Non-Fixed Investment in Voluntary Public Goods Games - Springer Link

Journal of the Korean Physical Society, Vol. 72, No. 9, May 2018, pp. 972∼979

Non-Fixed Investment in Voluntary Public Goods Games Shuai Wang School of Economics, Peking University, Beijing 100871, China

Zhaojin Xu∗ School of Science, Tianjin University of Technology, Tianjin 300384, China

Lianzhong Zhang† Department of Physics, Nankai University, Tianjin 300071, China (Received 20 June 2017, in final form 1 August 2017) In this work, we introduce a non-fixed investment ratio to the public goods games in which players can determine whether or not to participate and how much money to invest into the common pool, and with it a new mechanism has been established. We explicitly demonstrate a different rockscissors-paper dynamics which is a consequence of this model. Meanwhile, it is shown how the mechanism of non-fixed investment ratio influences the players’ decision. In addition, we found that the length of memory has an important effect on the average payoff of the population by this introduction. PACS numbers: 02.50.Le, 87.23.Ge, 87.23.Kg Keywords: Public goods games, Non-fixed investment, Memory DOI: 10.3938/jkps.72.972

I. INTRODUCTION Cooperative and competitive behaviors exist widely in natural or social systems [1, 2]. Game theory can well describe and explain these phenomena. The Prisoner’s Dilemma model is one of the earlier game models, which can help people understand the process of game [3, 4]. “The Tragedy of the Commons” leads to an inevitable collapse of cooperation among unrelated individuals, such as our fail to reduce pollution and sustain the global climate [5].The public goods game(PGG) model extends the prisoner’s dilemma model into a multiplayer game model. It can also be regarded as a very useful tool to study qualitatively interactions among human beings [6,7]. In this model, each individual has the motivation to make null contributions and exploit other cooperators. Voluntary PGG model provides a mechanism to achieve sizeable levels of cooperation among non-kin populations [8–10]. In a voluntary PGG, one sample group of N players is selected from a large population randomly. All the players within such a group must simultaneously decide whether to participate in the PGG or not. Each individual can be a cooperator who will contribute a fixed amount of money c to the common pool, a defector who ∗ E-mail: † E-mail:

[email protected] [email protected]

pISSN:0374-4884/eISSN:1976-8524

participate but attempt to exploit, as a free rider, the contributions of the common pool, or a loner who are unwilling to participate in the PGG but gain a small but fixed payoff σc. The amount of money in the common pool is then multiplied by an interest rate r and divided equally among all the participants irrespective of their contributions. If all players cooperate, they will end up with (r − 1)c dollars each. But each individual is faced with the temptation to be a free rider. If all the players are perfectly rational, they will invest nothing. Such behavior prescribed to “homo oeconomicus” leads to economic stalemate [11]. A number of theories have been suggested to maintain and facilitate cooperation. For example, direct reciprocity [12, 13] and indirect reciprocity [14, 15], punishment [16, 17], reward [18], spatially structured populations [19], voluntary participation [8–10], and tolerance [20–22]. Traditionally, most researchers pay attention to the imitation rules about the cooperation in social dilemmas, with the central argument that an individual could compare his payoff with those of the others [23–26]. But the imitation rules will fail if an individual knows only his own payoff in each round. In this work, we consider the voluntary PGG with nonfixed amount of investment. Players can decide not only whether to participate in the public goods game or not, but also how much money to invest. Cooperators who are willing to participate in the PGG can contribute a

-972-

c 2018 The Korean Physical Society

Non-Fixed Investment in Voluntary Public Goods Games· · · – Shuai Wang et al.

-973-

non-fixed amount of money λc to the common pool. In one model, the cooperators can adjust the parameters λ in each round.

II. THE MODEL AND DYNAMICS In this model, we consider the voluntary PGG on square lattice with periodic boundary conditions. Every player is confined to a site x on the lattice and interacts only with his four nearest neighbors. The payoff to the player then depends on its strategy as well as the composition of its four neighbors. The players who play in the PGG consist of cooperators C, defectors D, and loners L. So, if nc , nd , and nl denote the number of C, D, and L (with nc + nd + nl = N = 5), then the net payoff for cooperators PC , defectors PD , and loners PL is given by rλcnc − λc, n c + nd rλcnc = , n c + nd = σc.

PC =

(1)

PD

(2)

PL

(3)

In particular, if only one player participates in the PGG (i.e., nc + nd = 1), he should be accounted as a loner and gain a fixed payoff σc. For a public goods game deserving its name, from Eq. (1) and Eq. (2), we must have 1 < r < N.

(4)

The first inequality states that all cooperators are better than all defectors; the second inequality states that it is better to betray than to cooperate for an individual. Because defection is the dominating strategy, so players can choose not to participate in the game and become a loner. These players prefer to rely on a small but fixed payoff PL = σc with 0 < σ < λ(r − 1),

(5)

such that the players in a group where all cooperate are better off than loners, but loners are better off than defectors. Combining Eq. (4) and Eq. (5) we can get 1

σ . r−1

(7)

Equation (1) and Eq. (2) imply that the difference between the payoff of the defector and the cooperator is PD − PC = λc.

(8)

Fig. 1. (Color online) The percentage of all players desired investments (black squares) and the percentage of high willingness investors (red circles) at dynamic equilibrium state in the spatial voluntary public goods games as a function of λmax . The parameters are r = 3, σ = 1, ω1 = ω2 = 0.02, and μ = 0.001.

The above expression indicates that a domination pure strategy in voluntary PGG does not exist, and there is a rock-scissors-paper cycling dynamics of the three strategies in the system. If there are too many cooperators, the payoffs of defectors may increase and the number of defectors would increase. If there are too many defectors in the system, the payoffs of cooperators and defectors would become very low, increasing the number of loners. If there are too many loners, players who participate in the PGG will get more benefits in small groups, which leads to a growth of the number of cooperators. In order to allow the coexistence of the three strategies, Eq. (6) must be satisfied. Supposing that the players in spatial PGG can play mixed strategies. Then the mixed strategy of player x can be characterized by sx (pc , pd , pl , λ) (with i=c,d,l pi = 1 and pi ≤ 0, λ > 0), where pi represents the probability of i−strategy, λ represents the investment ratio of cooperators. In other words, at the beginning of the game, all players will choose a feasible strategy with a pre-specified probability and be assigned an initial investment ratio. As the game progressed, player x updates his mixed strategy and adjusts his investment ratio only if he is a cooperator according to his payoff and strategy in last round. For simplicity and without loss of generality, we set the fixed cost c equal to 1, so a cooperator contribute a non-fixed amount of money λ to the common pool, and a loner can obtain a small but fixed income σ. Player x increases the probability of the last round strategy if the payoff satisfied him, and decreases it if not. If he is a cooperator in the last round, he should adjust his investment ratio at the same time, so he will contribute to the common pool using his latest investment ratio when he becomes a cooperator in the later rounds. All players will realize that they can ob-

-974-

Journal of the Korean Physical Society, Vol. 72, No. 9, May 2018

Fig. 2. (Color online) The frequency of cooperators (a) and the average payoffs of population (b) in the spatial voluntary public goods games as a function of λmax . The parameters are σ = 1, ω1 = 2, ω2 = 1, and μ = 0.001.

tain at least a small but fixed income σ when they refuse to participate in the PGG at each round. So we assume that each player will compare his last round payoff with σ to determine whether the last round strategy is satisfactory. Assuming that player x chooses i−strategy and obtains payoff Pi in the last round, the evolution of his mixed strategy is then given by sx (p l , p c , pd , λ ) if i = C , (9) sx (pl , pc , pd ) → sx (p l , pc , p d , λ ) if i = D with p l = max [0, min [1 − pj , pl − ω1 (Pi − σ)]] , p i = 1 − p l − pj , λ = max [0, λ + ω2 (Pi − σ)] ,

(10) (11) (12)

where ω1 , ω2 denotes the magnitude of memory (that is, the player’s mixed strategy is determined by the lastround payoff only when the values of ω1 and ω2 are large), i = c, d and j = d, c (the opposite of i), respectively. Therefore, players in our model tend to choose the strategy that performed better in the history of the game. And they will increase the proportion of investment if their earnings through cooperation are greater than σ. The mechanism of non-fixed investment can promote cooperation in the cooperative environment and hinder cooperation in non-cooperative environment. In addition, following Ref. 26, we assume that players have a possibility μ to explore strategies, namely, choose random strategy.

III. RESULTS AND DISCUSSION Now, let us start analyzing the dynamical behaviors of the system in our model. Eq. (7) implies that there is

Fig. 3. (Color online) The rock-scissors-paper cycle of the frequency of three strategies in our model. The lines start in various points but all end up at the same fixed point Q. The arrows show the varying states of cooperators, defectors and loners. The parameters: N = 5, σ = 1, r = 3.0, ω1 = ω2 = 0.1, λmax = 1.5, and μ = 0.001.

a minimum value λmin to ensure the rock-scissors-paper cycling dynamics. But for λ → ∞, the term σλ → 0. The first inequality of Eq. (6) implies that L becomes the worst strategy, so we should remove the L strategy in the game. In other words, in our next analysis, the parameter λ must have a maximum value λmax , to ensure that L is not deleted and the minimum value λmin , to ensure that L is not a dominating strategy. Our simulations were carried out for a population of N = 100 × 100 individuals. Eventually, the system reaches a dynamic equilibrium state. The simulation results were obtained by averaging over the last 50000 Monte Carlo steps of the total 60000. In the game, players who gained a satisfactory payoff will increase its investment ratio λ. So they can increase their investment ratio more with an increase of λmax . The average percentage of all players desired investments (i.e., the average investment ratio λ of 100 × 100 individuals) and the percentage of high willingness investors (the players whose investment ratio λ is greater than 0.9λmax ) will rise (see Fig. 1). When λmax is equal to 4.5, everyone in the system will have a high willing investment ratio which is close to λmax , because the average payoffs of cooperators are higher than loners. As a result, most players who have become cooperators have increased their investment ratios. We can reasonably predict that when λmax → ∞ and r = 3, the frequency of loners will be close to zero, the sum of frequencies of cooperators and defectors will be close to one, and the average investment ratio of population will approach infinity. But it does not mean that the frequency of cooperators will certainly rise (see Fig. 2(a)). Equation (8) shows that the difference between the payoffs of defectors and those of cooperators is increasing synchronously with the increasing of average investment ratio of pop-


Fig. 4. (Color online) The number of players in 100 × 100 individuals at dynamic equilibrium state in the spatial voluntary public goods games as a function of λ for parameters N = 5, σ = 1, ω1 = ω2 = 0.1, λmax = 1.5, μ = 0.001, and (a) r = 2.6, (b) r = 2.8, (c) r = 3, (d) r = 4.8.

ulation, which encourages defection and hinders cooperation. We can see that for low r value, the frequency of cooperators increases initially and decreases eventually with increasing λmax . But for high r value, the frequency of cooperators always decreases with increasing λmax . Equation (1) and Eq. (2) imply that the preconditions for the formation of cooperation are large enough r or large enough λ. Equation (8) implies that a large λ is not conducive to cooperation. So for both small r and small λ, the level of cooperation is very low. Increasing λ can facilitate the emergence of cooperation for a lower r, for example, when λ approaches infinity, r only needs to be greater than one to facilitate the emergence of cooperation. However, after the emergence of cooperation, Eq. (8) implies that increasing λ may lead to a decline in the level of cooperation. The average payoffs of population rise with increasing r or λ although the frequency of cooperators does not necessarily rise (see Fig. 2(b)). This is the same as the results of Ref. 27, which predicted that loner will decrease with the increasing of r. So there are more players to participate in the PGG at this time which increases the average population income as well as the average investment ratio of population. Figure 3 shows that our mechanism would drive the system to end up at a fixed point Q, which is the same as the results of Ref. 28 although the dynamics of our model are different from it. Meanwhile, Fig. 3 also shows that when no cooperators exist or no loners exist initially, the system will reach a more loners position first, and then reach a more cooperators position, and ultimately reach an equilibrium state. This is because the initial conditions that are detrimental to the cooperators will lead to a decrease in the number of cooperators and a decrease in the average investment ratio of population simultaneously. However, when the initial defectors are less than cooperators, the system will reach a

-975-

Fig. 5. (Color online) The position of the center Q depends on the values of the parameters r, σ and λmax : The intersection of the three lines corresponds to N = 5, r = 3, σ = 1, λmax = 1, and μ = 0.001. Each line indicates the displacement of the center when varying a single parameter. The arrows represent the direction in which the three parameters change from small to large.

more cooperators position first, and reach the equilibrium state finally. This shows that individuals who live in an aggregate of cooperators are prone to cooperate with the increasing of the average investment ratio of population. Although previous theories provided a rockscissors-paper cycle [28,29], such a trend is obviously observed in our model and is different from Ref. 28. Note that the dynamics would always drive the system to end up at a same equilibrium no matter what the initial conditions are according to our simulations. The investment ratio λ represents the amount of money that a cooperator invests in a common pool. It can be adjusted according to comparing the payoff of the cooperator with the payoff of the loner. If a cooperator gains more than a loner, he will increase the probability of becoming a cooperator, and increase his investment ratio λ. In the next round, he will have a higher probability to be a cooperator and will invest more if he cooperates. If a cooperator gains less than a loner, he will decrease the probability of becoming a cooperator and decrease his investment ratio λ. In order to quantitatively analyze the influence mechanism of investment ratio λ, it would be helpful to study the information gained from Fig. 4. We can see that for low r value, the payoffs of cooperators and defectors are relatively low, which can reduce both the number of cooperators and the average investment ratio of the population and finally lead to further decline in the level of cooperation. At this time, even if a player becomes a cooperator via mutation, he will only invest a small amount of money into the common pool. Because the interest rate r is low, he will only gain a low payoff, and the phenomena of large cooperative clusters do not occur in this case consequently. But for high r value, the

-976-


Fig. 6. (Color online) The frequency of cooperators at dynamic equilibrium state in the spatial voluntary public goods games as a function of ω1 and ω2 for parameters N = 5, σ = 1, λmax = 1.5, μ = 0.001, and (a) r = 2.4, (b) r = 2.5, (c) r = 2.6, (d) r = 2.7, (e) r = 2.8, (f) r = 3.0.

payoffs of the cooperators are higher than those of the loners, which will increase the payoffs of players participated in the game. So more and more players are willing to participate in the game regardless of whether they are cooperators or defectors. They will also increase their willingness to invest. Therefore, as r rises, the difference of payoffs between participants and loners (i.e., PC − PL and PD − PL ) is increasing. Fewer players choose L and the consequence will be a growth of the number of players participated in the game with increasing willingness to invest. Variations of the three parameters r, σ, λmax allow to position Q variously in the interior of the simplex (see Fig. 5). Increasing the payoffs of loners σ shifts the center along the green dashed line in the direction indicated by the arrow, i.e. towards the corner that all are loners. Similarly, increasing λmax shifts the center upwards on the red solid line and increasing r moves the center to the left, along the blue dotted line. For the limiting cases r = N and σ = 0, Q approaches the left edge and bottom

edge respectively. Although the difference between the payoffs of players participated in the game and those of loners is increasing with the increasing of r, Eq. (8) shows that the difference between the payoffs of defectors and those of cooperators is increasing synchronously because of the increasing of average investment ratio of population (see Fig. 4. The increasing of λmax also may lead to a decline in the level of cooperation (see Fig. 2). But the increasing of r or λmax can lead to a reduction in the number of loners. When the loners payoff σ is close to zero, a player becomes a defector or a loner is no difference, so half of the players in the system are defectors, half of them are loners. When the loners payoff σ is very large, the interest rate r or the maximum value of investment ratio λmax is small, all players in the system will become loners. Apparently, the memory of individuals has an important effect on the behavior of individuals, and cannot be neglected in real-life situation [27]. Players tend to choose the action that has brought them higher incomes


-977-

Fig. 7. (Color online) The frequency of cooperators at dynamic equilibrium state in the spatial voluntary public goods games as a function of ω1 and ω2 for parameters N = 5, r = 2.5, λmax = 1.5, μ = 0.001, and (a) σ = 1.0, (b) σ = 0.8, (c) σ = 0.6, (d) σ = 0.4.

in the earlier history of the game, which means that the heterogeneity of individuals may come from the different environments they live. In real life, individuals living in selfish groups for a long time may not have a high level of cooperation, while individuals living in cooperative groups tend to cooperate. In the remaining text, we discuss how the magnitude of memory ω1 and ω2 affect on the behaviors of individuals. For ω1 → ∞, the present round payoffs have a significant impact on the strategies of individuals, i.e., the individuals are myopic, which is similar to the Win-StayLose-Shift rule. The individuals’ strategies are mainly affected by the current payoff, and the C clusters will collapse immediately once invaded by defectors. That is to say an accidental defection would lead to a breakdown in cooperation. For ω1 → 0, the evolution of mixed strategies are hardly affected by the present round payoffs, i.e., the individuals have longer memory in this case. Individuals will slowly change their strategies even when individuals can or cannot make more money than lon-

ers, so they would still keep their original action for a very long time. For ω2 → ∞, the current payoffs have a significant impact on the individuals’ investment ratios, i.e., the individuals are high-speculative investors, which is similar to the short-term speculative hot money. Individuals will invest more because of current high payoffs, and withdraw their investments because of current low payoffs. For ω2 → 0, the investment ratios of individuals are changed slowly. Even if the cooperators are invaded by defectors, they will not significantly decrease their investment ratios. Likewise, if both the level of cooperation and the cooperators’ payoffs are high, they will not significantly increase their investment ratios. In other words, the magnitude of memory represents the individual’s sensitivity to his surrounding environment. The lower the ω1,2 value is, the more slowly a player reacts to his surrounding, and vice versa. Hence, we argue that an appropriate magnitude of memory is helpful to promote the outcomes of the collectives. In general, it may result in more efficient outcome to

-978-


Fig. 8. (Color online) The frequency of cooperators at dynamic equilibrium state in the spatial voluntary public goods games as a function of ω1 and ω2 for parameters N = 5, r = 2.5, σ = 1.0, μ = 0.001, and (a) λmax = 1.5, (b) λmax = 2.0, (c) λmax = 2.5, (d) λmax = 3.0.

the population when ω1 approximates to 2 and ω2 approximates to magnitude between 1 and 10 (see Figs. 6 8). In fact, this indicates that both the behavior of reacting slowly and focusing on short-term goals will be harmful to the development of the population in reallife situation. Figures 6 - 8 exhibit the evolution of the frequency of cooperators as the interest rate increases (r becomes larger), the payoff of loners decreases (σ becomes smaller) and the maximum value of investment ratio increases (λmax becomes larger) respectively. For the low level of cooperation, the conditions in which the cooperation occurs in the system are moderate sensitivities (see Figs. 6(a) and 6(b)). However, as the level of cooperation rises, the sensitivity of the investment ratio ω2 becomes more and more unimportant. The reason is that the improvement of the cooperation level makes the investment ratio of each individual in the system approach to the maximum value (see Fig. 4). In this case, regardless of the sensitivity of the investment ratio, the system will have a large average investment ratio of pop-

ulation eventually in steady state. We have found that raising the maximum value of the investment ratio, as well as raising interest rates and reducing the payoff of loners, can raise the level of cooperation if the level of cooperation is low.

IV. SUMMARY In this work, we introduce the non-fixed investment ratio in the anonymous PGG and then find a different dynamic mechanism. In this model, people can select to participate in the PGG as a cooperator or a defector, or not to participate in the game as a loner. In addition, a cooperator can also decide how much money to invest into the common pool. The mechanism of non-fixed investment can promote cooperation in the cooperative environment and hinder cooperation in non-cooperative environment. The increasing of λ first facilitates the


emergence of cooperation, but Eq. (8) implies that continuing to increase λ may lead to a decline in the level of cooperation. In one model, players only know their payoffs in each round. They can adjust their mixed strategies and investment ratios continually according to their last round payoffs. We have found that the magnitude of individuals’ memory has an important effect on the evolution of cooperation at the low level of cooperation and the appropriate magnitude of memory is helpful to promote the average payoffs of the population. Besides, our work also indicates that regardless of the initial state of the system, the system would reach to a stable equilibrium in the rock-scissors-paper way according to our dynamic mechanism.

REFERENCES [1] L. A. Dugatkin, Cooperation among Animals: An Evolutionary Perspective (Oxford University Press, Oxford, 1997). [2] K. G. Binmore, Playing fair: game theory and the social contract (MIT Press, Cambridge, MA, 1994). [3] A. V. M. Herz, J. Theor. Biol. 169, 65 (1994). [4] G. B. Pollock, Social Networks 11, 175 (1989). [5] G. Hardin, Science 162, 1243 (1968). [6] A. M. Colman, Game Theory and Its Applications in the Social and Biological Sciences (Butterworth-Heinemann, Oxford, 1995). [7] J. H. Kagel and A. E. Roth, The handbook of Experimental Economics (Princeton University Press, Princeton, 1995). [8] C. Hauert, S. de Monte, J. Hofbauer and K. Sigmund, J. theor. Biol. 218, 187 (2002). [9] D. Semmann, H-J. Krambeck and M. Mininski, Nature 425, 390 (2003).

-979-

[10] G. Szab´ o and C. Hauert, Phys. Rev. Lett. 89, 118101 (2002). [11] E. Fehr and S. G¨ achter, Nature 415, 137 (2002). [12] R. Axelrod and W. D. Hamilton, Science 211, 1390 (1981). [13] R. L. Trivers, Q. Rev. Biol. 46, 35 (1971). [14] M. A. Nowak and K. Sigmund, J. Theor. Biol. 194, 561 (1998). [15] M. A. Nowak and K. Sigmund, Nature 393, 573 (1998). [16] C. Wedekind and M. Milinski, Science 288, 850 (2000). [17] X-J. Chen, A. Szolnoki and M. Perc, Phys. Rev. E 92, 012819 (2015). [18] M. Milinski, D. Semmann and H-J. Krambeck, Nature 415, 424 (2002). [19] C. Hauert and M. Doebeli, Nature 428, 643 (2004). [20] A. Szolnoki and X-J. Chen, Phys. Rev. E 92, 042813 (2015). [21] J. Gao, Z. Li, R. Cong and L. Wang, Physica A 391, 4111 (2012). [22] X-P. C. and DR. G. Bachrach, Organizational Behavior and Human Decision Processes 90, 139 (2003). [23] M. A. Nowak and R. M. May, Int. J. Bifurcat. Chaos Appl. Sci. Eng. 4, 33 (1994). [24] M. A. Nowak, S. Bonhoeffer and R. M. May, Proc. Natl. Acad. Sci. USA 91, 4877 (1994). [25] D. Funenberg and D. Levine, The theory of learning in games (MIT Press, Cambridge, 1998). [26] A. Traulsen, C. Hauert, H. de Silva, M. A. Nowak and K. Sigmund, Proc. Natl. Acad. Sci. USA 102, 10797 (2009). [27] Z-J. Xu, Z. Wang, H-P. Song and L-Z. Zhang, EPL 90, 20001 (2010). [28] Z-J. Xu, Z. Wang and L-Z. Zhang, Phys. Rev. E 80, 061104 (2009). [29] C. Hauert, S. de Monte, J. Hofbauer and K. Sigmund, Science 296, 1129 (2002).