Evolution of cooperation with joint liability

Abstract

‘Personal responsibility’, one of the basic principles of social governance, requires one to be accountable for what one does. However, personal responsibility is far from the only norm ruling human interactions, especially in social and economic activities. In many collective communities such as among enterprise colleagues and family members, one’s personal interests are often bound to others’—once one member breaks the rule, a group of people have to bear the punishment or sanction. Such a mechanism is termed ‘joint liability’. Although many real-world cases have evidenced that joint liability can help to maintain collective collaboration, a deep and systematic theoretical analysis on how and when it promotes cooperation remains lacking. Here, we use evolutionary game theory to model an interacting system with joint liability, where one’s losing credit could deteriorate the reputation of the whole group. We provide the analytical condition to predict when cooperation evolves and analytically prove that in the presence of punishment, being jointly liable greatly promotes cooperation. Our work stresses that joint liability is of great significance in promoting current economic prosperity.

1. Introduction

Cooperation, an altruistic behaviour of bearing a cost to benefit others, is ubiquitous in human society and various living systems. Understanding the emergence and maintenance of cooperation has long been a challenge in evolutionary biology and ecology. Past decades have seen many powerful mechanisms proposed to explain the evolution of cooperation, such as direct reciprocity [1,2], indirect reciprocity [3–6], spatial reciprocity [7–11] and costly punishment (also called altruistic punishment) [12–19]. Among them, both mechanisms of direct and indirect reciprocity indicate the principle of personal responsibility that once taking defection, one would incur others’ retaliations, whether in the repeated interactions with the same opponent repeatedly (direct reciprocity) or with different opponents (indirect reciprocity). However, despite its reasonability and prevalence in human society, personal responsibility is far from the only norm ruling human social interactions and economic activities. In fact, joint liability, which means once one breaks the rule, not the one but other group members have to bear the retaliations, abounds in collective communities, such as among enterprise colleagues, family members, friends and even in primate societies [20–25]. It actually closely binds one’s personal interests to others’. One representative example is the case of Grameen Bank, which has been widely studied in economics [26,27]. Due to the lack of valuable collateral, the poor often have less chance to receive loans from the bank. Muhammad Yunus, the founder of the Grameen Bank, introduced the mechanism of joint liability—to get the loan, each borrower must have at least four guarantors (could be other poor); while once a borrower fails to pay the debt back, all including the guarantors lose the credit and chance to get loans any more. Other examples include: misconduct of a fraction of foreigners leaving the natives with bad impressions of all foreigners [24]; the poor quantity of some products in a shop leads to the trust deficit to all other products [25]. Besides human societies, joint liability widely exists in various animal communities. For example, after food sources were looted by other vervet monkeys, adult females would attack looters’ relatives [23]. In the macaque society, members of different matrilineal groups form alliances, and once being attacked, they often retaliate against a vulnerable member of the attacker’s group [23].

In the example of Grameen Bank, after the poor get the loan from the bank, to maximize their temporary interests, they are expected to refuse to pay the debt back. From the perspective of evolution and competition, all guarantors and the borrower lose the credit, while the borrower has the extra benefit of not paying the debt back. Therefore, theoretically, all individuals eventually evolve into being credit-losers. However, history has seen the success of joint liability mechanism in promoting the prosperity and maintaining cooperation in economic activities. The success of Grameen Bank inspires us to study why and how joint liability can be facilitative to the evolution of cooperation. However, the prior studies about group scoring, a model similar to joint liability where all group members share a common reputation, have found that group scoring does not support cooperation [28,29]. This conclusion contradicts the case of Grameen Bank. Actually, the effectiveness of joint liability relies on peer pressure (see details in Discussion). That is, when an individual does not pay the debt back, other group members will exert pressures on that individual to ensure they can obtain loans in the future. Here, we use the framework of costly punishment to capture the peer pressure. In the framework of costly punishment, besides cooperators and non-cooperators (called ‘defectors’), a third type of individuals, called ‘punishers’, is introduced. Punishers, on the one hand, behave like cooperators and take the altruistic behaviour, and on the other hand, they pay a cost γ to impose a punishment β on the defector. Many empirical studies have shown that humans are willing to pay the cost in order to punish defectors, and costly punishment can sustain cooperation [17–19,30]. However, it still remains to be explained how costly punishment itself emerges as an altruistic behaviour. In fact, the introduction of punishers leads to a ‘second-order’ social dilemma—compared with cooperators, they pay an extra cost to punish defectors and thus put themselves at a disadvantage. Theoretically, without other mechanisms, costly punishment can hardly evolve. The emergence of costly punishing is acknowledged to be a major puzzle in the evolution of cooperation [12,16,31].

Here, we use evolutionary game theory to model an evolving system incorporating both costly punishment and joint liability. We derive a mathematical formula to predict the evolution of cooperation, which agrees well with the results obtained by simulations. In particular, in the absence of joint liability, cooperation evolves only if β − γ > 4c, where c is the cost of cooperation. When joint liability is present, the condition is relaxed to β > γ. In other words, as long as the punishment β exceeds the cost of punishment γ, cooperation evolves. This simple rule supports that joint liability is of great significance in establishing collective cooperation.

2. Model

2.1. Lending stage

Inspired by the example of bank loans, we propose a model of joint liability as follows. The system consists of N borrowers and one lender (e.g. a bank). The lender interacts with the borrower and decides whether or not to provide a loan c. The borrower, if receiving the loan, makes an investment and derives a return b (b > c > 0). For simplicity, we consider three types of borrowers, namely, punisher, cooperator and defector. The punishers and cooperators return the loan c after deriving benefits, which gives them a net benefit b − c. Defectors, however, choose to keep the loan and have a benefit b.

In each generation, the lender interacts with all borrowers, one by one and in a random order. In general, the lender has a tolerability of m (1 ≤ m ≤ N)—after encountering m defectors, the lender turns to reject providing loans to the rest. Here, we assume that after receiving the loan, cooperators and punishers immediately make investments and then return the loan to the lender, and therefore the lender can tell the type of borrower before the next interaction. m = 1 means the lowest tolerability—once encountering a defective individual, the lender rejects lending money to the rest. When the tolerability m exceeds the number of defectors, the lender lends money to all borrowers, which means joint liability does not occur. If m = N, it corresponds to the maximal tolerability, which means no matter how many defectors are among the group, the lender always lends money to all borrowers.

2.2. Punishment stage

Following the loan lending stage, each punisher pays a cost γ to impose a fine β on each defector (figure 1). To ensure that the total fine for a defector and the total cost for a punisher is bounded as the population size grows, we normalize γ and β by setting γ/(N − 1) and β/(N − 1). In the subsequent discussion, we call γ the punishing cost and β the fine.

Figure 1.

Illustrations of the two stages (lending and punishment) of the model. We illustrate a system consisting of six borrowers and one lender with tolerability m = 1. Blue, green and red circles represent punishers (P), cooperators (C) and defectors (D), respectively. Our model consists of two stages. (a) Money lending stage. The lender interacts with all borrowers, one by one. After encountering a defective borrower, the lender rejects lending money to the rest. (b) Intragroup punishment stage. Each punisher pays a cost γ to impose on every defector a fine β.

Based on these settings, we can calculate the expected payoff for each borrower (see electronic supplementary material for derivations). The payoffs for punisher, cooperator and defector are, respectively

$${\pi }_P=\text{min}\{\frac{m}{N_D+1},1\}(b-c)-\frac{\gamma }{N-1}{N}_D,$$ 2.1

$${\pi }_C=\text{min}\{\frac{m}{N_D+1},1\}(b-c)$$ 2.2

$$\text{and}{\pi }_D=\text{min}\{\frac{m}{N_D},1\}b-\frac{\beta }{N-1}{N}_P.$$ 2.3

Here, N is the population size, and N_P, N_C, N_D are the numbers of punishers, cooperators, defectors respectively. We have N_P + N_C + N_D = N.

2.3. Strategy update

After both lending and intragroup punishment stages, borrowers update their strategies according to the classic pairwise comparison rule [32]. Here, we take account of both random strategy exploration and imitation of successful strategies. Specifically, a random borrower x is selected. With probability μ, x switches to a random type, i.e. punisher, cooperator or defector. With probability 1 − μ, another random borrower y is selected among the rest. x compares its payoff (π_x) with y’s (π_y) and imitates y’s strategy with probability

$$\frac{1}{1+\exp (-\kappa ({\pi }_y-{\pi }_x))}.$$ 2.4

Here κ denotes the selection strength. A larger value of κ means an increasing likelihood of imitating the individual with a higher payoff [33].

2.4. Model extension

So far we have presented the minimal model, which helps to deliver the idea of joint liability. We can further extend the model to capture more realistic interaction scenarios. For example, besides the peer punishment, borrowers might interact with each other and defectors might enforce extra selection pressure on cooperators. In §3.3, we investigate the case that borrowers play donation games, where a cooperator/punisher pays a cost of c′ to bring his opponent a benefit b′, b′ > c′ > 0, while defectors do nothing. The donation game is a classic Prisoner’s Dilemma, where defection dominates cooperation [34]. We have also investigated the case where punishers punish only a certain fraction of defectors, or they punish those defectors who are identified by the bank (see §S4.1 in electronic supplementary material). We have demonstrated that all results are qualitatively consistent regardless of the model complexity.

3. Results

3.1. Finite population

We begin with the setup of rare mutation, in which the system is bounded to evolve into a homogeneous state (i.e. all-cooperator, all-defector or all-punisher state) before the occurrence of a new mutant. We calculate the fixation probability that the mutant of one type (cooperator, defector or punisher) takes over the whole population of another type [35]. The fixation probability measures the competition between two types, namely, one’s ability to invade the other type. Using the method of Fudenberg & Imhof [36], we can compute the average abundances of the three types throughout the evolutionary process (see §S2 in electronic supplementary material).

We investigate four representative interaction situations, namely, one with neither punishment mechanism nor joint liability, one with either only punishment mechanism or only joint liability, and one with both of them. Figure 2a shows that without punishment mechanism and joint liability, defectors easily invade both populations of cooperators and of punishers, and they dominate the population across the evolutionary process. The introduction of punishment mechanism leads to a slight fluctuation of the evolutionary outcome, but the effects are negligible (figure 2b). Therefore, the punishment mechanism alone does not rescue cooperation at all. Intriguingly, joint liability alone is capable of maintaining a slightly higher level of cooperation and weakens defectors’ invasion to cooperators or punishers, highlighting more notable effects on cooperation relative to punishment mechanism (figure 2c). We see that the combination of punishment mechanism and joint liability elevates the cooperation remarkably—makes it possible for punishers to invade the population of defectors, and reduces defectors’ abundance to lower than $20\text{\%}$ (figure 2d). We further confirm the theoretical predictions by Monte Carlo simulations (figure 2e–h).

Figure 2.

Joint liability boosts cooperation markedly. For a sufficiently small mutation rate, the system is bounded to evolve into the homogeneous states, full punishers (P), full cooperators (C) or full defectors (D). ρ denotes the fixation probability that a single individual of one type takes over the whole population of the other type (only fixation probability exceeding 1/N is presented here). The size of circles and values alongside represent the abundance of strategies. (a) Case with neither joint liability nor punishment mechanism. Defectors dominate. (b) Case with punishment while no joint liability. Punishment mechanism alone does not rescue cooperation. (c) Case with joint liability while no punishment. Joint liability weakens the advantages of defectors over cooperators/punishers. (d) Case with both joint liability and punishment mechanism. Cooperators/punishers dominate. (e–h) Monte Carlo simulations for the four cases, corresponding to (a–d). Each panel is obtained by averaging over 50 simulations. Parameters: N = 100, κ = 0.2, b = 3, c = 1, γ = 0.3, β = 2, m = 1 and μ = 0.001 (e–h).

Since both punishers and cooperators return the loans borrowed from the lender, they are categorized as altruists. When the selection intensity is zero, the abundances of all the three types are 1/3 and the altruists therefore have abundance 2/3. Selection favours altruists over defectors only if the former’s abundance exceeds 2/3. Here, we derive the exact condition for the evolution of altruists (see electronic supplementary material): under weak selection (κ → 0) and in sufficiently large populations (N → ∞), altruists are favoured if and only if the fine to defectors, β, exceeds the punishing cost, γ, by a tolerability-based threshold, given by

$$\beta -\gamma >4c(\delta -\delta \ln \delta ),$$ 3.1

where δ = m/N, denoting the ratio of defectors the lender can tolerate. Function δ − δlnδ is monotone increasing (figure 3a). So, the more tolerant the lender is, the more difficult it is for altruistic behaviour to evolve. For the extreme case with tolerability m = 1 and a sufficiently large population size N, the condition for altruists to evolve is

$$\beta >\gamma .$$ 3.2

Therefore, for sufficiently small tolerability, the evolutionary outcome is determined by fine β and punishing cost γ, while independent of benefit b and loan c. The case of m = N means the largest tolerability—the lender lends to every individual even when all of them are defectors. The largest tolerability in fact is equivalent to the absence of joint liability. And we have the condition for the evolution of altruists, given by

$$\beta >\gamma +4c.$$ 3.3

Equations (3.2) and (3.3) also support that joint liability relaxes the threshold for altruists to evolve and thus promotes cooperation. We further verify the theoretical analysis by Monte Carlo simulations. As in figure 3b, we fix three sets of parameters of punishment (i.e. β − γ) and calculate the respective critical tolerability m below which the abundance of defectors is lower than 1/3. For different choices of population size N, our analysis (equation (3.1)) exactly predicts the evolutionary outcome.

Figure 3.

The decreasing tolerability provides more benefits to cooperation. (a) Condition for the evolution of cooperation. The decreasing tolerability, i.e. δ = m/N, makes it possible for cooperation to evolve in a broader range of parameters, i.e. β − γ. (b) Results by simulations (coloured squares) agree well with analytical predictions (dashed lines). We choose three sets of punishment parameters (i.e. β − γ): 1.32, 2.09 and 2.64. Under these settings, we perform simulations by searching for the value of m with which the stationary abundance of defectors is close to 1/3 (coloured squares). On the other hand, the critical δ can be calculated by equation (3.1) analytically, which are 0.1, 0.2 and 0.3, respectively (dashed lines). Parameters: μ = 2.5 × 10⁻³, κ = 0.04, b = 3, c = 1, γ = 0.3.

We proceed with investigations into how selection intensity, population size and mutation rate affect the evolutionary dynamics. We study a broad parameter range by both theoretical analysis and Monte Carlo simulations. It turns out that the introduction of joint liability can maintain the advantages of cooperators and punishers over defectors for a wide range of selection intensity (figure 4a,c). An extremely small selection intensity leads to an evolutionary process close to that under neutral drift, while an extremely large value leads to a nearly deterministic process [37], which does not support the evolution of cooperation. An intermediate value of selection intensity, therefore, sees the highest level of altruists, or say, the lowest level of defectors. Analogously, an intermediate population size N is most beneficial to the evolution of cooperators and punishers (figure 4b,d).

Figure 4.

Medium-sized selection intensity, population size and mutation rate benefit cooperation most. We illustrate the abundance of three types of borrowers as a function of selection intensity κ (a,c), population size N (b,d) and mutation rate μ (e). Panels (a,b) present analytical results and (c–e) show results by individual-based simulations. The analytical predictions qualitatively agree with the results of simulations, i.e. panels (a,c) and (b,d). An intermediate level of selection intensity, population size and mutation rate are most beneficial to cooperation. Parameters: b = 3, c = 1, γ = 0.3, β = 2.

For an increasing mutation rate μ, the evolution sees an initial abundance increase in cooperators and punishers, and a following decrease (figure 4e). Note that extremely small mutation rates reduce the randomness of the evolving process, and extremely large mutation rates lead to a totally random system. When the mutation rate lies between the two cases, the evolution often sees abundant cooperation. The optimality of intermediate selection intensity and mutation rate both indicate that appropriate randomness is of significance to foster cooperation. Here, we provide a few intuitions about the underlying mechanisms. Since paying an extra cost to punish defectors, punishers put themselves at disadvantageous positions compared with cooperators. The increasing selection intensity leads to the extinction of punishers. Once punishers are eliminated, defectors are free of punishment and dominate the population. However, an intermediate selection intensity enables punishers to eliminate defectors first. The evolution, therefore, ends up with the coexistence of punishers and cooperation because of the equal payoff. Also, when the mutation rate is properly large, once the punishers become extinct, a new punisher mutant is likely to appear, which is adverse to defectors. Therefore, a proper mutation rate decreases defectors’ advantages throughout the evolutionary process. The following study of infinite populations further confirms our analysis here based on a finite number of individuals.

3.2. Infinite population

The setup of an infinite population could lead to totally different evolutionary dynamics from a finite population [38,39]. Here to prove the generality of the joint liability’s cooperation-promoting effects, we study the infinite population by means of the classic replicator equations. Let x, y, z denote the proportions of punishers, cooperators and defectors, respectively, which gives x + y + z = 1. According to the payoff expressions in the case of finite population, for N → ∞, we obtain

$${\pi }_P=\text{min}\{\frac{\delta }{z},1\}(b-c)-\gamma z,$$ 3.4

$${\pi }_C=\text{min}\{\frac{\delta }{z},1\}(b-c)$$ 3.5

$$\text{and}{\pi }_D=\text{min}\{\frac{\delta }{z},1\}b-\beta x.$$ 3.6

The average payoff is then given by: $\overline{\pi }=x{\pi }_P+y{\pi }_C+z{\pi }_D$. We assume that every player with strategy i ∈ {P, C, D} adopts another randomly selected player’s strategy, with a probability proportional to the difference between the two players’ payoffs. Let x_i denote the proportion of players who adopt strategy i in the population. The replicator equation can be written as

$${\dot{x}}_i=x_i({\pi }_i-\overline{\pi }).$$ 3.7

Substituting π_P, π_C and π_D into the replicator equation, we can obtain three differential equations. With the relation x + y + z = 1, the system is actually governed by two independent variables and equations. Choosing x and z as the independent variables, the replicator equations can be simplified to

$$\begin{array}{rl}& \dot{x}=x[(\gamma +\beta )xz-\gamma z-\text{min}\{\frac{\delta }{z},1\}cz]\\ \text{and}& \dot{z}=z[(\gamma +\beta )xz-\beta x-\text{min}\{\frac{\delta }{z},1\}(cz-c)].\end{array}\}$$ 3.8

We then analyse equilibrium points of the above equation, which is obtained by taking the right-hand side of equation (3.8) to be 0. We denote the equilibrium point by e = (x*, y*, z*). There are five solutions: four points

$$\begin{array}{rl}& e_1=(0,0,1),e_2=(\frac{2\gamma +\beta -\sqrt{{\beta }^2-4\delta c(\gamma +\beta )}}{2(\gamma +\beta )},0,\frac{\beta +\sqrt{{\beta }^2-4\delta c(\gamma +\beta )}}{2(\gamma +\beta )}),\\ & e_3=(\frac{2\gamma +\beta +\sqrt{{\beta }^2-4\delta c(\gamma +\beta )}}{2(\gamma +\beta )},0,\frac{\beta -\sqrt{{\beta }^2-4\delta c(\gamma +\beta )}}{2(\gamma +\beta )}),e_4=(\frac{\gamma +c}{\gamma +\beta },0,\frac{\beta -c}{\gamma +\beta }),\end{array}$$

and a line l : z = 0. e₁ and l always exist. However, e₂ and e₃ exist only when their third elements z* satisfy z* > δ; e₄ exists only when its element z* satisfies z* < δ. In the electronic supplementary material, we present all derivative details.

Furthermore, we use the method of linearization to analyse the stability of each equilibrium solution (see electronic supplementary material). All conditionally existing equilibrium points (i.e. e₂, e₃ and e₄) are unstable. Only e₁ and a subset of l are stable. Specifically, only the points in χ = {e = (x, 1 − x, 0)|c/β < x ≤ 1{ ⊂ l are stable. For a better understanding, we use simplex to illustrate the trajectories (figure 5). Throughout the evolutionary process, all trajectories starting from the interior of the simplex converge to e₁ or χ. The point e₁ represents the state where all individuals are defectors, and the set of points on χ means the population consists of only punishers and cooperators. We call the attraction basin of e₁ (χ) the defective region (cooperative region).

Figure 5.

The cooperation-promoting effects of joint liability in infinite populations. Let e = (x, y, z) denote the frequency of punishers, cooperators and defectors, with x + y + z = 1. The triangles represent the frequencies of the three types of individuals. Each arrow line represents a trajectory. The equilibrium points are marked in red. The dark green region is the attraction domain of the state where all individuals are defectors, and the rest of the region is the attraction domain of the invariant manifold χ = {e = (x, 1 − x, 0)|c/β < x < 1}. All points on manifold χ are cooperative states. As tolerability δ decreases, the attraction domain of χ becomes larger. It means small δ can boost cooperation. Even though in the dark green region trajectories do not converge to cooperative states, small δ can also change their directions and make them close to cooperative states. For example, let the trajectories start from the same point (the green point) in (a–c); the smaller δ is, the closer the trajectory will get to the line z = 0. Parameter settings: β = 2, γ = 0.3, c = 1.

Figure 5 shows that the decreasing tolerance ratio δ is more beneficial to cooperation, which can be understood from two perspectives. Firstly, the cooperative region enlarges as the tolerance ratio δ decreases. Therefore, when beginning with a random initial configuration, the system is increasingly likely to evolve into the cooperator–punisher state. In addition, the decreasing δ can change the evolutionary trajectories. For example, although the blue trajectories in figure 5a–c have the same initial configuration (green dots) and the stationary state (i.e. e₁), the evolutionary trajectories are remarkably different. The trajectory for small δ (figure 5a) could be greatly close to the cooperator–punisher state (i.e. l) during the evolutionary process. With any noise (e.g. demographic noise), the trajectory might fluctuate to the cooperator–punisher state and then become fixed, which is easier than those far away from the cooperator–punisher state. Once again, randomness is expected to play a critical role for joint liability to promote cooperation.

Furthermore, e₁ is asymptotically stable, while points in χ are not. With some perturbations, the system resting in χ (the right part of l (i.e. CP) in figure 5) might fluctuate to the left side of l. Then, with some mutations, it may leave l and converge to the full-defector state. Thus, in the long-term evolutionary process, the system is expected to converge to the state of full-defector, which is consistent with figure 4b,d where the large population size weakens the cooperation-promoting effects of joint liability. On the other hand, for sufficiently small δ, with some perturbations, the system starting from e₁ may fluctuate to the white region and then eventually evolve into the state in χ.

3.3. Intragroup interactions

Finally, we extend the research by enabling intragroup interactions—individuals within the lending group interact with each other and derive payoffs. We assume they play donation games (other games can be studied analogously), a classic Prisoner’s Dilemma game where defection dominates. In a donation game, a cooperator/punisher offers a benefit b′ to its opponent and incurs a cost c′; a defector pays nothing. Individuals’ payoff is given by

$$\begin{array}{rl}{\pi }_P& =\text{min}\{\frac{m}{N_D+1},1\}(b-c)-\gamma \frac{N_D}{N-1}\\ & +\frac{N-N_D-1}{N-1}{b}^{\prime }-c^{\prime },\end{array}\hspace{0.17em}$$ 3.9

$${\pi }_C=\text{min}\{\frac{m}{N_D+1},1\}(b-c)+\frac{N-N_D-1}{N-1}{b}^{\prime }-c^{\prime }$$ 3.10

$$\text{and}\hspace{0.17em}\hspace{0.17em}{\pi }_D=\text{min}\{\frac{m}{N_D},1\}b-\beta \frac{N_P}{N-1}+\frac{N-N_D}{N-1}{b}^{\prime }.$$ 3.11

The analogous analysis into four cases (figure 6) further confirms that cooperation thrives only when both punishment and joint liability work together. Using a similar method, we can obtain the explicit condition for the evolution of cooperation with intragroup interactions, which is

$$\beta -\gamma >4c(\delta -\delta \ln \delta )+4c^{\prime }.$$ 3.12

Figure 6.

Joint liability stabilizes cooperation when intragroup interactions are considered. Here, all intragroup borrowers play donation games. We investigate four cases, with neither punishment nor joint liability (a), with punishment alone (b), with joint liability alone (c), with both punishment and joint liability (d). Punishment or joint liability alone could slightly increase cooperation. And they together can elevate cooperation to a remarkably high level. Parameters: N = 100, m = 1, s = 0.05, γ = 0.5, β = 6, b = b′ = 3, c = c′ = 1.

The above analytical result supports that joint liability promotes the evolution of cooperation, and the decreasing tolerability further enhances the positive effects. On the other hand, the comparison between equations (3.1) and (3.12) shows that the intragroup interactions of playing donations games weaken the cooperation-promoting effects of joint liability. The detrimental effect of intragroup interactions is understandable since the payoff structure provides defectors chances to exploit cooperators. In the electronic supplementary material, we have investigated the general payoff structure, which captures all two-player two-strategy symmetric games (see §S4.3).

4. Discussion

How cooperators evolve from a population of defectors has long been an enigma in the field of evolution, and has received decades of studies by researchers from different disciplines [7,8,12,34,40–43]. Here, we introduce the mechanism of joint liability that binds one’s interests with others’ to explain the evolution of cooperation. More generally, joint liability can be interpreted as a kind of fitness interdependence, namely, one’s evolutionary fitness is related to others’, which has attracted much attention recently [44–48]. Here, we show that joint liability, as a specific case of fitness interdependence, is an effective promoter for the evolution of cooperation. Given the prevalence in human daily and economic activities, our work is of great significance in theoretically clarifying the positive effects of joint liability on long-term societal prosperity.

Indeed, in well-developed countries or regions, the wide applications of advanced technologies such as social media make one’s reputation more visible to the public, and banks therefore can easily make a decision whether or not to provide a loan by evaluating one’s personal credit. But in communities lacking this public information, it is unlikely to accurately evaluate a specific person [49]. A feasible approach is to refer to the reputation of the whole community [28,29]. When a sufficiently large proportion of members have low credits, the rest are also categorized into the class of credit-losers. Banks, despite the lack of borrowers’ information, can make use of the joint liability to mitigate the risks of providing loans to those with low credits.

We observe that the combination of joint liability and costly punishment contributes to cooperation most and boosts cooperation to a remarkably high level, which is difficult for costly punishment alone to achieve [12,31]. Costly punishment requires the punishers to pay an extra cost of implementing punishment, which thus reduces the punishers’ advantages. Therefore, without other mechanisms, players do not have incentives to punish defectors. However, joint liability provides individuals with incentives to punish defectors, since defectors’ behaviours seriously damage their interests. We conclude that joint liability gives individuals incentives to punish defectors and costly punishment offers individuals tools to punish them. It turns out that either incentives or tools are indispensable in maintaining large-scale cooperation. Actually, besides the costly punishment, many other mechanisms that enable the punishment to defectors in some ways, once combined with joint liability, also promote cooperation remarkably. A representative example we have studied is the social exclusion of defectors, where individuals within a group pay a cost to exclude defectors from the group [50]. We show that the combined mechanism can maintain cooperation to a higher level compared with social exclusion alone (see §S4.2 in electronic supplementary material).

Furthermore, by both extensive simulations and theoretical analysis, we prove that the fewer defectors the bank tolerates, the more beneficial it is for cooperation to evolve. Due to the cooperation-promoting effects of joint liability, one might take the monotonic effect for granted. However, we argue that although the smaller tolerability means less chance for defectors to cheat the banks (weakening the advantages of defectors), it also makes it harder for cooperators to increase their benefits (decreasing the interests of cooperators). The evolutionary outcome actually depends on the balance between the influence on defectors and cooperators. The evolution has seen more detrimental effects to defectors and therefore relative advantages to cooperators.

Although there are some prior studies about implicated punishment and reputation mechanism [4,28,29,51–54], we stress that models therein are different from ours. The implicated punishment assumes that an external supervisor directly imposes a fine on each player when there are defectors in the group [51]. And prior studies about reputation mechanism mainly consider personal reputation—once taking defection, one loses one's reputation, and then all other players cease to cooperate with that individual [4,52–54]. Joint liability, however, is similar to collective reputation—one’s personal behaviour can affect the whole group’s reputation, which in turn affects the personal reputation of other members [28,29,55]. In this setting, a player affiliated to a group with good (bad) reputation can obtain an identification of cooperation (defection) even though that player never shows the preference to cooperating or to defecting before. Despite these pioneering studies about group reputation, our work differs from them in research motivations and results. These studies aim to explore how group reputation affects the evolution of cooperation, and they do not find beneficial effects of group reputation on promoting cooperation. However, we are motivated by the success of many real cases of joint liability in promoting cooperation. Thus, we aim to study why joint liability in these cases can support cooperation and provide a new explanation for the evolution of cooperation. We find that the cooperation-promoting effect actually relies on the combination of joint liability and a punishing mechanism. From another perspective, we would like to stress the relation and difference between our model and interdependent networks [56–58]. Both the models stress the interdependency of individuals’ payoffs. In an interdependent network, one’s payoff is related to one's behaviours in different social contexts. However, in our model, one’s payoff is associated with other individuals’ behaviours, although they may not interact directly.

Given the prevalence of joint liability in all walks of life, a variety of realistic scenarios can be studied. For instance, human society consists of communities of different cultures and religions. Often, one’s impression of a community is based on the limited information and survey to a small number of members, or one’s behaviour when interacting with a specific member is determined by one's impression of the whole community. A promising research direction is to model multiple communities with joint liability. Furthermore, a prior study has considered the case where the observation of one’s reputation may be subject to noise [4], and such observation error is vital for the evolutionary stability. Here, an extension worthy of study is that the lender may judge the borrower by mistake. An occasional error in judging a cooperator may lead to all borrowers losing opportunities of leading loans, which may have a great impact on the evolving process. The realistic population is often structured and a large amount of the literature has proved that the population structure can notably influence the evolutionary outcome [7,9,10,59–61]. Analogously, the population structure might play an important role in the establishment of joint liability, i.e. only those socially connected sharing joint liability. Our work, here, presents the initial efforts, which show joint liability is powerful in promoting cooperation. We expect many deeper insights along this direction.

Data accessibility

The computer codes used in this work are available at https://github.com/cjbbtwgc/joint_liability.

Authors' contributions

G.W.: conceptualization, formal analysis, investigation, methodology, software, visualization, writing—original draft. Q.S.: conceptualization, methodology, writing—original draft, writing—review and editing. L.W.: conceptualization, funding acquisition, methodology, project administration, supervision, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

G.W. and L.W. gratefully acknowledge the support from the National Natural Science Foundation of China (NSFC 62036002) and PKU-Baidu Fund (2020BD017). Q.S. acknowledges support by the Simons Foundation Math+X grant to the University of Pennsylvania.