Okada, Isamu
Article
A review of theoretical studies on indirect reciprocity
Games
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Suggested Citation: Okada, Isamu (2020) : A review of theoretical studies on indirect reciprocity,
Games, ISSN 2073-4336, MDPI, Basel, Vol. 11, Iss. 3, pp. 1-17,
https://doi.org/10.3390/g11030027
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games
Review
A Review of Theoretical Studies on Indirect Reciprocity
Isamu Okada
1,2,†
1
Faculty of Business Administration, Soka University, Hachioji City, Tokyo 192-8577, Japan; [email protected];
Tel.: +81-42-691-8904
2
Department of Information Systems and Operations, Vienna University of Economics, 1020 Vienna, Austria
† Current address: Tangi 1-236, Hachioji City, Tokyo 192-8577, Japan
Received: 21 May 2020; Accepted: 17 July 2020 ; Published: 20 July 2020
Abstract:
Despite the accumulation of research on indirect reciprocity over the past 30 years and the
publication of over 100,000 related papers, there are still many issues to be addressed. Here, we look
back on the research that has been done on indirect reciprocity and identify the issues that have been
resolved and the ones that remain to be resolved. This manuscript introduces indirect reciprocity
in the context of the evolution of cooperation, basic models of social dilemma situations, the path
taken in the elaboration of mathematical analysis using evolutionary game theory, the discovery
of image scoring norms, and the breakthroughs brought about by the analysis of the evolutionary
instability of the norms. Moreover, it presents key results obtained by refining the assessment function,
resolving the punishment dilemma, and presenting a complete solution to the social dilemma problem.
Finally, it discusses the application of indirect reciprocity in various disciplines.
Keywords: indirect reciprocity; social dilemma; evolution of cooperation; evolutionary game
PACS: 02.70.-c; 05.45.-a; 87.23.Cc; 87.23.Ge
JEL Classification: C62; C72; C73
1. Introduction
While studies on indirect reciprocity over the past 30 years have led to the publication of more
than 100,000 related papers, there are still many issues to be addressed. Indirect reciprocity is a major
mechanism that answers the question of how cooperation evolves in social dilemma situations. Here,
cooperation refers to the provision of a wide range of resources, including effort, time, and money,
to the benefit of others or organizations, and by its definition is altruistic [
1
,
2
]. While classical game
theory, which is based on the assumption of rational decision-making, teaches that cooperation does
not emerge naturally, cooperative behavior is widespread not only in humans but also in animals.
Mechanisms of cooperative behaviors include kinship, direct and indirect reciprocity, and group
cohesiveness. Therefore, the evolution of cooperation has been studied not only in biology but also in a
wide range of fields such as sociology, economics, politics, psychology, mathematics, and physics [
3
–
7
].
The analysis of social dilemma situations is an important task in the study of the evolution
of cooperation. A situation constitutes a social dilemma if an individual rationally chooses not
to cooperate with someone due to an economic or psychological incentive despite the fact that
social welfare is maximized if all members of the group cooperate [
8
–
13
]. The cumulative effect
of selfish behaviors is a society with low social welfare. Social dilemma situations are widely
observed, from deciding whether to donate blood to deciding whether to take actions that protect the
environment. The evolution of cooperation in social dilemma situations has attracted widespread
social and academic interest because cooperation is maximized social welfare.
Games 2020, 11, 27; doi:10.3390/g11030027 www.mdpi.com/journal/games
Games 2020, 11, 27 2 of 17
Indirect reciprocity is a well-known mechanism for maintaining cooperation among unrelated
individuals in social dilemma situations. It involves cooperators requesting information about potential
recipients. This cooperation mechanism prevents free riders, i.e., individuals who do not cooperate
but receive benefits from naive cooperators, from invading the population of individuals. It works by
imposing a discrimination criterion that prevents individuals from cooperating with individuals who
do not have a good reputation.
Indirect reciprocity is considered to be one of five mechanisms by which cooperation evolves
in the context of social dilemmas and is considered to be particularly versatile compared with
the other mechanisms [
6
]. Because kin selection requires relative relationships, direct reciprocity
requires repeated game play, while network reciprocity and group reciprocity require heterogeneous
structures for interactions. If there are no such conditions, cooperation does not emerge. In contrast,
indirect reciprocity does not require such conditions. It is thus a factor in the evolution of cooperation
among individuals without direct interactions, so the breadth of its application area has attracted
academic interest. Here, we look back on the studies of indirect reciprocity and identify the issues that
have been resolved and the ones remaining to be resolved. Figure 1 presents an overview of indirect
reciprocity research.
Figure 1. Historical overview of indirect reciprocity research and milestone papers.
2. Development of Research Areas and Research Methods
Indirect reciprocity was first identified in folklore research in the 1960s and was explicitly defined
in relation to cooperative behavior in psychology in the late 1970s. For example, Eisenberg–Berg [
14
]
showed through interviews with children who had faced moral judgment dilemmas that indirect
reciprocity falls into the moral consideration category. Following a period of observing indirect
reciprocity phenomena, efforts to understand the mechanism scientifically began in biology.
Alexander [
15
–
17
] first discussed indirect reciprocity as a principle of human behavior and defined
it as the provision of benefits among unrelated persons, using the framework of evolutionary game
theory developed by Maynard Smith [
18
,
19
]. He focused on indirect reciprocity as a principle for
explaining cooperative behaviors like altruism [1,2].
In direct reciprocity [
2
,
20
,
21
], the reciprocal cooperation attained by repeatedly playing with the
same player maintains stable cooperation. Indirect reciprocity, in contrast, is based on the assumption
of an indirect situation in which a player cooperates with a different player rather than the same
Games 2020, 11, 27 3 of 17
player. In other words, a player contributes to a player who was not the player from whom he or she
received a contribution. There are two possible timings of the cooperative actions among these three
players [
22
–
24
]. In upstream indirect reciprocity, Player X first cooperates with Player Y, and Player
Y subsequently cooperates with Player Z. This timing is often observed in pay-it-forward situations.
While economic experiments have attempted to reveal why upstream indirect reciprocity is often
observed [
25
–
27
], explaining this timing theoretically is difficult because Player X does not receive any
benefit from her or his cooperation [24,28–30].
In downstream indirect reciprocity [
23
,
24
], on the other hand, Player Y contributes to Player Z
first and then Player X contributes to Player Y. This type of reciprocity can be theoretically rationalized
because there is an economic incentive for Player Y to cooperate. Player Y expects to gain future
cooperation by helping Player Z. Therefore, most theoretical analyses target downstream indirect
reciprocity, and this manuscript reviews this type of indirect reciprocity. For the downstream type to
work, cooperative actions must be communicated to a third party. In other words, as a precondition
for establishing indirect reciprocity, society must have a system for distributing reputation information.
This is because direct observation of all games is an extreme assumption, so it is necessary to
assume a situation in which the behavior of a third party is instantly transmitted to society as a
whole by reputation and gossip. Figure 2 summarizes the basic mechanisms of both direct and
indirect reciprocity.
Figure 2.
Basic mechanisms of direct and indirect reciprocity. This figure is based on Figure 1 of Nowak
and Sigmund (2005) [23].
Boyd and Richerson [
22
] were the first to model indirect reciprocity in the framework of
evolutionary games. They used indirect reciprocity as a principle to explain cooperation among
large numbers of genetically unrelated individuals. Their idea was an extension on the TFT (tit-for-tat)
strategy in the iterated prisoner’s dilemma game introduced by Anatol Rapoport around 1980.
The strategy is simple because a player who simply imitates the opponents latest behavior gains
a relatively high payoff. Boyd and Richerson classified indirect reciprocity into two types, upstream
and downstream, and asserted that downstream reciprocity has a cooperative equilibrium. However,
they assumed a priori that information about all the other people in the group is available.
Nowak and Sigmund [
31
] introduced reputation information based on scores to explicitly utilize
information about other people in the group. They assumed that each player is given an image score,
the value of which increases when the player exhibits cooperative behavior and decreases when
Games 2020, 11, 27 4 of 17
the player does not cooperate when the opportunity to do so is presented. The players cooperate
with those players whose scores exceed a threshold. That is, they explicitly stated that the players’
reputation information, i.e., a summary of their previous behaviors, is necessary for cooperation to
work [
26
,
32
–
35
]. This was an important breakthrough for indirect reciprocity research. Sigmund [
24
]
later asserted that direct reciprocity requires repetition while indirect reciprocity requires reputation.
Nowak and Sigmund [
36
] also developed a method for analyzing stability by incorporating
replicator dynamics [
37
] into their theoretical model of indirect reciprocity. They introduced into their
model players who unconditionally do not cooperate (i.e., they defect) and players who cooperate
unconditionally in addition to the players who adopt the norm
1
of indirect reciprocity. Nowak and
Sigmund [
36
] developed a method for calculating the expected payoff for each player in any population.
Moreover, they pioneeringly analyzed the population dynamics when the population ratio of the group
was updated in accordance with the replicator dynamics from the expected payoffs. The development
of this method enabled a rigorous mathematical analysis of stability.
3. Analytical Models of Indirect Reciprocity
Due to the pioneers’ efforts, a basic analytical model of indirect reciprocity has been developed
and shared. In the model, players play “giving games” (or “donation games”). A donor playing a
game decides whether to cooperate (
C
) or defect (
D
). If the donor chooses
C
, she or he must pay
a fixed amount
c
to a recipient. The recipient receives a fixed benefit
b
if and only if the donor
contributes. In this game, the benefit cost ratio of a game satisfies
r ≡ b/c >
1. If the donor chooses
D
,
nothing happens. Self-interested myopic players choose
D
because contributors do not benefit from
their own contribution, and thus the model reveals a social dilemma. Researchers in this field have
typically focused on two types of errors. One is implementation error
e
1
, which means unintentional
behavior. A player intending to cooperate may not actually cooperate due to this type of error. Strictly
speaking, this type of implementation error is called a “unilateral implementation error”. A “bilateral
implementation error” is two types of errors: A player intending cooperation mistakenly defects and
a player intending defect mistakenly cooperates. The second type of error is assessment error
e
2
,
which means misperception of an assessment. This type of error can lead to a player being given an
incorrect reputation.
Several developments have contributed to refined analysis of indirect reciprocity models.
Nowak and Sigmund [
31
] introduced the use of the probability that a player knows (either through
direct observation or via gossip) what a randomly assigned recipient did in the past. Brandt and
Sigmund
[38–40]
developed a continuous-entry model to balance reputation values among the different
types of groups. Ohtsuki et al. [
41
] established a two-time scale assumption, the analytical assumption
that reputation dynamics are independent of strategy dynamics. It is based on the assumption that
reputation information updating is much faster than strategy changing. Okada et al. [
42
] developed a
solitary observation method that guarantees that a solution is obtained for reputation dynamics in a
private assessment scheme. These developments have enabled a more systematic and comprehensive
analysis of indirect reciprocity.
While there are many analytical models that deal with indirect reciprocity, here we introduce a typical
replicator dynamics model consisting of three types of players. There are unconditional cooperators (
X
)
who always cooperate, unconditional defectors (
Y
) who never cooperate, and discriminators (
Z
) who
cooperate selectively on the basis of indirect reciprocity. To do so, they act in accordance with a rule that
determines the action to be taken, either
C
or
D
. To analyze replicator dynamics, we must assume that
1
“Norm” has often been used in theoretical studies of indirect reciprocity. In psychology and sociology, it means shared
societal rules. In the studies of the evolution of cooperation, it means assessment rules for determining reputation and
action rules for selecting cooperative behaviors.
Games 2020, 11, 27 5 of 17
an infinite number of well-mixed players play games repeatedly. Let
x
,
y
, and
z
be the proportions
(population ratios) of X, Y, and Z, respectively, where x + y + z = 1 is always satisfied.
In a simple model, it is assumed that the discriminators cooperate only with those players who
have good reputations. Let
g
be the fraction of good players. This fraction is decomposed into
g
X
,
g
Y
, and
g
Z
, where
g
s
is the fraction of good players with strategy
s
in the set
S = {X
,
Y
,
Z}
.
g = xg
X
+ yg
Y
+ zg
Z
is always satisfied.
Let
P
s
be the expected payoff of an
s
type player, where
s ∈ S
. The expected payoff functions of
the three strategists are:
P
X
= b(x + zg
x
) − c
P
Y
= b(x + zg
y
)
P
Z
= b(x + zg
z
) − cg,
where the factor (1 − e
1
) must be added if an implementation error is introduced.
We are ready to analyze a replicator dynamics model using these payoff functions. The dynamics
are described as
˙
x = x(P
X
−
¯
P)
,
˙
y = y(P
Y
−
¯
P)
and
˙
z = z(P
Z
−
¯
P)
, where
¯
P = xP
X
+ yP
Y
+ zP
Z
is the average payoff over the population. By drawing the dynamics completely, we can identify
evolutionary stable equilibria in the population, locally asymptotic points, cyclical dynamics, neutral
fixed points, neutral mutants, and so on. For example, an analysis shows that a discriminator (
Z
) is
evolutionary stable because neither perfect cooperators (
X
) nor perfect defectors (
Y
) invade into a
population consisting of only one type (
Z
) of player. Another analysis shows that the expected payoffs
for a
Z
player and an
X
player in a group without any
Y
players are the same, so
X
and
Z
are able to
invade to each other’s population as neutral mutants.
4. Problems with Image Scoring Norm
Many analyses have been carried out on indirect reciprocity models by using evolutionary game
theory. The first target was the image scoring norm [
31
,
39
,
43
–
45
], which follows very simple rules:
Increase the score of a player who cooperates, decrease the score of a player who defects, and cooperate
only with players with scores exceeding a certain threshold. This idea is simplified by using binary
scoring based on the assumption that there are only two kinds of scores, good and bad, and by
updating the scores in accordance with only the last behavior. A rigorous analysis of evolutionary
stability revealed that the image scoring norm does not have evolutionary stability. This finding was
important for the subsequent progress in research.
Rigorous analyses of the image scoring norm revealed three theoretical problems that prevent
it from having evolutionary stability. The first is that a second-order free rider can become a neutral
mutant [
46
–
53
], whose expected payoff is same as the other residents. The non-cooperators are called
first-order free riders because they do not pay the costs of providing public goods while those who do
not pay the costs of excluding first-order free riders are called second-order free riders. In the evolution
of cooperation based on reciprocity, it is necessary to execute selective cooperation in order to exclude
non-cooperators (first-order free riders). In other words, it is necessary to distinguish a cooperator
from a defector and to cooperate with only the former. Perfect cooperators are free riders in this regard
and are therefore called second-order free riders.
In a cooperative regime, players adopting the image scoring norm and those adopting the perfect
cooperator norm behave in exactly the same way because a player who adopts the image scoring norm
always cooperate in a cooperative regime, so they cannot be distinguished. Therefore, there is no
difference in payoff between them, meaning that the perfect cooperators are neutral mutants against
players who adopt the image scoring norm, and thus the perfect cooperators can invade the population
of the image scoring norm players. In other words, even if image scoring eliminates the first-order
free riders and results in the creation of a cooperative regime, the regime may still be invaded by
second-order free riders (the perfect cooperators). The cooperative regime of the second-order free
Games 2020, 11, 27 6 of 17
riders is subsequently invaded by the first-order free riders, so cooperation cannot be maintained.
This feature of the image scoring norm has been called its Achilles heel [54–56].
The second theoretical problem with the image scoring norm is the occurrence of bad reputation
chains due to errors [
57
]. The third one is punisher downgrading. That is, with the image
scoring norm, the score for a player who displays an uncooperative behavior because its partner
is identified as having a bad reputation is downgraded [
24
,
31
]. This problem is called the scoring
dilemma
[23,24,44,45,55,58]
, a newly identified dilemma in trying to resolve the social dilemma in the
framework of indirect reciprocity. Attempts to resolve this dilemma have brought a great breakthrough
to indirect reciprocity research.
5. Refinement of Assessment Function
The solution to the scoring dilemma brought a new type of dilemma, and methods to overcome it
have been sought. The direct solution is to identify non-cooperative behavior. How can we discriminate
between non-cooperative behaviors that are selfish and those that are intended as punishment?
Since the image scoring norm give scores to all players, it is natural to think that behaviors are
discriminated on the basis of the recipient’s score. To analyze this idea with a simple model, players
are assigned binary reputation labels of good and bad
2
. Punishment is considered justified if it is
non-cooperation with bad players and not non-cooperation with good players [
44
,
58
,
61
]. Many studies
have analyzed norms in which justified punishment does not downgrade the punisher’s reputation.
Many studies have looked for norms to replace the image scoring norm since refinement of the
assessment function used to determine reputation became the main concern in the 2000s. Given a
set of actions (
Action = {C
,
D}
) and a set of reputations (
Reputation = {G
,
B}
), the image scoring
norm uses an assessment function,
Assessment
IS
: Donor’s Action → Donor’s Reputation
, to assess
the behaviors of each player. It can be assumed that (
Assessment
IS
(C)
,
Assessment
IS
(D)) = (G
,
B)
.
By extending this formulation, we can take into account justified punishment by defining a new
assessment function:
Donor’s action × Recipient’s reputation → Donor’s reputation
. Consideration
was given to extending the domain of the assessment function in this way. In an exhaustive analysis,
Ohtsuki and Iwasa [
62
] used
Donor’s action × Recipient’s reputation × Donor’s reputation →
Donor’s reputation
as the assessment function. This formulation has been widely used in many
studies. Santos et al. [
63
] conducted an analysis that included past information in the domain of the
assessment function.
Consideration was also given to using an action function to extend the assessment function:
(Action
IS
(G)
,
Action
IS
(B)) = (C
,
D)
using the action function
Action
IS
:
Recipient’s reputation →
Donor’s Action
. Expanding on this, Panchanathan and Boyd [
58
] analyzed two action functions in
which the formula is
Donor’s reputation × Recipient’s reputation → Donor’s Action
. Figure 3 shows
the various configuration of assessment and action rules when binary reputation labels are used.
While many assessment and action functions have been formulated, determining how to maintain
stable cooperation is an important task. Ohtsuki and Iwasa [
62
] provided a solution in their exhaustive
analysis. By analyzing all combinations of 256 types of assessment functions and 16 types of action
functions, they identified 8 types of norms, the “leading eight norms”, that can maintain stable
cooperation. A feature having all eight of these norms is justified punishment because, if the donor’s
reputation is
Good
, the donor’s action is
D
, and if the recipient’s reputation is
Bad
, the donor’s
reputation is updated to
Good
in the assessment functions of all eight norms. Moreover, if the donor’s
reputation is
Good
and the recipient’s reputation is
Bad
, the donor’s action is
D
in the action rules
of all eight norms. These insights have been confirmed by their follow-up analyses [
64
,
65
] and other
2
While many studies have dealt with only the binary reputation label, other studies have dealt with three or more reputation
labels [59,60], as explained in detail by Rutte and Taborsly [25].
Games 2020, 11, 27 7 of 17
analyses [
24
,
66
,
67
] as well as by experimental validation [
68
,
69
] and analyses in other fields [
70
].
Studies on indirect reciprocity have made great progress due to their findings.
Figure 3.
Assessment and action rules for binary reputation labeling. (G: “Good label”; B: “Bad label”;
C: “Cooperation”; D: “Defect”).
Several of the leading eight norms have been analyzed individually. The standing (or
consistent-standing, L2) norm and the simple-standing (or L3) norm [
71
], which originated
from Sugden(1986) [
72
], have been well analyzed. Their performance has been shown to be
excellent [42,58,67,73,74].
The judging (or L8) norm and the stern-judging (or L6) norm [
56
,
71
,
75
–
79
],
which was derived from Kandori(1992) [
80
], has also been well analyzed. While not included
in the leading eight norms, the shunning norm [
81
], which is relatively simple, has also been
analyzed individually.
6. Resolving the Punishment Dilemma
While Otsuki and Iwasa’s exhaustive analysis [
62
] revealed the characteristics of the norms
supporting indirect reciprocity, it has been pointed out by several authors over the past decade that
the simple assumptions on which their model is based have revealed a new issue: The “punishment
dilemma”. Many theoretical analyses of indirect reciprocity have been, because of the ease of theoretical
analysis, based on the assumption that all reputations are public information. However, it is unlikely
that everyone will have the same impression of an individual.
If the assumption that all reputations are public information had not been adopted, the fact that
“justified punishment” would not always be justified would have been overlooked in most studies.
The punishment is justified if and only if both the punisher and the observer consider the recipient
to be bad. If the observer considers the recipient to be good, the punishment is deemed unjustified.
The issue has not been resolved yet in the evolution of cooperation based on indirect reciprocity, so the
social dilemma problem has not been completely solved. Models of private assessment that addressed
the punishment dilemma began to be reported a decade ago [42,67,82–88].
Analyses using these private assessment models clarified that several of the leading eight norms,
which are considered to be cooperative norms in a public assessment scheme, cannot maintain stable
Games 2020, 11, 27 8 of 17
cooperation due to the punishment dilemma. In particular, the breakdown of the stern-judging norm
3
has been demonstrated in several studies [
42
,
55
,
56
,
67
,
89
,
90
]. In addition, the required infinite system
of simultaneous equations makes the analysis quite difficult in the theoretical analysis of a private
assessment scheme
4
. Okada et al. developed a method for deriving an analytical solution without any
approximation that is based on the assumption of solitary observation [
42
]. An exhaustive analysis of
a private assessment scheme [
90
] has clarified the features of norms that resolve all three dilemmas:
Social, scoring, and punishment as shown in Figure 4.
Figure 4.
Resolving social dilemmas using indirect reciprocity. This figure is based on Figure 2 of
Okada (2020) [90].
7. Remaining Issues in Study of Indirect Reciprocity Using Evolutionary Game Theory
There remain several important issues in the study of indirect reciprocity using evolutionary
game theory. Generally, the greater the benefit cost ratio (
= r = b/c
) for cooperation, the greater the
maintenance of cooperative regimes [
91
].Whitaker et al. (2016) [
92
] analyzed the relationship between
the ratio and the emergence of cooperation and found that norms using second-order information such
as the simple-standing and stern-judging norms require
r
to be around or greater than 1.2, whereas
the image-scoring norm (using only first-order information) requires
r
to be around or greater than 5.
Furthermore, analyses of the effects of errors have shown that the higher the error rate, the lower the
level of cooperation [49,58,87].
According to the findings of studies that refined the assessment function, several norms maintain
stable cooperation, but how the order of usage emerges in a situation with a mixture of norms remained
unclear due to complexity of the required analytical methods. Yamamoto et al. [
93
] succeeded in
revealing the role of the norms by comparing the state of activating a specific norm and the state
of deactivating it by developing a norm knockout method. Uchida et al. [
94
] analyzed the norm
3
Of the leading eight norms, two (simple-standing and stern-judging) use up to second-order information and the other six
use up to third-order information. Of the first two, only the stern-judging norm breaks cooperative regimes in a private
assessment scheme.
4
This is because the definition of the conjunctive probability of
v
players whose images of a specified player are the same
needs the conjunctive probability of
v +
1 players. Therefore, the definition of the conjunctive probability of two players
whose images of a specified player are the same requires an infinite system of simultaneous equations when the number of
game observers is infinite.
Games 2020, 11, 27 9 of 17
ecosystem using equation systems representing the probabilities that a player adopting any norm
has a good image of another player adopting any norm. Gaudeul et al. [
95
] fully characterized the
evolutionary stable equilibria and analyzed their comparative statistics with respect to the benefit
cost ratio.
Many studies have explored other mechanisms for establishing indirect reciprocity. One approach
is to use a two-stage model in which punishment is administered separately from non-cooperative
action; that is, the decision as to whether to administer punishment is made after playing a social
dilemma game [
96
–
98
]. This approach to punishment has taken several branches. As well as
peer punishment [
99
–
103
], researchers investigated methods for proactively collecting the costs of
punishment [
49
] that overcome the evolutionary instability of peer punishment. Other extensions
include institutional and pool punishment systems [104–110].
Separating the punishment system from the game opened the door to generalization of
punishment to rewards and incentives [
111
]. Theoretical models of reward for cooperative behavior in
contrast with punishment for non-cooperative behavior have been analyzed [
100
,
112
–
117
]. In addition,
both anti-social punishment (punishment for cooperative behavior) [
118
,
119
] and anti-social reward
(reward for non-cooperative behavior) [
120
] have been theoretically analyzed. However, Li et al. [
121
]
argued that costly punishment has little effect on cooperation despite many the theoretical analyses
that have shown that it does. This topic needs further discussion. The effects of incentive systems
on the evolution of cooperation have been widely focused upon. A framework for introducing
rewards and punishment is being actively researched in the field of social psychology [
122
–
127
].
The combination of punishment with indirect reciprocity has also been analyzed. Jordan et al. [
128
]
conducted a game-theoretical analysis and their experiments showed that third-party punishment,
i.e., punishment for behavior by unaffected observers, has positive effects on cooperation. Several
groups have considered punishment to be a social norm and have explored conditions under which
punishment emerges and its functions [122,124,129].
Another mechanism for establishing indirect reciprocity is to selectively exclude the second-order
free riders from the population [
46
]. While indirect reciprocity drives selective cooperation, social
exclusion is an alternative means [
130
]. A combination of reputation and ostracism supporting
cooperation in groups of different sizes has been analyzed [131].
Clarifying effects of interaction structures on cooperation is also an important task. Theoretical
analysis favors a well-mixed population due to analytical simplicity. However, network theory has
revealed that real social networks have several features. Studies of the effect of the interaction structures
on the evolution of indirect reciprocity have produced many findings [64,132–136].
Other tasks include clarifying the effects of mutation [
137
], the effects of large groups [
61
,
138
,
139
],
and the effects of incomplete information [
41
,
87
,
138
,
140
]. Several studies compared cooperation rates
based on indirect reciprocity with that based on direct reciprocity [61,141–143].
8. Indirect Reciprocity in Various Disciplines
While we have focused on theoretical analysis of indirect reciprocity in game theory, its application
has been widely studied in many disciplines. Figure 5 shows a bird’s eye view of these disciplines.
In social psychology, several studies have revealed that the findings of theoretical research are
inconsistent with those of experimental research. Milinski et al. (2001) [
144
] compared the image
scoring norm, which takes into account only action information (first-order information), with a norm
in which the reputation information (second-order information) of the recipient is also taken into
account
5
. They showed that people prefer decision-making based on image scoring and thus concluded
5
In this paper, we use the terms ’first-order’ and ’second-order’ in two different contexts: ’Free riders’ and ’information used
for rules assessment’. First-order free riders are players who do not cooperate while second-order free riders are players
who do not punish first-order free riders. First-order information refers to donor action while second-order information
refers to recipient reputation.
Games 2020, 11, 27 10 of 17
that people do not make decisions as complexly as theory suggests. With this as a starting point,
theorists and experimentalists have argued about which domain of the assessment function should be
adopted as the norm for the indirect reciprocity mechanism. Swakman et al. [
68
] presented results
showing that people may actually take into account second-order information even of they must pay
a cost for doing so. Okada et al. [
69
] demonstrated that there is a decision-making method in which
second-order information is taken into account before first-order information. The debate continues.
Figure 5. Study of indirect reciprocity in various disciplines.
The origin of indirect reciprocity has also been studied in social psychology. A field study
investigated the development of indirect reciprocity in 18-24-month-old toddlers and in infants aged
around six months [145]. The results showed that toddlers reciprocated prosocial behavior indirectly.
The behavior of adolescence has also been observed. In upstream indirect reciprocity, receiving an
equal (vs. unequal) distribution led adolescents to become fairer to a third person. In downstream
indirect reciprocity, older adolescents were more likely to devote their own resources to enforce fairness
norms than younger adolescents [146].
Natural field experiments in behavioral science performed over the past decade have explored
evidence of indirect reciprocity. Studies include investigation of indirect reciprocity and charitable
work in a hair salon [
147
], the power of indirect reciprocity to promote prosocial behavior in one-shot
interactions among drivers [
148
], large-scale cooperation in real world settings [
149
], and whether a
service request is more likely to be rewarded for those with a profile history of offering the service
(to third parties) in an online environment where members can repeatedly ask for and offer services to
each other, free of charge [150].
In economics, several studies investigated the effects of conforming behavior [
129
] while
others have investigated the connection to signaling theory [
151
,
152
]. A theoretical study used the
mechanisms of indirect reciprocity to explain cognitive distortion predicted by prospect theory [153].
In sociology, the effect of indirect reciprocity on partner choice has been analyzed. Since indirect
reciprocity separates a group of people into good people and bad people who cooperate selectively,
it is a natural extension to play games with only good friends. One study examined game playing
in combination with partner choice [
54
,
154
]. This direction led to a study of the connection of
indirect reciprocity with group selection and with in-group favouritism [
155
]. A simulation analysis
demonstrated the results of introducing an indirect reciprocity mechanism into partner selection in a
group [156,157].
Games 2020, 11, 27 11 of 17
Application of indirect reciprocity to information science and the Internet is a hot topic.
By modeling an agent’s identity using traits, that can be shared with other agents, Bedewi et al.
presented a basis for agents to change their identity [
158
]. They expect that indirect reciprocity will
be effective for future technology and autonomous machines that need to function as a coalition.
Tian et al. focused on how spatial reciprocity aids indirect reciprocity in sustaining cooperation in
practical P2P (person-to-person) environments [
159
]. Their simulations showed that an incentive
mechanism enhances cooperation better among structured peers than among unstructured peers.
Toriumi et al. used simulation and the indirect reciprocity framework to investigate the relationships
between social media posts and their responses[
160
]. Their findings would be useful for designing
incentive systems in social media. Wang et al. used simulation of indirect group formation in a sparse
network to demonstrate the group formation process [161].
Many disciplines have explored indirect reciprocity. In behavioral science, a statistical analysis
of privately owned firms in China showed that philanthropic giving initiates and amplifies
indirect reciprocity between visiting officials and local businesses, thereby increasing corporate
investment [
162
]. In philosophy, indirect reciprocity has been explored in relation to moral dilemmas
since indirect reciprocity research deals with the evolution of norms [
63
,
163
]. In cognitive science,
the relationship between neuroscience and indirect reciprocity has been examined. One study
used neuroimaging experiments to reveal the functional and anatomical neural bases of indirect
reciprocity [74].
9. Conclusions
We have overviewed theoretical studies on indirect reciprocity. The history of these studies shows
how indirect reciprocity came to be established as an academic field [
6
,
23
]. Moreover, we introduced
many studies that refined the assessment function in an effort to enable a cooperative regime to be
maintained. Academic efforts in this field have identified the features of assessment functions that
resolve the scoring dilemma and the punishment dilemma that emerged simultaneously from the
social dilemma [
90
]. While theoretical studies on indirect reciprocity have been ongoing, studies in
many related disciplines have also investigated the mechanism of indirect reciprocity.
While indirect reciprocity requires the diffusion of reputation information, private assessment
schemes that relax this requirement have been actively studied in recent years. In reality, selective
cooperation is considered to be achievable through proper use of reputation information obtained
through public assessment and impressions obtained through private assessment. Extension along
these lines has begun [56], and future developments are expected.
The idea of selective cooperation as an explanation of how cooperation is rationalized in order to
resolve social dilemmas is based on the concepts of direct reciprocity, indirect reciprocity, and network
reciprocity. It comes from a kin selection framework, which is a common ancestor of these concepts.
Therefore, they cannot be applied to public goods games as they are. This is because, in public
goods games, the actions of individual players are equally distributed as public goods, so selective
cooperation is impossible. Future extensions may bring a breakthrough that resolves this dilemma.
Author Contributions:
This study was performed by a single researcher. The author has read and agreed to the
published version of the manuscript.
Funding:
Part of this work was supported by JSPS (Grants-in-Aid for Scientific Research) 17KK0055, 17H02044,
18H03498, and 19H02376.
Acknowledgments:
The author is grateful to Yutaka Nakai, Hitoshi Yamamoto, Satoshi Uchida, Tatsuya Sasaki,
Christian Hilbe, and Hannelore De Silva for their comments.
Conflicts of Interest: The author declares no conflicts of interest.
Games 2020, 11, 27 12 of 17
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