EFTA00803978.pdf

1 a Indirect reciprocity with private, noisy, and incomplete information Christian Hilbe11, Laura Schmid', Josef Tkadleca, Krishnendu Chatterjeee, and Martin A. Nowak" 'Institute of Science and Technology Austria, 3400 Klostemeuburg, Austria; °Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138; 'Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138; and °Department of Mathematics, Harvard University, Cambridge, MA 02138 Edited by Brian Skyrms, University of California, Irvine, CA, and approved October 16, 2018 (received for review June 19, 2018) Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is trans- mitted publicly, such that all community members agree on each others reputation, previous research has highlighted eight cru- an impaired reputation (30). According to "third-order norms,- observers need to additionally take the donor's reputation into account. In this way, assessment rules of higher order are increas- ingly able to give a more nuanced interpretation of a donor's action, but they also require observers to store and process more information. When subjects are restricted to binary norms, such that repu- cial moral systems. These "leading-eight" strategies can maintain tations are either "good" or "bad,- an exhaustive search shows cooperation and resist invasion by defectors. However, in real there are eight third-order norms that maintain cooperation (20, populations individuals often hold their own private views of oth- 31). These "leading-eight strategies" are summarized in Table ers. Once two individuals disagree about their opinion of some 1, and we refer to them as LI-IS. None of them is exclu- third party, they may also see its subsequent actions in a different sively based on first-order information, whereas two of them light Their opinions may further diverge over time. Herein, we (called "simple standing" and "stem judging," refs. 32 and 33) explore indirect reciprocity when information transmission is pri- require second-order information only. There are several uni- vate and noisy. We find that in the presence of perception errors, versal characteristics that all leading-eight strategies share. For most leading-eight strategies cease to be stable. Even if a leading- example, against a recipient with a good reputation, a donor who eight strategy evolves, cooperation rates may drop considerably cooperates should always obtain a good reputation, whereas a when errors are common. Our research highlights the role of reli- donor who defects should gain a bad reputation. The norms dif- able information and synchronized reputations to maintain stable fer, however, in how they assess actions toward bad recipients. moral systems. Whereas some norms allow good donors to preserve their good standing when they cooperate with a bad recipient, other norms cooperation I indirect reciprocity I social norms I evolutionary disincentivize such behaviors. game theory Ohtsuki and Iwasa (20, 31) have shown that if all members of a population adopt a leading-eight strategy, stable cooperation can umans treat their reputations as a form of social capital H (1-3). They strategically invest into their good reputa- tion when their benevolent actions are widely observed (4-6), emerge. Their model, however, assumes that the players' images are synchronized; two population members would always agree on the current reputation of some third population member. The which in turn makes them more likely to receive benefits in assumption of publicly available and synchronized information subsequent interactions (7-12). Reputations undergo constant changes in time. They are affected by rumors and gossip (13), which themselves can spread in a population and develop a Significance life of their own. Evolutionary game theory explores how good reputations are acquired and how they affect subsequent behav- Indirect reciprocity explores how humans act when their rep- iors, using the framework of indirect reciprocity (14-17). This utation is at stake, and which social norms they use to framework assumes that members of a population routinely assess the actions of others. A crucial question in indirect 1 observe and assess each other's social interactions. Whether reciprocity is which social norms can maintain stable cooper- a given action is perceived as good depends on the action ation in a society. Past research has highlighted eight such itself, the context, and the social norm used by the population. norms, called "leading-eight" strategies. This past research, Behaviors that yield a good reputation in one society may be however, is based on the assumption that all relevant infor- condemned in others. A crucial question thus becomes: Which mation about other population members is publicly available social norms are most conducive to maintain cooperation in a and that everyone agrees on who is good or bad. Instead, population? here we explore the reputation dynamics when information Different social norms can be ordered according to their com- is private and noisy. We show that under these conditions, plexity (18) and according to the information that is required most leading-eight strategies fail to evolve. Those leading- to assess a given action (19, 20). According to "first-order eight strategies that do evolve are unable to sustain full norms," the interpretation of an action depends only on the cooperation. action itself. When a donor interacts with a recipient in a social dilemma, the donor's reputation improves if she cooperates, Author contributions: C.M., LS, J.T., K.C., and M.A.N. designed research performed whereas her reputation drops if she defects (21-26). Accord- esearch analyzed data. and wrote the paper. ing to "second-order norms," the interpretation of an action The authors declare no conflict of interest. additionally depends on the reputation of the recipient. The This article is a PNAS Direct Submission. recipient's reputation provides the context of the interaction. It Published under the PNAS license. allows observers to distinguish between justified and unjustified 'To whom correspondence should be addressed. Email: christian.hilbellistac.at. defections (27-29). For example, the standing strategy consid- ThIsarlide contains supporting informationonlineat vwnvpnasorgAookuprsuPPIldoi:I6 ers it wrongful only to defect against well-reputed recipients; 1073/peas.111105651I9-itiCSupplementat donors who defect against bad recipients do not suffer from viww.pnas.orgicgildoV10.10731pnas.1810S6S1 S PNAS Latest Articles I t et 6 EFTA00803978 Table 1. The leading-eight strategies of indirect reciprocity a player misinterprets the donor's cooperation as defection or, CCs al conversely, the donor's defection as cooperation. After observ- C L ing an interaction, population members independently update a 2 a-2 their image of the donor according to the information they 5 co B have (Fig. 1). UN (nu) Assessment rule LI L2 L3 L4 L7 L8 To do so, we assume that each individual is equipped with a strategy that consists of an assessment rule and an action rule. Good cooperates with Good 9 9 9 9 9 9 9 9 The player's assessment rule governs how players update the rep- Good cooperates with Bad 9 b g g b b g b utation they assign to the donor. Here we consider third-order Bad cooperates with Good g g g g g g g g assessment rules. That is, when updating the donor's reputa- Bad cooperates with Bad g g g b g b b b tion, a player takes the donor's action into account, as well as Good defects against Good b b b b b b b the donor's and the recipient's previous reputation. Importantly, Good defects against Bad when two observers differ in their initial assessment of a given 9 9 9 9 9 9 9 9 Bad defects against Good b b b b b b b donor, they may also disagree on the donor's updated reputa- Bad defects against Bad b b o n g g b b tion, even if both apply the same assessment rule and observe the same interaction (Fig. IC). The second component of a player's strategy, the action rule, determines which action to take when Action rule LI L2 L3 L4 LS 16 L7 L8 chosen to be the donor. This action may depend on the player's own reputation, as well as on the reputation of the recipient. A Good meets Good C C C C C C C C player's payoff for this indirect reciprocity game is defined as the Good meets Bad D D D D D expected benefit obtained as a recipient, reduced by the expected Bad meets Good C C C C C C C C costs paid when acting as a donor, averaged over many rounds Bad meets Bad C C D D D (see Materials andMethods for details). There are eight strategies, called the Pleading eight* that have been shown to maintain cooperation under public assessment (20, 31). Each such Analysis of the Reputation Dynamics. We first explore how differ- strategy consists of an assessment rule and of an action rule. The assessment ent social norms affect the dynamics of reputations, keeping the rule determines whether a donor is deemed good (9) or bad (b). This assess- strategies of all players fixed. To this end, we use the concept of ment depends on the context of the interaction (on the reputations of the image matrices (34-36). These matrices record, at any point in donor and the recipient) and on the donor's action (C or D). The action rule time, which reputations players assign to each other. In Fig. 2 determines whether to cooperate with a given recipient when in the role of the donor. A donor's action may depend on her own reputation, as well as A-H, we show a snapshot of these image matrices for eight dif- on the reputation of the recipient. All of the leading-eight strategies agree ferent scenarios. In all scenarios, the population consists in equal that cooperation against a good player should be deemed as good, whereas proportions of a leading-eight strategy, of unconditional cooper- defection against a good player should be deemed bad. They disagree in ators who regard everyone as good (ALLC) and of unconditional how they evaluate actions toward bad recipients. defectors who regard everyone as bad (ALLD). Depending on the leading-eight strategy considered, the reputation dynamics in these scenarios can differ considerably. greatly facilitates a rigorous analysis of the reputation dynam- First, for four of the eight scenarios, a substantial propor- ics. Yet in most real populations, different individuals may have tion of leading-eight players assigns a good reputation to ALLD access to different kinds of information, and thus they might players. The average proportion of ALLD players considered disagree on how they assess others. Their opinions may well be correlated, but they will not be correlated perfectly. Once individuals disagree in their initial evaluation of some person, B Poem mega A I3,I4 li C Uptlean1notitelicee a;;:- kale r•pgilitlaill their views may further diverge over time. How such initial dis- agreements spread may itself depend on the social norm used I a) 191 , 29 24 Maw I by the population. While some norms can maintain coopera- 3g coopleatOS 30 30 1 tion even in the presence of rare disagreements, other norms are more susceptible to deviations from the public information Play- I Payer 2 Pleiyy Mayer 2 Pest 2 assumption (34-37). Here, we explore systematically how the leading-eight strategies fare when information is private, noisy, and incomplete. We show that under these conditions, most leading-eight strategies cease to be stable. Even if a leading- Pivot 3 Km% 3 eight strategy evolves, the resulting cooperation rate may be drastically reduced. Fig.'. Under indirect reciprocity, individual actions are continually assessed by all population members. (A) We consider a population of different Results players. All players hold a private repository where they store which of their coplayers they deem as either good (g) or bad (b). Different play- Model Setup. We consider a well-mixed population of size N. en may hold different views on the same coplayer. In this example, player The members of this population are engaged in a series of 2 is considered to be good from the perspective of the first two play- cooperative interactions. In each round, two individuals are ran- en, but he is considered to be bad by player 3. (8) In the action stage, domly drawn, a donor and a recipient. The donor can then two players are randomly chosen, a donor (here, player 1) and a recipient decide whether to transfer a benefit b to the recipient at own (here, player 2). The donor can then decide whether or not to cooper- cost c, with 0C c< b. We refer to the donor's two possible ate with the recipient. The donor's decision may depend on the stored actions as cooperation (transferring the benefit) and defection reputations in her own private repository. (C) After the action stage, all players who observe the interaction update the donors reputation. The (not doing anything). Whereas the donor and the recipient newly assigned reputation may differ across the population even if all always learn the donor's decision, each other population mem- players apply the same social norm. This can occur when individuals ber independently learns the donor's decision with probability already disagreed on their initial assessments of the involved players, (if) q> 0. Observations may be subject to noise: We assume that when some subjects do not observe the interaction and hence do not all players who learn the donor's action may misperceive it with update the donors reputation accordingly, or (iii) when there are percep- probability c > 0, independently of the other players. In that case, tion errors. 2 of 6 I vavw.pnas.orgrcgi/dolff0.I073/0nas.18105651t5 Hilbe et al. EFTA00803979 as good by L.3, L4, 15, and 1.6 is given by 31%, 31%, 42%, single disagreement, and they have the longest recovery time. and 50%, respectively (SI Appendix, Fig. SI). In terms of these This finding is also reflected in Fig. 2, which shows that these four leading-eight strategies, a bad player who defects against two strategies are unable to maintain cooperation. L6 eventually another bad player deserves a good reputation (Table 1). In assigns random reputations to all coplayers, whereas 1.8 assigns particular, ALLD players can easily gain a good reputation a bad reputation to everyone (SI Appendix, Fig. S4). While 1.6 whenever they encounter another ALLD player. Moreover, ("stern") has been found to be particularly successful under pub- the higher the proportion of ALLD players in a population, lic information (18, 32, 33), our results confirm that this strategy the more readily they obtain a good reputation. This finding is too strict and unforgiving when information is private and suggests that while 13-Lb might be stable when these strate- noisy (34-36). gies are common in the population (20, 38), they have prob- lems in restraining the payoff of ALLD when defectors are Evolutionary Dynamics. Next we explore how likely a leading-eight predominant. strategy would evolve when population members can change Second, leading-eight players may sometimes collectively their strategies over time. We first consider a minimalistic sce- judge a player of their own kind as bad. In Fig. 2, such cases nario, where players can choose among three strategies only, a are represented by white vertical lines in the upper left square leading-eight strategy L„ ALLC, and ALLD. To model how play- of an image matrix. In SI Appendix, Fig. S2 we show that such ers adopt new strategies, we consider simple imitation dynamics apparent misjudgments are typically introduced by perception (39-42). In each time step of the evolutionary process, one player errors. They occur, for example, when a leading-eight donor is picked at random. With probability p (the mutation rate), defects against an ALLC recipient, who is mistakenly considered this player then adopts some random strategy, corresponding as bad by the donor. Other leading-eight players who witness this to the case of undirected learning. With the remaining prob- interaction will then collectively assign a bad reputation to the ability 1— p, the player randomly chooses a role model from donor—in their eyes, a good recipient has not obtained the help the population. The higher the payoff of the role model, the he deserves. This example highlights that under private infor- more likely it is that the focal player adopts the role model's mation, an isolated disagreement about the reputation of some strategy (Materials and Methods). Overall, the two modes of population member can have considerable consequences on the updating, mutation and imitation, give rise to an ergodic process further reputation dynamics. on the space of all population compositions. In the following, To gain a better understanding of such cases, we analytically we present results for the case when mutations are relatively explored the consequences of a single disagreement in a homo- rare (43, 44). geneous population of leading-eight players (see SI Appendix First, we calculated for a fixed benefit-to-cost ratio of 61e= S for all details). There we assume that initially, all players con- how often each strategy is played over the course of evolu- sider each other as good, with the exception of one player who tion, for each of the eight possible scenarios (Fig. 3). In four considers a random coplayer as bad. Assuming that no further cases, the leading-eight strategy is played in less than 1% of the errors occur, we study how likely the population recovers from time. These cases correspond to the four leading-eight strate- this single disagreement (i.e., how likely the population reverts gies L3—L6 that frequently assign a good reputation to ALLD to a state where everyone is considered good) and how long players. For these leading-eight strategies, once everyone in a it takes until recovery. While some leading-eight strategies are population has learned to be a defector, players have difficul- guaranteed to recover from single disagreements, we find that ties in reestablishing a cooperative regime (in Fig. 3 C—F, once other strategies may reach an absorbing state where players ALLD is reached, every other strategy has a fixation probabil- mutually assign a bad reputation to each other. Moreover, even ity smaller than 0.001). In contrast, the strategy 1.8 is played if recovery occurs, for some strategies it may take a consider- in substantial proportions. But in the presence of noise, players able time (SI Appendix, Fig. S3). Two strategies fare particularly with this strategy always defect, because they deem everyone as badly: 1.6 and 1.8 have the lowest probability to recover from a bad (Fig. 2). A LI ALLC ALLD B La ALLC ALLD C L3 M1C MID D L4 MSC NW LI Lx OLLC AMC AILC ALLC NW ALM MID E LS ALLC ALLD F L6 ALLC ALLD H La ALLC OLID LS L6 LE NW ALLC ARLO LO ALLO ALLD Fig. 2. (A-H) When individuals base their decisions on noisy private information, their assessments may diverge. Models of private information need to keep track of which player assigns which reputation to which coplayer at any given time. These pairwise assessments are represented by image matrices. Here, we represent these image matrices graphically, assuming that the population consist of equal parts of a leading-eight strategy, of unconditional cooperators (ALLC) and unconditional defectors (ALLD). A colored dot means that the corresponding row player assigns a good reputation to the column player. Without loss of generality, we assume that ALLC players assign a good reputation to everyone, whereas ALLD players deem everyone as bad. The assessment of the leading-eight players depend on the coplayer's strategy and on the frequency of perception errors. We observe that two of the leading- eight strategies are particularly prone to errors: L6 ("stem judging") eventually assigns a random reputation to any coplayer, while 18 ("judging') eventually considers everyone as bad. Only the other six strategies separate between conditionally cooperative strategies and unconditional defectors. Each box shows the image matrix after 2 .104 simulated interactions in a population of size N = 3.30 = 90. Perception errors occur at rate e = 0.05, and interactions are observed with high probability, q = 0.9. Hilbe et al. PROS Latest Articles I 3 el 6 EFTA00803980  Consistent Stancing SMDIO Staicirg LI 13 A C D 001? 0000 4691 0090 eStel T 0.,21 c0501 0.130 <0.00, 0012\ coil 0409 h _-!. / \ 400. . 4.3%': COM 0.3% 4 •401. 0.1% 'MI, 0,0% AlLD ALLC MID ALLC ALLD NLC MID MSC F Stern Judong Sia)Ing Jul;ing E 1 ts L8 <0.001. .0.031 • ..001 0152 0020; WON .0001 aide 0A0,/ 10.020, 0,169 1 \ V • 99.5% <0001. 1000% `000‘. 00% 1% COI.. 55% '50.0% `00/... 0,0% N. OW Ot69 0%9_ /IUD ALLC ALLD MAC ALLD 3010 ALL0 ALIO Fig. 3. Most of the leading-eight strategies are disfavored in the presence of perception errors. We simulated the evolutionary dynamics when each of the leading-eight strategies competes with ALLC and ALLO. These simulations assume that, over time, players tend to imitate coplayers with more profitable strategies and that they occasionally explore random strategies (Materials and Methods). The numbers within the circles represent the abundance of the respective strategy in the selection-mutation equilibrium. The numbers close to the arrows represent the fixation probability of a single mutant into the given resident strategy. We use solid lines for the arrows to depict a fixation probability that exceeds the neutral probability 1/N, and we use dotted lines if the fixation probability is smaller than 1/N. In four cases, we find that ALL° is predominant (C4). In one case (H), the leading-eight strategy coexists with ALM. but without any cooperation. In the remaining cases (A, 8, and G), we find that LI and L7 are played with moderate frequencies, but only populations that have access to 12 (*consistent standing') settle at the leading-eight strategy. Parameters: Population size N= 50, benefit b = S, cost c = 1, strength of selection s =1, error rate e = 0.05. observation probability q = 0.9, in the limit of rare mutations p 0. There are only three scenarios in Fig. 3 that allow for positive which further errors may accumulate. As a result, whereas L2 cooperation rates. The corresponding leading-eight strategies seems most robust when coevolving with ALLC and ALLD, it are LI, 12 ("consistent standing"), and L7 ("staying,- ref. 45). is unable to maintain full cooperation. Furthermore, additional For LI and L7, the evolutionary dynamics take the form of simulation results suggest that even if L.2 is able to resist invasion a rock-scissors-paper cycle (46-50). The leading-eight strategy by ALLC and ALLD, it may be invaded by mutant strategies that can be invaded by ALLC, which gives rise to ALLD, which in differ in only one bit from L2 (SI Appendix, Fig. S6). turn leads back to the leading-eight strategy. Because ALLD is So far, we have assumed that mutations are rare, such that most robust in this cycle, the leading-eight strategies are played populations are typically homogeneous. Experimental evidence, in less than one-third of the time (Fig. 3A and C). however, suggests that there is considerable variation in the Only consistent standing, I.2, is able to compete with ALLC social norms used by subjects (4, 7-11). While some subjects are and ALLD in a direct comparison (Fig. 38). Under consistent best classified as unconditional defectors, others act as uncon- standing, there is a unique action in each possible situation that ditional cooperators or use more sophisticated higher-order allows a donor to obtain a good standing. For example, when a strategies (I I). In agreement with these experimental studies, good donor meets a bad recipient, the donor keepsv her good there is theoretical evidence that some leading-eight strategies standing by defecting, but loses it by cooperating. Compared like L7 may form stable coexistences with ALLC (36). In SI with stem judging, which has a similar property (18), consis- Appendix, Figs. S7-59, we present further evolutionary results for tent standing incentivizes cooperation more strongly. When two higher mutation rates, in which such coexistences are possible. bad players interact, the correct decision according to consistent standing is to cooperate, whereas a stern player would defect (Table I). 0 • Nevertheless, we find that even when consistent standing is LI L2 1.3 a L4 L5 16 L7 18 common, the average cooperation rate in the population rarely A B Ci exceeds 65%. To show this, we repeated the previous evolution- 1.0 ary simulations for the eight scenarios while varying the benefit- e 0.8 9 to-cost ratio, the error rate, and the observation probability 10.6 (Fig. 4). These simulations confirm that five of the leading-eight 0.4 • 30 ' • strategies cannot maintain any cooperation when competing with 99$ Os 0 8 t S 9 ALLC and ALLD. Only for LI, L2, and L7 are average coop- 0.0 0 0.0 0 0_ 0 0 0 0 0 eration rates positive, reaching a maximum for intermediate 1 3 5 7 9 0.01 0.05 0.09 0.1 0.3 0.5 0.7 0.9 Benefit b Error probstaity e Observation probability ct benefit-to-cost ratios (Fig. 44). If the benefit-to-cost ratio is too '44* low, we find that each of these leading-eight strategies can be Flg. 4. Noise can prevent the evolution of full cooperation even if leading- invaded by ALLD, whereas if the ratio is too high, ALLC can eight strategies evolve. We repeated the evolutionary simulations in Fig. 3, invade (SI Appendix, Fig. S5). In between, consistent standing but varying (A) the benefit of cooperation. (8) the error rate, and (C) the may outperform ALLC and ALLD, but in the presence of noise observation probability. The graph shows the average cooperation rate for each scenario in the selection-mutation equilibrium. This cooperation rate it does not yield high cooperation rates against itself. Even if all depends on how abundant each strategy is in equilibrium and on how much interactions are observed (q = I), cooperation rates in a homoge- cooperation each strategy yields against itself in the presence of noise. For neous L2 population drop below 70% once the error rate exceeds five of the eight scenarios, cooperation rates remain low across the con- 5% (SI Appendix, Fig. S4). Our analytical results in SI Appendix sidered parameter range. Only the three other leading-eight strategies can suggest that while L2 populations always recover from single dis- persist in the population, but even then cooperation rates typically remain agreements, it may take them a substantial time to do so, during below 70%. We use the same baseline parameters as in Fig. 3. Oaf 6 I www.pnas.orgrcgi/dol/10.10734mas.0310565115 MI6e et al. EFTA00803981 Them we show that in the three cases LI, L2, and L7, popula- a =(npcp• ago. aKg. naa, nen. a0 b. ooOsp nom). tions may consist of a mixture of the leading-eight strategy and ALLC for a considerable time. However, in agreement with our corresponds to the player's assessment rule. An entry nµy is equal to one if ram-mutation results, we find for LI and L7 that this mixture the player assigns a good reputation to a donor of reputation x who chooses action A against a recipient with reputation y. Otherwise, if such a donor is of leading-eight strategy and ALLC is susceptible to stochastic considered as bad, the corresponding entry is zero. The second component invasion by ALLD. of the strategy, Discussion = 099. .9,0, /So, 4 4,), (2) gives the player's action rule. An entry 9y is equal to one if the focal player Indirect reciprocity explores how cooperation can be maintained with reputation x cooperates with a recipient with reputation y; otherwise when individuals assess and act on each other's reputations. Sim- it is zero. The assessment and action rules of the leading-eight strategies ple strategies of indirect reciprocity like image scoring (21, 22) are shown in Table 1. We define ALLC as the strategy with assessment rule have been suspected to be unstable, because players may abstain a = (I 1) and action rule $ = (1 1). ALLD is the strategy with n = from punishing defectors to maintain their own good score (27). (0 0) and =(0 0). In contrast, the leading-eight strategies additionally take the con- text of an interaction into account. They have been considered Reputation Dynamks. To simulate the reputation dynamics for players with to be prime candidates for stable norms that maintain coop- fixed strategies, we consider the image matrix (34-36) Mit) = (NO) of a eration (20, 31). Corresponding models, however, assume that population at time t. Its entries satisfy mii(t)= 1 if player i deems player j as good at time t and mg(t)=0 otherwise. We assume that initially, all each pairwise interaction is witnessed only by one observer, who players have a good reputation, rev(0)= 1 for all 1, j. However, our results disseminates the outcome of the interaction to all other popula- are unchanged if the players' initial reputations are assigned randomly. We tion members. As a consequence, the resulting opinions within get only slightly different results if all initial reputations are bad; in that a population will be perfectly synchronized. Even if donors are case, L7 players are unable to acquire a good reputation over the course of subject to implementation errors, or if the observer misperceives the game (for details, see SI Appendix). an interaction, all players will have the same image of the donor 1 In each round t, two players i and j are drawn from the population at ran- after the interaction has taken place. dom, a donor and a recipient. The donor then decides whether to cooperate. While the assumption of perfectly synchronized reputations is Her choice is uniquely determined by her action rule y9 and by the reputations a useful idealization, we believe that it may be too strict in some she assigns to herself and to the recipient, me(t)and:7),(0. The donor andthe recipient alwaysobservethedonor's decision; all other players independently applications. Subjects often differ in the prior information they observe it with probability q. With probability e, a player who observes the have, and even if everyone has access to the same information [as donor's action misperceives it, independent of the other players. All players is often the case in online platforms (51,52)], individuals differ in who observe the interaction update their assessment of the donor according how much weight they attribute to different pieces of evidence. to their assessment rule. This yields the image matrix M(t + 1). As a result, individuals might disagree on each other's reputa- We iterate the above elementary process over many rounds (our num- tions. These disagreements can proliferate over time. Herein, bers are based on 106 rounds or more). Based on these simulations, we can we have thus systematically compared the performance of the now calculate how often player i considers j to be good on average and leading-eight strategies when information is incomplete, private, how often player i cooperates with j on average. If the estimated painvise and noisy. The leading-eight strategies differ in how they are cooperation rate of i against j is given by we define player i's payoff as S s f; fxtP —cup. affected by the noise introduced by private perception errors. = I Strategies like stem judging, that have been shown to be highly Evolutionary Dynamks. On a larger timescale, we assume that players can successful under public information (18, 32, 33), fail to distin- change their strategies (n, .3). To model the strategy dynamics, we consider guish between friend and foe when information is private. While a pairwise comparison process (39-41). In each time step of this process, we have considered well-mixed populations in which all play- one individual is randomly chosen from the population. With probability ers are connected, this effect might be even more pronounced this individual then adopts a random strategy, with all other available when games take place on a network (53, 54). If players are able strategies having the same probability to be picked. With the remaining only to observe interactions between players in their immediate probability 1 -;a the focal individual i chooses a random role model j neighborhood, network-structured populations may amplify the from the population. If the players' payoffs are *; and cu, player i adopts problem of incomplete information. Pairwise interactions that fs strategy with probability P(*i. fra = (1+ exn( -Art; - 100' (SS). The parameters > 0 is the "strength of selection." It measures how strongly imi- one player is able to observe may be systematically hidden from tation events are biased in favor of strategies with higher payoffs. For s = 0 his neighbor's view. Thus, the study of indirect reciprocity on we obtain P(ti, = 1/2, and imitation occurs at random. Ass increases, networks points to an interesting direction for future research. payoffs become increasingly relevant when i considers imitating Ps strategy. The individuals in our model are completely independent In the main text, we assume players can choose only between a leading- when forming their beliefs. In particular, they are not affected eight strategy L,, ALLC, and ALLO. As we show in SI Appendix, Fig. 56, the by the opinions of others, swayed by gossip and rumors, or stability of a leading-eight strategy may be further undermined if additional engaged in communication. Experimental evidence suggests that mutant strategies are available. Moreover, in the main text we report only even when all subjects witness the same social interaction, gos- results when mutations are comparably rare (43, 44). In SI Appendix, Figs. sip can greatly modify beliefs and align the subjects' subsequent 57-59 we show further results for substantial mutation rates. Given the players' payoffs for each possible population composition, the selection- behaviors (13). Seen from this angle, our study highlights the mutation equilibrium can be calculated explicitly. All details are provided in importance of coordination and communication for the stability