1. Introduction
Cooperation is essential for the progression of human societies. This is especially true for large-scale cooperation, which is one of the most notable characteristics of human society, because large-scale cooperation benefits are shared amongst all members of a group. However, such cooperation should be vulnerable to the force of natural selection because individuals can free ride to receive the fruits of others’ efforts. In these settings, individuals cannot prohibit defectors from enjoying the benefits produced by other members. In the social sciences, this problem is referred to as a social dilemma (SD), a public goods problem or an n-person prisoner’s dilemma problem and it has been studied extensively in many disciplines, including economics, psychology, anthropology, biology and physiology [
1,
2,
3,
4,
5].
Other than kin selection [
6,
7,
8,
9], network reciprocity [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19] and direct reciprocity [
20,
21,
22,
23,
24,
25,
26,
27], introducing a punishment is one of the most thoroughly investigated solutions to SDs [
28,
29,
30,
31,
32]. Punishments transform the payoff structure of the SD so that free-riders cannot receive a higher payoff. With a punishment, the penalty imposed on a defector either can cancel out the benefit of free riding or can even make free-riders worse off than cooperators. However, the imposition of punishments generates another problem, which is called the second-order free-rider problem [
29,
32,
33]. Since punishment inflicts a personal cost onto the punisher and given that it is impossible to prohibit nonpunishers from enjoying the benefit of punishments (i.e., a higher level of cooperation), the provision of punishment also creates another SD problem. Several answers to this problem have been proposed, such as meta-punishment (i.e., punishment toward second- or higher-order free-riders) [
34,
35,
36,
37], signaling (i.e., cooperators receive relational benefits later because cooperation signals one’s quality as a worthy individual) [
38,
39,
40,
41], coevolution of punishment and other traits [
42,
43] and spatial structures [
44].
In the current study, we focus on the “linkage” between SDs and indirect social exchanges (ISEs) as a solution to the SD problem that does not include the second-order free-rider problem [
45,
46,
47,
48]. The SD problem does not exist in isolation. It is embedded in various kinds of social interactions, such as when members within a society try to supply public goods while simultaneously trying to conduct ISEs with another type of good based on unilateral giving between individuals. Cooperation in an ISE can be sustained by indirect reciprocity [
49,
50,
51,
52,
53,
54,
55,
56,
57,
58,
59,
60,
61]. In a situation where an SD game and an ISE coexist, it is possible to design new strategies that cannot have been realized when each game is considered separately. One such strategy is the
linkage strategy, which excludes people who defected in the SD game from the ISE. Past studies have shown that the linkage strategy does not inflict additional costs when compared to strategies that do not link two games, indicating that the second-order free-rider problem will not occur when using this strategy [
46,
47,
48]. However, its evolutionary stability is not robust in situations where perception errors exist [
48]. Therefore, in this current study, we devise several variations of linkage strategies in order to find a linkage strategy that is evolutionarily stable even in the presence of perception errors.
Panchanathan and Boyd [
46] showed that a linkage strategy called
Shunner, which does not offer help to an SD defector in ISEs, could be an evolutionarily stable strategy (ESS). In their model, players engaged in two social interaction stages. In the first stage, players engaged in a one-shot SD game in which each player decided whether or not to cooperate with members in the group. There was a cooperation cost
C to the player and a benefit
B that was shared equally amongst the
n − 1 other group members (
C <
B). The second stage was an ISE in which players engaged in one form of an ISE called a multi-period
mutual aid game. In each period, one player was randomly selected from a group and denoted as “needy” while each other member decided whether to help the needy player or not. There was a helping cost
c incurred by the helper, which then yielded a benefit
b to the needy player (
0 < c < b). In this model, there was a possibility of implementation errors occurring in the ISE. For instance, such an error could occur when a player intended to help the needy but failed to do so. Everyone in the population knew the behavioral history of all group members in both stages.
The strategies in Panchanathan and Boyd’s model consisted of how to behave in the SD and in the ISE. There were two SD strategies, which were to defect or to cooperate. There were four possible strategies in the ISE, which were the following—All-D (never helps needy players), All-C (always helps needy players) and the final two strategies instruct that players should use others’ behavioral histories to assign reputation in the ISE. These two strategies using reputation based on the
standing (explained below) [
55,
56,
62], which is one of the norms for assigning reputation scores that makes mutual cooperation possible in an ISE (
Table 1). From the viewpoint of standing, all players have a good score at the beginning of the game. Players lose their good scores if they fail to help a needy player who has a good score during the ISE. Those whose scores are bad can restore their good scores by helping a needy player in the future. The standing strategy only helps a needy player who has a good score. In Panchanathan & Boyd [
46], the
original discriminator strategy required that players ignore other players’ behaviors in the SD when they assign reputations. This original discriminator strategy stated that only the players’ ISE behaviors matter. In contrast, the
linkage discriminator strategy regarded SD defectors as being permanently bad even if the defectors helped a needy player. The linkage discriminator applied their norm only to SD cooperators and this strategy therefore never helped SD defectors in the ISE. Their study includes eight possible strategies (see
Table 2), four of which were investigated by Panchanathan and Boyd [
46]. 1.
Defectors defected in both games; 2.
Cooperators cooperated in both games; 3.
Reciprocators defected in the SD and played as original discriminators in the ISE; 4. Shunners (who applied the linkage strategy) cooperated in the SD and played as linkage discriminators in the ISE. Their results showed that the Shunners resisted being invaded by the other three strategies.
Inaba, Takahashi, and Ohtsuki [
48] investigated the evolutionary stability of a Shunner against a new strategy called the
Cooperating-Reciprocator (
Table 2) using the same game structure as Panchanathan & Boyd [
46]. They showed that, in the model in which the possibility of an implementation error could only occur in the ISE, Shunners were not evolutionarily stable against Cooperating-Reciprocators. Because both Cooperating-Reciprocators and Shunners cooperated in the SD, there was no chance that Shunners took action against players who defected in the SD. Consequently, these two strategies took the exact same action and were evolutionarily neutral. In addition, Inaba, Takahashi, and Ohtsuki [
48] conducted a series of computer simulations under the assumption that various error types could occur. They derived eight types of errors based on three dimensions—whether errors occurred only in the ISE or both in the ISE and the SD, whether perception errors could occur and whether errors could occur one way or two ways (
Table 3). Each player might misperceive an action performed by another player, which was called a perception error based on a direct observation model [
56]. Players who were willing to cooperate might mistakenly defect because of an error, which was called a
one-way error. Players might cooperate or defect randomly ignoring the action that was prescribed by their employing strategy. It was called a
two-way error. As a result, the existence of errors significantly affected the evolutionary stability of the Shunner. Their results showed that the linkage strategy was an ESS only when there were implementation errors in both the SD and the ISE (conditions c and d in
Table 3). However, the linkage strategy was less unstable when perception errors existed (conditions g and h). In terms of the direction of the error, they found that the linkage strategy was more stable under one-way implementation error conditions than under two-way implementation error conditions. On the other hand, the linkage strategy was more stable under two-way perception error conditions than under one-way perception errors conditions.
When subjective perception errors are small, a linkage strategy in an ISE can be used as a solution for a group to achieve cooperation. However, people are hardly free from perception errors in their social lives. Does this mean that the linkage strategy can be evolutionarily stable under implausible situations only? We expect that there are some linkage strategies that are evolutionarily stable even in conditions when perception errors do exist. In previous studies, Shunners assigned a reputation to others based on standing. However, several other strategies with a different norm have been proposed in the indirect reciprocity literature [
49,
51,
52,
56,
58,
63,
64,
65,
66]. Moreover, several of these strategies are known to be robust against subjective perception errors [
63].
Ohtsuki and Iwasa [
51,
52] investigated the evolutionary stability of 4,096 strategies in ISEs. Each strategy included a norm and an action rule. A norm told a player how to judge the reputations of others and an action rule told a player which behavior to take depending on the recipient’s score and the player’s own score. Of the investigated 4,096 strategies, the researchers identified the most effective set of ESS strategies, which they called the
leading eight. The common features of these strategies’ norms specified that (1) cooperation toward a good recipient is good while defection toward a good recipient is bad and (2) defection toward a bad recipient is good. Standing is included in the leading eight.
Ohtsuki and Iwasa’s model [
51,
52] was based on the indirect observation model, which specifies that everyone in the population has the same opinion of a focal player and therefore players did not suffer from subjective perception errors. Takahashi and Mashima [
63] explored the effect of subjective perception errors on the evolutionary dynamism of indirect reciprocity. They investigated the evolutionary stability of 16 norms with the assumption that all individuals employ one kind of action rule in which the donor helps when they assess the recipient as good and refuse to help otherwise. They found that two norms, the
strict discriminator (i.e., shunning [
49,
53]) and
kandori (i.e., extra standing [
63] or stern-judging [
64]), were not invaded by either an All-C or an All-D strategy (see
Table 1 for the norms). Under the strict discriminator, helping a good individual is the only action that can lead to a player developing a good reputation. Since strict discriminators assign bad reputations to players who defected toward bad recipients, it does not satisfy the criterion of the leading eight. On the other hand,
kandori is a member of the leading eight. Under this rule, helping a good individual or refusing help to a bad individual leads to a good score, whereas refusing help to a good individual or helping a bad one leads to a bad score. Moreover, Takahashi and Mashima [
36] showed that the
sugden strategy was invaded by an All-D strategy when the cost-benefit ratio was low.
Sugden is sometimes called simple-standing [
53] because it is slightly different from standing in d
4b (
Table 1).
Sugden is also a member of the leading eight. Ohtsuki and Iwasa [
53] also investigated the evolutionary stability of norms under perception errors in which a player makes a mistake in interpreting the reputations of others where one-way implementation errors existed. They showed that the
sugden,
kandori, and strict discriminator strategies were evolutionarily stable when perception errors did not exist, but they found that these strategies were all unstable when perception errors did exist.
1 We summarized the results of previous studies in
Table 4. These results suggest that the evolutionary stabilities of strategies that adopt promising norms are affected by perception errors.
In this research, we examined the robustness of several types of linkage strategies that adopt different norms in order to examine which type of linkage strategies can firmly attain cooperation in SD games. We investigated standing,
sugden, strict discriminator and
kandori. Considering that multiple action rules make the model more complex, we only examined one type of action rule in which the donor helps if he/she assesses the recipient as good and refuses to help otherwise. The details of our evolutionary simulation were based on Inaba, Takahashi, and Ohtsuki [
48].
4. Discussion
Previous research has proposed that linkage is a promising solution of SDs only if players do not suffer from perception errors [
48]. However, since perception errors usually exist in our society, it is necessary that people overcome these errors so that groups can achieve cooperation. Therefore, we searched for other linkage strategies that are not affected by perception errors. We found several strategies that could sustain cooperation in SDs by linkage with ISEs even when perception errors existed. Linkage strategies with
kandori in ISEs were most robust against perception errors. Thus, our study shows that cooperation in SDs can be maintained by linkage strategies with ISEs even when perception errors exist. We conclude that our results strongly support the view that a linkage strategy is one of the most effective solutions to the SD problem.
Kandori evaluates as bad a donor who helped a bad recipient. Consequently, one who adopts a linkage strategy based on
kandori judges cooperation in the nonlinkage strategy as bad because the nonlinkage strategy helps defectors in SDs. This is the source of the linkage strategy’s advantage against the nonlinkage strategies under
kandori. As a result, linkage was robust when perception errors exist. In order to identify violators of the linkage norm early in the game, it was important to assign bad scores to those who helped bad recipients (d
3* in
Table 1). It is known that
kandori has an evolutionary advantage over other strategies [
53,
64,
65,
71]. The reason for this advantage is that
kandori adjusts reputations quickly. It has one good action and one bad action for each type of encounter so that it becomes possible for players to readily punish bad players and to quickly forgive good ones [
64]. Our results show that even in a linkage situation, this feature appeared as applying a direct punishment to others who had different norms (i.e., nonlinkage norm) and this led
kandori players to their striking success.
As mentioned in the introduction, Ohtsuki and Iwasa [
53] showed that kandori is not evolutionarily stable when perception errors exist. This result seems to contradict both the results of Takahashi and Mashima [
63] and also our own results. There are two possible reasons for this inconsistency. First, Ohtsuki and Iwasa [
53] investigated the robustness of norms against perception errors in a situation where implementation errors occur in one way. Second, Ohtsuki and Iwasa [
53] stated that an agent who made a perception error mistakenly perceived the reputation score (i.e., good or bad), whereas in the current study and in that of Takahashi and Mashima [
63], a perception error caused an agent to misperceive the action of the target (i.e., helped or not). Future studies should address how these differences affect the evolutionary stability of each norm.
We used the giving game as an ISE instead of the mutual aid game that had been used in previous linkage studies [
46,
48]. This alteration did not affect the results. Therefore, we assert that the results of our study can be generalized for other indirect reciprocity exchanges.
The limitation of this study is that we did not consider evolutionary dynamics between different norms [e.g., S[ST] vs. C-R[KAN]]. Pacheko, Santos, and Chalub [
64] showed that
kandori has an evolutionary advantage over other norm in ISE. On the other hand, Uchida and Sigmund [
66] showed that
sugden and
kandori can coexist in ISE. Therefore, it is not clear what rule evolves when different norms exist within a society where an SD and an ISE coexist.
On the other hand, we may not need to consider the situation in which different norms coexist. Since cooperation cannot be achieved in SD without a supporting mechanisms such as sanction or linkage, it may be more appropriate to conceive that an ISE is formed first and then an SD can be established by linkage during the course of the development of human society. In that case, it is natural to consider whether the linkage strategy whose norm was originally evolved in ISEs can invade the society of the nonlinkage strategy employing the same norm.
It follows, then, the next step is to examine whether the linkage strategy can invade the nonlinkage strategy. Our simulation started with a population in which all players adopted a linkage strategy. If a population starts with a nonlinkage norm, the linkage strategies cannot invade that population. As previous linkage studies have mentioned [
46,
48], there is a need to examine the equilibrium selection process that enables the transition from a nonlinkage norm to a linkage norm. There are several candidates that might be useful for examining this issue, such as interactions with similar strategies [
46], small population sizes [
72], higher mutation rates [
73], between-group competitions [
74], selective immigration [
75,
76] and learning [
77,
78]. Future research should investigate whether a linkage norm could spontaneously emerge from a nonlinkage equilibrium.