aa r X i v : . [ ec on . T H ] A ug Pricing group membership ∗ Siddhartha Bandyopadhyay † ,Antonio Cabrales ‡ August 2020
Abstract
We consider a model where agents differ in their ‘types’ which determines theirvoluntary contribution towards a public good. We analyze what the equilibrium com-position of groups are under centralized and centralized choice. We show that thereexists a top-down sorting equilibrium i.e. an equilibrium where there exists a set ofprices which leads to groups that can be ordered by level of types, with the first k typesin the group with the highest price and so on. This exists both under decentralized andcentralized choosing. We also analyze the model with endogenous group size and ex-amine under what conditions is top-down sorting socially efficient. We illustrate whenintegration (i.e. mixing types so that each group’s average type if the same) is sociallybetter than top-down sorting. Finally, we show that top down sorting is efficient evenwhen groups compete among themselves.
JEL Classification : D02, D64, D71, H41
Keywords : Top down sorting, Group formation, Public good, Segregation, Integra-tion. ∗ We thank Kaustav Das, Aditya Goenka and seminar participants at the University of Birmingham forcomments. † [email protected] ‡ [email protected] Introduction
In this paper we analyze whether groups can ‘price’ group membership to screen who becomesa member. Joining many groups and clubs is not costless. Many of them require monetaryfees, or non-monetary ‘sacrifices’ to become a member Winslow (1999). Examples abound,new members of a criminal gang or a terrorist organization have to undergo hazing rituals(Vigil, 1996). Similarly, new recruits into many military units and college fraternities alsohave to go through painful or shameful activities to be accepted into them (Ostvik andRudmin 2001, Mercuro, Merritt and Fiumefreddo 2014, De Klerk 2013, Groves, Griggsand Leflay 2012, Keating et al 2005). Even religious groups have elaborate initiation rites(Berman 2000, Iannacone 1992). Exclusive clubs, or elite schools require the payment of veryexpensive fees to become members (Jenkins, Micklewright and Schnepf 2008). The questionof interest in this paper is the rationale for these practices, the equilibrium value of suchfees, and their welfare implications.In all those groups, the value of being a member depends on the extent to which otherparticipants contribute to the common cause or public good they provide. At the same time,individuals differ in their inclination and ability to dedicate themselves to the cause, i.e. tocontribute to the public good provided by the group. For this reason, the utility obtainedby any member in belonging to a group will depend on the types of all the other members.As a consequence, the group will be interested in making sure that those who gain entry asmembers are of the right type. But these types are not necessarily observable, and the entryfee is a way to select members in an incentive compatible way.Our model assumes there are a number of participants in a game which has two stages.The first stage develops as follows. A set of ‘entrepreneurs’, one for each of a fixed numberof groups, posts a price for belonging to their group. The participants then decide, inde-pendently and simultaneously, whether to pay the price to one of those groups, or noneat all. In the second stage, the participants who are in a group decide on their level ofeffort/contribution to a public good for the group to which they belong.The participants differ in the way they benefit from the public good. Each individual ischaracterized by a type that is a multiple of the amount of the public good. This hetero-geneity can be variously interpreted as either a different intrinsic personal enjoyment of thegood, or a difference in the degree of altruism (i.e. some individuals internalize the benefit ofthe public good on others to a larger extent). One important implication of this assumptionis that the group members do not care care directly about the types of others, or aboutthe group size. This is because we assume there is no direct externality caused by others’2ypes or the number of individuals in the group. Individuals do care about the actions takenby other group members, because those actions affect the amount of public good providedwhich they enjoy, and contributions to the public good are indeed affected by their types.This assumption distinguishes our model from others in the literature of club games (startingwith Buchanan, 1965) and congestion games (Oakland 1972, Baumol and Oates 1988).Clearly, since the groups provide a public good, we would in general have an under-provision of the public good in the group if the provision is voluntary and decided on anindividual basis. But many of the groups that we have used to motivate our model have thepossibility of imposing a contribution level within the group. We can think for example ofthe tightly hierarchical organization of most armies, and even gangs. For this reason, we alsostudy group formation when contributions within groups are decided by a planner withinthe group.Whether the individual or the group leader (social planner) decides the contributionlevel, our first result is that in both cases we can establish the existence of a top-downsorting equilibrium.
This equilibrium has the following characteristic. The ‘entrepreneurs’announce a list of distinct prices, which thus can be ordered from highest to lowest. Theparticipants sort themselves into groups by their types. A set composed of the highest typeschooses to belong to the group with the highest price. Another set of types just below thefirst chooses to belong to the group with the second highes price. This continues until thelast set of types, who belong to a group with price zero.We analyze the equilibrium under both centralized and decentralized decision-making.One difference between the equilibrium under centralized and decentralized choice of effortis that under decentralized effort choice every individual gains whenever the average typeof the group increases (although relatively higher types gain more, which is the basis forsegregation). On the other hand, with centralized effort choice within the group an increasein average type improves the utility of above average types within the group, but decreases forbelow average types. This happens because the group planner does not know the individualtypes within the group, and so she requires the same level of contribution from every one.We next turn our attention to some welfare properties of the decentralized process forgroup formation. For this we need to take into account both group composition and groupsize. In terms of group composition, we compare the welfare of the top-down sorting equilib-rium with the one arising from an equilibrium in which there is no sorting, and all groupshave equal average types. It turns out that the answer depends on the curvature of theoutput function. For example, when the concavity is such the risk aversion parameter in the3RRA function is bigger than one (i.e. α > top-down sorting equilibrium is lesssocially beneficial than one where the population is sorted accross groups so that all groupshave the same average type. We call this ‘integration’. A corollary of this result is that inthis case, the segregation occurring in equilibrium is socially inefficient. On the other hand,when α <
1, then the top-down sorting equilibrium has a larger total welfare than the oneachieved when groups are ‘integrated’ and have the same average types.In terms of size, for α >
1, the optimal size of the group is N , the whole population,since individual payoff increases with the size of the coalition. For α <
1, the optimal size ofthe group has to trade off the benefits from segregation (which are highest if average groupquality increases, say by dropping the lowest types in the group), with those arising fromhaving larger groups, because group size affects equilibrium contribution levels.Finally, up to now we have assumed that there is no inter-group conflict. However, insome of our applications the groups compete with one another after they are formed (again,it is easier to think of armies and gangs, but even educational institutions compete ex post fortop-paying jobs). We show that our top-down sorting equilibrium also exists in this context,so our conclusions are robust to this kind of setting.Our paper has important implications for policy. There are relevant environments ( α >
1) where the social planner would like an ‘integrated’ society, which might not arise inequilibrium. She wants to achieve this purely on social surplus maximization grounds. Thisprovides a new rationale for integration in various social domains, like education and housing,without having to resort to preference about equality. We will discuss this in more depth inthe conclusion.
Our paper develops a novel theory of group formation. Clearly, this paper links to theclassical literature on club theory (Buchanan 1965, Berglas 1976). This literature does notconsider differences in information about preferences, and therefore it does not provide arationale for undertaking actions to signal preferences. Somewhat closer to our work isBen Porath and Dekel (1992), who use the potential for self sacrifice as a way to signalfuture intentions. However, they do it because of equilibrium selection problems, and theself sacrifice does not actually occur in equilibrium. Helsley and Strange (1991), in turn,model club formation in a context with homogeneous tastes and costs. In their model feesand prices are used to obtain second best usage when congestion within a club is a problem.In terms of theory, two close papers to our’s are Jaramillo and Moizeau (2002), and4ornes and Silva (2013). Jaramillo and Moizeau (2003) study a model where individualsdiffer in income, and higher income people desire a different level of the local public good.Since information about income is private, individuals use costly signalling to join a club withothers who have similar income (and, hence, preferences for the local public good). Differentfrom us, the group formation is uncoordinated (there is no entrepreneur creating the clubs),there are only two types, and they do not contemplate the possibility of internal coordinationof contributions. Also, in their model there is no reason for individuals of different types togroup together in the social optimum. In Cornes and Silva (2013) some people love prestige(they have higher utility from contributing relative to the average) and others are purists.Fees serve to sort them into clubs. The model, unlike ours has both positive and negativeexternalities, but the competitive equilibrium is inefficient. Competition yields co-ordinationbenefits with the formation of prestige clubs.Our paper also has some connection to a literature that is related to group formation incontexts were types are differentiated horizontally. For example, Levy and Razin (2012) ana-lyze explicit displays of religious beliefs and cooperation within the religious group. Baccaraand Yariv (2013), also analyze group formation and contribution to horizontally differenti-ated tasks.Another relevant literature for us relates to assortative matching. Durlauf and Seshadri(2003) examines when assortative matching is efficient. Legros and Newman (2007) studysufficient conditions for assortative matching in equilibrium. Hoppe, Moldovanu and Sela(2008) analyze when costly signaling is necessary for assortative matching under incompleteinformation. Importantly, they check when gains form assortative matching are offset by sig-nalling costs. There is also a large literature studying sorting into schools, to take advantageof peer effects (Epple Romano 1998, Cullen, Jacob, and Levitt 2003, Hsieh and Urquiola2006 or MacLeod and Urquiola 2015).There is a body of work in the experimental literature exploring how endogenous sortinginto groups can help solve the problem of free riding. For example, costly rituals have beenshown to promote greater co-operation (Sosis 2004, Ruffle and Sosis 2007). Page, Puttermanand Unel (2005) show that endogenous segregation helps avoid free-riding and provides somesupport for the efficiency of top down sorting. Cimino (2011) does a survey based experimentwhere participants are asked about an initiation activity for a group, which itself providesbenefits to members. Those participants randomly allocated to a group providing a goodwith a more public component were significantly more likely to choose a stressful initiationtask. This sort of hazing can be considered as an entry fee to prevent free riding. This would5e very much in the spirit of our model. Aimone et al. (2013) show that something akin toour endogenous segregation equilibrium happens in the lab. In their experiment participantscan choose to participate in a voluntary contribution public good game in groups havingdifferent rates of return. People that are more prone to contribute to public goods join thosegroups with lower rates of return, thus signalling their type. As in our model, a costly choice(in this case, choosing a less efficient technology) is a credible signal of a type wishing tocontribute more to the public good.In addition to experimental results, there is a significant ethnographic evidence that isconnected to our issues. For example, Vigil (1996) analyzes gang initiation. This study isconsistent with our model, i.e. initiation rites are used to screen potential members. Sosis,Kress and Boster (2007) show the importance of such costly male rituals in signalling com-mitment and promoting solidarity among men who need to organise themselves for warfare.Soler (2012) rationalises the existence of wasteful religious rituals. It analyzes the practiceof Candomble, an Afro Brazilian ritual. It shows that participation is correlated with highercontributions to public goods, which is consistent with our model. Cleaver (2004) shows thatentry fees are used in clubs to exclude the “non elite” .
This model has two stages. There are a large and finite number of agents playing the game N . Each agent j ∈ N is characterized by a level of preference for the public good ξ j . In thefirst stage the agents form coalitions of a fixed number of players N , in a way we will describein section 2.1. In the second stage, once a coalition is formed, every agent simultaneouslydecides how much to contribute towards a public good, x j taking as given the coalitionstructure.The total amount of the public good is given by V (cid:16)P Ni =1 x i (cid:17) , with V (.) being a strictlyconcave twice continuously differentiable function. The personal cost of the contribution x j is given by x j / . The utility of every agent after the coalition is formed is U j = ξ j V N X i =1 x i ! − x j The parameter ξ j indicates that the individual may care for more than only his own utilitybut she internalizes the benefits of other players. A player with a ξ j > ξ j = 1 is a selfish player. The value of ξ j is obviously relevant for allplayers in any coalition, as it increases the marginal value of contributions of their owners.6t is also the private information of the players, which can be (partially) solved using apre-game costly signaling exercise which we now describe. Consider a finite set of coalitions l ∈ { , . . . , L } , each with N slots. Let ¯ ξ l = P i ∈ l ξ i .Assume as well that each N is large enough so that the compositional impact of changingone member’s type on the ¯ ξ l of coalition l is negligible. The coalition formation game is asfollows. A set of ‘entrepreneurs’ post a set of prices p l for l ∈ { , . . . , L } . We think of pricesthat need not be ‘monetary’. Any costly action whose effect in utility is separable wouldwork. Then all individuals in the population decide which coalition to join (and thus paythe price of joining). If a coalition is oversubscribed (it has M > N candidates applyingto it), then a fair lottery decides which M − N candidates get allocated to not being in acoalition, which is free and obviously has enough capacity for the full group.Order arbitrarily the available coalitions. We denote by top-down sorting the followingassignment of members into coalitions according to their type. Coalition 1 gets assignedthe N highest type members, coalition 2 the N highest type members among the remainingones, and so on until all members are assigned to one (and only one) coalition. The top-downsorting leads to a coalition structure with types stratified from higher to lower. Namely, giventwo coalitions l > k and two members i, j that are assigned to either coalition by top-downsorting, then, ξ i > ξ j and ¯ ξ l ≥ ¯ ξ k . To ensure that this inequality is strict for at least onepair of players in two different coalitions, we assume that two successive coalitions cannotbe fully occupied by players of the same type. As mentioned above to join a coalition l , theymust choose an action with cost p l . The last group is the option not being in a coalition.We say that an assignment of members to coalitions and a vector of entry costs forms anequilibrium when, given the costs, no individual prefers to change coalitions and either acoalition is full or its associated cost is zero.Denote by ∂U j ∂ ¯ ξ l the derivative of the equilibrium utility of individual j with respect to changes in the averagetype ¯ ξ l of the coalition to which they belong. The differential sensitivity of different typesof potential coalition members to the composition of coalitions has implications for coalitionformation that we now analyze. We assume for simplicity that all coalitions are equally sized. Nothing substantial changes if they arestill exogenous but differently sized. Later on we address endogenously sized coalitions. roposition There exists an assignment equilibrium with top-down sorting if whenever i and j are such that ξ i > ξ j we have that ∂U i ∂ ¯ ξ l − ∂U j ∂ ¯ ξ l > Letting ξ i ∗ ( l ) be the type of the lowest member in coalition l , the fee for a full coalition l isdefined recursively as: p l = ¯ ξ l Z ¯ ξ l +1 ∂U i ∗ ( l ) ∂ ¯ ξ j d ¯ ξ j + p l +1 , l = 1 , . . . , L − , (2) with p L = 0 . Proof.
See Appendix.This condition provides a test for the existence of endogenous sorting.
Given the coalitions that have formed, we now analyze the equilbria in the subgames wherethe coalitions are present. Remember that U j = ξ j V N X i =1 x i ! − x j The FOC of the second stage problem are ∂U j ∂x j = ξ j V ′ N X i =1 x i ! − x j = 0This implies that for all i, j x j ξ j = x i ξ i So normalizing ξ = 1 ξ i x = x i and hence in equilibrium V ′ x N X i =1 ξ i ! = x (3)which yields a unique equilibrium given the composition of the coalition. Let a particularcoalition A composed of a group of people with qualities ( ξ , ξ , . . . , ξ n ) with¯ ξ l = 1 N N X i =1 ξ i ξ l we get; V ′′ (cid:0) N x ¯ ξ l (cid:1) ∂x ∂ ¯ ξ l N ¯ ξ l + V ′′ (cid:0) N x ¯ ξ l (cid:1) N x = ∂x ∂ ¯ ξ l ∂x ∂ ¯ ξ l = V ′′ (cid:0) N x ¯ ξ l (cid:1) N x − V ′′ (cid:0) N x ¯ ξ l (cid:1) N ¯ ξ l < U j = ξ j V (cid:0) N x ¯ ξ l (cid:1) − ξ j x so that ∂U j ∂ ¯ ξ l = ξ j V ′ (cid:0) N x ¯ ξ l (cid:1) N ¯ ξ l ∂x ∂ ¯ ξ l + ξ j N x V ′ (cid:0) N x ¯ ξ l (cid:1) − ξ j x ∂x ∂ ¯ ξ l = ξ j N x (cid:18) − V ′′ (cid:0) N x ¯ ξ l (cid:1) ξ j − V ′′ (cid:0) N x ¯ ξ l (cid:1) N ¯ ξ l (cid:19) > ξ l changes little with ξ j , say because everyindividual is negligible, then ∂ U j ∂ ¯ ξ l ∂ξ j > Proposition There exists an assignment equilibrium with top-down sorting under decen-tralized effort choice.
Suppose instead that the level of contributions in a coalition are decided centrally by autilitarian social planner. U j = ξ j V N X i =1 x i ! − x j U = N X j =1 U j = N X j =1 ξ j V N X i =1 x i ! − x j ! = N ¯ ξ l V N X i =1 x i ! − N X j =1 x j The FOC of the second stage problem are now ∂ U ∂x j = N ¯ ξ l V ′ N X i =1 x i ! − x j = 09o for all i, j within a coalition x i = x j N ¯ ξ l V ′ ( N x ) − x = 0 (5)Totally differentiating 5 with respect to ¯ ξ l we get; N ∂x ∂ ¯ ξ l V ′′ ( N x ) N ¯ ξ l + N V ′ ( N x ) = ∂x ∂ ¯ ξ l ∂x ∂ ¯ ξ l = V ′ (cid:0) N x ¯ ξ l (cid:1) N − V ′′ ( N x ) N ¯ ξ l > U j = ξ j V ( N x ) − x j N ¯ ξ l V ′ ( N x ) − x = 0, so that ∂U j ∂ ¯ ξ l = ξ j V ′ ( N x ) N ∂x ∂ ¯ ξ l − x ∂x ∂ ¯ ξ l = ξ j x ¯ ξ l ∂x ∂ ¯ ξ l − x ∂x ∂ ¯ ξ l = (cid:18) ξ j − ¯ ξ l ¯ ξ l (cid:19) x ∂x ∂ ¯ ξ l = (cid:18) ξ j − ¯ ξ l ¯ ξ l (cid:19) x V ′ (cid:0) N x ¯ ξ l (cid:1) N − V ′′ ( N x ) N ¯ ξ l where the sign is positive for j with ξ j − ¯ ξ l > Remark It is clear that with respect to the decentralized equilibrium some types of players,i.e. those with a higher than average type within a coalition, have a higher utility while others,i.e. those with lower than average type, have a lower utility.
Furthermore, from inspection it is clear that if changing a single ξ j does not change muchthe average type of a coalition, then ∂ U j ∂ ¯ ξ l ∂ξ j > Proposition There exists an assignment equilibrium with top-down sorting under cen-tralized effort choice. Endogenous size and social optima
In the previous sections, the size of coalitions has been exogenously fixed at N . But giventhe environment considered, it would natural to consider the equilibrium when the coalitionsize is also endogenous. This could have important implications both for the equilibriumcontributions, and for the efficient composition and size of the groups. One main tradeoffis the following. In a larger group, the free-riding can become more problematic. On theother hand, in a larger group, the marginal benefits of an action positively affect a larger setof people. What is optimal will likely depend on specific features of the technology.In this section we show that when the group size can be centrally chosen a utilitariansocial planner would sometimes prefer top-down sorting (we call this ‘segregation’) whileunder other conditions she would prefer types to mix so that all groups have the sameaverage type (we call this ‘integration’).We now use a CRRA V (.) function to analyze this problem. V N X i =1 x i ! = (cid:16)P Ni =1 x i (cid:17) − a − − a In order for this V ( · ) function to make sense as a production function, we assume x i ≥ . Note that there is no risk in this problem, so we use the CRRA function as a convenient wayto parameterize concavity.Then, for a given N the FOC for coalition efforts in this case are N ¯ ξ l (cid:0) ( N x ) − a (cid:1) − x = 0¯ ξ l N − a = x a Remember that in coalition all members choose the same value under centralized decision-making and so x i = x j = x within the coalition. Thus the total utility within the coalitionis U l = N ¯ ξ l (cid:16) N (cid:0) ¯ ξ l N − a (cid:1) a (cid:17) − a − − a − N (cid:16)(cid:0) ¯ ξ l N − a (cid:1) a (cid:17) so that U l = (cid:18) − a − (cid:19) ¯ ξ a l N − a a − N ¯ ξ l − a and society welfare is U = L X l =1 U l = L X l =1 (cid:18)(cid:18) − a − (cid:19) ¯ ξ a l N − a a − N ¯ ξ l − a (cid:19) (7)11efine U T D as the total utility obtained in society when coalitions are formed with top-down sorting and V (cid:16)P Ni =1 x i (cid:17) = (cid:18)(cid:16)P Ni =1 x i (cid:17) − a − (cid:19) / (1 − α ) .Suppose that it is possible to organize coalitions so that all coalitions have the same mean¯ ξ l and define U I as the total utility obtained in society when all coalitions have the samemean ¯ ξ ∗ l = P Ll =1 ¯ ξ l /L where ¯ ξ l is the group l mean under top-down sorting. Because all thegroups have the same average type, the optimally chosen contributions are the same in allof them and thus no one has an incentive to choose the group to which they belong.Remember that the central planner does not know the types of any of the players. Thus,we assume groups are sufficiently large that simply allocating individuals randomly to groupsgenerates groups with equal average types in expectation. Proposition Suppose coalitions size is exogenously set to N . Then U T D > U I if and onlyif a < . Proof.
This follows from (7) since U T D = L X l =1 (cid:18)(cid:18) − a − (cid:19) ¯ ξ ∗ a l N − a a − N ¯ ξ ∗ l − a (cid:19) = (cid:18) − a − (cid:19) N − a a L ¯ ξ ∗ a l − LN ¯ ξ ∗ l − a = (cid:18) − a − (cid:19) N − a a L (cid:18)P Ll =1 ¯ ξ l L (cid:19) a − N P Ll =1 ¯ ξ l − a U I = L X l =1 (cid:18)(cid:18) − a − (cid:19) ¯ ξ a l N − a a − N ¯ ξ l − a (cid:19) = (cid:18) − a − (cid:19) N − a a L X l =1 ¯ ξ a l − N P Ll =1 ¯ ξ l − a so that by applying Jensen’s inequality, we have that U T D > U I U l N = 1 + a − a ) ¯ ξ a l N − a a − ¯ ξ l − a (8) ∂ (cid:0) U l N (cid:1) ∂N = ¯ ξ a l N − a a > ξ l the average payoff in a group increases in N independentlyof the concavity of the individual functional form within the CRRA class.Given that proposition 4 establishes that average payoff in the coalition is concave inaverage type for a > roposition For a > the social planner would like a single group of the maximal size. When, on the other hand a <
1, there is a tradeoff for the “high” quality groups. Onthe one hand, they would prefer to have total segregation to increase payoff, since averagepayoff increases with average type as in equation 8 we see that it increases in ¯ ξ a l . On theother hand, a bigger size increases payoff within the group, as in equation 8 we see that itincreases in N − a a . Clearly this tradeoff pushes for at least some degree of mixing. So far, we have studied the problem of coalition formation and activities as if the activitiesof those coalitions did not interact with one another. But in our motivation we discussedthe evidence that group formation often occurs in contexts where the groups compete, suchas gangs (Vigil 1996) or warfare (Sosis, Kress and Boster 2007). For this reason we willnow study the case where, after the groups form, they compete. The main insight of thissection, is that our earlier results also apply in this case. In particular, let us assume thatthe coalitions compete, after having formed, in a contest. We will show that the incentivesfor coalition formation that allow for a top-down sorting assignment (as in Proposition 2)still hold in this case. To be more precise, take two coalitions with respective sizes M and N but otherwise identical utility functions.Then assume that the payoffs are given by U j = ξ j N V (cid:16)P Ni =1 x i (cid:17) V (cid:16)P Ni =1 x i (cid:17) + V (cid:16)P Mi =1 y i (cid:17) − x j and V N X i =1 x i ! = N X i =1 x i ! b ; V M X i =1 y i ! = M X i =1 y i ! b and b < Proposition There exists an assignment equilibrium with top-down sorting under decen-tralized effort choice when coalitions compete.
Proof.
See Appendix. We assume that payoffs are the outcomes of a contest, and given by a contest success function, as inSkaperdas (1996). Conclusion
We have studied a model of public good provision within groups. The group members haveheterogeneous preferences for the public good, and high types contribute more towards itsprovision. Thus, all individuals prefer to be in groups with higher average types. We allowfor the existence of ‘entrepreneurs’ who create the groups and demand a (possibly non-monetary) ‘fee’ to enter the group. We study the case where the ‘entrepreneurs’ can enforcea contribution level within the group and also when once in the group, contributions arevoluntary.We first show that both under centralized and decentralized provision there is top-downsorting equilibrium in which groups are organized assortatively by type. Under centralizedcontributions every participant is better off if average types increase. When contributionsare centralized that is not true and only above average types are better off.There are interesting results in terms of welfare. Under more concave utility functions,utilitarian welfare is higher when average types are the same across groups ( ‘integration’)than under top-down sorting ( ‘segregation’). With less concave utilities, the opposite istrue. Concavity also favors large groups. Under less concave functions there is a trade-off, asvery homogeneous groups are good, but large groups are also good, so one could be willingto sacrifice in terms of homogeneity to increase contributions because of group size. Theresults are robust to environments in which groups compete among themselves.While our results provide important insights, there are of course limitations to our anal-ysis. For example, we have not studied repeated interactions, which, through reciprocity,could lead to different results. We conjecture that in a repeated environment we would bemore likely to observe the results we posit under ‘centralized’ group management. We thinkthe reasons for assortative matching will survive in that case.We have studied an environment where all the agents are ‘symmetric’ even if they havedifferent types. However, there are important applications where matching is bilateral, likethe marriage market. We think that some of our insights will carry over in those applications.Indeed Greenwood et al. (2014) document an increase in marriage market assortativityas an important source of inequality. However, it is not immediately obvious that ourwelfare results on sorting will carry over in that context. Another intriguing question inthat environment has to do with the fact that in many species, it is only one of the sidesof the marriage market that engages in costly signaling before mating (Jiang, Bolnick andKirkpatrick 2014).Our paper also provides interesting insights for public policy. Most extant foundations for14integration’ policies, either in housing or school settings have to do either with concerns toreduce inequality (Ananat 2011, Reardon 2016) or with direct spillovers from higher abilityindividual on other individuals (Duflo, Dupas, Kremer 2011, Graham, Imbens, Ridder 2014).We provide a new foundation for integration that is based on indirect spillovers. Individualsdo not care directly about the type of others’, they care because high types provide highereffort towards public good provision.It is worth commenting that the integration result in our model comes from the objec-tive of social surplus maximization rather than equity considerations. Nonetheless, it hasequity implications as low types benefit from the higher provision of public goods in theirgroup. Thus, the efficient distribution is also equitable and may provide a new rationale forintegration efforts in different societies. Our model may also provide a different rationaleto whether the attainment gap of children (of lower socio-economic status) in segregatedneighborhoods (see for instance Ananat, 2011) may have something to do with the lowerprovision of public good in such neighborhoods. If so, this may have positive implications,particularly across generations, which may further strengthen case for integration. We leavethis interesting issue for future work.
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Proof of Proposition 1
A member of coalition l with type ξ i does not want to move to coalition l + 1 providedthat: U i (cid:0) ¯ ξ l (cid:1) − p l ≥ U i (cid:0) ¯ ξ l +1 (cid:1) − p l +1 U i (cid:0) ¯ ξ l (cid:1) − U i (cid:0) ¯ ξ l +1 (cid:1) ≥ p l − p l +1 .Such person will have a type such that ξ i ∗ ( l − ≥ ξ i ≥ ξ i ∗ ( l ) . Then we have that: U i (cid:0) ¯ ξ l (cid:1) − U i (cid:0) ¯ ξ l +1 (cid:1) = ¯ ξ l Z ¯ ξ l +1 ∂U i ∂ ¯ ξ j d ¯ ξ j ≥ ¯ ξ l Z ¯ ξ l +1 ∂U i ∗ ( l ) ∂ ¯ ξ j d ¯ ξ j = p l − p l +1 ,where the inequality is true by (1). Similarly a member of coalition l with type ξ i does notwant to move to coalition l − U i (cid:0) ¯ ξ l (cid:1) − p l ≥ U i (cid:0) ¯ ξ l − (cid:1) − p l − p l − − p l ≥ U i (cid:0) ¯ ξ l − (cid:1) − U i (cid:0) ¯ ξ l (cid:1) Remember that ξ i ∗ ( l − ≥ ξ i ≥ ξ i ∗ ( l ) . Thus: p l − − p l = ¯ ξ l − Z ¯ ξ l ∂U i ∗ ( l − ∂ ¯ ξ j d ¯ ξ j ≥ ¯ ξ l − Z ¯ ξ l ∂U i ∂ ¯ ξ j d ¯ ξ j = U i (cid:0) ¯ ξ l − (cid:1) − U i (cid:0) ¯ ξ l (cid:1) ,where, again, the inequality is true by (1). (cid:4) roof of Proposition 6 The FOC for the different players are now ξ j N b (cid:16)P Mi =1 y i (cid:17) b (cid:16)P Ni =1 x i (cid:17) b − (cid:18)(cid:16)P Ni =1 x i (cid:17) b + (cid:16)P Mi =1 y i (cid:17) b (cid:19) = x j N b (cid:16)P Mi =1 y i (cid:17) b (cid:16)P Ni =1 x i (cid:17) b − (cid:18)(cid:16)P Ni =1 x i (cid:17) b + (cid:16)P Mi =1 y i (cid:17) b (cid:19) = x j ξ j = x N b (cid:16) y P Mi =1 ξ i (cid:17) b (cid:16) x P Ni =1 ξ i (cid:17) b − (cid:18)(cid:16) x P Ni =1 ξ i (cid:17) b + (cid:16) y P Mi =1 ξ i (cid:17) b (cid:19) = x N b (cid:0) y M ¯ ξ k (cid:1) b (cid:0) x N ¯ ξ l (cid:1) b − (cid:16)(cid:0) x N ¯ ξ l (cid:1) b + (cid:0) y M ¯ ξ k (cid:1) b (cid:17) = x (9)1 M b (cid:0) y M ¯ ξ k (cid:1) b − (cid:0) x N ¯ ξ l (cid:1) b (cid:16)(cid:0) x N ¯ ξ l (cid:1) b + (cid:0) y M ¯ ξ k (cid:1) b (cid:17) = y ¯ ξ k M y ¯ ξ l N x = x y q ¯ ξ k M y = q ¯ ξ l N x ¯ ξ k M y = q ¯ ξ k q ¯ ξ l N x (10)and substituting (10) into (9) we get x = 1 N b (cid:0) y M ¯ ξ k (cid:1) b (cid:0) x N ¯ ξ l (cid:1) b − (cid:16)(cid:0) x N ¯ ξ l (cid:1) b + (cid:0) y M ¯ ξ k (cid:1) b (cid:17) = 1 x N b (cid:16)p ¯ ξ k p ¯ ξ l N (cid:17) b (cid:0) N ¯ ξ l (cid:1) b − (cid:18)(cid:0) N ¯ ξ l (cid:1) b + (cid:16)p ¯ ξ k p ¯ ξ l N (cid:17) b (cid:19) x = 1 N b (cid:16)p ¯ ξ k p ¯ ξ l (cid:17) b (cid:0) ¯ ξ l (cid:1) b − (cid:18)(cid:0) ¯ ξ l (cid:1) b + (cid:16)p ¯ ξ k p ¯ ξ l (cid:17) b (cid:19) = 1 N ξ l b (cid:16)p ¯ ξ k p ¯ ξ l (cid:17) b (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) y = ¯ ξ k ¯ ξ l ( N x ) (cid:0) ¯ ξ k M (cid:1) = 1 M ξ k b (cid:16)p ¯ ξ k p ¯ ξ l (cid:17) b (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) j = ξ j N − (cid:0) y M ¯ ξ k (cid:1) b (cid:0) x N ¯ ξ l (cid:1) b + (cid:0) y M ¯ ξ k (cid:1) b ! − x j = ξ j N − (cid:16)p ¯ ξ k (cid:17) b (cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b − ξ j N ξ l b (cid:16)p ¯ ξ k p ¯ ξ l (cid:17) b (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) ∂U j ∂ ¯ ξ l = 12 N b ¯ ξ l ξ j (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b + 12 N b p ¯ ξ l ξ j (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) (cid:18)q ¯ ξ k (cid:19) b (cid:18)q ¯ ξ l (cid:19) b − + 12 N b ¯ ξ l ξ j (cid:18)(cid:16)p ¯ ξ k (cid:17) b + (cid:16)p ¯ ξ l (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b (cid:18)q ¯ ξ l (cid:19) b − − N b ¯ ξ l (cid:18)q ¯ ξ k (cid:19) ξ j (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b − which implies that ∂ U bj ∂ ¯ ξ B ∂ξ j = 12 N b ¯ ξ l ξ j (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b + 12 N b ¯ ξ l ξ j (cid:18)(cid:16)p ¯ ξ k (cid:17) b + (cid:16)p ¯ ξ l (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b (cid:18)q ¯ ξ l (cid:19) b − + 14 N b ¯ ξ l ξ j (cid:16)p ¯ ξ l (cid:17) b − (cid:16)p ¯ ξ k (cid:17) b (cid:18)(cid:16)p ¯ ξ l (cid:17) b + (cid:16)p ¯ ξ k (cid:17) b (cid:19) (cid:18)q ¯ ξ k q ¯ ξ l (cid:19) b and thus the condition 7 is satisfied so that we can have an analog of Proposition 2. (cid:4)(cid:4)