Anonymous rituals

Journal of Economic Behavior & Organization 81 (2012) 478–489

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Anonymous rituals David Hugh-Jones a , David Reinstein b,∗ a b

Centre for Competitive Advantage in the Global Economy, Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom Department of Economics, University of Essex, United Kingdom

a r t i c l e

i n f o

Article history: Received 17 May 2010 Received in revised form 9 June 2011 Accepted 13 July 2011 Available online 3 August 2011 JEL classification: H41 Z12 D82 Keywords: Signaling Anonymity Club goods Religion Social capital Trust

a b s t r a c t Some social institutions reveal participants’ behavior in the aggregate, while concealing the identities of the participants. For example, individual church donations may be kept anonymous, while the total amount raised is publicized. This presents a puzzle in light of recent evidence that anonymity reduces contributions. We offer an explanation for this puzzle in the context of a model of costly signaling with two types of agents: conditionally cooperative (“good”) and uncooperative (“bad”). We consider costly participation in a community activity (e.g., tithing) as a signal of an individual’s type. By signaling the presence of one more good type, this may lead other good types to contribute more in future, more important, collective goods problems (CGP’s). Thus, if good types also value others’ contributions more than bad types, good types gain more from sending the signal. But if those who do not signal face exclusion, the signal would need to be made very costly to dissuade bad types from signaling. In contrast, if the institution is anonymous, so that it reveals only the total number of signals, then while signals cannot be used to exclude bad types, even an inexpensive signal may succeed in revealing the total number of good types. This information helps good types maximize the conditional cooperation component of their utility in the CGP, and under specified conditions, can increase expected CGP contributions. We characterize conditions under which an anonymous signaling institution increases expected welfare. We provide examples of institutions that may yield such benefits, including religion, music and dance, voting, charitable donations, and military traditions. © 2011 Elsevier B.V. All rights reserved.

1. Introduction People are seen to act more selfishly and contribute less to public goods when their choices are hidden from the public eye (Harbaugh, 1998; Glazer and Konrad, 1996; Milinski et al., 2002; Andreoni and Petrie, 2004; Soetevent, 2005; Alpizar et al., 2008). However, many real-world institutions allow anonymity. For example, churches often encourage anonymous contributions, announcing only the total amounts raised. If the amount collected could be increased by removing anonymity, it is a puzzle why the authorities do not make the change. This paper offers an explanation for that puzzle. Our explanation can also be used to analyze some institutions which do not usually fall within economists’ purview, such as religious rituals and mass demonstrations. Our story considers the “burned money” signaling of Austen-Smith and Banks (2000) in advance of a collective goods problem. Donations to public goods, as well as activities, such as religious rituals, that have little or no direct material benefit, can be used to show that participants are truly committed to a social group or a future collective project (Weber, 1946; Ruffle and Sosis, 2003; Bird and Smith, 2005). For example, a long and challenging initiation ceremony will be undergone only by those who are willing to pay a high cost to become a member of a religious order. Suppose that, as suggested by a substantial

∗ Corresponding author. Tel.: +44 01206 87 3518. E-mail addresses: [email protected] (D. Hugh-Jones), [email protected] (D. Reinstein). 0167-2681/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2011.07.005

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body of lab and field evidence (discussed below), there are two kinds of participants: self-interested types who prefer not to cooperate on a group project, and conditional cooperators, who prefer to contribute to a collective good if they expect others do so too. Suppose also that conditional cooperators gain more from inducing others’ future cooperation. Then, performing the ritual may send a signal that “there is one additional cooperator in the group”, and only conditional cooperators may perform the ritual, justifying this belief. This signaling has two potential social benefits. First, it may allow the group to exclude less committed people from the group or from the project. Second, even if no exclusion takes place, a high level of signaling may increase group members’ trust in each other and make them more willing to work together. Both of these benefits help explain findings like those of Sosis and Bressler (2003), who show that 19th century religious communes that imposed stricter requirements on members’ behavior survived for longer, and Sosis and Ruffle (2003), who find that members of religious kibbutzim (communes) cooperate more than others in an anonymous game. In general, the first benefit requires more information than the second. Group trust may be increased if members learn only the total number of committed players. However, for uncommitted players to be excluded, the signal must reveal their identity. This is the key difference between anonymous and public signaling: anonymous signaling reveals only the total number of committed players, but public signaling also shows who they are. To explain why anonymous rituals may sometimes be socially preferable to public ones, consider the issue of signaling cost. Ritual behavior must be costly enough to deter the uncommitted from participating: otherwise it will fail to provide information about the group’s composition. In particular, if non-participants are to be excluded from the group, then participating in the ritual must cost more than being excluded. These costs are then a burden on all group members. While a commune with too lax rules might be invaded by free-riders, a too strict commune may be unpleasant for everyone. The group may then increase its welfare by making the ritual anonymous, so that only the total number of participants can be observed, and not the participation of any individual. This will reduce the incentive for “free-riders” to participate in the ritual, since anonymity means they cannot be punished for shirking. While this removes the first social benefit of signaling – free-riders cannot be excluded from the group – the second social benefit may be preserved. If the overall level of participation in the ritual is made public, participants learn about the level of commitment in the group, and this may encourage them to cooperate with each other by building their trust. For example, a village which raises a large amount of funds to provide Christmas lighting probably contains many public-spirited people, and knowing this may in turn encourage villagers to put effort into neighborhood watch and beautification projects, confident that many others will support them.1 Why not simply guarantee that no one will be excluded? The answer is that this promise would not be credible. Groups often have good reasons to keep free-riders out. However, if anonymity is built into the technology of the institution, then the commitment not to exclude people is unbreakable. In the next section we give some examples of institutions which seem to fulfill this purpose. In Section 3 we present our theory in more detail, developing a model of anonymous rituals and showing how and when they might survive, in terms of some key parameters: the benefit from reducing uncertainty about the overall composition of the group; the benefit from learning about particular individuals’ types, and the cost to individuals who are exposed as uncommitted. Before this, we motivate our model with some empirical examples, and survey the related literature.

2. Examples of anonymity-preserving institutions Costly signaling explanations for religious behavior have been widely advanced, so we look first here for examples of anonymity-preserving institutions. Notably, donations to churches are often anonymous. Supporters for anonymity find a Biblical justification in Matthew 6:2–4, which references the different motivations of public and private givers: Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men.... But when thou doest alms, let not thy left hand know what thy right hand doeth...2 Some churches use “pledge cards” and collection plates, which are generally thought to increase the visibility of donations, but many denominations do not (Hoge et al., 1996).The Dutch Reformed Church typically passes around a “collection bag” rather than an open collection plate, as in many Catholic churches. Where collection plates are used, many churches allow donations in sealed envelopes. Furthermore, religious institutions are increasingly encouraging members to make regular anonymous automatic bank deductions. As a modern Baptist says, “many of our lay leadership like the idea of anonymity... In small rural churches, it is a difficult thing to get commitment cards signed” (quoted in Hoge et al., 1996). Churches with pledge cards appear to raise more donations (Hoge et al., 1996) so why the resistance? We theorize that a key good provided by churches is mutual support.3 Donations not only raise funds, but signal individuals’ commitment to the congregation, and thereby their likely future behavior. When individual donations are private but total levels are

1 Of course, if the village fails to raise much money, this will have the opposite effect. In Section 3 we give examples where knowing the overall level of commitment raises contribution levels in expectation. 2 Islam and Judaism also encourage anonymous charity; see, in particular, the Koran section 2.271, and Maimonides’ Mishneh Torah, Laws of Gifts to the Poor 10:7-14. 3 That is, churches are clubs (Iannaccone, 1992). For an in-depth examination of one church from the club goods perspective, see McBride (2007).

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public, as in many churches,4 the overall level of “community spirit” (that is, the number of congregation members who are truly committed to the church community) is known. Making donations public could lead either to a pooling equilibrium in which both committed and uncommitted members contribute, losing information about community spirit, or to a separating equilibrium (in which only committed members donate) requiring a higher level of donations than was required under anonymity. From the perspective of church members, this higher level of donations may seem excessive. Anonymous giving has puzzled several economists (e.g., Andreoni and Petrie, 2004), since much existing work has focused on the disadvantages of anonymity: generosity and pro-social behavior increase when reputation is at stake (Harbaugh, 1998; Glazer and Konrad, 1996; Milinski et al., 2002; Cooter and Broughman, 2005; Andreoni and Petrie, 2004). Although anonymity may reduce giving by reputation-seekers, it may help to create mutual trust. Furthermore, early anonymous donations may possess greater signaling value in encouraging others to come forward (see List and Lucking-Reiley, 2002; List and Rondeau, 2003; Karlan and List, 2007; Reinstein and Riener, 2011 for related experimental evidence, albeit with a different interpretation). Our theory is also relevant to collective action in politics. Voting can be a reliable signal of public support (Stigler, 1972; Londregan and Vindigni, 2006). Strike ballots are particularly likely to be a signal of resolve in industrial disputes. In the UK, legislation requiring a secret ballot before a strike was introduced during the 1980s. Unions found an unexpected advantage to the secret ballot: as voting is a costly activity, negotiators could use the level of turnout in such elections to persuade management of the strength of feeling among their members. The secret ballot may have provided more credible information than public demonstrations of support, in which some workers might have felt pressured to take part. It may also have increased members’ willingness to take action after a successful ballot (Martin et al., 1991), and allowed union leaders to call off a strike when it was unlikely to succeed.5 Anonymous signaling can also be found in military culture and institutions. As noted by Akerlof and Kranton (2000), “in the military, it is hard to observe effort, especially when it is most crucial—in battle.” Furthermore, even if a soldier’s effort is observed from amidst the fog of war, it is hard to reward a soldier who has been killed. A soldier who values being hailed as a hero may be willing to fight bravely – but only if he expects others to join the fight in numbers strong enough to make victory likely. Hence, soldiers who value personal glory may be conditional cooperators, while those who do not are our model’s “selfish” types. Kellett (1982) notes “Israelis regard fighting as very much a social act based on collective activity, cooperation, and mutual support.” According to King (2006) (citing Randall, 2004) “British forces have deliberately sought to engage in ritualized forms of movement to encourage collective action.” Some such forms of movement, for instance war chants, can be seen as anonymous signals, in which individual soldiers’ physically costly effort is not observable, but the general cohesiveness and power of the exercise signals the unit’s overall level of commitment to themselves and to others. According to an Israeli Defense Force spokesperson, a successful company commander must have “the ability to create mutual trust between the sub-commanders and the soldier.”6 An officer who is confident that his men are all “conditional cooperators” may employ such exercises to make this fact common knowledge. Similarly, company managers may need to limit individual accountability in order to get high performance from teams. The management literature on team-building emphasizes this point: “In a real team effort... individuals do not fail; only the team can fail. Pointing the finger of blame at one person invariably diminishes mutual accountability” (Katzenbach and Smith, 2001). Through this “mutual accountability” both the manager and the employees can learn whether they are in a group full of “team players”; this confidence may be necessary in order to induce high effort from conditional cooperators. Our model thus provides one motivation for why managers may foster a culture in which individual blame is discouraged, and use information systems that report collective rather than individual performance. Several cultural “technologies” seem well-suited to preserve anonymity. Hagen and Bryant (2003) suggest that song and music evolved as “coalitional signaling”. Many traditional rituals include communal singing and dancing. Communal singing can be judged by its overall volume, harmony, and enthusiasm, while it is difficult to discover which individuals are out of tune or keeping quiet. In communal dances, if one person makes a mistake, the entire group may lose the rhythm, and the guilty party cannot always be identified. The collective benefits of anonymous signaling may help to explain the evolution and persistence of these human universals. In a modern context, applause, cheers and jeers are all reliable signals of collective appreciation, since contribution levels cannot be distinguished. We are not claiming that information gain is the sole reason that anonymity is preserved in the above examples. For instance, people may be concerned to protect low contributors from embarrassment, not only to avoid pooling as our theory suggests, but also out of empathy or a desire not to drive others away from the group. We are merely proposing a previously unexplored motive which may complement other explanations. Indeed, institutions may persist because of their adaptive side-effects, even if they were introduced for other reasons (Parsons, 1968). As we discuss in the conclusion, we hope that future empirical work, including lab and field experiments (such as our concurrent paper), will help identify the importance

4 A scan of online weekly church newsletters suggests that many announce the totals raised in recent fundraisers, but they rarely publish individual names and individual amounts contributed. 5 Card and Olson (1995) note that “an increase in the fraction of the firm’s employees involved in the strike increases both the probability of success and the wage increase conditional on a success.” They also point out that in the early days of the labor movement, union leaders seemed to want “to prevent a strike when the likelihood of success was ‘too low’.” 6 Soldiers’ Rights Commissioner, IDF Spokesperson’s Unit, “The Human Being’s Human Doctrine,” http://www.idf.il/english/organization/nakhal/kavod.stm.

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of signaling relative to other factors (such as avoiding embarrassment) in the survival of anonymous institutions. It is often difficult to categorize particular institutions as pure “anonymous signaling rituals.” As an example, consider a group of people going out to a restaurant, who agree to contribute “what they think they owe.” This avoids working out each individual’s debt. But, although it is not designed to measure the cooperativeness of the group, when contributions are counted up and compared to the total bill, the diners still learn something about whether their companions are generous or stingy on average. When all present learn that their peers are “nice,” this may lead to future dinners out, and the group may eventually become a close circle of friends. To the best of our knowledge, our work is the first to analyze the benefit of anonymity in a signaling context. The advantages of anonymity have been discussed in the literature on principal-agent relationships (Holmstrom, 1999), as well as in a legislative context (Prat, 2005; Levy, 2007b,a), though not to our knowledge in the context of large elections. With incomplete contracts and “career concerns” principals may benefit from committing not to learn too much; this will better align incentives and induce the agents to make more productive choices (Acemoglu, 2007). In our model the benefit of anonymity is more indirect: revelation of types can be achieved at a lower cost, leading in some cases to more efficient decisions by other agents. 3. Theory Our basic situation is a group of N people who can contribute to a collective good. Cooperation in such situations can be assured by binding agreements, or by an equilibrium in a repeated game (Fudenberg and Tirole, 1991). But in many cases contracts are unenforceable – for instance, in illegal enterprises, stateless societies or weak states, chaotic situations such as wars and humanitarian disasters, or when performance cannot reasonably be verified by third parties. Similarly, in a stochastic environment where the immediate temptation of defection sometimes outweighs the “shadow of the future”, or where actions are imperfectly observable, high levels of cooperation cannot be sustained in a repeated game. In these situations, it is a reasonable simplification to model collective good provision as a one-shot game. We will sometimes call this the “basic collective good problem (CGP)”. We first analyze the CGP on its own, taking players’ information about the number of good types and the exclusions as given. Then, we introduce a first “ritual” stage, in which players may signal their types. The good is not itself excludable – one’s consumption level cannot be made conditional on one’s contribution level. However, before the CGP, it is possible for people to be excluded from the group. Lastly, there is some crowding: the collective good becomes less valuable if more people consume it. For this reason, all group members will wish to exclude free-riders. If all players are selfish, then the collective good will always be underprovided. Instead, we assume that there are two types of players, the conditionally cooperative (henceforth “good”) and self-interested (henceforth “bad”) types mentioned in the introduction. Essentially, holding others’ contributions constant, good types get a higher payoff from their own contributions than do bad types. Let xi represent the amount to the CGP, P ∈ [0, 1] the proportion of  of wealth an individual i contributes  agents included in the collective good, X = xj total contributions, and X−i = j= / i xj contributions from players other than i. Unless otherwise specified the choice set is xi ∈ R+ . An individual i of type  ∈ {G, B} receives utility w (xi , X−i , P) if he is included in the collective good. Excluded players get 0. We assume that wG (xi , X−i , P) =

(X)

wB (xi , X−i , P) =

(X)

Pı Pı

− xi + ˇ (xi , X−i ), and

(1)

− xi .

(2)

This specification allows clear proofs and intuition, without any interesting loss of generality.  is a bounded, weakly increasing production function for the collective good, with the normalization (0) = 0. ı ≥ 0 measures crowding. When ı = 0 the good is fully non-rival; when ı = 1 the good resembles a pie to be divided among the group’s members; when ı > 1 the total benefit of the good actually declines in the group’s size. We sometimes call ((X)/Pı ) − xi “material utility”. By contrast, , which we label “cooperation utility,” is received only by good types. ˇ > 0 measures the strength of good types’ cooperative motivations. We assume is differentiable and concave, and to ensure an interior equilibrium, that 1 (x, X)→ ∞ as x → 0, 7 1 (x, X) → 0 as x→ ∞. Finally, define T(X−i ) = argmaxx (x, X−i ), the contribution that maximizes cooperation utility given the contributions of others. All players benefit from others’ contributions, i.e., w2B > 0 and w2G > 0. Good types are more willing to provide some of the G (x, X, P) is non-negative everywhere. collective good if others do so as well: that is, the cross-partial of their total utility w12 Pure public goods are often assumed to have diminishing returns, and other people’s contributions perfectly substitute for G < 0. However, in many economic experiments, some players one’s own (see Bergstrom et al., 1986): this would imply w12 demonstrate “conditionally cooperative” or “reciprocal” preferences: they will give more if they expect others to do so too (Keser and van Winden, 2000; Simpson and Willer, 2008; Fischbacher et al., 2001). Experiments also show that some players

7

Subscripts on function arguments denote partial derivatives, e.g.,

1 (x,

X) = (∂/∂ x) (x, X).

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are more materially self-interested, like our bad types (Berg et al., 1995; Kreps et al., 2001; Schram, 2000; Ostrom, 2000). Alternatively, good types may simply have a higher value for a material collective good which has increasing returns to total G (x, gx) + gw G (x, gx) < 0.8 contributions. For technical reasons, we also impose the condition that for any g ≤ N, w11 12 The conditional cooperators in the group will wish to learn the other players’ types, since it will help them decide how much to cooperate. We assume that each of the N player types are drawn by chance. Before observing others’ behavior, an agent’s prior belief is that there are g (other) good types with probability (g) > 0 for g ∈ {0, . . . , N − 1} .9 3.1. Equilibrium in the CGP We first examine the case without exclusion, so that P = 1 always. For convenience, we writew (x, X) ≡ w (x, X, 1) for welfare when no agent is excluded. Before introducing signaling institutions, we examine equilibria in the basic CGP, first when the number of good types g is unknown, second when it is common knowledge. Suppose  is concave  with  < 1. Then bad types never contribute. A good type’s interior best response xi will satisfy the first order condition w1G (x, X−i )d(X−i ) = 0, where (·) is the probability distribution of others’ total contributions, given their types and strategies. Plugging in the derivative of (1) we have





 (X−i + xi ) + ˇ



1 (xi , X−i )

d(X−i ) = 1.

When X−i is certain (i.e.,  puts 100% probability on X−i ) we write b(X−i ) for the best response satisfying G > 0 everywhere then b(·) is weakly increasing by monotone comparative statics  (X−i + b(X−i )) + ˇ 1 (b(X−i ), X−i ) = 1. If w12 (Milgrom and Roberts, 1990): other people’s contributions make good types want to contribute more. Where the number of good types is unknown, a symmetric interior equilibrium, in which each good type contributes x*, satisfies N−1 



(g)  ((g + 1)x∗ ) + ˇ

1 (x

∗



, gx∗ )

= 1,

(3)

g=0

recalling that (·) gives the probability distribution of the number of other good types. Suppose now that the number of good types is known to be g + 1. An interior equilibrium in which good types contribute xg has total “others”’ contributions of gxg and must simultaneously satisfy the best response condition x = b(X); hence w1G (xg , gxg ) =  ((g + 1)xg ) + ˇ

1 (xg , gxg ) − 1

= 0.

(4)

G (x, gx) + gw G (x, gx) < 0 guarantees that x will be unique. The condition that (d/dx)w1 (x, gx) = w11 g 12

3.2. Effect of complete information Anonymous signaling is intended to reveal the number (but not the identity) of good types: thus, we want to know when this revelation will be socially beneficial. We examine the effects on good and bad type welfare separately. This knowledge is double-edged. It enables good types to increase their cooperation utility, since they will have more accurate expectations of others’ donations. When there are many good types, complete information over types will also raise total contributions. On the other hand, when there are few good types, complete information will lower all good types’ contributions and exacerbate the CGP. We offer two conditions under which complete information over the number of good types is beneficial for good types (and for all types, in the second case). Condition 1: Complete information over the number of good types is certainly beneficial when cooperation utility is important enough to outweigh any possible decrease in contributions.10 As an illustration, consider the additively separable case where good types’ cooperation utility is purely a function of T (X−i ) – representing perhaps guilt, the difference between own donations and some target function of others’ contributions  reciprocity, or inequity aversion. (xi , X−i ) = ˆ (xi −  T (X−i )) with ˆ strictly concave and single  peaked at 0, and T strictly increasing. If ˇ is high enough (holding other parameters fixed), then common knowledge of the Example 1.

Suppose that  is concave with  < 1, and that

number of good types increases good types’ welfare.11

8

This restriction guarantees a unique equilibrium in the CGP. It could be loosened at the cost of some complexity. This belief is prior to observing others’ behavior, but after observing her own type. In a previous draft, we allowed this prior belief to depend on a player’s type, either because of dependence in the way nature draws types, or because the distribution of types itself is drawn from a distribution. This extension complicates our results but adds little insight, and the assumption of identical independent draws is standard in the signaling literature. 10 A general proof of this claim is provided in Appendix B. 11 All proofs are in Appendices A and B. 9

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Condition 2: Complete information over the number of good types will also be beneficial (for both types) if, by reducing uncertainty, it never decreases contributions, and it sometimes (i.e. for some realizations of the number of good types) increases contributions. In the next example, we show this with linear material utility and a cooperation utility term for which the marginal benefit of good types’ contributions is proportional to others’ total contributions. When expectations of the number of good types are low enough, contributions will be 0 without complete information; complete information enables good types to contribute when there are enough of them.12

N−1

(xi , X−i ) = X−i xi , 0 < ˛ < 1, and xi ∈ [0, 1]. Let g¨ ≡ (g)g be the expected number of good g=0  (1−˛)/ˇ types conditional on at least one good type, and let gˆ ≡ (g)/Pr(g ≤ (1 − ˛)/ˇ) be the expected number of good g=0 types given 1 ≤ g ≤ (1 − ˛)/ˇ. (Here (1 − ˛)/ˇ is the minimum share of contributions such that good types prefer to contribute.) Then complete information improves good types’ welfare iff either g¨ ≤ (1 − ˛)/ˇ or gˆ ≤ (1 − ˛)/˛ + ˇ, and improves bad types’ welfare iff g¨ ≤ (1 − ˛)/ˇ.

Example 2.

Let (X) = ˛X,

Intuitively, if g¨ ≤ (1 − ˛)/ˇ then no one will contribute given uninformed prior beliefs, and thus information can only increase welfare, for both types. If g¨ ≥ (1 − ˛)/ˇ, good types will contribute 1 given uninformed prior beliefs; however under complete information, they will give 0 if there are fewer than (1 − ˛)/ˇ good types. This harms them because other good types reduce their contributions, but benefits them since they lose less themselves; when gˆ ≤ (1 − ˛)/˛ + ˇ, the benefit dominates since their cooperation utility is small in expectation. Of course, in this case bad types lose from the reduction in good types’ contributions. 3.3. Signaling institutions As these examples show, welfare in the CGP may increase if group members can learn each others’ types. However, they will not reveal their types through cheap talk: selfish types would pretend to be cooperative so as to induce more cooperation from the others. Instead, type revelation requires a costly action which only cooperative types choose to take. While we assume that this action is equally costly for both types, the good types may benefit more from its value in signaling the presence of “one more good type.” The benefit is the resulting increase in others’ cooperation; if good types gain more from this than bad types, then separation can be achieved (as in Austen-Smith and Banks, 2000).13 Thus, rituals may be useful as costly signaling devices even where exclusion from the group is impossible. We model the ritual as follows. Definition. In a revealed signaling institution, players choose whether to participate in the ritual by making a fixed payment of  ≥ 0 before the basic CGP. Each player’s choice to participate is publicly observable by all other players.14 We seek conditions for an equilibrium in which only good types pay the signaling cost, since without this separation, the ritual cannot improve welfare. These are given in Lemma 1: Lemma 1. Suppose that wG and wB are bounded and strictly concave with w1B < 0. 15 Then a separating equilibrium in a signaling institution (without exclusion) exists if w2G (b(X−i ), X−i ) > w2B (0, X−i ) for all X−i ∈ [0, NxN ]. In other words, under the conditions given and our maintained assumptions, a separating equilibrium exists if good types value a marginal increase in others’ contributions more than bad types do. This ensures that the good types gain more from sending the signal than do the bad types, making a separating equilibrium possible. The signaling institution may be beneficial even in large groups. When N is large, signaling the presence of an additional good type will make little difference to the beliefs about the share of good types, and will likely have little effect on others’ behavior, so the cost of signaling need not be high to deter bad types. Indeed, we might expect that as N increases, this minimum signal size will become vanishingly small.. For example, this limit property will hold when welfare is a function of ˆ  (xi , X−i /N). Several of our examples the average contribution level, that is, when welfare can be written as w (xi , X−i ) = w above, such as applause and voting, use low-cost actions which are nevertheless informative in the aggregate.

12 We give a simple example at the cost of violating concavity of wG ; therefore, there may be multiple equilibria in this case. For the sake of the example, we assume that the group can use cheap talk to coordinate on the equilibrium with the most contributions. 13 This condition, though often plausible, need not hold. Selfish players could benefit more from others’ cooperation, while simultaneously getting less utility from cooperating themselves. Cf. Lemma 1 below. Separation may also be achievable if (as in the well known Spence, 1973 model) the good types find it less costly or more directly beneficial to take this “ritual” action (see [authors’ other paper]). 14 The ritual need not represent “burning money”; we choose this extreme case for simplicity. Our leading example, church contributions, obviously provides tangible benefits to congregants. If good types gain more than bad types from contributing to the ritual, a separating equilibrium may be achievable even if both gain the same amount from others’ cooperation in the CGP. In a previous draft of [authors’ other paper] we modeled a case where the “ritual” is a smaller version of the basic game, and gave conditions under which this “minigame” could only separate types if it was anonymous. This paper also examines this case in a laboratory experiment, and finds evidence that the anonymous minigame yielded higher overall payoffs. We also speculate that the insights of our model could be extended to a case with a continuum of player types, and a separating equilibrium in which the costliness of the signal is an increasing function of a player’s “goodness.” 15 This condition could be loosened. The proof requires only that good types gain more from than bad types from the increase in equilibrium donations xg caused by their signal. The conditions in the text are one way to guarantee that.

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Lemma 2. If welfare is a differentiable function of the average contribution level, and, for all g, (g) → F(g/N) as N→ ∞ with F continuous, then for high enough N, if a signaling equilibrium is possible, the cheapest possible signal  approaches 0. When the conditions for the above lemmas hold, and when knowledge is welfare-improving, then a signaling institution can be beneficial, since it allows learning at a low cost. For example, with utility functions as in Example 2, the conditions for Lemmas 1 and 2 are met and the signaling institution must be beneficial for groups of a sufficient size. 3.4. Exclusion and the advantage of anonymous rituals Rituals that reveal group members’ types may have further consequences. If the public good suffers from crowding, then the group has an incentive to exclude selfish types, who will not contribute. For example, those not conforming to the rules of a religious commune may be expelled, and face bleak prospects in the outside labor market. Selfish types may also be punished in other ways. Workers who break trade union rules have sometimes been “sent to Coventry”, that is, socially ostracized (Time Magazine, 1960). Similarly, hunter-gatherer societies exclude those are seen as antisocial by gossip and avoidance (Boehm, 1999). We suppose that after the signaling mechanism, but before the CGP, it is possible to exclude some or all players, at zero direct cost. This could happen in a number of ways, for example by a formal voting process, or by informal action such as social ostracism.16 We do not model the exclusion process directly. Instead we simply assume that exclusions will be done so as to maximize good type’s welfare ex post (that is, after group members’ choices to participate in the ritual are known). This is plausible since, in any decision process, bad types will wish to “argue as” good types in order to conceal their type. Furthermore, no bad type has an incentive to prevent the exclusion of another bad type: in the CGP, bad types increase crowding but make no contributions. As a result, if the institution separates types, then bad types will be excluded.17 Now, selfish players who reveal their type will suffer by being excluded, rather than merely by reducing others’ cooperation. As failing to participate in a ritual now brings a greater penalty, the cost of participation must be made more costly in order to deter selfish types.18 In fact, the cost may be so high that group members would be better off ex ante by committing not to exclude bad types, so that types can separate at lower cost. However, after a ritual has revealed types, they will face the temptation to exclude them ex post. In this context, an anonymous signaling institution, in which only the number of agents who paid  is visible, may serve as a commitment device. The group can tie its hands by using an anonymizing technology that reveals the overall level of participation in the ritual, but keeps individual behavior hidden. The cost of revealing one’s type is then limited to its effect on others’ cooperation. Definition. In an anonymous signaling institution, group members choose whether to participate in the ritual by paying  ≥ 0 before the basic CGP. Only the total number of members participating is publicly observable. The social welfare comparison between anonymous and non-anonymous rituals will depend on the trade-off between the signaling cost and the benefits of excluding individuals. If crowding is extremely costly, then anonymity will give lower welfare than a revealed signaling institution in which non-participating agents are excluded from the collective good. Proposition 1. Suppose that the function  is strictly concave with  < 1, and the good types’ total welfare function wG is continuously differentiable. Then, when crowding is less important (i.e., for ı close enough to 0), if there is a revealed signaling institution that separates the types, then there is also an anonymous signaling institution that separates the types and gives strictly higher welfare to all group members.19 Conversely, there are conditions under which good types prefer revealed institutions. In particular, if bad types get little welfare from the basic game, the signaling cost to exclude them will be negligible in both revealed and anonymous institutions. It will then be cheap to exclude them, and this will benefit the good types by decreasing crowding. Proposition 2. Suppose the conditions of Lemma 1 hold, implying that a separating equilibrium exists. If bad type welfare from being included in the collective good is minimal, i.e., if |wB (x, X, P)| < ε for all x, X and P and for ε small enough, then good types will attain higher welfare under revealed institutions than under anonymous institutions.

16 For example, suppose a player is chosen at random to exclude or include all players. All players will include themselves, but when N is large this will have a negligible effect; otherwise, all players will include only good types. Alternatively, inclusion might be decided on a case-by-case basis by a majority vote. In this case, if there is a separating equilibrium, there will be unanimous agreement to include every good type, so long as their expected contributions outweigh crowding, and a N − 1 against 1 vote to exclude every bad type, since even bad types wish to exclude other bad types. 17 Some good types might also be excluded, if their contributions would not compensate for the increased crowding. This issue introduces cumbersome technicalities and is not central to our argument, so we simply assume that inclusion decisions cannot be probabilistic: either all those who pay the cost are included, or none are. In general, we expect our qualitative results will carry over to any institution that is less likely to exclude an individual who participates in the ritual. 18 For an example of how costly rituals can be, consider the followers of Saint Simeon Stylites, who would spend months or years living on top of a pillar (Brock, 1973). 19 Bad types will always strictly prefer an anonymous institution to a revealed one. In the former case, they gain from the collective good, while in the latter case they are excluded. Bad types will also prefer the anonymous signaling institution to no signaling at all, whenever common knowledge of types increases the expected level of contributions, since they do not pay the cost of the signal and are not excluded, but types are made common knowledge.

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Thus, under the conditions of Proposition 2, when there are few bad types, or bad types gain little from the collective good, revealed institutions will lead to higher overall welfare, and will be preferred by a benevolent social planner. This situation might be expected where bad types represent people with a high outside option and therefore a low level of commitment to the group. One also might expect the ritual institution that benefits good types to be the one chosen, for the reasons discussed in Section 3.4.20 Considering our results as a whole, our model suggests that anonymous institutions will be particularly favorable, and thus frequently adopted, when the cost of crowding is not too large, and when there are strong incentives for bad types to avoid exclusion. For example, in small communities where people are highly mutually interdependent, exclusion is likely to carry severe costs, so that bad types will have strong motives to avoid it. These communities will then improve total welfare by adopting anonymous signaling institutions. This seems likely to be true to life: the charitable fund for a small village hall is less likely to publicize donor names than the fund for a city’s park. Similarly, ritual forms such as music and dance developed in small and primitive societies. 4. Conclusion In interactions outside of competitive markets, where enforcement is difficult and the “shadow of the future” is not strong enough to discipline behavior, others’ character and intentions towards us may be vital for our success and survival. In these contexts it is crucial to be able to gauge others’ character, so we can choose who to interact with and how much to invest in these interactions. This in turn gives some individuals a strong incentive to conceal their true character. Certain rituals can be seen as institutions which provide a forgiving environment, getting people to reveal their character by letting their identity be hidden. In this paper, we developed a model showing first, how rituals can foster cooperation by revealing the overall type composition of a group, and second, how anonymous rituals can do so at less cost than revealed rituals. In general, when no one prefers to contribute without particular “good news” about the group’s composition then the revelation of types can only help (Example 2). Signaling institutions can reveal types when increasing the number of apparent good types increases contributions and when good types value others’ contributions more than bad types (Lemma 1). Finally, Proposition 2 suggests that in the presence of an exclusion mechanism maximizing good types’ welfare, an anonymous signal will tend to be preferred to one that identifies each player’s types when crowding is less important and when bad types get significant utility from the public good. We note that cultural forms such as song seem especially well-suited for preserving the anonymity of participants and shirkers. We hope this approach will be of interest to anthropologists and sociologists of religion. However, as noted above, further empirical work is needed to establish whether our theory explains the survival of particular rituals and institutions. Future work can test whether such anonymous instutitions do serve to build trust in particular contexts. For example, researchers can measure the level of participation in a particular “ritual” (e.g., tithing in a church), varying the anonymity and the information disclosed. This could be connected to data on behavior in subsequent social dilemmas (e.g., success in future collaborative projects or in a “framed field experiment” [Harrison and List, 2004]). If the correlation between participation in and later cooperation is greater for an an anonymous institution than for a revealed one, this may be taken as support for our theory (if not, our theory is falsified as an explanation for this institution). However, attaining evidence on whether the signaling advantages account for the survival of anonymous rituals will be a more difficult task. As a first step, the fitness and survival of institutions may be studied in repeated “evolutionary game” laboratory experiments; these may provide suggestive, but not definitive evidence for analogous real-world environments. Complementary evidence may come from careful cliometrics on the success of groups and institutions (taking the endogeneity problem into account), and perhaps from more long-term field experiments (e.g., on the survival of social and business groups in the presence of exogenously introduced institutions). As well as explaining existing institutions, we believe that our theory has practical implications for the design of new ones. As an example, consider the finding by Karlan (2005) that behavior in a trust game predicts default by borrowers in a microcredit scheme. Bank lenders might wish to use this result to predict default risk. But if it became known that trustworthiness in a trust game was the way to get a loan, the game would quickly lose its predictive power (or would have to be played for such high stakes that it deterred many borrowers). However, many microcredit schemes rely on informal group enforcement to avoid default. So, groups of potential borrowers could be asked to play an anonymous public goods game or take on a small collective project in advance of the loan. If the group size and incentives were rightly balanced, this would be an incentive-compatible way to revealing participants’ character in the aggregate. The game’s overall results could be announced, and this could both aid banks in deciding how much to lend, and allow trust to develop among group members in cases where that trust is warranted. More generally, anonymous institutions may be useful tools for policy-makers and managers who wish to predict whether cooperation will be sufficient to undertake an ambitious project, or to reduce agents’ uncertainty and thus foster voluntary cooperation. In designing these “trust-measuring” and “trust-building” exercises it may be important not to expose indi-

20

However, as our focus is on the advantages of anonymity, we do not emphasize these arguments.

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vidual performance too much, since this can lead to uninformative pooling or pandering behavior. Incentive-compatible mechanisms which allow for collective achievement may work better. Appendix A. Proofs For ease of reading we use the notation w(x, X, P) ≡ wG (x, X, P). Proof of Example 1 Proof. Rewriting (4) as ((g + 1)xg )/ˇ + ˆ (xg − T (gxg )) = 1/ˇ, as ˇ increases xg approaches the solution to ˆ (xg − T (gxg )) = 0, in other words xg approaches T(gxg ). On the other hand, contributions without knowledge approach x* satN−1 (g) ˆ (x∗ − T (gx∗ )) = 0. Since ˆ is single-peaked, x* will give weakly lower cooperation utility than xg for all isfying g=0

g, and since T is single-valued and strictly increasing, x* will give strictly lower cooperation utility than xg for at least some g ∈ {1, . . . , N} (which will occur with positive probability (g) > 0). For ˇ high enough the resulting loss will dominate any change in the amount of the public good provided, since  is bounded.  Proof of Example 2 Proof.

N−1

xg = 1 is possible in equilibrium if ˛ − 1 + ˇg ≥ 0 ⇔ g ≥ (1 − ˛)/ˇ; otherwise xg = 0. Ex ante good type welfare is



g= (1−˛)/ˇ



(g) ˛(g + 1) − 1 + ˇg . When g is unknown, by the FOC x* = 1 if g¨ ≥ (1 − ˛)/ˇ, and 0 otherwise; the cor-

responding good type welfare is either

N−1 g=0



(g) ˛(g + 1) − 1 + ˇg



or 0. In the latter case, knowledge always improves

good type welfare since for g ≥ (1 − ˛)/ˇ, ˛(g + 1) − 1 + ˇg = (˛ + ˇ)g + ˛ − 1 ≥ (˛ + ˇ)((1 − ˛)/(ˇ)) + ˛ − 1 = (˛(1 − ˛))/ˇ > 0. (The increase in contributions also improves bad type welfare.) In the former case, the difference in favor of ignorance is    (1−˛)/ˇ  (1−˛)/ˇ (g) ˛(g + 1) − 1 + ˇg = (˛ + ˇ) g=0 (g)g + Pr(g ≤ (1 − ˛)/ˇ)(˛ − 1). Rearranging gives that this is g=0 positive iff gˆ > (1 − ˛)/˛ + ˇ. On the other hand, bad type welfare is always lowered since contributions are decreased.  Proof of Lemma 1 Proof. Since w1B < 0 everywhere, bad types contribute nothing, so in an equilibrium with g good types, w1 (xg , gxg ) = 0. (The Inada condition on ensures the equilibrium is interior.) Define V (X−i ) as individual i of type ’s value from known contributions of X−i , thus VG (X−i ) = wG (b(X−i ), X−i ), and since bad types contribute nothing, VB (X−i ) = wB (0, X−i ). Recall that xg is good type equilibrium contributions when there are g + 1 good types in total. The condition for good types to prefer to pay the costly signal is N−1 

(g)VG (gxg ) −  ≥

g=0

N−1 

(g)VG (gxg−1 ).

g=0

That is, when there are g other good types, by signaling one induces them to play the correct equilibrium for a total of g + 1 good types; by not signaling one misleads them. Similarly, the condition for bad types to prefer not to pay the signal cost is N−1 

(g)VB (gxg−1 ) ≥

g=0

N−1 

(g)VB (gxg ) − .

(5)

g=0

where the honest signal leads good types to believe correctly that there are only g good types. N−1 N−1 (g)VG (gxg ) ≥ (g)VG (gxg−1 ), and secondly that For these both to hold for some  ≥ 0 it must be firstly that g=0 g=0 N−1  g=0





(g) VG (gxg ) − VG (gxg−1 )

≥

N−1 





(g) VB (gxg ) − VB (gxg−1 ) .

(6)

g=0

The first condition will hold if VG is increasing and total contributions gxg are increasing in g. By the Envelope Theorem, VG (X−i ) = w2G (b(X−i ), X−i ) which is non-negative by assumption. That total contributions are increasing in g follows from the Implicit Function Theorem: (dxg )/dg = −(w11 + gw12 )/w12 and by assumption w11 + gw12 < 0. Thus xg increases in g, and total contributions gxg do so also.  gx  gx To show that (6) holds, simply write V (gxg ) − V (gxg−1 ) = gx g V  (y)dy = gx g w2 (b (y), y)dy, for all g and for  ∈ {G, g−1

g−1

B}, where bG = b and bB = 0, and apply the assumption that w2G > w2B along the equilibrium path. 

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Proof of Lemma 2 ˆ 1 (xg , gxg /N) = 0, or writing = g/N and = x N , Proof. When welfare is a function of average contributions, xg solves w ˆ 1 ( , ) = 0. Thus as N increases, equilibrium contributions for a fixed proportion of good types will be a constant and w average contributions will be the constant . The minimum signaling cost is given by  satisfying (5) with equality, i.e. by N−1 

(g){VB (gxg ) − VB (gxg−1 )}.

(7)

g=0

To show that this goes to 0 when N grows large, observe that it approaches



1

ˆ ˆ B (0, −1/N )}dF( ). {w(0, ) − w 0

Now since = x N is increasing in by w12 ≥ 0, it is continuous almost everywhere. Thus the term inside curlies goes to 0 almost everywhere as N→ ∞, and since F is continuous the whole expression goes to 0.  Proof of Proposition 1 Proof. The analysis of anonymous signaling proceeds exactly as in Lemma 1, since nothing there depended on the identity of any participant being known. In particular, good type welfare from the cheapest possible anonymous signaling institution is N−1 



(g)

 N−1

((g + 1)xg ) − xg + ˇ (xg , gxg ) 1

−

g=0





(g) (gxg ) − (gxg−1 )

(8)

g=0

where the second term is the lowest possible signaling cost that separates the types, from (7). In a separating equilibrium in a revealed institution, bad types who do not pay the cost will be excluded from the basic CGP. For, in the basic CGP with g + 1 good types and P included players, good type donations solve  ((g + 1)x)

+ˇ

Pı

1 (x, gx)

=1

and the solution is decreasing in P (as an application of the Implicit Function Theorem shows). Thus, included bad types decrease welfare by increasing crowding, and by decreasing contributions (by assumption, w2G (xi , X−i , P) ≥ 0 and in equilibrium w1G (xg , gxg , P) = 0). After all bad types are excluded, and g + 1 known good types remain, equilibrium good type contributions are given by x˜ g solving  ((g + 1)˜xg ) [(g + 1)/N]ı

+ˇ

xg , g x˜ g ) 1 (˜

=1

(9)

(compare Eq. (4)). If, in a separating equilibrium, a bad type were to pay the signaling cost and be included in the basic CGP, his expected welfare would be N−1 

(g)

g=0

(g x˜ g ) [(g + 1)/N]ı

.

(10)

(With probability (g) there are g good types; the misleading signal leads them to believe there are g + 1 good types and so to play x˜ g ; g + 1 players are included so P = (g + 1)/N.) This is therefore the lowest possible revealed signaling cost, since the bad type’s alternative is to pay nothing, be excluded and receive a subsequent payoff of 0. Ex ante good type welfare is thus N−1  g=0

(g)

((g + 1)˜xg ) [(g + 1)/N]ı

 N−1

− x˜ g + ˇ (˜xg , g x˜ g )

−

g=0

(g)

(g x˜ g ) [(g + 1)/N]ı

.

(11)

Note also that this expression must be non-negative for a separating equilibrium to be possible, since the first term gives welfare from inclusion for good types, the second term welfare for bad types; if the second term is larger than good types would rather be excluded and get 0 than pay the cost and be included. If this does not hold, then revealed signaling fails to separate the types and gives weakly lower welfare than ignorance of g. Comparing (11) to (8) shows that it will suffice for our proof if x˜ g → xg as ı → 0. For then the first term of each expression will be the same and the only difference will be the larger

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signaling cost in the second term – since the bad type must be paid his total loss from exclusion, rather than the marginal loss from revealing his type and lowering xg . Observe that x˜ g = xg when ı = 0, so it will suffice to show that x˜ g is continuous in ı at ı = 0. This can be done by the Implicit Function Theorem applied to the left hand side of (9). The requirements are that the LHS is continuously differentiable in x˜ g (which holds by assumption), that its slope in x˜ g is non-zero at ı = 0 (which follows from strict concavity of  and concavity of ), and that some (˜xg , ı) satisfying (9) exist in an open interval around (˜xg = xg , ı = 0), which holds since xg is interior.  Proof of Proposition 2 Proof. In both anonymous and revealed institutions, the signaling cost approaches 0 along with bad type welfare, as can be seen from (7) to (10) (more generally, in the revealed case, the minimum signaling cost is N−1 

(g)VB (gxg ),

g=0

since as before, bad types are excluded and receive 0 welfare if they do not pay the signaling cost). Thus, good type welfare for anonymous institutions approaches welfare from the basic game. Since, in the anonymous institution, bad types are included and decrease welfare, welfare is higher from the revealed institution.  Appendix B. (from footnote 10) – Proof that when “cooperation utility” matters enough, knowledge improves good type welfare Suppose w is strictly concave and differentiable with w2 > 0 and w12 > 0, and let b(X) ≡ x solving w1 (x, X) = 0 be positive for all X. Set wi (x, X) = w (b(X), X) + i[w (x, X) − w (b(X), X)] for i > 0 (and set wi ≡ wiG ). Then for i high enough, good type welfare is higher when g is known than when g is unknown. Proof. Set wi (x, X) = w(b(X), X) + i[w(x, X) − w(b(X), X)] for i > 0. It is easy to show that wi (b(X), X) = w(b(X), x), and that N w1i (x, X) = iw1 (x, X), for all x and X. Therefore, the sets of solutions xg and x* to w1i (xg , gxg ) = 0 and (g)w1i (x∗ , gx∗ ) = 0, g=0 i.e. of equilibria with and without knowledge, do not vary with i, and expected good type welfare when g is known is also i (x, X) = w (b(X), X) + i[w (x, X) + w (b(X), X)] = iw (x, X) > 0 for all i. unchanged. Also w12 12 12 12 12 We wish to show that N−1 

(g)[wi (xg , gxg ) − wi (x∗ , gx∗ )] > 0

(12)

g=0

for high enough i. Fix i and define b(·) temporarily as the best response function for wi . First observe that if xg ≥ x*, then wi (xg , gxg ) − wi (x∗ , gx∗ ) ≥ 0. For wi (xg , gxg ) − wi (x∗ , gx∗ ) =

wi (xg , gxg ) − wi (b(gx∗ ), gx∗ ) + wi (b(gx∗ ), gx∗ ) − wi (x∗ , gx∗ )



xg

= b(gx∗ )

d

dx



wi (x, b−1 (x))



dx + wi (b(gx∗ ), gx∗ ) − wi (x∗ , gx∗ ) .

i > 0.) Now the second term in brackets is non-negative by b(·) a best (b−1 is well defined since b is strictly increasing by w12 / xg . On the other hand, since w1i (x, b−1 (x)) = 0, ((d)/dx)wi (x, b−1 (x)) = w2i (x, b−1 (x)) response (and positive whenever x* = and this is positive, so the integral is non-negative. Thus the whole term is non-negative. (This whole step holds for any wi including w1 ≡ w.) Next for xg < x* we have

wi (xg , gxg ) − wi (x∗ , gx∗ ) = w(xg , gxg ) − wi (x∗ , gx∗ ) (sincexg = b(gxg )) = w(xg , gxg ) − w(b(gx∗ ), gx∗ ) + i [w(b(gx∗ ), gx∗ ) − w(x∗ , gx∗ )] and as i→ ∞ the sign of this depends on the sign of the term in square brackets, which is positive since b(gx*) is the unique best response to gx*. Thus for i high enough the terms of (12) are always non-negative, and are positive whenever xg = / x*. 

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