Theoretically, defectors will outperform cooperators within a well-mixed group. However, the emergence of cooperation in a group is widely observed by social and biological scientists. Until recently, the understanding of the emergence of cooperation still remains a fundamental conundrum in social and behavior sciences.^[1–6] The conundrum is a so-called social dilemma and has been extensively investigated as the public good game (PGG) in an enormous body of theoretical studies.^[7–9] In the PGG, cooperators confer benefits on other group members with some contributions to the public good, while defectors exploit the benefits without any contributions. From the perspective of evolution, a defector can obtain a higher individual payoff compared with a cooperator in the PGG. As a result, defection is regarded to be a rational strategy. Therefore, the natural selection process will drive the elimination of cooperation within a population. In order to resist defection and promote cooperation, many effective mechanisms have been put forward in evolutionary game studies, such as kin selection,^[10] direct reciprocity,^[11–13] group selection, reputation,^[14–16] reward,^[17–20] punishment,^[21–28] spatial structure,^[29–41] and so on.^[42–49]

To further reveal the underlying reasons of the emergence of cooperation in the PGG, in this paper, we explore a new mechanism, i.e., deposit mechanism, which is originated from our real life. In the realistic world, people often prefer to get a product or service at a relatively lower price. Therefore, many advertising strategies are invented by companies to promote the selling. One effective way is to provide some discounts under particular conditions. For example, it is widespread that a discount can be obtained by the consumers if they are willing to pay a deposit in advance.^[50] Intuitively, obtaining a public good at a lower contributions might motivate players to choose a cooperation strategy in the PGG. Nevertheless, it is still unclear whether such a deposit mechanism can elicit cooperation from the theoretical view. Considering the role of deposit played in the promotion of cooperation, it is valuable to investigate how such a new mechanism can affect the evolutionary dynamics of cooperation, and whether it can resist defection and promote cooperation.

In a theoretical model of the PGG with a deposit mechanism constructed in this paper, three different strategies are proposed:^[51] cooperators (C), deposit cooperators (DC), and defectors (D). For clarity, we divide each round of one-shot game into three stages and different types of players adopt different behaviors in different stages. Specifically, in the first stage, DC-players pay a deposit; in the second stage, DC- and C-players contribute to the public good, but defectors contribute nothing; in the third stage, the benefits of provided public good will be allocated among players in the one-shot game. In this paper, we assume that the public good will be provided only when the total contributions reach a certain threshold which is the minimum contributions to the provision of the public good. So if the total contributions provided by DC- and C-players are less than the threshold, the provision of the public good is failure. In this case, the one-shot game will end, and the total contributions other than deposit will be returned to DC- and C-players in the second stage. Otherwise, the public good can be provided, and the benefits of the public good will be shared among group members in the third stage. In addition, we consider two types of benefit-sharing ways, which differ on whether or not a contributor in the PGG may oneself benefit, namely “weak altruism” and “strong altruism”. Moreover, we also consider the contributions’ time value, which is different from the previous work.^[52] In view of the aforementioned description, we explore the evolution of cooperation in the PGG with deposit by means of the replicator dynamics.^[53,54]

Consider the PGG with deposit mechanism in an infinitely large and well-mixed population. For clarity, we divide each one-shot round game with this mechanism into three stages. In stage 1, n players are randomly selected from the well-mixed population and then form a group (where ). All players in the group are required to decide whether to pay a deposit or not. The players paying a deposit will provide c₁ resources in this stage and must contribute to the public good in next stage. In stage 2, the players who do not pay a deposit need to make a decision whether to contribute to the public good. Cooperators who decide to contribute will provide c resources and defectors contribute nothing. In addition, the players paying a deposit in stage 1 will be given a ( ) ( ) discount on c so that they only need to provide resources in stage 2. If the amount of the integrated contributions , where and are the numbers of the players who pay the deposit in stage 1 and players who contribute c in stage 2, are less than the threshold T ( ), an one-shot round game will end. As a result, the resources contributed in stage 2 will be returned intact to those who contribute, but the deposit will not be refunded to the players who pay the deposit. On the contrary, if the integrated contributions reach T, the public good will be successfully provided and the benefits of the public good will be shared among group members in stage 3. Furthermore, the benefits of the public good are distributed in the following different way: one is “weak altruism (W)”, in the case of which the benefits of it will be equally shared among all n players including themselves; the other is “strong altruism (S)”, in the case of which the benefits will be shared equally among other co-players except themselves. Combined with the above-mentioned description, we use Fig. 1 to describe the whole process of each one-shot public good game with deposit mechanism. To explore the evolution of cooperation, we consider three different strategies: deposit cooperators (DC) who pay c₁ deposit in stage 1 and contribute in stage 2, cooperators (C) who only contribute c in stage 2, and defectors (D) who neither pay deposit in stage 1 nor contribute in stage 2. In this paper, each round of one-shot public good game has multiple stages. In fact, it is true that money has a time cost in real life. In order to be closer to the reality, here we introduce a discount rate into the theoretical model and suppose that discount rate for each stage is δ . Therefore, when the integrated contributions reach the threshold T, the public good will be provided and all group members can obtain benefit in stage 3, where . In the case of W and S, the benefits of each player can be calculated as follows respectively:

Fig. 1.

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Fig. 1. The specific process of each one-shot round game with deposit mechanism. In stage 1, the randomly chosen n players have to decide whether to contribute to the public good. In stage 2, all players are divided into three different types, namely DC-player, C-player and D-player according to the behavior they adopt in stage 1 and 2. In stage 3, the benefits of the provided public good are allocated among all players.

For simplicity, let be the set that is the list of all possible numbers of three different types’ players in a group with n players, be the set that the list of the numbers of three types’ players that can successfully provide public good, and represent the set that is the list of the numbers of three types’ players that cannot provide public good. Also, the three sets satisfy condition . To investigate how the set varies with the threshold T, three different sceneries are considered. Presented results in Fig. 2 demonstrate that the set becomes smaller when the threshold T of public good increases from 6 to 12 from Figs. 2(a)–2(c). This tendency indicates that the higher the threshold T of public good, the more difficult the successful provision of the public good. So according to the above description, the cost of each player with different strategies can be written as follows:

Fig. 2.

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	Fig. 2. The changes of the magnitude of set and with increasing the threshold T. The red area represents the magnitude of set , while the blue area represents the magnitude of set . The higher the threshold T of public good, the larger the magnitude of the red area.

Next, we apply replicator equations in analyzing the evolutionary dynamics of three strategies in this model. We denote the frequencies of C-, DC-, and D-players in the infinite population with x, y, and z respectively . The expected payoff values for C-, DC-, and D-players are expressed as , , and respectively and can be calculated as

In this part, we will comprehensively explore how the evolutionary dynamics of C, DC, and D strategies are influenced by the strengths of variables and find out the stable equilibrium points in dynamical system. Within the three-strategy environment, the evolutionary dynamics take place in the state space . Indeed, the three homogeneous states in which all players of the population are C-players (x = 1), DC-players (y = 1), and D-players (z = 1) correspond to three vertices of the simplex S₃. For given N, T, θ, r, c which are parameters of the PGG, it can be observed that the number of equilibria in the evolutionary system is determined by parameters k, , and δ. When the values of parameters k, , and δ satisfy the condition , there will only exist one stable equilibrium point in the dynamic system. However, there will appear two stable equilibrium points in the evolutionary dynamics in the case of and (see Appendix A for detail). In what follows, we consider the evolutionary dynamics of these two typical cases under different values of parameters and discuss the resulting game dynamics one by one.

3.1. Case 1:

In this case, according to preliminary numerical simulations, presented results in Figs. 3(a)–3(f) show that there is no interior equilibrium point and all interior orbits in the simplex S₃ converge to the stable equilibrium point on the edge C–D, irrespective of the initial condition. The direction of evolution on the edge C–DC is from C to DC so that C-players always prevail in the competition between C-players and D-players, and from DC to D on the edge DC–D so that D-players dominate DC-players in the competition between them. But the edge C–D of S₃ is separated into two segments by the stable equilibrium point , and both orbits of the two segments converge to the equilibrium point. Moreover, all the three vertices of the simplex S₃ corresponding to full C-players (x = 1), DC-players (y = 1), and D-players (z = 1) are unstable equilibrium points in the evolutionary system. Thus, random drift and occasional invasion of the missing D-player will eventually bring the population to the stable state, which is the coexistence of cooperation and defection.

Fig. 3.

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Fig. 3. The evolutionary dynamics of the three strategies of T and in the public good game. Filled cycles in S3 denote stable equilibria and open cycles denote unstable equilibria. All orbits in the system converge to stable state which is a coexisting state of C-players and D-players. For given T values, the fraction of C-players in the stable state on the C–D edge increases as θ increases. Moreover, for given values the fraction of C-players decreases as T increases. Other parameter values are N = 20, , k = 0.8, c = 1, , and respectively.

Next, we study how the values of parameters T and affect the equilibrium point on the edge C–D. As shown in Fig. 3, for a fixed value of T, slightly increases when changes from 0 to 1. For a high T = 12, shows an upward trend, increasing from 0.49 to 0.5 when changes from 0 to 1. Besides, for T = 8 or T = 6, rises from 0.28 to 0.3, or from 0.13 to 0.18 respectively when increases from 0 to 1. Furthermore, for a fixed value of , increases when T increases. For or 1, when T increases from 6 to 12, significantly rises from 0.2 to 0.9 or from 0.19 to 0.49 respectively. Therefore, both adopting the “weak altruism” benefit-sharing way and increasing the threshold of public good are effective to enhance the level cooperation in the well-mixed population.

3.2. Case 2:

In this case, there will appear two stable equilibrium points in the evolutionary dynamics system. One of the two stable equilibrium points is on the edge C–D and the other on the edge DC–D. There also does not exist any interior equilibrium point and all interior orbits in the simplex S₃ converge to the stable equilibrium point on the edge C–D (Figs. 4(a)–4(f)). The direction of evolution on the edge C–DC is from C to DC so that C-players prevail in the competition between C-players and D-players. But the edge C–D of S₃ is separated into two segments by the stable equilibrium point on this edge, and all of the orbits of this edge converge to the stable equilibrium point . Similarly, the edge DC–D of S₃ is separated into two segments by the stable equilibrium point on the edge DC–D, all orbits of this edge converge to the stable equilibrium point . Besides, the three vertices of the simplex S₃ corresponding to full C-players , DC-players (y = 1), and D-players (z = 1) in the population are unstable equilibrium points in the evolutionary system. Thus, random drift and occasional invasion of the missing D-player will eventually bring the population to the stable equilibrium points, and which one stable state will be converged to mainly depends on the initial state of the population. Due to the irrelevance between the dynamics of the equilibrium point on the edge DC–D and θ, it is not necessary to further analyze the influence of θ on the evolutionary dynamics (see Appendix A). In this part, we further analyze the specifically evolutionary dynamics for different values of parameters T, c₁, and k. As depicted in Fig. 4, for fixed values of c₁ and k, we note that both and increase when T rises from 6 to 12. For example, for , k = 0.8, and increase from 0.23 to 0.53 and from 0.25 to 0.58 respectively when T rises from 6 to 12. Besides, for fixed values of T and k, we observe that does not change with the increase of c₁, but decreases when c₁ rises from 0.1 to 0.15. For example, for T = 12, k = 0.8 or T = 12, k = 0.9, decreases from 0.62 to 0.58 or 0.57 to 0.54 respectively when c₁ rises from 0.1 to 0.15. Furthermore, for fixed value of parameters c₁ and T, we can observe from Fig. 2 that remains unchanged with the increase of k, but decreases when k rises from 0.8 to 0.9. For example, for T = 12, or T = 12, , decreases from 0.62 to 0.57 or 0.58 to 0.54 respectively when k rises from 0.8 to 0.9. In conclusion, to elicit deposit cooperation in the large population, it is effective to raise the threshold of public good and lower k and c₁.

Fig. 4.

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Fig. 4. Evolutionary dynamics of C, DC, and D strategies for different values of T. Nodes C, DC, and D of the simplex denote homogeneous population of C-players, DC-players, and D-players respectively. And open cycles represent unstable equilibrium points and filled cycles means stable points in these pictures. For given T values and c1 values, the fraction of DC-players in the stable state on the D–DC edge decreases as k increases. For given T values and k values, the fraction of DC-players and C-players increase as T increases respectively. Besides, for given values of k and c1, the larger the value of T, the larger the fraction of C-players on the edge of C–D edge and DC-players in the edge of DC–D edge respectively. For given T values, the fraction of C-players in the stable state on the D–C edge remain unchanged irrespective of the values of k and c1. Other parameters values are N = 20, , c = 1, and respectively.

It is inevitable that conflict often appears between contributors and freeloaders in public goods interactions. So far, many theoretical works have revealed that many mechanisms can maintain sufficiently high levels of cooperation. To further explore the emergence and maintenance of cooperation in a population, in this paper, we have introduced the deposit cooperation strategy with which players need to pay a deposit before the PGG. Based on the evolutionary game theoretical models, we have studied the evolutionary dynamics in the infinitely well-mixed population with a new deposit mechanism, by focusing particularly on the role of deposit played in the evolutionary dynamics of cooperation. In order to be closer to reality, this paper has also considered the value of contributions and introduced two different benefit-sharing ways. In the view of above-mentioned discussion, we have demonstrated that a stable coexistence state of cooperators and defectors can appear when . Besides, two stable coexistence states can appear, one is the coexistence of cooperator and defectors, the other is the coexistence of deposit-cooperators and defectors when and . On the one hand, we also have found that adopting the “weak altruism” benefit-sharing way and increasing the threshold of public good T are effective to enhance the level cooperation in the well-mixed population. Interestingly, the level of cooperation is independent of k and c₁. On the other hand, the increase of the threshold T of public good and the reduction of discount k and deposit c₁ are also conducive to enhance the level of deposit cooperation in the well-mixed population. But we also have observed that the level of deposit cooperation within a population has no correlation with the benefit-sharing ways.