∗−1 > 0. Cooperators can gain more utility than defectors when nc = n − 1, so the only defector will transform his/her strategy and nc will change from
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B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796 Table 3 The analysis of F ∗ in different conditions. nc < A nc = A nc > A
hnc < 0
hn c ≥ 0
Fn∗c = f < 0 Fn∗c = f < 0 Fn∗c = f < 0
Fn∗c = F < 0 Fn∗c = F = 0 Fn∗c = F > 0
n − 1 to n. Accordingly, there is not punishment when there is no defector, and defection is the dominant strategy, so nc will change from n to n − 1 again. It is clear that full cooperation is one part of vibratory equilibrium actually. For the third stable equilibrium, FB∗ = 0 is one limitation when the equilibrium quantity of cooperators is B. We can know B = A and hnc ≥ 0 based on Table 3. Thus, it is obvious that FB∗−1 < 0 because B − 1 < A. However, FB∗−1 > 0 is a limitation of the third stable equilibrium as well. Therefore, this equilibrium cannot exist. The second stable equilibrium is a part of vibratory equilibrium, the third stable equilibrium cannot exist, and unstable equilibrium can be ignored because it is unstable. Thus, only the first stable equilibrium and vibratory equilibrium are possible outcomes. However, the first stable equilibrium is a dilemma, so the equilibrium we aim to form is vibratory equilibrium. Formation conditions of vibratory equilibrium are FB∗ < 0 and FB∗−1 > 0. Based on Table 3, FB∗−1 > 0 means hB−1 > 0 and B − 1 > A, and it must be B > A. Then hB < 0 is necessary when we hope to get FB∗ < 0. So, for the vibratory equilibrium, cooperators will choose to punish defectors when nc = B − 1. But when nc = B, cooperators will not choose punishment. 3.2.2. Game with unfixed scale Since there is no limitation on players, and they can freely leave or enter the cluster, the number of players n will be unfixed. After the analysis in Section 3.2.1, we can acquire an equilibrium between vibratory equilibrium and social dilemma when n is constant. If utilities of all players are positive at equilibrium, the cluster is attractive to foreign people and n will increase, n will decrease if one’s utility is not positive. After the change of n, we will get new equilibrium and work out new utilities of all players, then again use the utilities of all players to analyze the change of n. Until n does not change or fluctuates in a small range, it means the formation of stabilization. We must emphasize that the cost of punishment C will be varied with the change of n. Obviously, the difficulty of implementing punishment will increase faster and faster with cluster scale grows, because punishment efficiency and 2 > 0 and ddnC2 > 0, and simply set scale are negatively related when the cluster has been large-scale. So, we can obtain dC dn C = k3 n2 . After the setting on C , the difficulty of analyzing stabilization has increased as well, we will utilize simulation to solve it in the next section. 4. Simulation This section divides the evolution process of cluster into three stages based on the change of n. The first stage is cluster formation, in which two same-strategy groups start a game and cause the emergence of full cooperation, and the cluster scale should be small that proved in Section 4.1. Then the full cooperation makes players’ utility be k1 − 1 > 0 and attracts other player outside. Thus, the second stage is called cluster extension. However, it will be impossible to achieve that everyone chooses to join a same-strategy group, so we utilize multiplayer game to explain the extension process based on the interactions among four factors (k1 , k2 , C , δ ). Finally, the third stage is boom and bust. Understanding the extension process based on all players’ utility will help us obtain stabilization by simulation, then we can determine the cluster is boom or bust through the change of stabilization with different C . There we set several restrictions on key factors, k1 ∈ [3, 5], k2 ∈ [0.5, 1], C ∈ (0, 6], and δ ∈ (1, 2.5]. Except the restriction of k1 is set by referring some papers [22,34,47,53], others’ rationality will be explained in Section 4.2. The number of players n can change spontaneously in our model, the range of n will be certain by the combination of the above factors, which will be analyzed in Section 4.3. We study these stages and draw figures correspondingly using MATLAB. 4.1. Cluster formation We form same-strategy groups and let other factor values be k2 = 0.75, C = 2, and δ = 2. The equilibrium is denoted by the equilibrium utility of player in group I. By the curve below Fig. 1 which represents k1 = 3, we know equilibrium is full cooperation when n1 changes between 7 and 13, and each player’s utility is k1 − 1 = 2. We also acquire that the range of n2 is [7,13] as well because n = n1 + n2 , so each same-strategy group should have seven players at least, which is the condition of formation of full cooperation. However, it is possible to get full cooperation with over four players in each group when k1 = 5. Therefore, the increase of k1 is advantageous to form full cooperation. We also form two same-strategy groups and give k2 = 0.75 and δ = 2. However, we only set C = 10, because the following result will become more obvious if C increases faster and faster with n grows based on the analysis of C in Fig. 4. Compare the curve in Fig. 2 with that below Fig. 1, we discover that it is more difficult to form full cooperation when n is 100, because the number of players in each same-strategy group needs to reach over 30, such scenario is impossible with a variety of people. So, we acquire Proposition 3, and see full cooperation as basis for the next stage.
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Fig. 1. Two-player game equilibrium with different n1 when k1 is 3 or 5 and n = 20.
Fig. 2. Two-player game equilibrium with different n1 when k1 is 3 and n is 100.
Proposition 3. Same-strategy group can solve the public goods problem and form full cooperation, but it is only used in the small-scale cluster. 4.2. Cluster extension k1 = 3, C = 5, δ = 2, n = 50, and the initial value of nc is in the range of n/4 ≈ 13 to 3n/4 ≈ 38. Player will change strategy when other players’ utility is bigger, which will lead to the change of nc . If nc does not change or fluctuates in a small range, it means the formation of equilibrium. Thus, we obtain that the cooperation level will be improved as k2 increases because of Fig. 3. The Fig. 3 shows that equilibria all are social dilemmas with different initial value of nc when k2 = 0.5, and the correlation between k2 and cooperation level at equilibrium is positive, so social dilemma will be an inevitable outcome when k2 < 0.5. The analysis about k2 < 0.5 is meaningless. Also, we know k2 ≤ 1 because of the attribution of fairness mechanism, so k2 ∈ [0.5, 1]. The analysis of Fig. 4 is same to that of Fig. 3. The commonalities of these two subfigures are k1 = 3, k2 = 0.75, and n = 50, but the values of C and δ are different. The left settings are C = 1, 2, 3, 4, 5, 6 and δ = 2, it is clear that the lower C is, the easier it is to form full cooperation. Then we draw figure when C = 5 and δ = 1.5, 2, 2.5 on the right, and find that δ is negatively correlated with cooperation level at equilibrium too. It is obvious that equilibrium is a social dilemma
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Fig. 3. Multiplayer game equilibrium when k2 = 0.5, 0.75, 1 with 50 steps.
Fig. 4. Multiplayer game equilibrium when C = 1, 2, 3, 4, 5, 6 and δ = 1.5, 2, 2.5 with 50 steps.
when C = 6, then we combine the relationship between C and cooperators’ equilibrium number, and know it should be C ≤ 6, which is similar to k2 . Accordingly, C ∈ (0, 6]. We also can utilize the same means to prove δ ≤ 2.5, and we have calculated δ > 1 in Section 2.2, so δ ∈ (1, 2.5] is reasonable. This result about δ in Fig. 4 should undergo further analysis. When cooperators choose to punish defectors, the increase of δ is advantageous for cooperators, it has been proved in Section 2.2. But the increase of δ could lead to the ∂ (σ ′ )
2
nc losses of cooperators and defectors all, of course, defectors lose more. The reason is that ∂δ = −2g n− > 0 when n2 implementing punishment causes g < 0 based on (B.2) in Appendices, it is possible that cooperators and defectors all feel unfair. Cooperators could choose no punishment when U ′ c decreases but Uc is invariable, then it will cause nc decrease when without punishment based on the result in Section 2.1. So, the increase of δ is advantageous to cooperators with punishment, but cooperators will not punish when it forms Uc > U ′ c . The change of k1 will result in the utilities of all players increase or decrease together. Consequently, no change is expected for strategies. If we want to improve the number of cooperators in equilibrium, three ways are suggested, increasing k2 , decreasing C , and decreasing δ . But decreasing δ is ineffective because it improves defector’s utility greatly, which will reduce cooperator’s evolutionary advantage. Thus, increasing k2 and decreasing C are our choice. There we must emphasize that C will increase faster and faster with cluster scale grows, and use C = k3 n2 that set in Section 3.2.2 to analyze the extension process. n2 because of the ranges of C and n, and other factors are k1 = 3, k2 = 0.75, δ = 2. After the analysis We let C = 600 of roles of all factors, we can obtain equilibria with difference scale, and use cooperators’ equilibrium quantity to denote equilibrium in Fig. 5. When each player’s utility is positive, it will attract outside players pursuing maximal utility, and
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Fig. 5. Cooperator’s equilibrium quantity when n changes from 20 to 65.
Fig. 6. Cluster stabilization when initial n changes from 20 to 80 with 50 steps.
then cluster extends. If everyone is defector and it forms social dilemma, the cluster will not grow again, so we get the largest size of the cluster and it is 64. Next, we start an analysis of the third stage.
4.3. Boom and bust 2
n In Fig. 6 k1 = 3, k2 = 0.75, C = 600 , and δ = 2 as well. The change of n is determined by the positive and negative of all players’ utilities. When cluster scale is small initially, everyone’s utility is positive and the cluster will extend. Moreover, the cluster could start with too many players, and then the cluster scale will decrease. When the initial value of n changes between 40 and 64, n will fluctuate in a small range finally. It is the formation of stabilization, and the upper limit of stabilization is same to the limit of cluster in Fig. 5 (see Fig. 7). Other settings except k2 and C are same to Fig. 6. We obtain the change of cluster stabilization that shown in Fig. 7, and set the increase and decrease of stabilization as boom and bust respectively. It is obvious that the relationship of k2 and stabilization is positive, but the correlation of C and stabilization is negative. Thus, increasing k2 and decreasing k3 are two ways for cluster to form boom.
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B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
Fig. 7. Cluster stabilization when k2 = 0.5, 0.75, 1 and C =
2 2 n2 , n , n 400 600 800
with 50 steps.
5. Discussion In Section 2, we calculate the utilities of cooperator and defector based on Static Analysis. Then we utilize Comparative Static Analysis in Section 3. We compare the utilities of cooperator and defector, which is the basis of the change of cooperators’ number, and compare the utilities of players in the cluster with that of foreign players which is the reason why players’ number changes. Finally, we employ Dynamic Analysis to obtain the process of cluster evolution over time in Section 4. To summarize, we have introduced same-strategy group in PGG and found the public goods problem indeed can be effectively solved, full cooperation is the equilibrium. We also show that fairness mechanism with peer punishment is useful, in which we divide the player’s utility into two parts, profit utility and fairness utility, and set everyone pursues maximum utility. We find that coexistence of cooperators and defectors is the dominant outcome, increasing fairness factor and decreasing punishment cost are two effective ways. Therefore, the key to solving the public goods problem is to focus on fairness and difficulty of implementing punishment. Also, there is a limitation in our paper, and the setting of Eq. (2) has certain subjectivity. Finally, some issues such as the rock–paper–scissors type of cyclic dominance among cooperator, defector, and punisher are not covered in this study as we assume cooperators equally pay the punishment cost. We would like to explore this subject in the future work. Prior works have studied the strategic change when the number of players is n in well-mixed or structured population [3,19,22,53,54]. However, we focus on the change of players’ number n when players are free to enter and exit because public goods are nonexclusive. Different from the punishment mechanism used in prior papers [19,21–24], we here use a fairness mechanism and the equation determining fairness utility is set based on a previous study [57] and realistic experience. Although we also consider the peer punishment strategy in our model, such strategy is mainly to provide the possibility of rising fairness utility. We finally emphasize that such fairness mechanism is simple, but is demonstrated to be an effective approach for resisting public goods dilemma. Acknowledgments This research is supported by Humanities and Social Science Fund of Ministry of Education of China (No. 18YJA630137) and National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (No. 71303143). Appendix A. Utilities of cooperator and defector when cooperators do not punish defectors When qc = 1 and qd = 0, we can work out the mean and variance of total contributions.
µ= σ2 =
qc nc + qd nd n
=
nc n
.
(A.1)
(qc − µ)2 nc + (qd − µ)2 nd
=
nnc − n2c
.
n n2 Analyze players’ fairness utility Uif of two strategies (i = c , d) based on (2). Ucf = k2 (µ − qc − 1) σ 2 = −k2
nc (n − nc ) (2n − nc ) n3 nc (n − nc )
.
(A.2)
(A.3)
2
Udf = k2 (µ − qd − 1) σ 2 = −k2
n3
.
(A.4)
B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
11
Because of (1), (A.3) and (A.4), we obtain Ui (i = c , d) as follows
( Uc = Ud =
k 1 nc n
k1 nc
) nc (n − nc ) (2n − nc ) − 1 − k2 . 3
− k2
n
(A.5)
n
nc (n − nc )2 n3
.
(A.6)
Appendix B. Utilities of cooperator and defector when cooperators punish defectors
( )2
When cooperators punish defectors, we analyze the new mean µ′ and variance σ ′
( µ =
1+
′
C nc
)
nc + δ nd
=
n
(
1+ ( ′ )2 σ =
C nc
−
−δ
g n
)2
(1 − δ) nc + C n nc + δ −
(
g n
+δ =
−δ
)2
n
nd
g n
of total input as follows:
+ δ.
= g2
(B.1)
n − nc n2 nc
.
(B.2)
Then we study the new fairness utilities U ′ if of cooperator and defector based on (2) as well.
) ( ′ )2 n − nc (n − nc )2 σ = −k2 g 3 − k2 g 2 2 . 3 2
Ucf′ = k2 µ′ − q′c − 1
(
) ( ′ )2 σ = k2 g
′ Udf = k2 µ′ − q′d − 1
(
n nc 3 n − nc
n − nc
n3 nc
n2 nc
− k2 g 2
n nc
.
(B.3) (B.4)
After the consideration about (1), (B.3) and (B.4), we can determine the total utilities of cooperator and defector, respectively, as follows: ′,
Uc = U ′,
′ cp
+U
n − nc (n − nc )2 − k2 g 3 − 1+ − k2 g 2 2 . n nc n3 n2c n nc ) ( n − nc n − nc k 1 nc − δ + k2 g 3 3 − k2 g 2 2 . =
[
′ cf
Ud = U ′ dp + U ′ df
=
k1 nc
n
(
C
)]
n nc
n nc
(B.5) (B.6)
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Cluster evolution in public goods game with fairness mechanism ∗
Baojian Zhang a , Zeguang Cui a , , Xiaohang Yue b a b
School of Management Science and Engineering, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee 53211, WI, USA
highlights • • • •
The problem in public goods game is solved, and the process of cluster evolution is studied. Same-strategy group is utilized to obtain full cooperation in small-scale cluster. Punishment is the emphasis for the formation of cooperation, punishment cost is paid by cooperators evenly. Fairness mechanism that employs punishment is established.
article
info
Article history: Received 29 October 2018 Received in revised form 27 May 2019 Available online 18 June 2019 Keywords: Public goods game Fairness mechanism Punishment Cluster evolution
a b s t r a c t Defector contributes nothing but gains benefit in public goods game. This phenomenon creates an incentive for free riding and causes inefficient equilibrium in which everyone is defector. However, coexistence of cooperation and defection is pervasive for a large well-mixed population in nature society. Inspired by the fact that humans have strong but also emotional tendencies for fair play, we employ a fairness mechanism with peer punishment to solve the public goods problem of a well-mixed cluster and aim to analyze cluster evolutionary process. The process is divided into three stages. In the first stage called cluster formation, this problem can be effectively solved by same-strategy groups. The second stage is cluster extension, we utilize multiplayer game and prove that increasing fairness factor and decreasing punishment cost are two effective ways to improve cooperation. Finally, we obtain the coexistence of cooperation and defection in the third stage called boom and bust. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Each player contributes an amount to the common pool in public goods game (PGG), the total contributions will be multiplied by a factor, and then equally allocated by all players. PGG plays an important role in a wide range of clusters, including the management of rural public goods [1,2], the voluntary contribution of social public goods [3], and the supply of international public goods [4–6]. Obviously, coexistence of cooperation and defection can be often found for a large well-mixed population in human society [7]. It has aroused extensive attention of social scientists, economists, and biologists [8–13]. For the presence of public goods, in theory everyone will favor defective behavior because defector contributes nothing but gains benefit, which leads to social inefficient equilibrium. Individual self-interest is at odds with group interest. ∗ Corresponding author. E-mail address: [email protected] (Z. Cui). https://doi.org/10.1016/j.physa.2019.121796 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
However, the vast majority of studies show that total contributions will lie between 40% to 60% of group optimum in one-off experiments [14], while the number of experiments is not one, the total contributions are also not zero obviously. It is true that there is a free rider problem, but the strong free rider prediction is clearly wrong, a range of behavior exists from fully selfish to fully altruistic. It is same as business incubators, and participants in incubator share information. There must be information sharing to maintain operation of the incubator, but each participant will not fully share their core information to completely lose their competitiveness. Therefore, the coexistence of cooperation and defection is a logical outcome. In order to solve the free-rider problem, we utilize commitment and form same-strategy group. Commitment has an immediate positive effect [15–17] because it makes the interests of players more aligned when they commit to making a similar decision [18]. For instance, a player commits to contributing 0.5a when another contributes a. Then if certain players commit to contributing the same amount in PGG, such action represents the formation of same-strategy group, which is utilized to form full cooperation in our study. But we also prove that this way can be useful only when the cluster scale is small because of the limitation and difficulty of forming same-strategy groups. On the other hand, punishment has been verified to promote the evolution of cooperative behavior [19–33], and punishment entails paying a cost. Obviously, punishment is harmful to defectors because punishment compels all defectors to pay a fine. Therefore, cooperators paying the cost of punishment is logical [34–40]. Then we utilize the assumption that cooperators equally pay the cost together [34,41,42], which is an effective means without a second-order free-rider problem. However, the emergence of punishment remains a challenge. Punishment brings a loss of profit to cooperators while they pay the cost, and puts them at the risk of being retaliated by defectors [43]. So, why do cooperators choose to punish defectors? Inspired by the fact that humans have a strong but also emotional sense of fairness [44–49], we employ a fairness mechanism to solve the above problem. With the emergence of fairness mechanism, the individual utility will be divided into two parts, which are profit utility and fairness utility. We know that punishment brings a decrease of profit utility to cooperators, but it also causes an increase of fairness utility, then cooperator’s strategy depends on the change of total utility in the pursuit of maximizing utility level. For example, if the increase of fairness utility is higher than the decrease of profit utility, cooperators’ total utility increase and punishment is a dominant strategy. So, punishment can appear because of the emergence of fairness. After the above consideration, in this work, we propose same-strategy group and fairness mechanism with peer punishment to solve the public goods problem in a well-mixed cluster, and aim to study the process of cluster evolution. In particular, we will focus on the phenomenon of coexistence of cooperation and defection [50–56]. We divide player’s utility into profit utility and fairness utility, and set the numbers of cooperators and players to change based on that everyone pursues maximum utility. Then we find same-strategy groups can solve this public goods problem effectively and equilibrium is full cooperation, we also prove that this result is useful when the cluster scale is small. In addition, we find that the coexistence of cooperators and defectors is the dominant outcome after cluster scale grows, and increasing fairness factor and decreasing punishment cost are two effective ways. Our paper except Introduction is organized as follows. We compare the utilities of cooperator and defector with a different choice about punishment in Section 2, it is the reason why the number of cooperators changes. In Section 3, we obtain equilibrium after the change of cooperators’ number by two-player and multiplayer game, and analyze stabilization based on the positive and negative of utilities of all players at equilibrium. In Section 4, cluster evolutionary process is simulated, the process is divided into three stages. We conclude the paper in Section 5. 2. The simple model We analyze PGG in a well-mixed population, n individuals of the population are randomly chosen and formed a cluster. The notations related to the model are shown in Table 1. 2.1. Fairness mechanism when cooperators do not punishment In the model, each player in the cluster has two strategies, namely, cooperation and defection, so we utilize the initials c and d to distinguish everything of cooperator and defector at the subscript. For example, we describe the contributions of cooperator and defector as qc and qd respectively, and set qc is 1 and qd is 0. Thus, the quantity of total contributions is equal to the number of cooperators nc . The total contributions will be multiplied by an income factor k1 (1 < k1 < n), and then split evenly among all players. Therefore, the player’s profit utility is Up =
k 1 nc n
− qi (i = c , d) .
(1)
Huaizhou et al. [57] have described a Fairness Optimization Model and used two parameters to define unfairness, one parameter is the difference between the mean of total utility and individual utility, another is the variance of total utility. However, in our model, player’s fairness utility Uf is reflected by the comparison of contributions rather than utility, thus we set mean and variance of contributions as µ and σ 2 , and get fairness utility by using a combination of µ − qi and σ 2 . In human society, it is logical that Uf and µ − qi are positively related when σ 2 is fixed, but the correlation of Uf and σ 2
B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
3
Table 1 Notations list including all key symbols. Symbol
Description
Symbol
Description
n qc k1 C
The number of players The contributions of cooperator The income factor The punishment cost paid by all cooperators The mean of contributions of all players The total utility of cooperator The profit utility of player The threshold of cooperators’ number for cooperators to get more An intermediate variable for analyzing F The difference of utilities of cooperator and defector without punishment The difference of utilities of cooperator and defector after the choice about punishment when the number of cooperators is nc A symbol representing the existence of punishment, which could appear in the top right of other symbols
nc qd k2
The The The The The The The The
µ Uc Up A g f Fn∗c ′
δ σ2 Ud Uf B
k3 F hn c
∗
number of cooperators contributions of defector fairness factor fine paid by defector variance of contributions of all players total utility of defector fairness utility of player number of cooperators at equilibrium
The cost factor The difference of utilities of cooperator and defector with punishment The difference of utilities of cooperators with different choice about punishment when the number of cooperators is nc A symbol representing the finish of choice about punishment, which could appear in the top right of other symbols
is negative when µ − qi is constant. Then we combine the principle of ‘‘Efficiency is priority and fairness should also be taken into consideration’’ and set Uf = k2 (µ − qi − 1) σ 2 (i = c , d) .
(2)
where k2 is the fairness factor in a range of (0, 1). The principle can be embodied from two aspects. First, the player’s fairness utility is obviously not positive because k2 > 0, µ − qi − 1 < 0 and σ 2 ≥ 0. Accordingly, the total utility must be negative if one’s profit utility is negative. So, efficiency, represented by profit, is priority. Second, fairness is an influential factor but not decisive when k2 reflects the importance of fairness and 0 < k2 < 1. After the proof in Appendices, we can know players’ total utilities Ui of two strategies (i = c , d) as
( Uc =
k 1 nc n
k 1 nc
) nc (n − nc ) (2n − nc ) − 1 − k2 . 3 n
(3)
nc (n − nc )2
. (4) n n3 Then we set f = Uc − Ud , and determine whose total utility is bigger through the positive and negative of f . However, Ud =
− k2
it must be that f = −1 − k2 punishment.
nnc −n2c n2
< 0 in any condition. Therefore, the social dilemma is the inevitable outcome without
2.2. Fairness mechanism when cooperators do punishment If cooperators punish defectors by paying a cost C evenly, defectors all should pay a fine δ . After punishment, the new contribution of cooperator is q′c = 1 + nC , and defector’s is q′d = δ . Of course, punishment cannot appear if no cooperator or c defector exists, so the range of nc is (0, n) when punishment is implemented. In order to simplify the following formulas, we firstly set g = (1 − δ) nc + C .
(5)
After the analysis of new utilities of cooperator and defector in Appendices, we can work out n − nc (n − nc )2 − 1+ − k2 g 3 − k2 g 2 2 . n nc n3 n2c n nc ( ) k n n − n n − n 1 c c c Ud′ = − δ + k2 g 3 3 − k2 g 2 2 . ′
Uc =
[
k1 nc
(
C
)]
n
n nc
n nc
(6) (7)
Our paper sets an equation F = Uc′ − Ud′ . When cooperators implement punishment, we can determine the better strategy between cooperation and defection through the analysis of F , which is same to f . F = Uc′ − Ud′ = −g
[
1 nc
+
k2 ( nc
σ′
)2
]
.
(8)
( )2 1 + nk2c σ ′ > 0 by nc > 0, k2 > 0 nc ( ′ )2 and σ > 0, and Eq. (5) has given that g is dependent on C , δ and nc , so we can confirm that F is positive or negative Obviously, F and g have opposite positive and negative values when we know
4
B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796 Table 2 Game model of the two groups.
through the analysis about these three factors. We firstly analyze C and δ .
∂F 1 n − nc = − − 3k2 g 2 2 2 < 0. ∂C nc n nc n − n ∂F c = 1 + 3k2 g 2 2 > 0. ∂δ n nc
(9) (10)
It is clear that F and C are negatively related, and the correlation between F and δ is positive. Hence, decreasing C and increasing δ are two effective strategies to increase F when cooperator implement punishment, in other words, these two strategies are beneficial to cooperators with punishment. However, F and nc do not have such obvious relationship, we will utilize C and δ to get a new factor A for the next analysis about nc . g We assume that Cn + 1 ≥ δ , then it must be nC + 1 > δ because nc < n, and it is certain that nC + 1 − δ = n > 0. c c c Subsequently, we can know F < 0 because of the relationship between F and g. Therefore, the total utility of cooperator cannot be bigger than defector’s even punishment is implemented. The assumption Cn + 1 ≥ δ must be illogical, and it should be Cn + 1 < δ , so C + n (1 − δ) < 0. Also, it is obvious that C + 0 (1 − δ) > 0, we combine them and form C + n (1 − δ) < 0 < C + 0 (1 − δ) .
(11)
Then we can determine the rationality of existence of C + A (1 − δ) = 0, in which A can change from 0 to n, and it can C be transformed into A = δ− . There we must emphasize that 1 − δ < 0, if not, C + A (1 − δ) = 0 cannot be logical because 1 C > 0 and A > 0. Based on these, the positive and negative of F can be obtained by comparing nc and A. If nc < A, it must be g = (1 − δ) nc + C > (1 − δ) A + C = 0 because 1 − δ is negative. Then, F is negative because of the correlation of g and F . If nc ≥ A, then g ≤ 0 and F ≥ 0 can be proved in the same way. We get Proposition 1. Proposition 1. If the number of cooperators nc is less than A when cooperators punish defectors, g > 0 and F < 0 is certain, defection is the superior strategy. If nc ≥ A, we can obtain g ≤ 0 and F ≥ 0, then cooperation is better. 3. Game 3.1. Two-player game Cooperators can choose punishment or not, so each player in the well-mixed cluster has three strategies, which are defection, cooperation with punishment and cooperation without punishment. However, all cooperators equally pay the cost together when punishment is implemented, accordingly the three strategies cannot coexist, and two strategies can coexist at the most. Based on this argument, we divide all players into two groups when the size of cluster n is fixed, namely, Groups I and II, and their quantities are represented as n1 and n2 respectively. Players choose a same strategy in each group, it is the formation of same-strategy groups. Each group could choose a strategy in its own strategy space, then we obtain the following game model. To review our settings in this paper, punishment does not exist if there is no defector, and it means everyone’s strategy is cooperation without punishment. So, there are three blanks in this table, which is reasonable. We can work out Uc , Ud , Uc′ , and Ud′ when all the factors are certainly based on Eqs. (3) (4) (6) and (7), and utilize 1 them to analyze each utility level in Table 2. For example, U11 is the utility of player in Group I when all groups 1 choose defection together. Also, U11 is the utility of defector when no cooperator and punishment. So, it is obvious that 1 U11 = Ud (nc = 0) = 0 because of Eq. (4). Subsequently, we analyze equilibrium. First, if the equilibrium of this model is that all players are defectors, which becomes a social dilemma, everyone cannot obtain an increment of the total utility and thus tends to leave the cluster. This equilibrium should be avoided. Second, the coexistence of cooperator and defector can be equilibrium as well. The third equilibrium is that everyone chooses to
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cooperate and it is called full cooperation. Everyone’s total utility is Uc = k1 − 1 > 0 because of Eq. (3) when nc = n. This scenario is actually the ideal state that the two-player game hopes to achieve. Cluster scale is initially in a small state. It is easy to form same-strategy groups because the correlation between formation difficulty and cluster scale are positive, and it will be proved in Section 4.1. After the two-player game, equilibrium will be obtained as the basis to analyze large-scale cluster by a multiplayer game. 3.2. Multiplayer game In a large-scale cluster, it is difficult to form same-strategy groups, and each player should decide individually to be a cooperator or defector. Player may change strategy when other player’s utility is bigger, which will lead to a change in the number of cooperators nc . After the change of nc , all cooperators should reconfirm their strategies on whether to punish or not, then everyone will get a new total utility. However, the phenomenon that some players’ utility is bigger may also exist here, the above process will be repeated. The number of players n can also change when a player enters or leaves, which will be analyzed in Section 3.2.2. We study the cluster when n is constant. 3.2.1. Game with fixed scale The number of players n represents the scale of cluster, so n is constant when the scale is fixed. Then players can transform their strategies between cooperation and defection in the pursuit of maximizing utility level. If the number of cooperators nc does not change or fluctuates in a small range, it means the formation of equilibrium. Before the analysis of equilibrium, we set that hnc = Uc′ − Uc and Fn∗c = Uc∗ − Ud∗ , where Uc∗ is defined as the final utility of cooperator after the choice about punishment and Ud∗ is that of defector. We use hnc to judge whether to punish or not, and use Fn∗c to judge whose final utility is bigger. For example, hnc > 0 represents that cooperators can obtain additional utility by implementing punishment, punishment is the better strategy and Uc∗ = Uc , also, the final utility of defector Ud∗ equals to U ′ d , Fn∗c = F = Uc′ − Ud′ . When hnc ≤ 0, we can utilize the same way to confirm that cooperators will not punish defectors, it will be Uc∗ = Uc , Ud∗ = Ud , and Fn∗c = f = Uc − Ud . Based on these setting, we obtain these following equilibria, and set equilibrium quantity of cooperator as B (0 ≤ B ≤ n). (1) Stable equilibrium If all players will not transform their strategies in a state, the state is called stable equilibrium. Three stable equilibria exist. First, if defector’s final utility Ud∗ is bigger than cooperator’s when only one cooperator exists, the latter will become a defector. So, this scenario is a social dilemma when F1∗ < 0, and the equilibrium quantity of cooperator B is 0. Second, if Fn∗−1 > 0, it means that the quantity of cooperators is n − 1 and the only defector will definitely choose to be a cooperator. This scenario reaches an equilibrium called full cooperation, in which everyone chooses cooperation. We also know B = n. Third, the equilibrium conditions are FB∗ = 0, FB∗+1 < 0 and FB∗−1 > 0. When nc = B, it lets all players’ final utilities be same. In addition, defector’s final utility Ud∗ is better with the increase in B by 1, and cooperator’s final utility Uc∗ is larger if B decreases by 1. The equilibrium value of cooperator’s number must be B and is stable. (2) Vibratory equilibrium If FB∗ < 0 and FB∗−1 > 0, it will form the vibratory equilibrium. Defector is more satisfied with his/her final utility when nc = B, so nc will change from B to B − 1. Then it will let cooperator’s final utility Uc∗ be higher because FB∗−1 > 0, and cause that nc changes from B − 1 to B again. Therefore, we can determine nc will change continuously between B − 1 and B. It is the formation of vibratory equilibrium. We see B as the equilibrium quantity. If FB∗+1 < 0 and FB∗ > 0, it is the same as the above analysis, so we can use FB∗ < 0 and FB∗−1 > 0 as the only premise of vibratory equilibrium. (3) Unstable equilibrium This equilibrium is similar to the third stable equilibrium. When nc = B, all players’ final utilities are also same. But cooperator obtains more when nc = B + 1, it will promote nc to increase because FB∗+1 ≥ 0, or nc will decrease when nc = B − 1 because FB∗−1 ≤ 0. This state is an equilibrium, which will be broken with the change of nc , so it is unstable. We have known all equilibria and the formation conditions of them by now. However, several of them are illogical in our paper. Starting the analysis on feasibilities of all equilibria is the basis for next study. Based on the results about f and F in Section 2 and the setting of hnc , we can acquire the following proposition. Proposition 2. It must be Fn∗c = f < 0 when hnc < 0, because cooperators will not choose punishment. But if hnc ≥ 0, the positive and negative of Fn∗c is undetermined, it is related to the difference between nc and A. Then we must let hnc ≥ 0 and nc > A for obtaining Fn∗c = F > 0. We use the Table 3 to present the proposition. So, there should be enough cooperators to choose punishment and spare the cost, it is the sufficient and necessary condition of that cooperators can get more than defectors. Then we will prove that the second stable equilibrium, called full cooperation, is a part of vibratory equilibrium and the third stable equilibrium cannot exist. For full cooperation, its formation condition is Fn∗−1 > 0. Cooperators can gain more utility than defectors when nc = n − 1, so the only defector will transform his/her strategy and nc will change from
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B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796 Table 3 The analysis of F ∗ in different conditions. nc < A nc = A nc > A
hnc < 0
hn c ≥ 0
Fn∗c = f < 0 Fn∗c = f < 0 Fn∗c = f < 0
Fn∗c = F < 0 Fn∗c = F = 0 Fn∗c = F > 0
n − 1 to n. Accordingly, there is not punishment when there is no defector, and defection is the dominant strategy, so nc will change from n to n − 1 again. It is clear that full cooperation is one part of vibratory equilibrium actually. For the third stable equilibrium, FB∗ = 0 is one limitation when the equilibrium quantity of cooperators is B. We can know B = A and hnc ≥ 0 based on Table 3. Thus, it is obvious that FB∗−1 < 0 because B − 1 < A. However, FB∗−1 > 0 is a limitation of the third stable equilibrium as well. Therefore, this equilibrium cannot exist. The second stable equilibrium is a part of vibratory equilibrium, the third stable equilibrium cannot exist, and unstable equilibrium can be ignored because it is unstable. Thus, only the first stable equilibrium and vibratory equilibrium are possible outcomes. However, the first stable equilibrium is a dilemma, so the equilibrium we aim to form is vibratory equilibrium. Formation conditions of vibratory equilibrium are FB∗ < 0 and FB∗−1 > 0. Based on Table 3, FB∗−1 > 0 means hB−1 > 0 and B − 1 > A, and it must be B > A. Then hB < 0 is necessary when we hope to get FB∗ < 0. So, for the vibratory equilibrium, cooperators will choose to punish defectors when nc = B − 1. But when nc = B, cooperators will not choose punishment. 3.2.2. Game with unfixed scale Since there is no limitation on players, and they can freely leave or enter the cluster, the number of players n will be unfixed. After the analysis in Section 3.2.1, we can acquire an equilibrium between vibratory equilibrium and social dilemma when n is constant. If utilities of all players are positive at equilibrium, the cluster is attractive to foreign people and n will increase, n will decrease if one’s utility is not positive. After the change of n, we will get new equilibrium and work out new utilities of all players, then again use the utilities of all players to analyze the change of n. Until n does not change or fluctuates in a small range, it means the formation of stabilization. We must emphasize that the cost of punishment C will be varied with the change of n. Obviously, the difficulty of implementing punishment will increase faster and faster with cluster scale grows, because punishment efficiency and 2 > 0 and ddnC2 > 0, and simply set scale are negatively related when the cluster has been large-scale. So, we can obtain dC dn C = k3 n2 . After the setting on C , the difficulty of analyzing stabilization has increased as well, we will utilize simulation to solve it in the next section. 4. Simulation This section divides the evolution process of cluster into three stages based on the change of n. The first stage is cluster formation, in which two same-strategy groups start a game and cause the emergence of full cooperation, and the cluster scale should be small that proved in Section 4.1. Then the full cooperation makes players’ utility be k1 − 1 > 0 and attracts other player outside. Thus, the second stage is called cluster extension. However, it will be impossible to achieve that everyone chooses to join a same-strategy group, so we utilize multiplayer game to explain the extension process based on the interactions among four factors (k1 , k2 , C , δ ). Finally, the third stage is boom and bust. Understanding the extension process based on all players’ utility will help us obtain stabilization by simulation, then we can determine the cluster is boom or bust through the change of stabilization with different C . There we set several restrictions on key factors, k1 ∈ [3, 5], k2 ∈ [0.5, 1], C ∈ (0, 6], and δ ∈ (1, 2.5]. Except the restriction of k1 is set by referring some papers [22,34,47,53], others’ rationality will be explained in Section 4.2. The number of players n can change spontaneously in our model, the range of n will be certain by the combination of the above factors, which will be analyzed in Section 4.3. We study these stages and draw figures correspondingly using MATLAB. 4.1. Cluster formation We form same-strategy groups and let other factor values be k2 = 0.75, C = 2, and δ = 2. The equilibrium is denoted by the equilibrium utility of player in group I. By the curve below Fig. 1 which represents k1 = 3, we know equilibrium is full cooperation when n1 changes between 7 and 13, and each player’s utility is k1 − 1 = 2. We also acquire that the range of n2 is [7,13] as well because n = n1 + n2 , so each same-strategy group should have seven players at least, which is the condition of formation of full cooperation. However, it is possible to get full cooperation with over four players in each group when k1 = 5. Therefore, the increase of k1 is advantageous to form full cooperation. We also form two same-strategy groups and give k2 = 0.75 and δ = 2. However, we only set C = 10, because the following result will become more obvious if C increases faster and faster with n grows based on the analysis of C in Fig. 4. Compare the curve in Fig. 2 with that below Fig. 1, we discover that it is more difficult to form full cooperation when n is 100, because the number of players in each same-strategy group needs to reach over 30, such scenario is impossible with a variety of people. So, we acquire Proposition 3, and see full cooperation as basis for the next stage.
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Fig. 1. Two-player game equilibrium with different n1 when k1 is 3 or 5 and n = 20.
Fig. 2. Two-player game equilibrium with different n1 when k1 is 3 and n is 100.
Proposition 3. Same-strategy group can solve the public goods problem and form full cooperation, but it is only used in the small-scale cluster. 4.2. Cluster extension k1 = 3, C = 5, δ = 2, n = 50, and the initial value of nc is in the range of n/4 ≈ 13 to 3n/4 ≈ 38. Player will change strategy when other players’ utility is bigger, which will lead to the change of nc . If nc does not change or fluctuates in a small range, it means the formation of equilibrium. Thus, we obtain that the cooperation level will be improved as k2 increases because of Fig. 3. The Fig. 3 shows that equilibria all are social dilemmas with different initial value of nc when k2 = 0.5, and the correlation between k2 and cooperation level at equilibrium is positive, so social dilemma will be an inevitable outcome when k2 < 0.5. The analysis about k2 < 0.5 is meaningless. Also, we know k2 ≤ 1 because of the attribution of fairness mechanism, so k2 ∈ [0.5, 1]. The analysis of Fig. 4 is same to that of Fig. 3. The commonalities of these two subfigures are k1 = 3, k2 = 0.75, and n = 50, but the values of C and δ are different. The left settings are C = 1, 2, 3, 4, 5, 6 and δ = 2, it is clear that the lower C is, the easier it is to form full cooperation. Then we draw figure when C = 5 and δ = 1.5, 2, 2.5 on the right, and find that δ is negatively correlated with cooperation level at equilibrium too. It is obvious that equilibrium is a social dilemma
8
B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
Fig. 3. Multiplayer game equilibrium when k2 = 0.5, 0.75, 1 with 50 steps.
Fig. 4. Multiplayer game equilibrium when C = 1, 2, 3, 4, 5, 6 and δ = 1.5, 2, 2.5 with 50 steps.
when C = 6, then we combine the relationship between C and cooperators’ equilibrium number, and know it should be C ≤ 6, which is similar to k2 . Accordingly, C ∈ (0, 6]. We also can utilize the same means to prove δ ≤ 2.5, and we have calculated δ > 1 in Section 2.2, so δ ∈ (1, 2.5] is reasonable. This result about δ in Fig. 4 should undergo further analysis. When cooperators choose to punish defectors, the increase of δ is advantageous for cooperators, it has been proved in Section 2.2. But the increase of δ could lead to the ∂ (σ ′ )
2
nc losses of cooperators and defectors all, of course, defectors lose more. The reason is that ∂δ = −2g n− > 0 when n2 implementing punishment causes g < 0 based on (B.2) in Appendices, it is possible that cooperators and defectors all feel unfair. Cooperators could choose no punishment when U ′ c decreases but Uc is invariable, then it will cause nc decrease when without punishment based on the result in Section 2.1. So, the increase of δ is advantageous to cooperators with punishment, but cooperators will not punish when it forms Uc > U ′ c . The change of k1 will result in the utilities of all players increase or decrease together. Consequently, no change is expected for strategies. If we want to improve the number of cooperators in equilibrium, three ways are suggested, increasing k2 , decreasing C , and decreasing δ . But decreasing δ is ineffective because it improves defector’s utility greatly, which will reduce cooperator’s evolutionary advantage. Thus, increasing k2 and decreasing C are our choice. There we must emphasize that C will increase faster and faster with cluster scale grows, and use C = k3 n2 that set in Section 3.2.2 to analyze the extension process. n2 because of the ranges of C and n, and other factors are k1 = 3, k2 = 0.75, δ = 2. After the analysis We let C = 600 of roles of all factors, we can obtain equilibria with difference scale, and use cooperators’ equilibrium quantity to denote equilibrium in Fig. 5. When each player’s utility is positive, it will attract outside players pursuing maximal utility, and
B. Zhang, Z. Cui and X. Yue / Physica A 532 (2019) 121796
9
Fig. 5. Cooperator’s equilibrium quantity when n changes from 20 to 65.
Fig. 6. Cluster stabilization when initial n changes from 20 to 80 with 50 steps.
then cluster extends. If everyone is defector and it forms social dilemma, the cluster will not grow again, so we get the largest size of the cluster and it is 64. Next, we start an analysis of the third stage.
4.3. Boom and bust 2
n In Fig. 6 k1 = 3, k2 = 0.75, C = 600 , and δ = 2 as well. The change of n is determined by the positive and negative of all players’ utilities. When cluster scale is small initially, everyone’s utility is positive and the cluster will extend. Moreover, the cluster could start with too many players, and then the cluster scale will decrease. When the initial value of n changes between 40 and 64, n will fluctuate in a small range finally. It is the formation of stabilization, and the upper limit of stabilization is same to the limit of cluster in Fig. 5 (see Fig. 7). Other settings except k2 and C are same to Fig. 6. We obtain the change of cluster stabilization that shown in Fig. 7, and set the increase and decrease of stabilization as boom and bust respectively. It is obvious that the relationship of k2 and stabilization is positive, but the correlation of C and stabilization is negative. Thus, increasing k2 and decreasing k3 are two ways for cluster to form boom.
10
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Fig. 7. Cluster stabilization when k2 = 0.5, 0.75, 1 and C =
2 2 n2 , n , n 400 600 800
with 50 steps.
5. Discussion In Section 2, we calculate the utilities of cooperator and defector based on Static Analysis. Then we utilize Comparative Static Analysis in Section 3. We compare the utilities of cooperator and defector, which is the basis of the change of cooperators’ number, and compare the utilities of players in the cluster with that of foreign players which is the reason why players’ number changes. Finally, we employ Dynamic Analysis to obtain the process of cluster evolution over time in Section 4. To summarize, we have introduced same-strategy group in PGG and found the public goods problem indeed can be effectively solved, full cooperation is the equilibrium. We also show that fairness mechanism with peer punishment is useful, in which we divide the player’s utility into two parts, profit utility and fairness utility, and set everyone pursues maximum utility. We find that coexistence of cooperators and defectors is the dominant outcome, increasing fairness factor and decreasing punishment cost are two effective ways. Therefore, the key to solving the public goods problem is to focus on fairness and difficulty of implementing punishment. Also, there is a limitation in our paper, and the setting of Eq. (2) has certain subjectivity. Finally, some issues such as the rock–paper–scissors type of cyclic dominance among cooperator, defector, and punisher are not covered in this study as we assume cooperators equally pay the punishment cost. We would like to explore this subject in the future work. Prior works have studied the strategic change when the number of players is n in well-mixed or structured population [3,19,22,53,54]. However, we focus on the change of players’ number n when players are free to enter and exit because public goods are nonexclusive. Different from the punishment mechanism used in prior papers [19,21–24], we here use a fairness mechanism and the equation determining fairness utility is set based on a previous study [57] and realistic experience. Although we also consider the peer punishment strategy in our model, such strategy is mainly to provide the possibility of rising fairness utility. We finally emphasize that such fairness mechanism is simple, but is demonstrated to be an effective approach for resisting public goods dilemma. Acknowledgments This research is supported by Humanities and Social Science Fund of Ministry of Education of China (No. 18YJA630137) and National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (No. 71303143). Appendix A. Utilities of cooperator and defector when cooperators do not punish defectors When qc = 1 and qd = 0, we can work out the mean and variance of total contributions.
µ= σ2 =
qc nc + qd nd n
=
nc n
.
(A.1)
(qc − µ)2 nc + (qd − µ)2 nd
=
nnc − n2c
.
n n2 Analyze players’ fairness utility Uif of two strategies (i = c , d) based on (2). Ucf = k2 (µ − qc − 1) σ 2 = −k2
nc (n − nc ) (2n − nc ) n3 nc (n − nc )
.
(A.2)
(A.3)
2
Udf = k2 (µ − qd − 1) σ 2 = −k2
n3
.
(A.4)
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11
Because of (1), (A.3) and (A.4), we obtain Ui (i = c , d) as follows
( Uc = Ud =
k 1 nc n
k1 nc
) nc (n − nc ) (2n − nc ) − 1 − k2 . 3
− k2
n
(A.5)
n
nc (n − nc )2 n3
.
(A.6)
Appendix B. Utilities of cooperator and defector when cooperators punish defectors
( )2
When cooperators punish defectors, we analyze the new mean µ′ and variance σ ′
( µ =
1+
′
C nc
)
nc + δ nd
=
n
(
1+ ( ′ )2 σ =
C nc
−
−δ
g n
)2
(1 − δ) nc + C n nc + δ −
(
g n
+δ =
−δ
)2
n
nd
g n
of total input as follows:
+ δ.
= g2
(B.1)
n − nc n2 nc
.
(B.2)
Then we study the new fairness utilities U ′ if of cooperator and defector based on (2) as well.
) ( ′ )2 n − nc (n − nc )2 σ = −k2 g 3 − k2 g 2 2 . 3 2
Ucf′ = k2 µ′ − q′c − 1
(
) ( ′ )2 σ = k2 g
′ Udf = k2 µ′ − q′d − 1
(
n nc 3 n − nc
n − nc
n3 nc
n2 nc
− k2 g 2
n nc
.
(B.3) (B.4)
After the consideration about (1), (B.3) and (B.4), we can determine the total utilities of cooperator and defector, respectively, as follows: ′,
Uc = U ′,
′ cp
+U
n − nc (n − nc )2 − k2 g 3 − 1+ − k2 g 2 2 . n nc n3 n2c n nc ) ( n − nc n − nc k 1 nc − δ + k2 g 3 3 − k2 g 2 2 . =
[
′ cf
Ud = U ′ dp + U ′ df
=
k1 nc
n
(
C
)]
n nc
n nc
(B.5) (B.6)
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