Punish and perish?

Introduction

Even though the evolution of cooperation among humans has been extensively studied in the last decades (see, for example, the famous work by Axelrod, 1984), a crucial question remains largely open: how can large-scale cooperation endogenously emerge and be successfully sustained over time? Research on the theme has persuasively argued that invoking explanations based on more or less sophisticated forms of “enlightened self-interest” alone, such as kin selection (Hamilton, 1964), repeated encounters (Fudenberg and Maskin, 1986), and reputation formation is not sufficient for accounting for the evidence available about the relevance of cooperation within several significant human contexts where collective action problems naturally arise but interactions involve genetically unrelated individuals. As both theoretical contributions and empirical evidence confirm, insofar as altruists are grouped together and mainly interact among themselves, within a neighborhood structure (Eshel et al., 1998), exploitation on the part of free riders can be prevented by restricting access to the gains from cooperation. Unlike these studies, we depart from close-knit parochial communities and test the survival potential of pro-social behavior within a more “hostile” environment where neither group selection nor assortative interactions are allowed and develop an evolutionary game-theoretical analysis aimed at investigating the diffusion of cooperation when exogenous enforcement devices are not available.

In recent years, a growing body of experimental evidence has convincingly shown that the so-called strong reciprocity is a powerful device for the enforcement of cooperation, despite the presence of large proportions of selfish subjects (Fehr and Gächter, 2000; Gächter and Herrmann, 2011; Gintis et al., 2005; Reuben and Riedl, 2009). The key feature of strong reciprocators (SRs) is their willingness to incur costs in order to conditionally cooperate and punish non-cooperators. However, in a lively interdisciplinary debate currently involving economists, biologists, and social psychologists (see on this Fehr and Henrich, 2003), critics argue that strong reciprocity is maladaptive, in the sense that it is evolutionarily weak and has no adaptive power (Dreber et al., 2008).

Hence, the following question naturally arises: can strong reciprocity survive and favor the enforcement of cooperation, within a behaviorally heterogeneous population in which also non-reciprocating players are involved? The existence of heterogeneous types is being increasingly confirmed by experimental research. ¹ In particular, available lab evidence indicates that (a) a significant proportion are unconditionally cooperative (e.g. systematically cooperate in the prisoner’s dilemma or make positive contributions in public goods or dictator games) and (b) a significant proportion of subjects are self-interested and tend to free ride on others’ generosity (by defecting from the outset). Furthermore, (c) many subjects who act neither purely selfishly nor simply altruistically seem to be SRs (Carpenter et al., 2009).

However, it is worth pointing out that strong reciprocity is often viewed as something more than costly punishment of non-cooperators: in many existing works on the theme, a strongly reciprocal player is generically defined as a person who is willing to bear costs to be kind to those who are being kind (by cooperating and rewarding them; strong positive reciprocity) and to be mean to those who are being mean (by defecting and punishing them; strong negative reciprocity; Fehr et al., 2002). A relevant problem with this definition is that it takes for granted that if a person is willing to be kind to those who are being kind, she will also be mean to those who are being mean (or vice versa). By contrast, several experimental papers (see, for example, Abbink et al., 2000; Offerman, 2002; Reuben and Van Winden, 2010) show that strong positive reciprocity need not be the flip-side of strong negative reciprocity. Moreover, a further extension of the notion of strong reciprocity is needed as some new studies interestingly reveal that punishers target their sanctions not only to defectors but also, to a significant extent, to other cooperators (Abbink et al., 2010; Gächter and Herrmann, 2011; Goette et al., 2012; Herrmann et al., 2008). This suggests that, on both conceptual and empirical grounds, strong reciprocity has a plural nature. In light of this, in this article, we decompose such behavioral attitude by introducing a taxonomy of strongly reciprocal players. Next, we comparatively lay out the evolutionary foundations of the varieties of strong reciprocity we identify within a behaviorally heterogeneous social environment where also unconditional defectors (UDs) and unconditional cooperators (UCs) are initially present. This will allow us to explore the different medium-long run implications which can be drawn, for society at large, depending on the variety of strong reciprocity one specifically refers to.

The main reason why in our behaviorally heterogeneous populations we include UCs who avoid to sanction free riders is that, as confirmed by prior theoretical and experimental work, pro-social acts such as “cooperating” (to a public good) and “punishing” (free riders) need not work synergistically: as Bowles and Hwang (2012) note, laboratory evidence reveals that unconditional altruists among American students are significantly less likely to sanction low contributors in a public goods game (Carpenter et al., 2009). They also refer to Gächter and Herrmann’s (2011) finding indicating that, among Russian urban and rural adults, those who contributed more to the public good sanctioned low contributors significantly less. Insofar as UCs fail to sanction free riders, they act as “second-order free riders.” ² Therefore, it will be interesting to pay special attention, in our evolutionary analysis, to a population where strongly reciprocal players are supposed to be willing to sanction such second-order free riders, rather than classic first-order free riders.

The structure of the remainder of this article is as follows. Section “The prisoner’s dilemma with carrot and stick” illustrates the main features of the analytical model. Section “The evolutionary game-theoretical model: do punishers perish?” analyzes social dynamics for each of the variants of strong reciprocity we investigate and contains our basic mathematical results. It also extends our analysis to a large-scale population containing all the seven player types that we focus on in this article. Section “Discussion and concluding remarks” discusses the main findings and concludes. Our main propositions and the proofs are in Appendix 1.

The prisoner’s dilemma with carrot and stick

We consider a large-scale population of individuals enjoying the benefits of a given collective (i.e. non-rival and non-excludable) good. In this society, three player types are initially present: UDs (hereafter “defectors” only, for simplicity), UCs (hereafter “cooperators”), and SRs. The existence and quantitative relevance of these behavioral types has been increasingly confirmed in laboratory environments, within the framework of prisoner’s dilemma and public good game experiments (see, for example, Camera and Casari, 2009; Carpenter et al., 2009; Fehr and Fischbacher, 2005; Fischbacher and Gächter, 2010; Ones and Putterman, 2007).

As we anticipated above, we take the inherently plural nature of strong reciprocity into account and model it by introducing a taxonomy of SR types. Infinitely many random encounters occur between two individuals at a time and, whenever two players meet, their behavior affects each other’s enjoyment of the collective good. Besides, type recognition holds: players are supposed to be able to identify their co-player’s type in each pairwise matching. This feature of the model is in common with models of good standing (Panchanathan and Boyd, 2004) as well as with prior evolutionary work on altruistic punishment (Fowler, 2005). In each matching, we assume that the material consequences for the players are captured by a two-stage prisoner’s dilemma with carrot and stick (PDCS) game. In the first stage (the Cooperation stage), the material consequences depend on players’ choices between “Cooperate” (C) and “Defect” (D) only. Hence, each matching between two individuals will produce one of the following four outcomes: (D, D), (C, C), (C, D), and (D, C). We suppose that, from the viewpoint of the individual player, the material consequences of these four possible outcomes have the structure of the prisoner’s dilemma, with γ > α > β > δ (see the left side of Figure 1). Mutual cooperation pareto-dominates mutual defection and free riding, by exploiting the non-excludability of the good to be provided and the opponent’s cooperation, is the most individually rewarding outcome, in material terms. Players behave according to their type. Hence, while UCs play C and UDs play D in each matching (regardless of the opponent’s type), SRs play C when matched with another SR or with a UC player, and play D when matched with a UD player (see the right side of Figure 1).

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Figure 1.

PD game and material payoffs (Stage 1).

In the second stage (the Punishment/Reward stage), players have to choose among Punish, P (“stick”), Reward, R (“carrot”), and Neither, N (i.e. abstaining from both punishment and rewarding). Each SR chooses N if matched with a player of the same type. Furthermore, we suppose that while both UCs and UDs systematically abstain from punishing and/or rewarding others, SR types are classified according to their choices (P, R, or N) when matched with UCs and UDs (see the left side of Figure 2). In particular, we separately consider five types of players who act identically in the first stage—that is, they play C (resp., D) when matched with either a UC or another SR (resp., a UD)—but differ as to their strategic choice in the second stage. In particular, as the left side of Figure 2 shows, we specifically focus on (1) strong negative reciprocators (SNRs), who only punish defectors; (2) strong positive reciprocators (SPRs), who only reward cooperators; (3) symmetric strong reciprocators (SSRs), who both punish defectors and reward cooperators; (4) punishers of non-punishing cooperators (PNPs), who only punish cooperators and, finally, (5) hyper-strong negative reciprocators (HSNRs), who punish both cooperators and defectors. ³

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Figure 2.

SRs’ classification and material payoffs (Stage 2).

The matrix on the right of Figure 2 provides us with the material payoffs at Stage 2, where ε = cost of being punished, λ = cost of punishing, π = cost of rewarding, η = benefit from being rewarded, and λ , ε , π , a n d η are strictly positive parameters. This means that, as it is often true both in naturally occurring environments (Sethi and Somanathan, 1996) as well as in laboratory experiments (Gächter and Herrmann, 2011), punishing (resp., rewarding) is a costly activity for the punisher (resp., rewarder). ⁴

We claim that the two-stage structure of the PDCS allows us to go beyond a further limitation which characterizes existing studies on strong reciprocity, that is, their inability to sharply disentangle implicit from explicit forms of rewarding and punishment. With regard to rewarding, one may argue that, for example, in a prisoner’s dilemma game, deciding to cooperate with a cooperator entails in itself sacrificing resources to be kind toward (i.e. to reward) a person being kind (strong positive reciprocity), since the same person would have obtained a larger material benefit by defecting (rather than by cooperating) with a cooperating player. Analogously, defection can be seen as an implicit means of punishing defectors. The Folk Theorem literature provides us with two famous examples of implicit punishment via defection such as Tit-for-Tat and the Grim Trigger strategy. By contrast, the structure of the PDCS allows us to incorporate two levels of punishment and rewarding into the analysis, so that strong reciprocity turns out to be a behavioral attitude characterized by both conditional niceness (i.e. willingness to cooperate with cooperators and to defect with defectors) and costly acts of punishment and/or rewarding. ⁵

The evolutionary game-theoretical model: Do punishers perish?

As we made clear in the previous section, in our evolutionary game-theoretical model player types prescribe the behavioral patterns that, via pairwise matchings, determine specific material consequences. In turn, such material consequences drive social evolution, in the sense that the types which turn out to be more rewarding—in material terms—are imitated and, by replicating faster, manage to spread over at the expense of less rewarding ones. Time is continuous and the population is modeled as a continuum of players. As far as pairwise matchings are concerned, the material game that individuals play is the previously described two-stage PDCS game. We represent the state of the population of individuals by the vector x = ( x 1 , x 2 , x 3 ) ∈ R 3 , where x 1 , x 2 , and x 3 indicate the shares of individuals of the types UC, SR, and UD, respectively. Thus, x i ⩾ 0 , for all i, and ∑ i x i = 1 , so x belongs to the two-dimensional simplex S (see Figure 3). Let us indicate by A the payoff matrix of the PDCS game associated with the material payoffs related to both Stage 1 and Stage 2 (see Figures 1 and 2), whose entries a i j depend on the specific SR type considered and represent the row player’s payoffs corresponding to each pairwise interaction, in a UC–SR–UD population

A = U C S R U D U C a 11 a 12 a 13 S R a 21 a 22 a 23 U D a 31 a 32 a 33

(1)

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Figure 3.

The two-dimensional simplex S.

Given the pairwise random matching structure of the game, the (expected) material payoffs for UCs, SRs, and UDs are, respectively

Π U C = a 11 x 1 + a 12 x 2 + a 13 x 3 Π S R = a 21 x 1 + a 22 x 2 + a 23 x 3 Π U D = a 31 x 1 + a 32 x 2 + a 33 x 3

Following Taylor and Jonker (1978), we assume that the growth rates x ˙ i / x i = ( d x i / d t ) / x i of the shares are given by the well-known replicator dynamics (see also Weibull, 1995)

x ˙ 1 = x 1 ( Π U C − Π ¯ )

x ˙ 2 = x 2 ( Π S R − Π ¯ )

x ˙ 3 = x 3 ( Π U D − Π ¯ )

(2)

where Π ¯ = x 1 Π U C + x 2 Π S R + x 3 Π U D represents the population-wide average payoff.

Our formalization allows us to directly draw implications about the social dynamics taking place within large-scale three-type populations in which cooperators and defectors initially coexist with strong negative reciprocators (SNRs), strong positive reciprocators (SPRs), symmetric strong reciprocators (SSRs), punishers of non-punishing cooperators (PNPs) and hyper-strong negative reciprocators (HSNRs), respectively.

The dynamic system (equation (2)) is analyzed in Appendix 1 by using the classification due to Bomze (1983) for replicator dynamics. In the following subsections, we illustrate the basic features of dynamics generated by equation (2) by separately focusing on each of the five varieties of strong reciprocity under examination.

In Figures 4 –8, attractive stationary states are indicated by full dots, repulsive ones by open dots and saddle points by drawing their stable and unstable branches. The vertices e 1 = ( 1 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 ) , and e 3 = ( 0 , 0 , 1 ) of the simplex S (see Figure 3), which represent, respectively, the states where only types UC, SR, and UD are present in the population, are indicated by UC, SR and UD. These states are always stationary states under replicator dynamics.

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Figure 4.

Social dynamics in a population of cooperators, defectors and altruistic punishers.

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Figure 5.

Social dynamics in a population of cooperators, defectors and altruistic rewarders.

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Figure 6.

Social dynamics in a population of cooperators, defectors and players driven by symmetric strong reciprocity.

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Figure 7.

Social dynamics in a population of cooperators, defectors and punishers of non-punishing cooperators.

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Figure 8.

Social dynamics in a population of cooperators, defectors and hyper-strong reciprocators.

Altruistic punishers (SNRs)

Figure 4 illustrates the dynamics emerging when SRs display a willingness to costly punish defectors only (see, on this, also Wichardt’s (2011) evolutionary analysis of the prisoner’s dilemma game), consistently with many laboratory studies (see Gintis et al., 2005), where a sizeable proportion of SNRs is identified, as well as with recent evidence from the field (Balafoutas et al., 2014). In such a context, the payoff matrix A becomes

A = U C S N R U D U C α α δ S N R α α β − λ U D γ β − ε β

Let us observe that a UC–UD–SNR population may end up either in a “bad” stationary state (the vertex UD), where cooperators and SRs perish and all players are defectors, or in a “good” stationary state belonging to the edge joining the vertices UC and SNR (every point of such an edge is a stationary state) where defectors perish, with positive proportions of cooperators and SRs. However, the latter evolutionary outcome is fragile and the maintenance of cooperation may be jeopardized: if the share of SNRs falls below a certain threshold in the polymorphic stationary states of the edge UC–SNR, such polymorphic configurations can be invaded by defectors. This result is in line with past evolutionary work (Sethi and Somanathan, 1996) and experimental evidence (Carpenter et al., 2004) revealing that when “sufficiently many” punishers are initially present, free riders are likely to be matched with agents reducing their payoffs, so that the former will be driven out of the population. At that point, since there will be no selection pressure against punishing players, the population shares stabilize. In such a case, a polymorphism with a positive proportion of two pro-social behavioral types (cooperators and (a high enough number of) punishers) takes place and universal cooperation prevails. In our analysis, we also find that, other things being equal, as defectors’ costs of being punished increase, the basin of attraction of the vertex UD becomes smaller (see Appendix 1). This can be seen as an evolutionary confirmation of what Sethi and Somanathan (1996) refer to as the centerpiece of economic reasoning, that is, “the tendency of human behavior to adjust in response to persistent differential in material incentives.”

Altruistic rewarders (SPRs)

This case represents a scenario where cooperators and defectors coexist with SPRs, conditional cooperators who are willing to incur costs to reward cooperators (altruistic rewarding), but abstain from punishing defectors, unlike SNRs. In such case, the payoff matrix A is given by

A = U C S P R U D U C α α + η δ S P R α − π α β U D γ β β

In their public goods experiment on endogenous institutional choice (carrot vs stick), Sutter et al. (2010) find that subjects typically vote for the reward option. In this case, our analytical model shows that the three types coexist in positive, permanently fluctuating proportions (Figure 5). Such dynamics qualitatively resembles one of the findings obtained in the well-known evolutionary paper on indirect reciprocity by Nowak and Sigmund (1998), where it is shown that long-term simulations that incorporate mutations usually do not converge to a simple equilibrium distribution of strategies, but display endless cycles, with defectors, discriminators, and cooperators. This is the only coexistence outcome we obtain (although we do not get an attractive stationary state with coexistence), with reference to both behavioral types (as selfish and non-selfish players coexist) and behavioral outcomes (as we observe both cooperation and defection, within the overall population). By contrast, all the other four varieties of strong reciprocity we investigate lead to the survival of either selfish or non-selfish (i.e. cooperators and/or SRs) players only, which either universal defection or universal cooperation is associated with.

Symmetric strong reciprocators (SSRs)

Let us now consider the dynamics associated with the case in which cooperators and defectors initially coexist with conditional cooperators displaying symmetric strong reciprocity, that is, the combination of altruistic punishment (punishment of defectors) and altruistic rewarding (rewarding of cooperators; Fehr et al., 2002). In this case, the payoff matrix A is

A = U C S S R U D U C α α + η δ S S R α − π α β − λ U D γ β − ε β

Here, we find that the stationary state UD, where all players are defectors, is a global attractor in the interior of the simplex S (Figure 6). The strong result we obtain is that now complete free riding prevails regardless of the proportion of non-selfish players (SRs and UCs) initially present in the population. The intuition, in a nutshell, is that altruistic rewarding “crowds-out” altruistic punishment. What happens in this case resembles the well-known dynamics characterizing a classic prey–predator model, but within a cultural evolution framework in which different cultural orientations compete with one another and evolution is driven by material payoffs. Within the behaviorally heterogeneous framework under study, symmetric strong reciprocity is maladaptive due to the key negative role played by the group of cooperators, as such players, by so doing, make themselves vulnerable and exploitable on the part of UDs, so favoring their evolutionary success. As we have seen by analyzing strong negative reciprocity, such an unpleasant social outcome can be prevented—provided that “sufficiently many” punishers are initially present—as SRs in that case abstain from rewarding cooperators. By contrast, with symmetric strong reciprocity, universal defection prevails regardless of the initial share of SRs in the population. Since selection favors second-order free riders, strong reciprocity declines and eventually first-order free riders take over. This is a crucial point which, although speculatively made (Panchanathan and Boyd, 2004) or investigated by means of exploratory simulations (Fehr and Fischbacher, 2003), had not received specific attention so far at the analytical level.

Punishers of non-punishing cooperators (PNPs)

Let us now turn to the dynamics associated with the existence of the “new” form of strong reciprocity that we label punishment of non-punishing cooperators; in this case SRs are willing to incur costs in order to punish cooperators—who unconditionally cooperate but fail to punish defectors, therefore acting as second-order free riders—rather than defectors themselves (for a similar notion, see the seminal paper by Axelrod (1986)). Tax evasion provides us with a well-known real-world example of a social phenomenon where the extent of second-order free riding is likely to play a key role in a society’s long-run social battle against tax evasion. Over time, citizens have to decide not only whether to honestly comply with their fiscal duties or not, but also whether they are willing to costly sanction evaders or not. Failing to do so would mean acting as second-order free riders. Sanctioning of fiscal misconduct may come in many forms. For instance, honest citizens who are willing to sanction (first-order) free riders may report to fiscal authorities as they learn that other citizens (e.g. neighbors, colleagues, friends or relatives) are cheating on taxes (see on this Antoci et al., 2014). ⁶ Relatedly, in the South of Italy, with regard to another well-known problem such as the large diffusion of mafia and other criminal organizations, some citizens (especially young students) have been promoting public initiatives in order to make clear that the key to defeating mafia is effectively fighting against omertà, that is, the conspiracy of silence of the majority of citizens who do nothing to react. A similar dynamics took place during the Nazi regime in Germany, where a non-violent resistance group known as the “White Rose” organized a leaflet campaign in which they called for active opposition to the totalitarian regime on the part of the “silent majority” of German people.

The quantitative relevance of UCs who fail to punish defectors has been recently confirmed by experimental evidence showing that unconditional altruists are significantly less likely to punish low contributors in a public goods game (Carpenter et al., 2009; see also Bowles and Hwang, 2012). As Bowles and Hwang (2012) point out, while one often refers to individuals as being “cooperative” or “uncooperative,” the motives supporting high levels of cooperation are heterogeneous and need not work synergistically. The payoff matrix A, in this case, becomes

A = U C P N P U D U C α α − ε δ P N P α − λ α β U D γ β β

Recent evidence from experimental games confirms that cooperative subjects get heavily punished (see, for example, Abbink et al., 2010; Denant-Boemont et al., 2007; Goette et al., 2012; Herrmann et al., 2008). ⁷ Also Gächter and Herrmann (2011), in their large-scale experiment with subjects from urban and rural Russia, find a surprisingly high rate of punishment of cooperators: as they correctly point out

Punishment of cooperators has been largely neglected in previous research on social preferences because it was negligible compared to the punishment of free riders. Our results show that this neglect is not warranted because punishment of cooperators can be very significant in some subject pools.

Our dynamic analysis for this case shows that now the pure population stationary state where everyone is a PNP is a global attractor in the interior of S and universal cooperation arises (Figure 7). Hence, such a seemingly paradoxical form of sanctioning turns out to be successful in both endogenously enforcing cooperation and being sustainable over time.

Hyper-strong reciprocators (HSNRs)

We finally illustrate the dynamics in the context in which HSNRs are present in the population. HSNRs abstain from rewarding other agents (unlike SPRs) and incur costs to punish both defectors and cooperators, that is, both first-order and second-order free riders. The payoff matrix A associated with this case is

A = U C H S N R U D U C α α − ε δ H S N R α − λ α β − λ U D γ β − ε β

When strong reciprocity takes the form of hyper-strong negative reciprocity, so that SRs display both altruistic punishment of defectors and punishment of non-punishers, in equilibrium either all players become defectors, so that universal defection occurs, or all players become HSNRs, so that cooperation flourishes (Figure 8). This case resembles the case of SNRs, as also in such case we have found that initial conditions turn out to be crucial in order to determine the evolutionary outcome. The key difference, however, is that with HSNRs the cooperative equilibrium is less fragile than with SNRs, as it is associated with a monomorphic population rather than with a polymorphic population that can be invaded by defectors. ⁸

What happens when all seven player types are simultaneously present in the initial population?

In this final subsection, we extend our evolutionary analysis by exploring the social dynamics taking place within a large-scale seven-type population in which cooperators and defectors initially coexist with all the five types of strongly reciprocal players that we separately considered above, that is, strong negative reciprocators (SNRs), strong positive reciprocators (SPRs), symmetric strong reciprocators (SSRs), punishing of non-punishing cooperators (PNPs) and hyper-strong negative reciprocators (HSNRs). When all seven player types can meet and play the PDCS, we get the following payoff matrix

U C S N R S P R S S R P N P H S N R U D U C α α α + η α + η α − ε α − ε δ S N R α α α α α α β − λ S P R α − π α α α α α β S S R α − π α α α α α β − λ P N P α − λ α α α α α β H S N R α − λ α α α α α β − λ U D γ β − ε β β − ε β β − ε β

This matrix allows us to reach the following conclusions: (1) the PNP strategy always dominates the HSNR strategy; (2) the SPR strategy always dominates the SSR strategy; (3) the SNR strategy always dominates the SSR strategy; (4) the SPR strategy dominates the PNP strategy (and, therefore, also the HSNR strategy) if π < λ , when π > λ , the PNP strategy dominates the SPR strategy (and, therefore, also the SSR strategy); (5) we never observe a dominance relationship between the PNP strategy and the SNR strategy; (6) we never observe a dominance relationship between the SPR strategy and the SNR strategy. Conclusions (5) and (6) imply that it is never possible to find parameter conditions under which one specific SR-based strategy dominates all the other SR-based strategies. Let us further observe that, when we consider three-type dynamics with one type of strongly reciprocal player at a time, the most interesting cases are the ones in which strong reciprocity either takes the form of the SPR strategy (see Figure 5, where fluctuations take place) or takes the form of the PNP strategy (see Figure 7, where the PNP vertex is globally attractive). Let us note that the SPR and the PNP type are right the types of SRs that we find in conclusions (5) and (6) above. Therefore, the PNP strategy never dominates all the other SR-based types: the PNP strategy always dominates the HSNR strategy; it dominates the SPR strategy (and therefore also the SSR strategy, if π > λ ); however, insofar as UCs and UDs are there, there is never a dominance relationship between the PNP strategy and the SNR strategy. This implies that the dynamics depicted in Figure 7 can never be viewed as a limiting configuration which the dynamics where all seven types are present converges to as dominated strategies cease to be played. Similarly, the dynamics depicted in Figure 5 can never be observed when all seven types are present as the SPR strategy always dominates the SSR strategy, dominates the PNP strategy (and therefore the HSNR strategy if π < λ ), but there is never a dominance relationship between the SPR strategy and the SNR strategy.

On the whole, we claim that while it is worth wondering what would happen in a large-scale population where all the player types analyzed in this article are simultaneously present, in real-life situations as well as in laboratory environments it is arguably more likely to observe populations where a lower number of types are simultaneously present (see Sections “Introduction” and “The prisoner’s dilemma with carrot and stick,” on this).

However, it is also important to note that there are interesting relationships between the five 3-type dynamics analyzed in the previous subsections and the seven-type dynamics considered in this subsection. In particular, the analysis of the five 3-type populations provides us with important indications as to the more complex seven-type context where all types of SRs are simultaneously present. In this regard, it is worth stressing that the pure population stationary states where only one strategy gets played (i.e. UC, SNR, SPR, SSR, PNP, HSNR, and UD) can be locally attractive (for the seven-type dynamics) only insofar as they are locally attractive for all the three-type dynamics illustrated in Figures 4 –8. This implies that

The stationary state UD can never be locally attractive (and therefore it can never be a strict Nash equilibrium), as it is not attractive when only UCs, UDs, and SPRs are present (Figure 5) and when only UCs, UDs, and PNPs are present (Figure 7). Therefore, this state would have been locally attractive only if SNR, SSR, and HSNR types were (also simultaneously) present.

The stationary state UC can never be locally attractive (and therefore it can never be a strict Nash Equilibrium) as it is never attractive in the three-type populations we considered in our previous analysis (Figures 4 –8).

Analogously, the stationary states SPR (Figure 5) and SSR (Figure 6) can never be attractive.

Finally, if we consider that the submatrix 5 × 5 of matrix 7 × 7 , obtained by getting rid of the rows and columns that regard the types UC and UD, is composed of equal entries (all α ), we find that all population states (i.e. strategy distributions) in correspondence of which neither the UC type nor the UD type is present are (non-isolated) stationary states for the replicator dynamics with seven types. This implies that neither the stationary states SNR, PNP, and HSNR (like SPR and SSR states) can be locally attractive. However, unlike the states SPR and SSR, they can be stable in Liapunov’s sense (i.e. the trajectories that start sufficiently close to them, keep on being sufficiently close, even though they do not necessarily converge to them; an example is provided by the right portion of the UC–SNR edge in Figure 4).

Points 4 and 5 in this list imply that “almost all” (i.e. except for those belonging to a zero measure set) trajectories of the system where all seven types are present converge to either a stationary state or to a limit set (e.g. a limit cycle) in which at least a SR-based type is present. Therefore, in general terms, our analysis reveals that punishing doesn’t imply perishing for all players who are willing to punish: some SR-based types may disappear and it is plausible to believe (although this is not necessarily the case) that these will be the ones who turn out to be dominated by other SR-based types.

Discussion and concluding remarks

In recent years, the idea of strong reciprocity has been gaining more and more credit. The main reason of this seems to lie in the appeal of a notion of endogenous sanctioning based on people’s willingness to perform such actions despite the associated monetary costs. Experimental confirmations have generated even more interest toward this behavioral attitude. One of the major results of our main analysis, however, is that in the three-type population under study, where defectors, cooperators, and SRs initially coexist, if SRs display both altruistic punishment and altruistic rewarding (i.e. SSRs), in equilibrium all cooperators and punishers perish, so that universal defection eventually prevails. This “paradox of strong reciprocity,” making a behavioral attitude such as the SSR strategy maladaptive and ineffective as a cooperation enforcement device is due to the “crowding-out effect” dynamically produced by altruistic rewarding on altruistic punishment: rewarding second-order free riders (i.e. cooperators) makes the latter vulnerable and indirectly favors the expansion of first-order free riders (i.e. defectors), who effectively exploit cooperators. This makes the SSR strategy unsustainable and leads to the demise of cooperation. Fehr and Fischbacher (2003) claimed that when cooperation in a population is widespread, altruistic punishers have only a small or no selective disadvantage relative to pure cooperators who abstain from punishing. However, insofar as strong reciprocity means not only costly punishment of defectors but also (symmetrically) costly rewarding of cooperators, our study reveals that—when the proportion of cooperators is extremely large—the cost disadvantage of SRs is still relevant due to the fact that rewarding so many cooperators will be costly—although the costs of punishing defectors will be small, in such a circumstance. The second (causally related) problem is that such an increase in cooperators, together with the lack of a large number of strongly reciprocal players around, makes cooperators extremely vulnerable to the “attack” of defectors, who exploit them and derive relevant advantages from this. This allows them to grow at the expense of cooperators and, eventually, to take over and make the monomorphic pure population equilibrium where all agents defect globally attractive.

Withholding reward to cooperators significantly improves the situation: passing from the SSR strategy to the SNR strategy makes strong reciprocity less costly and cooperation sustainable through positive proportions of SRs and UCs. This is in line with what happens in an indirect reciprocity scenario, where individuals benefit from withholding help. We have also shown that when rewarding does not occur, punishing works better when both cooperators and defectors are sanctioned (under the HSNR strategy) than when defectors only are sanctioned (SNR strategy), as in the latter case, even if costly rewarding does not occur, the locally attractive cooperative equilibria are fragile. However, the adaptive power of strong reciprocity, as well as its capacity to favor the endogenous enforcement of cooperation, is even greater when such behavioral attitude takes the form of the PNP strategy only, that is, when SRs simply sanction (non-punishing) cooperators and abstain from costly punishing defectors. On the whole, then, our comparative analysis establishes the evolutionary superiority of some varieties of strong reciprocity over others. We have seen that the SNR, SPR, and SSR strategies perform badly and do not act as effective cooperation enforcement devices, when they have to compete evolutionarily with unconditional cooperation and unconditional defection. By contrast, the PNP and HSNR strategies survive and succeed in enforcing cooperation. Hence, once the inherently plural nature of strong reciprocity is taken into account, it is necessary to specify what is the variety of strong reciprocity one aims at incorporating in a theoretical model based on type heterogeneity, as it would be otherwise impossible to draw unambiguous conclusions about the medium-long run stability of this behavioral attitude. ⁹

Why is it the case that, within the same social environment and information scenario, some varieties of strong reciprocity are adaptive while others are not? In a nutshell, our study suggests the following unified answer: in a world in which defectors initially coexist with SRs and cooperators, the latter can (paradoxically) be an obstacle to the stability of cooperation. The existence of cooperators as prey provides benefits to defectors as predators. Hence, the best way for SRs to generate an environment of cooperation and avoid to perish is to try to drive the cooperators to extinction: we show that a strategy by which strongly reciprocal players punish cooperators is highly adaptive both on its own (PNP strategy) and when combined with punishment of first-order free riders (HSNR strategy). On the contrary, a strategy by which SRs reward cooperators is highly non-adaptive both on its own (SPR strategy) and, even more so, when combined with punishment of defectors (SSR strategy). The point is that due to both their being second-order free riders and their failing to reward others, seemingly nice guys such as unconditionally cooperative players are in fact not so nice and deserve the stick, rather than the carrot. These results are in line with recent theoretical work on indirect reciprocity (Ohtsuki et al., 2009) as well as with experimental evidence (Dreber et al., 2008), indicating that subjects who do not punish earn a lot (they are the “winners”), while punishers end up with low payoff levels (they are the “losers”). Hence, punishment of cooperators becomes itself socially beneficial and, therefore, “altruistic,” while rewarding cooperators is socially harmful and can be viewed as “antisocial.”

Our findings suggest that cooperation can emerge due to the crucial role played by strong reciprocity but also that, in societies with sizeable shares of UCs, strong reciprocity can be successful insofar as it takes the form of “punishment of cooperators.” Such an evolutionary account of cooperation is based on an individual selection framework and is compatible with the presence, in the population, of cooperative “good men” who, by doing nothing, risk favoring the “triumph of evil” (as the poet Burke famously put it): unlike theories of cooperation based on altruistic punishment of defectors only, this explanation takes into account the “dark” side of (seemingly) other-regarding behavior and sheds light on the potential role of a plural behavioral attitude such as strong reciprocity in effectively dealing with it.