We consider public goods games played iteratively over a fixed connected network. The vertices of the network represent the players and the edges represent neighboring connections5,10,11,12. The dynamics evolve in discrete rounds. In each round, each player chooses a behavior that minimizes its cost, where the player’s cost is affected by its own behavior and the behaviors of its neighbors.
Our main model includes three behavior types, namely, defection, hypocrisy, and cooperation, in which those who hardly contribute to the social welfare, i.e., defector and hypocritical players, face the risk of being caught and punished by their neighbors who are non-defectors. The level of risk together with the extent of punishment is captured by a notion that we call “social-pressure”. The main result is that adjusting the level of social-pressure employed against hypocritical players compared to the one employed against defectors can have a dramatic impact on the dynamics of the system. Specifically, letting the former level of social-pressure be within a certain range below the latter level, allows the system to quickly transform from being composed almost exclusively of defectors to being fully cooperative. Conversely, setting the level to be either too low or too high locks the system in a degenerate configuration.
As mentioned, our main model assumes that non-defectors induce mild social-pressure on the defectors among their neighbors. This implicitly assumes that inducing the corresponding social-pressure is beneficial (e.g., allows for a social-upgrade), although other explanations have also been proposed21. To remove this implicit assumption we also consider a generalized model, called the two-order model, which includes costly punishments. Consistent with previous work on the second-order problem, e.g.,23,25,26,27,36,40, this model distinguishes between first-order cooperation, that corresponds to actions that directly contribute to the social welfare, and second-order cooperation, that corresponds to applying (costly) social-pressure, or punishments, on others. As in the main model, the level of punishment employed against first-order defectors may differ from that employed against second-order defectors. We identify a simple criteria for the emergence of cooperation: For networks with minimal degree (Delta), cooperation emerges when two conditions hold. The first condition states that the cost (alpha _2) of employing punishments against second-order defectors should be smaller than the corresponding punishment (beta _2) itself, i.e., (alpha _2<beta _2). The second condition states that the cost (alpha _ 1) of employing punishments against first-order defectors should be smaller than the corresponding punishment (beta _1) times the minimal number of neighbors, i.e., (alpha _ 1<beta _1cdot Delta). The second condition is also a necessary condition for the emergence of cooperation in the two-order model.
The main model
The model considers two extreme behaviors, namely, cooperation (c) and defection (d), and an additional intermediate behavior, called hypocrisy (h). The system starts in a configuration in which almost all players, e.g., (99%), are defectors (see “Methods”). Execution proceeds in discrete rounds. The cost of a player depends on its own behavior and on the behavior of its neighbors. All costs are evaluated at the beginning of each round, and then, before the next round starts, each player chooses a behavior that minimizes its cost (breaking ties randomly), given the current behavior of its neighbors. In other words, we assume that players greedily choose their best behavior, given the current configuration. In our simulations, we also consider a relaxed version, where players choose the best behavior with high probability, and with small probability choose an arbitrary behavior. In contrast to many previous works on cooperation in networks5,6,7,8,9, we assume that benefits from altruistic acts are negligible (i.e., the MPCR is zero), so that a player does not gain anything when others cooperate.
The cost of a player u with a behavior type (iin {d,h,c}) is composed of two components: the energetic cost (E_i) associated with the contribution to the social welfare, and the social-pressure cost (S_i(u)) it faces, that is:
$$begin{aligned} {mathcal{{C}}}_i(u)=E_i+S_i(u). end{aligned}$$
We assume that the energetic cost of a defector is 0, and the energetic cost of a cooperator is 1, where the value of 1 is chosen for normalization:
$$begin{aligned} E_{d}=0quad {text{ and }}quad E_{c}=1. end{aligned}$$
A hypocritical player produces the minimal social welfare required to pretend to be cooperative. Hence, we assume that
$$begin{aligned} 0<E_h<1, end{aligned}$$
thinking of (E_h) as closer to 0 than to 1.
As mentioned above, we focus on relatively mild social-pressure induced by cooperative players, aiming to improve their social status. Since hypocritical players aim to appear similar to cooperators from the perspective of an external observer, we assume that they too induce social-pressure on their neighbors. Defectors, on the other hand, do not induce any social-pressure since such an enhancement of the social status is not justified for them. In principle, cooperators and hypocritical players might induce different levels of social-pressure, yet, for the sake of simplicity, we assume that they induce the same extent of social-pressure. This assumption is further justified by the fact that a player u cannot distinguish its hypocritical neighbors from its cooperative neighbors, hence, u’s calculation of the social-pressure is evaluated assuming all of its non-defector neighbors are cooperators.
Formally, we assume that the possible social-upgrade gain associated with cooperators or hypocritical players as a result of applying social-pressure is already taken into account when calculating the energetic costs (E_c) and (E_h). Since we assume that this gain is small, it hardly perturbs the cost, keeping the energy consumption as the dominant component.
Implicitly, we think of the social-pressure cost incurred by a player u as the product of two factors: (1) the risk of being caught, which is assumed to be proportional to the number of u’s neighbors inducing social-pressure, and (2) a fixed penalty paid when caught, which depends on u’s behavior. The product of the risk and penalty represents the expected punishment in the next round, if behaviors remain the same.
Cooperators are assumed to pay zero penalty, and are hence effectively immune to social-pressure:
$$begin{aligned} S_{c}(u)=0. end{aligned}$$
Conversely, the social-pressure induced over defectors and hypocritical players is non-zero. For a given round, let (Delta _{bar{d}}(u)) denote the number of neighbors of u which are non-defectors at that round. The social-pressure cost induced over a defector, and respectively, a hypocritical, player u is:
$$begin{aligned} S_{d}(u)=rho _{d}cdot Delta _{bar{d}}(u),quad {text{ respectively, }} quad S_{h}(u)=rho _{h}cdot Delta _{bar{d}}(u), end{aligned}$$
where (rho _{d} >0), respectively (rho _{h} >0), represents the social-pressure induced over a defector, respectively a hypocritical, from one neighboring non-defector. Note that when comparing the social-pressure incurred by defectors versus hypocritical players, both the risk of being caught and the extent of punishment are expected to be different. Indeed, since hypocritical players pretend to be cooperators, their risk of being caught is expected to be lower than that of defectors. Moreover, after being caught, the respected punishment of a defector might be different than that of a hypocritical player, depending on the social norms. Altogether, here we focus on the regime where (rho _h < rho _d), since otherwise, becoming a defector is always more beneficial than becoming a hypocritical.
To sum up, at a given round, the total cost incurred by a player u is:
$$begin{aligned} {mathcal{{C}}}(u) = {left{ begin{array}{ll} 1 &{}quad {text{ if }},, u,, {text{ is a cooperator, }} rho _{d} cdot Delta _{bar{d}}(u) &{}quad {text{ if }},, u,, {text{ is a defector, }} E_h + rho _{h} cdot Delta _{bar{d}}(u) &{}quad {text{ if }},, u,, {text{ is hypocritical. }} end{array}right. } end{aligned}$$
Before stating our main result, we recall few standard definitions in graph-theory41. The diameter of a network G, denoted ({text {diam}}(G)), is the maximal distance between any pair of players (see “Methods”). A network is (Delta)–regular, if every player has precisely (Delta) neighbors. Theorem 1 below assumes that the underlying network is (Delta)-regular. However, this theorem can be generalized to arbitrary networks with minimal degree (Delta) (see SI, Theorem 6).
Theorem 1
Consider a (Delta)-regular network G with n players. Assume that
$$begin{aligned} ({1-E_h})/{Delta }<rho _{h}<rho _{d}-E_h. end{aligned}$$
(1)
Then, with probability at least (1-frac{1}{c^n}), for some constant (c>1), in at most (3 cdot {text {diam}}(G)+1) rounds, the system will be in a configuration in which all players are cooperative, and will remain in this configuration forever.
The two stages of the dynamics. The direction of the red and blue arrows indicates the direction of the social-pressure applied on the player occupying the central vertex. Cooperative players pay an energetic cost of (E_c=1) and are immune to social-pressure. A defector player pays a social-pressure cost of (rho _{d}=0.5) per non-defector neighbor. A hypocritical player pays social-pressure cost of (rho _{h}=0.25) per non-defector neighbor, and an energetic cost of (E_h=0.2). (a) First stage: defectors become hypocritical players. A defector player (central vertex on the left) has one non-defector neighbor (in this case, a cooperator), implying that its social-pressure cost is (rho _{d}=0.5). Therefore, that player would prefer to be hypocritical (right), paying only (0.25+0.2=0.45). (b) Second stage: hypocritical players become cooperators. Here, a hypocritical player (central vertex on the left) is surrounded by four non-defector neighbors. In this case, the social-pressure accumulates to favor cooperation (right).
The formal proof of Theorem 1 appears in the SI, Section B. Intuitively, the main idea behind it is as follows. When the extent of social-pressure against hypocritical players is moderate, that is, when (rho _h) satisfies Eq. (1), the transition process can be divided into two stages. At the first stage, since the punishments of hypocritical players are sufficiently lower than those of defectors, specifically, (rho _{h}<rho _{d}-E_h), or equivalently (rho _{h}+E_h<rho _{d}), the presence of at least one neighboring non-defector u makes a hypocritical player pay less than a defector. In this case, u’s neighbors would become non-defectors at the next round (Fig. 1a). Although this does not necessarily imply that u itself remains a non-defector in the next round, it is nevertheless possible to show that the proportion of hypocritical players gradually increases on the expense of defectors. Note that at this point, the social welfare may still remain low, since hypocritical players hardly contribute to it. However, the abundance of non-defectors increases the overall social-pressure in the system. In particular, since the social-pressure on hypocritical players is also not too mild, specifically ((1-E_h)/{Delta }<rho _{h}), or equivalently (1<rho _{h}Delta +E_h), the presence of many neighboring non-defectors can magnify it up to the point that the total cost incurred by a hypocritical player surpasses the energetic cost of being a cooperator (Fig. 1b). At this second stage, cooperators prevail over both defectors and hypocritical players, and so the system converges to a cooperative configuration.
Conversely, severely punishing hypocritical players diminishes the prevalence of such players, preventing the system from escaping the initial degenerate configuration. Contrariwise, incurring too mild social-pressure towards hypocritical players would prevent the second stage of the dynamics. In particular, if (rho _h < (1-E_h)/Delta), or equivalently, if (E_h + rho _h Delta < 1), then a player would always prefer to be hypocritical over being cooperative (even when all its neighbors induce social-pressure). In this case, the system would remain degenerative since the population would consist of a combination of defectors and hypocritical players.
To illustrate the dynamics we conducted simulations over several types of networks. Figure 2 shows how the population evolves over time, when considering a grid network (Fig. 2a) and a random 10-regular network (Fig. 2b). The chosen parameters satisfy the assumption in Eq. (1). In both dynamics, the role of hypocritical behavior as a transitory state, essential to achieving cooperation, is well illustrated by the initial peak of hypocritical players, preceding the rise of cooperative players. Moreover, if hypocritical behavior is disabled (see “Methods”), then the system is unable to escape the defective state (insets).
Evolution of cooperation in grids and random 10-regular networks. (a) Corresponds to a (50times 50) grid network, and (b) corresponds to a random 10-regular network with 1000 vertices. Both simulations start with a configuration in which 99% of players are defectors. (a) and (b) show how the population evolves over time (number of rounds). The chosen parameters satisfy the assumption in Eq. (1). The insets show the population’s evolution when hypocritical behavior is not available to the agents. See “Methods” for more details.
Figure 3 depicts the steady-state configuration, when hypocritical players experience different levels of energetic cost ((E_h)) and social-pressure ((rho _h)). This is illustrated on a grid network (Fig. 3a), random 10-regular networks (Fig. 3b), Erdös–Rényi networks with average degree 10 (Fig. 3c), and a Barabasi Albert networks with median degree 7 (Fig. 3d). The figures indicate that for small values of (rho _h) and (E_h), hypocritical behavior is, unsurprisingly, dominant: punishments deter defectors, but are insufficient to incentivize cooperation. For moderate values of (E_h), this phenomenon changes when (rho _h) enters the range specified in Theorem 1. Then, when (rho _h) increases further, the system remain defective. The correspondence to Theorem 1 is striking in Fig. 3a–c, whereas it is slightly more moderate in Fig. 3d. Recall that Theorem 1 considers (Delta)-regular networks, and therefore directly applies to grid networks and random regular networks, as simulated in Fig. 3a,b, respectively. Moreover, although a typical Erdös–Rényi network is not regular, the degrees of its vertices are relatively concentrated around the average degree, justifying the similarity between the results in Fig. 3b,c. For Barabasi Albert networks (Fig. 3d) the average degree is not a good representative for the typical degree since these networks are power-law. Hence, we drew the line corresponding to (rho _h=(1-E_h)/Delta), taking (Delta) to be the median degree, which was in this case roughly 7. Even though many vertices in the network have a smaller degree than the median degree, high levels of cooperation emerge in the region specified by Theorem 1.
Consistent with Theorem 1, Fig. 3 considers the case that players behave in a fully greedy fashion while having perfect knowledge regarding their costs. To check if this assumption is impactful, we also simulated a more noisy variant of our model, in which each player chooses the behavior that minimizes its cost with probability 0.95, and otherwise chooses a behavior uniformly at random. This relaxed model yields more mixed populations at steady-state, as indicated in Fig. 4a regarding a grid network and in Fig. 4b regarding random 10-regular networks. As another relaxation, we also simulated the case that the initial configuration is not overwhelmingly composed of defectors. Specifically, in Fig. 4c (grid network) and 4d (random 10-regular networks) we assumed that initially (80%) of the players are defectors, instead of (99%) as used in Fig. 3. Not surprisingly, this relaxation enhances cooperation. Indeed, comparing Fig. 4c to Fig. 3a, and comparing Fig. 4d to Fig. 3b, we observe that for each of these networks, the corresponding regime of cooperation includes the one that emerges when there are more defectors initially. Overall, in all the relax versions in Fig. 4 we see that the necessity of the condition (rho _h > (1-E_h)/Delta) to the emergence of cooperation is still respected. However, the other condition mentioned in Theorem 1, namely, (rho _h < rho _d – E_h) appears to be more sensitive to randomness. Indeed, and especially for the cases of random (Delta)-regular networks, cooperation emerges also for larger values of (rho _h).
Emergence of cooperation on various networks. The figure depicts the steady-state levels of cooperation on different network families. (a) Corresponds to a (50times 50) grid network, (b) corresponds to random 10-regular networks with 1000 vertices, (c) corresponds to Erdös–Rényi networks with 1000 vertices and parameter (p=1/100), and (d) corresponds to Barabási–Albert networks with 1000 vertices and parameter (m=5). All simulations start with a configuration in which (99%) of players are defectors. In all figures, for each couple ((rho _h,E_h)), a pixel is being drawn, whose red (resp. green, blue) component corresponds to the average proportion of defectors (resp. cooperators, hypocrites) at steady state. See “Methods” for more details.
Relaxed model on grids and random 10-regular networks. The figure depicts simulation results using relaxed versions of the main model. In (a) and (b) the greediness assumption in the decision making process is relaxed, allowing for some “irrationality” (see “Methods” for more details). In (c) and (d), the initial configuration contains (80%) defectors, instead of (99%) as used in Fig. 3. The plots in (a) and (c) correspond to a (50times 50) grid network and should be compared with Fig. 3a, whereas (b) and (d) correspond to random 10-regular networks with 1000 vertices and should be compared with Fig. 3b.
A generalized model with costly punishments
We next describe a different, more general model, termed the two-order model, that includes costly punishments. We then show how the second-order problem is solved in this model for a certain regime of parameters.
As the name suggests, the two-order model includes two levels of cooperation. Players engaged in first-order cooperation incur an energetic cost to produce a benefit for other individuals, whereas players engaged in second-order cooperation induce costly punishments on other individuals whenever they fail to cooperate (on any order). The two orders of cooperation are not mutually exclusive, that is, a player can cooperate (or not cooperate) on one of the two orders or on both.
Similarly to the main model, players are organized over a connected network G. A behavior for Player u is defined as a couple of indicator functions ((chi _1(u),chi _2(u))), with the convention that (chi _1(u) = 1) if u cooperates on the first-order (and 0 if it defects), and (chi _2(u) = 1) if u cooperates on the second-order (and 0 if it defects).
The cost incurred by a player is divided into two components. We denote by (alpha _1 > 0) the cost associated with first-order cooperation (this is analogues to the energetic cost in the main model), while (alpha _2 > 0) refers to the cost of second-order cooperation, that is, the cost of incurring punishments. A player u such that (chi _2(u) = 1) induces a social-pressure cost on each of its neighbors, whenever these fail to cooperate, at any order. As in the main model, the extent of this social-pressure may differ depending on whether it is applied against first-order defectors or second-order defectors. Specifically, we denote by (beta _1) the social-pressure cost paid by a first-order defector, and by (beta _2) the social-pressure cost paid by a second-order defector (fully defecting players pay both). Formally, denoting by (Delta _2(u)) the number of neighbors of u which are cooperating on the second-order, that is, (Delta _2(u) = |{v, { is, a, neighbor, of}, u, chi _2(v) = 1 }|), the total cost paid by u equals:
$$begin{aligned} {mathcal{{C}}}(u) = chi _1(u) alpha _1 + chi _2(u) alpha _2 + (1-chi _1(u)) Delta _2(u) beta _1+ (1-chi _2(u)) Delta _2(u) beta _2. end{aligned}$$
(2)
Let us name each of the four behaviors, and recap their cost:
a cooperator ((chi _1(u) = 1 , chi _2(u) = 1)) pays (alpha _1 + alpha _2),
a defector ((chi _1(u) = 0 , chi _2(u) = 0)) pays (Delta _2(u) (beta _1+ beta _2)),
a hypocritical ((chi _1(u) = 0 , chi _2(u) = 1)) pays (alpha _2 + Delta _2(u) beta _1),
a private cooperator ((chi _1(u) = 1 , chi _2(u) = 0)) pays (alpha _1 + Delta _2(u) beta _2).
As in the main model, the system starts in a configuration in which almost all players, e.g., (99%), are defectors (see “Methods”). The execution proceeds in discrete synchronous rounds. The costs of each player are evaluated at the beginning of each round, and then, before the next round starts, each player chooses a behavior that minimizes its cost (breaking ties randomly), given the current behavior of its neighbors.
The theorem below assumes that the underlying network is (Delta)-regular. However, as in the case of Theorem 1, the theorem can be generalized to arbitrary networks with minimal degree (Delta) (SI, Theorem 13).
Theorem 2
Consider a (Delta)–regular network G with n players undergoing the two-order model. Assume that the following two conditions hold.
Condition (i) (alpha _2 < beta _2), and
Condition (ii) (alpha _1 < Delta beta _1).
Then, with probability at least (1-frac{1}{c^n}), for some constant (c>1), in at most (3 cdot {text {diam}}(G)+1) rounds, the system will be in a configuration in which all players are cooperative, and will remain in this configuration forever.
The formal proof of Theorem 2 appears in the SI, Section C. Intuitively, the proof starts by showing that for the regime of parameters satisfying Conditions (i) and (ii), after the first round, no player ever chooses to be a private cooperator. The proof proceeds by showing that for this regime of parameters, the dynamics of the two-order model can be translated to the dynamics of the main model for the regime of parameters satisfying Eq. (1). In other words, the proof of Theorem 2 is based on a reduction to Theorem 1.
Source: Ecology - nature.com
