Social Division with Endogenous Hierarchy∗
James P. Choy†
May 2, 2016
Abstract
Many societies are divided into multiple smaller groups. Certain kinds of interaction are more likely to take
place within a group than across groups. I model a reputation effect that enforces these divisions. Agents
who are observed to interact with members of different groups can support lower levels of cooperation
with members of their own groups. A hierarchical relationship between groups appears endogenously in
equilibrium. The information requirements for my equilibrium to exist are much less demanding than
the information requirements in related models in the literature. Using anthropological evidence, I argue
that these different information requirements correspond to concrete differences between the institutions of
different Indian castes. I discuss the factors that affect when and where social divisions are likely to appear.
Keywords: Cooperation, Caste, Social Institution
JEL Classification Numbers: C73, O12, O17
∗ I am grateful to Mark Rosenzweig, Larry Samuelson, and Chris Udry for their advice and support throughout this project.
Treb Allen, Priyanka Anand, David Berger, Gharad Bryan, Avinash Dixit, Tim Guinnane, Melanie Morten, Sharun Mukand,
Motty Perry, Joe Vavra, and various seminar participants provided helpful comments. I acknowledge research funding from the
Yale Economic Growth Center.
† Department of Economics, University of Warwick and CAGE. E-mail: j.choy@warwick.ac.uk
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1
Introduction
Many societies are divided into multiple smaller groups. These divisions are especially salient in many
developing countries, where the groups have names such as castes, tribes, or clans, but developed countries
are divided as well, for example by race and religion. One stylized fact about group divisions is that people
are more likely to interact in certain ways with members of their own groups than with members of different
groups. Interactions that take place primarily within groups include trade (Greif 1993, Anderson 2011),
mutual insurance (Grimard 1997, Munshi and Rosenzweig 2009, Mazzocco and Saini 2012), and job referrals
(Munshi and Rosenzweig 2006). At first glance the lack of interaction between groups is puzzling, since
the argument from the gains from trade suggests that people should seek to interact with the most diverse
possible range of partners. In this paper, I describe a mechanism that can generate such social divisions in
equilibrium through a reputation effect, even when there are no fundamental barriers to interaction between
members of different groups.
An example of social division of the type I wish to describe comes from Mayer’s (1960) description of the
caste system in the village of Ramkheri in central India. The central fact of the caste system, according to
Mayer, is what he refers to as the commensal hierarchy, which prescribes who may eat with whom. There are
five major caste groupings in the village, and members of higher ranked castes refuse to eat with or accept
food from members of lower ranked castes, although members of lower ranking castes are willing to accept
food from members of higher ranking castes. Mayer writes, “Eating the food cooked or served by a member
of another caste denotes equality with it, or inferiority, and not to eat denotes equality or superiority.” As
eating together is one of the main ways to develop friendships, friendships are less likely to form across caste
lines than within castes.
Whether people follow the rules of the hierarchy depends to some extent on whether other members of
their caste can observe them. Mayer describes a member of an upper caste who was born in the village but
who is working in the city of Indore. On a visit to the village, he is offered tea by a member of a lower caste,
but he refuses, saying “I would willingly drink in Indore, but I must be careful not to offend anyone here.”
Similarly, Mayer describes a meal at a training camp for development workers held in the village, which is
attended by delegates from many other villages. The delegates from other villages all eat together, while the
delegates from Ramkheri sit separately in accordance with the caste rules. The Ramkheri delegates explain
the situation, saying, “We could not sit with them here; but they, being away from their villages, were able
to sit next to Muslims and even Harijans [members of the lowest Hindu caste].” According to Mayer this
phenomenon is due to the greater difficulty in observing violations of caste rules that take place outside the
village. Mayer writes, ”The orthodox in Ramkheri know that the rules are being broken outside, [but] they
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are content not to investigate, so long as the matter is not given open recognition.” Finally, after breaking the
rules regarding caste contact, caste members are obliged to perform a ritual purification. However, whether
the purification is in fact performed depends on whether the violation is observed. Mayer writes, “Touching
a Tanner [one of the lowest castes] is a more generally acknowledged matter for purification..., though it is
admitted that many people would not do anything if they were not seen to touch.” Thus people seem to
follow the rules of the hierarchy in part to preserve their reputations with members of their own castes.
Not all interactions between castes are penalized in Ramkheri. The Ramkheri caste system distinguishes
between the sharing of different kinds of foods between castes. Kacca foods are foods cooked with water
or salt. They include most daily staples. Pakka foods are foods cooked with butter. They are served at
ceremonial occasions. The rules regarding kacca foods are much more stringent than the rules regarding
pakka foods, and people are willing to accept pakka foods from members of lower castes from whom they
would not be willing to accept kacca foods. My interpretation of this distinction is that sharing kacca food,
which is eaten every day, is much more likely to lead to a deep, cooperative relationship than sharing pakka
food, which is eaten only rarely.
To summarize, the Ramkheri caste system exhibits four important features. First, members of different
castes do not interact in certain ways. Second, there is a hierarchy over castes, and members of higher
ranking castes refuse to interact with members of lower ranking castes but not vice versa. Third, caste
members follow the rules about non-interaction with other castes in part to preserve their reputations with
members of their own castes. Fourth, the reputational penalties for interacting with members of other castes
are more severe for those interactions which are most likely to lead to deep, cooperative relationships. I now
outline a model that accounts for all of these features.
In the model, agents search over the community to find partners for cooperative relationships. If an
agent cheats in any relationship, then the relationship breaks up and each partner to the relationship must
search for a new partner. Search requires effort and hence is costly. Cooperation is maintained by the
threat that any cheating agent will have to pay the cost of search, and the level of cooperation that any
agent can support is inversely related to the search cost that the agent is expected to incur at the end of
the relationship. Agents who expect to form matches with a larger fraction of potential partners pay lower
search costs in expectation. Thus an agent who is expected to form matches with a larger proportion of the
community can support a lower level of cooperation in any given relationship. Each agent is also a member
of a payoff irrelevant group, and in equilibrium each agent interacts only with members of her own group.
If an agent is observed to have formed a match with a member of a different group in the past, then it
is believed that the agent will continue to accept matches both with members of her own group and with
members of the other group in the future. Thus, agents who are observed to have interacted with members
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of different groups in the past are able to support lower levels of cooperation. This penalty for interacting
with members of different groups is sufficient to prevent members of different groups from interacting in
equilibrium. I refer to this state of affairs as group segregation. Group segregation increases the level of
cooperation that each agent can support compared to the situation without segregation, and if the benefits
of cooperation are sufficiently important, then group segregation is welfare improving for the community as
a whole.
The reputation mechanism yields novel theoretical insights. The first insight is that people may lose
reputation with members of their own group by interacting with members of different groups. Specifically,
people who interact with members of different groups are believed to be less trustworthy by members of
their own group.
A second insight is that the reputation mechanism endogenously generates an asymmetry between different groups. Consider two groups, group 1 and group 2, and suppose that the reputation effect prevents
members of group 1 from interacting with members of group 2. Members of group 1 do not interact with
members of group 2 because it is believed that a member of group 1 who has interacted with a member of
group 2 in the past will continue to interact with members of group 2 in the future. However, this belief
is rational only if members of group 2 are willing to interact with members of group 1. The groups are
thus organized in a hierarchical structure, with higher ranking groups being unwilling to interact with lower
ranking groups, but not vice-versa.
In the literature the model most closely related to mine is Eeckhout (2006). Eeckhout also constructs
a model in which agents refuse to interact with members of other groups, and in which group segregation
increases the level of cooperation that can be sustained within each group. Eeckhout’s model does not feature
the reputation effect or the group hierarchy that characterize my model. Without these features, Eeckhout’s
model fails to satisfy an equilibrium refinement called bilateral rationality, first introduced by Ghosh and Ray
(1996). Bilateral rationality requires that an equilibrium should be robust to the possibility of simultaneous
joint deviations by matched pairs of agents. It is related to the various renegotiation-proofness concepts
discussed by Farrell and Maskin (1989) and Bernheim and Ray (1989). The introduction of the reputation
effect and the group hierarchy ensure that my model does satisfy bilateral rationality.
My paper is also related to the larger literature on community enforcement. A central concern in this
literature is the information structure of the community. Early community enforcement models, including
Kandori (1992) and Greif (1993), assume that agents can costlessly observe the actions that their current
relationship partners have taken within their previous relationships. More recent additions to this literature,
such as Bowen, Kreps and Skrzypacz (2013) and Ali and Miller (2015), consider the conditions under which
agents will truthfully report what happened in their previous relationships to their current relationship
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partners. In both classes of models agents are able to find out what happened within their partners’ past
relationships. I depart from this tradition by assuming that agents have no way of obtaining information
about what their current relationship partners have done in their previous relationships. However, agents
do observe with whom their current relationship partners have interacted in the past.
Two other papers with a related information structure are Akerlof (1976) and Pe˛ ski and Szentes (2013).
In both of these papers, agents can observe not only with whom their current relationship partners have
interacted in the past, but also with whom their partners’ partners have interacted in the past, with whom
their partners’ partners’ partners have interacted in the past, and so on to infinity. These papers then
construct equilibria that enforce group segregation through a kind of infinite regress, in which there is
a punishment for interacting with a member of a different group, a punishment for failing to inflict this
punishment, a punishment for failure to punish failure to punish, and so on. In contrast, in my model
people observe with whom their current relationship partners have interacted in the past, but that is all.
Moreover, in my model there is a punishment for interacting with a member of a different group but no
punishment for failure to punish. Hence the information requirements for my equilibrium to exist are much
less demanding than the information requirements for the equilibria in Akerlof or Pe˛ ski and Szentes. I
argue that the very demanding information requirements of the Akerlof and Pe˛ ski and Szentes models can
be satisfied in real societies only if there is a specialized institution dedicated to collecting the necessary
information and broadcasting this information throughout the society. In contrast, the much less demanding
information requirements of my model can be satisfied in a society in which information spreads through
uncoordinated gossip. Using evidence from the anthropological literature, I argue that some Indian castes
have in fact developed such a specialized information broadcasting institution, and that these castes have
social punishments that correspond to the punishments in the Akerlof and Pe˛ ski and Szentes models. In other
castes, in contrast, no information broadcasting institution exists, and in these castes social punishments
correspond more closely to the punishments in my model.
Finally, my model is related to a literature explaining the causes of ethnic and cultural division. Political
scientists classify such divisions as either “primordial” or “socially constructed” (see, e.g., Chandra (2012)).
Primordial divisions are due to fundamental differences between members different groups, while socially
constructed divisions appear in societies of people who are ex ante similar. In a developing country context,
the paradigmatic kind primordial division is division between ethnic groups that speak different languages.
Lazear (1999) develops a model of linguistic divisions, and Michalopoulos (2012) explains the conditions
under which linguistic divisions are likely to arise. In contrast, my model describes socially constructed
divisions between groups like castes that speak the same language. The conditions under which socially
constructed divisions are likely to arise are different from the conditions under which primordial divisions
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are likely to arise. I describe some of the conditions that may lead to socially constructed division in the
conclusion.
2
Model
2.1
Setup
Time is discrete, starts at period 0 and continues forever. A mass 1 of agents are born at the beginning of
each period. Each agent is a member of one of G groups, and the mass of newborn agents from each group
in each period is 1/G. These groups are payoff irrelevant, but group membership is observable. Each agent
has N relationship “slots”. Newborn agents come into existence already matched with N − 1 partners who
are members of the same group. Thus each newborn agent has one relationship slot open. All agents have
a fixed discount factor δ. In each period the following things happen:
1. Each agent with an open relationship slot pays a search cost c and is provisionally matched with
another agent with an open slot. Agents are provisionally matched according to a uniform probability
distribution over the set of agents with open relationship slots.1 An agent can search for at most one
new partner in any period, even if she has more than one open relationship slot.
2. Provisionally matched agents observe their partners’ groups and past match sets. Each agent may then
choose to accept or reject the match. If either agent rejects the match, then the match is dissolved
and both agents return to step 1. Otherwise a match forms and both agents continue to step 3.
3. All agents play a stage game with each of their partners, described below. The total payoff for each
agent for the period is the sum of the payoffs from the stage game in each relationship, minus any
search costs.
4. For each matched pair of agents i and j, let ai and aj be the actions chosen in the stage game for that
match. If ai 6= aj , then the match breaks up and both players begin the next period with an open
relationship slot. Otherwise the match continues to the next period.
The stage game is as follows.2 Both partners in the relationship simultaneously choose a stage game
action a ∈ [0, ∞). An agent’s payoff is Π(a, a′ ), where the agent chooses action a and her partner chooses
action a′ . Define v(a) = Π(a, a) and d(a) = Π(0, a). I make the following assumptions on Π, v, and d:
1 More precisely, as will be seen below an agent can be completely characterized by her group g and a what I call the agent’s
past match set, H. There are a finite number of possible tuples (g, H). The probability that an agent is provisionally matched
with a partner with group and past match set (g, H) is just the proportion of agents with group and past match set (g, H)
within the population of all agents with open relationship slots.
2 This stage game was first described in Ghosh and Ray (1996).
6
Assumption 1.
1. For all a > 0 and all a′ , Π(0, a′ ) > Π(a, a′ ).
2. v(a) is bounded.
3. v(0) = d(0) = 0
4. v(a) and d(a) are continuous, twice differentiable, and strictly increasing in a.
5. v(a) is strictly concave in a and d(a) is strictly convex in a.
Part 1 of the assumption states that 0 is the strictly dominant action in the stage game, which can be
interpreted as a generalized prisoner’s dilemma with a continuum of actions. If both players play a then
both receive a payoff v(a), and I will sometimes refer to this as the value of cooperation at level a. If one
player plays a and the other plays 0, then the player who plays 0 gets d(a), and I will sometimes refer to
this as the value of cheating at level a. Part 2 is required to rule out Ponzi schemes, in which any level of
cooperation can be attained through the promise of ever higher levels of cooperation in the future. Parts 3
through 5 imply that the temptation to cheat is small for a small, and that the temptation to cheat grows
large as a gets large. These assumptions ensure that the solution to each agent’s maximization problem is
interior.
Each agent can observe her group and the group of any other agent with whom she is matched. Each
agent can also observe the history of play within each current match, but she cannot observe the history of
play in any match in which she does not participate. However, each agent can observe something about with
whom each of her partners has matched in the past. Specifically, for each group g, an agent can observe
whether any of her current partners have ever been matched with any agent in group g. Let Hi ⊆ {1, ...G}
be the set of groups g such that agent i has been matched with a member of group g in the past. I refer to
the set Hi as agent i’s past match set. Note that for all groups g, if agent i is in group g then g ∈ Hi , since
agents are born matched to N − 1 members of their own groups.
A (pure) strategy si for an agent i is a choice of whether to accept a match with any given partner, and
a choice of action to take within the stage game, conditional player i’s information. I consider strategies in
which each agent conditions her choice of whether to accept a match and her stage game action within each
match only on her own and her partner’s groups and past match sets. That is, I consider strategies of the
form
si = s(gi , Hi , gj , Hj ) = (m(gi , Hi , gj , Hj ), a(gi , Hi , gj , Hj ))
where gj and Hj are the group and past match set of some partner j of agent i, m ∈ {A, R} is the decision
to accept or reject a match with that partner, and a is the stage game action chosen in a match with that
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partner. Notice in particular that strategies of this form do not allow players to condition the actions that
they take with one partner on any of the characteristics of their other partners, or on the histories of any
other contemporaneous or previous match. Since the function s mapping groups and past match sets to
strategies is the same for all agents, I also use s to denote the strategy profile consisting of each agent’s
strategy.
Given a strategy profile, an agent’s utility depends on her group, her past match set, the groups and past
match sets of the agents with whom she is matched in the current period, and the expected distribution
of groups and past match sets in the population in all future periods. I will be interested in stationary
strategy profiles, that is, strategy profiles with the property that if all agents follow the strategy profile,
then the distribution of groups and past match sets in the pool of agents with unfilled relationship slots is
constant over time. Let Γ be a vector containing the measure of agents in the pool of agents with unfilled
relationship slots with group and past match set (g, H) for each (g, H) at some point in time. Let µi be
a vector containing the groups and past match sets of the agents who are currently matched with agent i.
Then if s is a stationary strategy profile and Γ is the associated distribution of groups and past match sets,
we can write agent i’s expected utility as:
EUi (s, gi , Hi , µi , Γ).
It is also helpful to have notation for the utility that agents receive from various deviations from strategy
profile s. Suppose that agent i deviates to strategy s′i , while all other agents continue to follow strategy s−i .
I denote i’s utility from this deviation by
EUi (s′i , s−i , gi , Hi , µi , Γ).
Finally, suppose that agent i deviates to strategy s′i , and agent j deviates to strategy s′j , while all other
agents continue to follow strategy profile s−ij . I denote agent i’s utility from this joint deviation by
EUi (s′i , s′j , s−ij , gi , Hi , µi , Γ).
2.2
Discussion of Modelling Assumptions
The setup of the model incorporates seven major assumptions, all of which are necessary to produce the main
result. First, society is divided into groups, and group membership is observable. This could be because
group membership is defined by some physical characteristic such as skin colour. Alternatively, it could be
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because there is a commonly known social convention that assigns agents to groups. For example, Indian
castes are usually not physically distinguishable. However, people’s names signal their castes, and caste
membership is common knowledge within a village.3
Second, agents are provisionally matched with partners according to a uniform distribution over the
population. The precise probability distribution here is not essential, and all the results of my model
still go through if agents are more likely to be provisionally matched with members of their own groups
than with members of other groups. What is essential is that there is some positive probability that each
agent is provisionally matched with a partner from a different group in each period. The idea here is that
opportunities for profitable exchange with members of other groups arise with positive probability, and that
there is some cost for refusing to take up these opportunities.
Third, a relationship is represented as a repeated prisoner’s dilemma with a continuum of actions, with
higher actions representing greater levels of cooperation. This game seems to capture in a stylized way many
interactions that take place in village environments. For example, consider a trade relationship, in which the
quality of goods traded is not immediately observable. In this case the level of cooperation a represents the
quality of the goods that the agents agree to trade. Each agent then has the opportunity cheat by secretly
choosing to provide a lower quality good. Alternatively, consider a mutual insurance relationship, in which
each agent agrees to transfer a units of money to her partner in the case that her partner suffers a negative
shock. Either partner can cheat by refusing to transfer the agreed upon amount if a negative shock in fact
occurs. In both of these examples the incentive to cheat increases as the level of cooperation a increases,
which is the main implication of the concavity of the v(·) function and the convexity of the d(·) function.
Fourth, agents can observe with whom their partners have interacted in the past, but not what happened
within their partners’ past relationships. This seems like a plausible information structure for many kinds of
interactions in village environments. For example, Udry (1990) notes that villagers have good information
about their neighbours’ social and economic situations. In particular, he writes that villagers are able to
report accurately about the ceremonies that their neighbours have given and attended, which supports my
claim that villagers keep track of with whom their neighbours interact. In contrast, it is much harder to
observe what happens within many kinds of relationships. For example, consider a credit relationship. In
this kind of relationship, the stage game actions correspond to financial transactions between the relationship
3 Interestingly, migrants from one Indian village to another sometimes use the occasion of travelling to a new village where
they are not known to claim a different caste membership. Mayer writes, “[A] way to assume a different caste affiliation is to
go to a place where you are not known, and assert that you belong to that caste.” Of course, people are aware of this trick and
may be reluctant to believe the caste claims of new migrants. Mayer writes, “I have known immigrants from north India who
have lived in a village for two generations, and of whose status the villagers were still uncertain. In one case the newcomers said
they were Brahmans [the highest caste], but nobody would eat from them lest their claim be false.” These anecdotes highlight
the idea that group membership is determined through commonly known conventions and is not necessarily related to physical
characteristics.
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partners. However, various papers have shown that it is frequently very difficult for people to observe their
neighbours’ financial transactions. For example, Anderson and Baland (2002) show that people in village
environments are frequently unaware even of their own spouses’ financial transactions. So my information
assumption seems like a reasonable approximation to the true information environment in villages, at least
for many kinds of relationships. The assumption that agents cannot observe whether their partners have
cheated in the past rules out ostracism as a strategy to promote cooperation. Ostracism is a strategy under
which agents refuse to cooperate with partners who have cheated in the past, as in Greif (1993). Since agents
do not observe whether their partners have cheated, this strategy cannot be used.
Fifth, there is a continuum of agents. This assumption rules out the possibility of contagion strategies,
such as the strategies described by Kandori (1992), to promote cooperation. Contagion strategies are equilibria only with a finite number of agents. More generally, with a large but finite number of agents contagion
strategies are equilibria only if the discount factor falls into a very small range, and there is no reason to
believe that parameter values are in fact precisely calibrated in this way. Indian villages may have several
hundred to more than a thousand inhabitants, and so my assumption seems like a reasonable approximation
to the conditions in these fairly large societies.
Sixth, agents have more than one relationship at each point in time. This opens up the possibility that
actions taken by an agent in one relationship slot can affect the agent’s payoffs in her other relationship
slots. In particular, if an agent’s past match set changes due to her actions in one relationship slot, this may
affect the level of cooperation that the agent can achieve in all of her other relationship slots. This potential
externality across matches is necessary for the construction of the equilibrium.
Seventh, agents are born matched with (N − 1) partners from their own groups. The assumption that
agents are born with existing matches captures the idea that people begin life with connections to their
families and childhood friends, and that they must later search for additional relationships as adults. The
assumption that agents are born with connections only to other members of their own groups captures the
hereditary nature of divisions like the Indian caste system, in which children are members of the same castes
as their parents. The heritability of group membership implies that children’s initial relationships are likely
to be with other members of their own groups. Upon reaching adulthood agents have the option to begin
interacting with members of other groups, although, as will be seen, in equilibrium they may choose not to
do so. As will be seen, in equilibrium agents do not want to form new relationships with members of different
groups because doing so would damage their existing relationships with members of their own group. In
order for this effect to function, agents must be born with existing relationships to members of their own
group.
Each of the previous six assumptions has economic content, and all six assumptions are necessary to
10
produce the main result of my model. The setup of the model also includes two further assumptions that are
not essential and are made only for analytical convenience. First, the assumption that new agents are born
every period is necessary to ensure that there are always agents with empty relationship slots, which in turn
is necessary to ensure that the cost of searching for a new partner is well-defined. An alternative modelling
strategy would be to assume that each relationship breaks up with some exogenous probability p each period,
refilling the pool of agents with empty relationship slots. This model would be more complicated, because I
would have to account for the possibility that two of an agent’s relationships might break up simultaneously,
but the results would be the same.
A second assumption made for analytical convenience is that relationships break up automatically if
either partner deviates from the agreed level of cooperation. An alternative would be to allow for asymmetric
strategies within relationships, so that an agent could be punished for a deviation while remaining within her
current relationship. Under either modelling assumption, the most severe punishment that can be imposed
on an agent who cheats in some relationship slot is to reduce her continuation payoff in the relationship slot
to the payoff that she would receive from an empty relationship slot, and so under either assumption the
level of cooperation that can be supported on the equilibrium path is the same. Since the modelling choice
does not affect the results, for simplicity I assume that relationships break up automatically if either partner
cheats.
2.3
Equilibrium Concept
In the literature, the usual equilibrium concept for repeated games like the one in this paper is subgame
perfect Nash equilibrium. However, subgame perfection is in one sense too strong, and in another sense too
weak, as an equilibrium concept for the phenomena I wish to describe. In this section, I present an alternative
equilibrium concept, and I explain the sense in which my equilibrium concept is in one way weaker and in
another way stronger than subgame perfection.
Before explaining my equilibrium concept, I first set out some definitions. I say that a pair (g, H) is
reachable given a strategy profile s if it would be possible for an agent to have group g and past match set
H in some period when all agents are following strategy profile s. A vector of partners µi for agent i is a
minimally deviant match if agent i has at most one partner under µi whose group and past match set (g, H)
is not reachable under s. Given the concept of a minimally deviant match, I define individual incentive
compatibility as follows:
Individual incentive compatibility: A stationary strategy profile s and the associated distribution Γ
satisfy individual incentive compatibility if, for any agent i with any group and past match set (gi , Hi ),
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for any minimally deviant match µi , and for any deviant strategy s′i ,
EUi (si , s−i , gi , Hi , µi , Γ) ≥ EUi (s′i , s−i , gi , Hi , µi , Γ)
I also define bilateral rationality as follows:
Bilateral rationality: A stationary strategy profile s and the associated distribution Γ satisfy bilateral
rationality if, for any agents i and j with any groups and past match sets (gi , Hi ), (gj , Hj ), and for any
minimally deviant matches µi and µj , there does not exist a deviant strategy profile s′ such that
EUi (s′i , s′j , s−ij , gi , Hi , µi , Γ) ≥ EUi (si , sj , s−ij , gi , Hi , µi , Γ) and
EUj (s′j , s′i , s−ij , gj , Hj , µj , Γ) ≥ EUi (sj , si , s−ij , gj , Hj , µj , Γ)
with at least one of the two previous inequalities strict, and such that for all s′′ ,
EUi (s′i , s′j , s−ij , gi , Hi , µi , Γ) ≥ EUi (s′′i , s′j , s−ij , gi , Hi , µi , Γ) and
EUj (s′j , s′i , s−ij , gj , Hj , µj , Γ) ≥ EUi (s′′j , s′i , s−ij , gj , Hj , µj , Γ)
A stationary strategy profile s and the associated distribution Γ form an equilibrium if they satisfy both
individual incentive compatibility and bilateral rationality.
Several comments are in order regarding this equilibrium concept. Individual incentive compatibility is
the familiar requirement that an equilibrium strategy profile should be robust to the possibility of unilateral
deviations. As such, individual incentive compatibility is similar to, but weaker than, subgame perfection.
Individual incentive compatibility is weaker than subgame perfection because I do not require that each agent
prefer to follow the strategy profile after all histories, but only after histories in which each agent has at most
one partner who has deviated from the equilibrium in the past, and in which the distribution Γ of groups and
past match sets in the pool of unmatched agents is unchanged from the equilibrium path. Individual incentive
compatibility is closely related to the equilibrium concept in Okuno-Fujiwara and Postlewaite (1995).
A strategy profile that satisfies individual incentive compatibility is stable in the sense that after a single
deviation from the strategy profile by any agent, it is not optimal for any agent to deviate again in any
future period, assuming that all other agents continue to follow the strategy profile. A subgame perfect
strategy profile is stable in the stronger sense that even after multiple deviations by multiple agents, it is
not optimal for any agent to deviate again. In the context of this paper, in which a strategy profile describes
a social norm, individual incentive compatibility captures the idea that all members of society should want
12
to follow the norm even if a small number of members of society have deviated from the norm in the past.
In contrast, subgame perfection captures the much stronger idea that all members of society should want
to follow the norm even if every member of society has deviated from the norm in the past. This second
requirement seems like too much to ask of a social norm; intuitively it seems that a social norm that is
universally disobeyed would cease to exist. For this reason I use the weaker concept of individual incentive
compatibility. I elaborate further on the difference between individual incentive compatibility and subgame
perfection after I have presented my main equilibrium.
Subgame perfection is too strong an equilibrium concept to describe social norms because it requires that
social norms continue to have power even when they are universally disobeyed. In another sense, however,
subgame perfection is too weak as an equilibrium concept. Consider two matched agents who are trying
to decide whether to follow a strategy profile. Subgame perfection requires that neither agent should want
to deviate individually, taking her partner’s action as fixed. However, suppose that the two agents can
communicate before deciding how to act. In this case, the two agents might agree to deviate simultaneously
to some joint set of actions that is better for both them than the set of actions prescribed by the strategy
profile. Subgame perfection does not rule out the possibility of jointly profitable deviations of this form. I
introduce the bilateral rationality requirement to ensure that under an equilibrium strategy profile there are
no profitable joint deviations.
Not all possible joint deviations are credible. In order for a proposed joint deviation to be credible, it
must be the case that neither partner to the deviation would prefer to deviate again individually from the
proposed joint deviation. This consideration motivates the definition of bilateral rationality above. The first
two inequalities in the definition of bilateral rationality describe a strategy profile s′ that matched agents i
and j both prefer to the proposed equilibrium strategy profile s. The second two inequalities in the definition
of bilateral rationality state that there is no further deviation s′′ such that either i or j prefer to follow s′′ ,
given that the other partner is following s′ (and given that all other members of society continue to follow s).
A strategy profile s satisfies bilateral rationality if there do not exist any such jointly profitable and credible
deviations.
In the same way that I require that there not be any profitable individual deviations from a strategy
profile only after histories in which not too many deviations have happened in the past, I also require that
there not be an profitable joint deviations from a strategy profile only after histories in which not too many
joint deviations have happened in the past. Formally, I require that there be no profitable joint deviations
only for agents who are matched with at most one partner who has deviated from the strategy profile in
the past, and only when the distribution Γ of groups and past match sets in the pool of unmatched agents
is unchanged from the equilibrium path. The motivation for limiting the bilateral rationality requirement
13
to this subset of all possible histories is the same as the motivation for limiting the individual incentive
compatibility requirement to the same subset of all possible histories.
The definition of bilateral rationality is adapted from Ghosh and Ray (1996). As will be seen, the
requirement that equilibrium satisfy bilateral rationality is the main innovation of this paper, and the need
to satisfy the bilateral rationality requirement drives essentially all of my results.
One problem with my equilibrium concept is that, as will be seen, an equilibrium does not necessarily
exist for all parameter values. This is a problem common to many of the renegotiation-proofness concepts
in the literature. For example, equilibrium also fails to exist for some parameter values in Ghosh and Ray
(1996). The fact that equilibrium fails to exist for some parameter values suggests that my equilibrium
concept is still too strong. However, my goal is primarily to demonstrate the possibility of an equilibrium
strategy profile that supports segregation, and the fact that such an equilibrium exists even under strong
equilibrium conditions supports the claim that such a strategy profile might in fact be chosen in real life.
2.4
A Benchmark Equilibrium
I will begin my analysis by discussing a benchmark strategy profile in which agents do not condition their
actions on their own or their partner’s group membership or past match set. If the benchmark strategy
profile is part of an equilibrium, I will refer to that equilibrium as a benchmark equilibrium.
A benchmark strategy profile is as follows:
Benchmark strategy profile All agents accept all matches, and choose a level of cooperation āB within
each match.
A benchmark strategy profile is an equilibrium if there are no profitable individual or joint deviations.
We must check that no agent can profit individually by cheating in any relationship, and also that no pair of
matched agents can jointly profit by deviating to a higher level of cooperation that is individually incentive
compatible for both agents. In principle, we also need to check that it is optimal for all agents to accept
matches with all other members of the community. However, this last condition is trivial in the benchmark
equilibrium, since all match partners are identical.
Because actions taken in one relationship do not affect any other relationship under the benchmark
strategy profile, it is possible to analyse each relationship slot separately. Let VBu be the value that an agent
expects to receive from an open relationship slot at the beginning of any period. Let VBm be the value that
an agent expects to receive from a relationship slot that is filled at the beginning of a period. I also define
VBf to be the expected value to each agent of having a filled relationship slot at the beginning of any future
period. In the proof of proposition 1 it is helpful to distinguish VBf from VBm because agents may be able to
affect VBm through renegotiation, but they cannot affect VBf . Bilateral rationality dictates that each pair of
14
matched agents chooses the level of cooperation that maximizes their joint utility, subject to the constraint
that no agent can profit individually by choosing to cheat. That is, VBm must satisfy:
VBm = max v(a)
(1)
VBm ≥ (1 − δ)d(a) + δVBu
(2)
a
subject to the constraint
Equation (1) says that an agent gets v(a) from a match both in the current period and in all future
periods. The constraint (2) is the individual incentive compatibility constraint. It states that the value of
cooperating must be greater than the payoff that the agent receives from cheating. If the agent cheats she
receives d(a) in the current period and then gets the value of an empty relationship slot in the next period.
The payoff to having an empty relationship slot VBu is defined by:
VBu = −(1 − δ)c + VBf
(3)
Equation (3) says that an agent with an empty relationship slot must pay the search cost in the current
period before being matched with a new partner and receiving the payoff to that future match.
A benchmark equilibrium is a benchmark strategy profile such that VBm , VBu , and VBf satisfy equation (1)
subject to (2) and equation (3), such that āB maximizes (1) subject to (2), and such that VBm = VBf .
Define â to be the value of a that solves
max v(a) − (1 − δ)d(a).
a
The following proposition provides conditions under which a benchmark equilibrium exists, and derives
the level of cooperation in a benchmark equilibrium:
Proposition 1. A benchmark equilibrium exists if and only if c satisfies
c≥
1
[d(â) − v(â)].
δ
(4)
If a benchmark equilibrium exists, then the equilibrium level of cooperation āB solves
d(āB ) − v(āB ) = δc
15
(5)
Omitted proofs are in appendix A.
The interpretation of the expression for the level of cooperation in the benchmark equilibrium is straightforward. If an agent cheats in the current period, her net gain in the period is the difference between the
value of cheating d(āB ) and the value of cooperating v(āB ). The cost of cheating is that the cheating agent’s
match will break up, so that in the next period she will have to pay the search cost to find a new partner.
Discounted for one period, this cost is δc. The maximum level of cooperation that can be sustained is the
level of cooperation such that the net cost of cheating is equal to the net benefit. The bilateral rationality
condition ensures that all agents will renegotiate up to the highest possible level of cooperation, so only the
maximum sustainable level of cooperation is consistent with equilibrium.
I briefly discuss the intuition for the fact that no bilaterally rational equilibrium exists unless c is sufficiently large. I consider strategy profiles in which all agents choose the same level of cooperation every
period. Since all agents accept all matches, any agent can cheat in her current relationship, break up the
relationship at the end of the period, pay the search cost c, and find a new partner in the next period.
Since all agents choose the same level of cooperation, the deviating agent will be able to cooperate at the
same level in her new relationship as she did in the old relationship. Thus, if c is low, then the penalty
for cheating in any given relationship is low, and so the common sustainable level of cooperation is low.
However, if all agents are cooperating at some common low level, then any two matched agents can jointly
deviate to a higher level of cooperation. This higher level of cooperation does not violate the individual
incentive compatibility constraint, so long as only two agents are cooperating at the high level, because
the penalty for breaking up this deviant relationship is high: if either agent breaks the relationship, both
agents must go back to cooperating at the low common level of cooperation. Thus the individual incentive
compatibility requirement rules out all strategy profiles except those strategy profiles with a low common
level of cooperation, and the bilateral rationality requirement rules out strategy profiles with a low common
level of cooperation, so that there are no remaining equilibrium strategy profiles. As c gets larger, higher
levels of cooperation become compatible with the individual incentive compatibility constraint, and for c
sufficiently large there exist levels of cooperation that are high enough to satisfy the bilateral rationality
requirement while still satisfying the individual incentive compatibility constraint.4
2.5
Motivating the Segregated Equilibrium
My goal is to construct an equilibrium that supports higher levels of cooperation than the benchmark
equilibrium. I do this by constructing an equilibrium in which agents reject some matches, instead of
accepting all matches as in the benchmark equilibrium. If agents reject some matches, then the expected
4A
similar issue arises in Ghosh and Ray (1996), and the proof of proposition 1 draws on ideas from the proofs in that paper.
16
cost of search for an unmatched agent is higher than in the benchmark equilibrium, and so the penalty for
cheating and the level of cooperation that can be supported in each match are also higher.
The main barrier to constructing an equilibrium in which agents reject some potential matches is the
bilateral rationality requirement. To build intuition for why bilateral rationality makes it difficult to construct
such an equilibrium, consider the following strategy profile, which is a simplified version of the strategy profile
considered by Eeckhout (2006), and which I will refer to as strategy profile E:
Strategy profile E: Agents accept matches with members of their own group, and reject matches with
members of any other group, regardless of past match histories. Within each match all agents choose
action āE .
As in the benchmark equilibrium, under strategy profile E actions taken in one relationship slot do not
affect the optimal action in any other relationship slot. Thus it is possible to analyse each relationship slot
separately. Let VEm be the value to an agent from having a filled relationship slot in a period, and let VEu
be the value to an agent from having an empty relationship slot in a period. Under strategy profile E the
composition of the pool of agents with unfilled relationship slots is strategically relevant. Fortunately the
composition is easy to describe. If all agents follow strategy profile E, then at the beginning of each period
there are
1
G
agents from each group in the pool of agents with unfilled relationship slots. Thus a searching
agent meets a partner from her own group with probability
1
G.
Using this probability I can write expressions
for VEm and VEu as follows:
VEm = v(āE )
VEu = −(1 − δ)c +
1 m G−1 u
V +
VE
G E
G
The first equation says that an agent who is matched cooperates forever at level āE . The second equation
says that an unmatched agent pays the search cost and is matched with a partner with probability
1
G,
and
otherwise remains unmatched and must pay the search cost again.
Strategy profile E satisfies the individual incentive compatibility condition if
VEm ≥ (1 − δ)d(āE ) + δVEu
Substituting in the definitions of VEm and VEu and rearranging yields that strategy profile E satisfies the
individual incentive compatibility condition if
d(āE ) − v(āE ) ≤ δGc
17
Comparing this expression to the expression defining the benchmark level of cooperation āB yields the
following:
Lemma 1. If G > 1, then there exist values of āE such that āE > āB and such that strategy profile E
satisfies the individual incentive compatibility condition.
Proof. Since d(·) is strictly convex and v(·) is strictly concave, the difference d(a) − v(a) is increasing in a.
Thus for G > 1 there exist values of āE that solve
d(āE ) − v(āE ) ≤ δGc
and such that āE > āB , where āB is defined as in proposition 1.
Higher levels of cooperation are individually incentive compatible under strategy profile E than in the
benchmark equilibrium because agents expect to form matches with only 1/G of their potential partners
under strategy profile E, while they expect to form matches with all of their potential partners in the
benchmark equilibrium. Thus, the expected cost of breaking up a relationship is higher under strategy profile
E than in the benchmark equilibrium, and so the individually incentive compatible level of cooperation is
higher under strategy profile E than in the benchmark equilibrium.
Although higher levels of cooepration are individually incentive compatible under strategy profile E than
under the benchmark strategy profile, we also have the following:
Lemma 2. Strategy profile E is not an equilibrium, because it does not satisfy the bilateral rationality
requirement.
Proof. Let āE be an action such that d(āE ) − v(āE ) ≤ δGc, which implies that strategy profile E satisfies the
individual incentive compatibility condition. Consider two agents from different groups who are provisionally
matched. By rejecting the match both agents get utility VEu , while by jointly deviating to accept the match
both agents get VEm > VEu . Moreover, the incentives in this deviant relationship are exactly the same as the
incentives in relationships entered into by following strategy profile E, and so since strategy profile E satisfies
the individual incentive compatibility condition so does the joint deviation to accepting this match. Thus
the joint deviation to accepting the match is individually incentive compatible and makes both partners to
the match strictly better off, and so strategy profile E does not satisfy the bilateral rationality condition
and is not an equilibrium.
The problem with strategy profile E is that under the strategy profile relationships between members of
different groups are just as profitable as relationships between members of the same group, and yet members
18
of different groups do not interact. Intuitively it seems implausible that people would consistently fail to
seize opportunities for profitable interaction in this way. The bilateral rationality requirement formalizes this
intuition. A more plausible theory of group segregation would provide a reason why relationships between
members of different groups are less profitable than relationships between members of the same group. In
the next section I construct a strategy profile that contains just such a reason, and which therefore does
satisfy the bilateral rationality requirement.
2.6
The Segregated Equilibrium
In this subsection I propose what I will call the segregated strategy profile. As before, if the segregated
strategy profile is part of an equilibrium, I refer to the equilibrium as a segregated equilibrium. In the
segregated equilibrium agents interact only with members of their own groups on the equilibrium path, which
increases the cost of breaking up any match and thereby allows matched agents to support higher levels of
cooperation than can be supported in the benchmark equilibrium. In addition, there is a reputational penalty
for agents who interact with members of certain other groups. This reputational penalty makes interactions
between members of different groups less profitable than interactions between members of the same group
and thus ensures that the segregated strategy profile is bilaterally rational.
Under the segregated strategy, groups are ranked in a hierarchy. I label the groups so that group 1 is
ranked highest in the hierarchy and group G is ranked lowest. Thus g < g ′ means that g is ranked above g ′ .
An agent accepts matches with members of a group if and only if 1) the group is included in the agent’s past
match set, or 2) the group is of equal or higher rank to the agent’s group. Matched agents choose a level of
cooperation in each period that depends on the groups and past match histories of each of the partners to
the relationship. Formally, the segregated strategy profile is as follows:
Segregated strategy profile: The matching strategies are m(g, H, g ′ , H′ ) = A if and only if g ′ ≤ g or
g ′ ∈ H. The level of cooperation chosen between an agent with group and past match set (g, H) and an
agent with group and past match set (g ′ , H′ ) can be written as āS (g, H, g ′ , H′ )
Let ΓS be the distribution of agents in the pool of agents with unfilled relationship slots induced by the
segregated strategy profile. That is, ΓS is the distribution such that the measure of agents who have group
and past match set (g, {g}) is
1
G
for all g, and the measure agents who have group and past match set (g, H)
for all H 6= {g} is zero. A segregated strategy profile is an equilibrium if there are no profitable individual or
joint deviations, when all agents are matched with at most one partner whose past match set is not reachable
under the segregated strategy profile, and when the distribution of agents in the pool of unmatched agents
is ΓS . More specifically, a segregated equilibrium must satisfy four conditions. To cut down on notation I
state these conditions informally:
19
If each agent has at most one partner whose past match set is not reachable under the segregated strategy
profile, and if the distribution of agents in the pool of unmatched agents is ΓS , then
1. No agent can profit by cheating in any relationship.
2. No pair of matched agents can jointly profit by deviating to a higher level of cooperation.
3. All agents prefer to accept matches with members of their own or higher ranking groups, or with
members of groups that are in their past match sets.
4. All agents prefer to reject matches with members of lower ranking groups that are not in their past
match sets.
My goal is to find parameter values under which a segregated equilibrium satisfying these conditions
exists. I begin my analysis by describing the levels of cooperation that are chosen under the segregated
equilibrium. Define γ(g, H) by
γ(g, H) = |{g ′ : m(g, H, g ′ , {g ′ }) = A and m(g ′ , {g ′ }, g, H) = A}|
That is, γ(g, H) is the number of groups with whom an agent with group and past match set (g, H)
expects to form matches if all other agents follow the segregated strategy profile. For example, a member
of group 1 with past match set {1, 2} expects to form matches with members of groups 1 and 2 under the
segregated strategy profile, so γ(1, {1, 2}) = 2.
Implicitly define ā(γ) as the solution to the following set of equations:
V m (γ) = v(ā(γ))
= (1 − δ)d(ā(γ)) + δV u (γ)
V u (γ) = −(1 − δ)c +
γ m
G−γ u
V (γ) +
V (γ)
G
G
The previous equations define V m (γ) to be the value of a filled relationship slot for an agent who expects
to cooperate at level ā(γ) in both her current and all future matches. The value V u (γ) is defined to be the
value of an empty relationship slot for an agent who expects to form matches with γ groups and to cooperate
at level ā(γ) in all future matches. The level of cooperation ā(γ) is defined implicitly as the highest level
of cooperation that is individually incentive compatible for an agent who expects to form matches with γ
groups and who expects to cooperate at level ā(γ) in each match. Rearranging these equations yields
d(ā(γ)) − v(ā(γ)) = δ
G
c
γ
(6)
Comparing this equation to (5) shows that ā(γ) > āB for all γ < G. This is because an agent who expects to
form matches with γ < G groups expects to pay a higher search cost upon breaking up the relationship and
20
so can support a higher level of cooperation than an agent who expects to form matches with all possible
partners.
I can now describe the level of cooperation that is chosen in the segregated equilibrium:
Lemma 3. If a segregated equilibrium exists, then the equilibrium level of cooperation chosen in a match
between an agent with group and past match set (g, H) and an agent with group and past match set (g ′ , H′ )
is
āS (g, H, g ′ , H′ ) = min{ā(γ(g, H)), ā(γ(g ′ , H′ ))}
In the segregated equilibrium, ā(γ(g, H)) is the maximum level of cooperation that is individually incentive compatible for an agent with group and past match set (g, H). Lemma 3 states that each pair of
matched agents chooses the highest level of cooperation that is incentive compatible for both agents individually. Intuitively, if the level of cooperation were higher, then at least one agent would be able to profit by
cheating, while if the level of cooperation were lower than both agents would have a profitable joint deviation
to a higher level of cooperation that would still be individually incentive compatible for each of them.
It is useful know how the values V u (γ) and V m (γ) depend on γ. The following lemma describes these
relationships when the search cost c is sufficiently large:
Lemma 4. Suppose that
c≥
1
[d(â) − v(â)].
δ
Then
1. V m (γ) is strictly decreasing in γ.
2. V u (γ) is strictly increasing in γ.
3. V m (G) > V u (G).
The first part of lemma 4 is true for any search cost c, and follows from equation (6) and the fact that
v(a) is concave and d(a) is convex. The third part of lemma 4 is also true for any search cost, and follows
directly from the definition of V u (G). The second part of lemma 4 is more subtle. To see why it is true,
suppose to the contrary that for some γ > γ ′ , V u (γ) ≤ V u (γ ′ ). From the definition of ā(·), we have that
v(ā(γ ′ )) ≥ (1 − δ)d(ā(γ ′ )) + δV u (γ ′ )
which implies that
v(ā(γ ′ )) ≥ (1 − δ)d(ā(γ ′ )) + δV u (γ)
21
So, given two matched agents who expect to receive V u (γ) when unmatched, it is individually incentive
compatible to cooperate at level ā(γ ′ ) > ā(γ) when matched. In other words, given a strategy profile in
which the members of γ groups choose to accept matches only with each other and to cooperate at level
ā(γ) in each period, any two matched agents have a jointly profitable and individually incentive compatible
joint deviation to a higher level of cooperation. However, using logic similar to the logic in proposition 1, it
can be shown that if the search cost c is sufficiently large then there is no such joint deviation. So when the
search cost c is sufficiently large, it must be the case that V u (γ) > V u (γ ′ ) for all γ > γ ′ .
Putting together the pieces of lemma 4 yields the following corollary:
Corollary 1. Suppose that
c≥
1
[d(â) − v(â)].
δ
(7)
Then
V u (1) < ... < V u (G) < V m (G) < ... < V m (1)
Using lemmas 3 and 4, it is possible to prove the following proposition:
Proposition 2. Fix v(·), d(·), c, and δ, and suppose that
c≥
1
[d(â) − v(â)].
1−δ
(8)
Then there exists N̄ such that for all N > N̄ , a segregated equilibrium exists. If a segregated equilibrium
exists, then the equilibrium level of cooperation chosen by an agent with group and past match set (g, H)
matched with a partner with group and past match set (g ′ , H′ ) is
āS (g, H, g ′ , H′ ) = min{ā(γ(g, H)), ā(γ(g ′ , H′ ))}.
(9)
The complete proof of proposition 2 is in the appendix, but I outline the main steps of the proof here.
A segregated equilibrium exists if conditions 1 through 4 above are satisfied. If agents choose the level of
cooperation in (9), then condition 1 is satisfied, since the level of cooperation in (9) is the highest level of
cooperation that is individually incentive compatible for all agents. If the search cost is sufficiently large,
as defined by (8), then condition 2 is satisfied. The reason that the search cost must be sufficiently large
for condition 2 to be satisfied is the same as the reason why the search cost must be sufficiently large for a
benchmark equilibrium to exist. Thus only conditions 3 and 4 remain.
Condition 3 for the existence of a segregated equilibrium is satisfied if, for all (g, H),
22
V u (G) + (N − 1)V m (γ(g, H)) ≤ N V m (γ(g, H))
(10)
Consider an agent i with group and past match set (g, H) who is matched with N − 1 partners, each
with group and past match set (g, {g}), and who is provisionally matched with an N th partner with group
and past match set (g ′ , H′ ) for some g ′ such that g ′ ≤ g or g ′ ∈ H. Suppose that the distribution of agents
in the pool of agents with unfilled relationship slots is ΓS . The left hand side of (10) is the maximum utility
that agent i can get from rejecting the match with her N th partner. The right hand side is the utility that
agent i gets from accepting the match with her N th partner. Since V u (G) < V m (γ(g, H)) for all (g, H) by
lemma 4, (10) is satisfied. Therefore, condition 3 for the existence of a segregated equilibrium holds.
Condition 4 for the existence of a segregated equilibrium is satisfied if, for all (g, H),
V u (1) + (N − 1)V m (γ(g, H)) ≥ N V m (γ(g, H) + 1)
(11)
Consider again an agent i with group and past match set (g, H) who is matched with N − 1 partners,
each with group and past match set (g, {g}), and who is provisionally matched with an N th partner with
group and past match set (g ′ , H′ ). This time, suppose that g ′ > g and that g ′ ∈
/ H. Suppose again
that the distribution of agents in the pool of agents with unfilled relationship slots is ΓS . The left hand
side of (11) is the minimum utility that agent i can get from rejecting the match with her N th partner.
The right hand side is the utility that agent i gets from accepting the match with her N th partner. Since
V m (γ(g, H+1)) < V m (γ(g, H)) for all (g, H) by lemma 4, (11) is satisfied if N is sufficiently large. Therefore,
for sufficiently large N condition 4 for the existence of a segregated equilibrium holds.
The intuition behind proposition 2 is as follows. In order for a segregated equilibrium to exist, it must
be the case that agents are willing to accept matches with members of higher ranking groups but not with
members of lower ranking groups. An agent who accepts a match with a member of a higher ranking group
achieves a lower level of cooperation in that relationship slot than she could achieve if she were matched with
a member of her own group. However, accepting a match with a member of a higher ranking group does not
affect the level of cooperation that an agent can achieve in any of her other relationship slots. In contrast,
accepting a match with a member of a lower ranking group reduces the level of cooperation that the agent
can achieve in all of her relationship slots. Thus the penalty for accepting a match with a member of a lower
ranking group is greater than the penalty for accepting a match with a member of a higher ranking group.
As the number of relationship slots N grows, the penalty for accepting a match with a member of a lower
ranking group increases while the penalty for accepting a match with a member of a higher ranking group
stays the same, and so for N sufficiently large the conditions for the existence of the segregated equilibrium
23
are satisfied.
I briefly discuss the role of my non-standard equilibrium concept in making the result in proposition 2
possible. An equilibrium in my model is a strategy profile with the property that even if one agent has
deviated in the past, no other agent or pair of matched agents wants to deviate again in the future. In
contrast, subgame perfection requires that even if every agent has deviated in the past, no agent wants to
deviate in the future. The segregated strategy profile does not satisfy this stronger requirement. For example,
consider a history in which all agents have interacted with members of all groups, so that Hi = {1, ..., G} for
all i. After this history, there cannot be any further reputational penalty for interacting with a member of a
different group, and so the only continuation strategy profile that is consistent with individual optimization
is the benchmark strategy profile. Thus, the segregated strategy profile is not subgame perfect. This fact
suggests that temporary shocks that cause many agents to deviate from the segregated strategy profile may
cause segregation to break down permanently.
This discussion also explains the necessity of the assumption that agents are born matched with N − 1
members of their own groups. As explained in section 2.2, this assumption reflects the hereditary nature
of groups like Indian castes, in which people are members of the same castes as their parents and inherit
some of their parents’ relationships. These inherited relationships act as a kind of collateral that enforces
segregation, because agents do not want to damage their inherited relationships by forming new relationships
with members of lower ranking groups. If agents were born with no relationships, then there would be no
downside for newborn agents to forming relationships with members of other groups and so there would be
no way to enforce segregation.
2.7
Welfare
In the benchmark equilibrium, each agent’s utility at birth is
VBu + (N − 1)VBm = V u (G) + (N − 1)V m (G)
In the segregated equilibrium, each agent’s expected utility at birth is
V u (1) + (N − 1)V m (1)
Since V m (1) > V m (G) whenever the segregated equilibrium exists by lemma 4, we have the following:
Proposition 3. If the segregated equilibrium exists, then there exists N̂ such that for all N > N̂ , an agent’s
expected utility at birth is greater under the segregated equilibrium than under the benchmark equilibrium.
24
For sufficiently large N , the increased welfare from more cooperation under segregation outweighs the
reduced welfare from higher search costs. This can explain why the segregated equilibrium would be selected
over the benchmark equilibrium without segregation.
I note that the threshold N̂ above which the segregated equilibrium is welfare improving relative to the
benchmark equilibrium need not be the same as the threshold N̄ above which the segregated equilibrium
exists. This opens up the possibility that segregation may persist even in situations in which it reduces
welfare.
2.8
Other Equilibria
The segregated equilibrium imposes a hierarchical relationship between groups, in which members of higher
ranked groups refuse matches with members of lower ranked groups but members of lower ranked groups
accept matches with members of higher ranked groups. This linear hierarchy seems to correspond to the
real hierarchy that characterizes the Indian caste system. However, the equilibrium is not unique. One way
in which the equilibrium fails to be unique is that the segregated equilibrium provides no guidance as to
which groups are ranked higher and which groups are ranked lower in the hierarchy. If there is a segregated
equilibrium in which group 1 is ranked higher than group 2, then there is also a segregated equilibrium in
which group 2 is ranked higher than group 1. This indeterminacy seems to correspond to real indeterminacy
in the Indian caste system, in which the relative caste rankings may differ in different places. For example,
Mayer (1960) writes “The Potter, for example, is of much lower standing in Uttar Pradesh than he is in
Malwa.” The relative ranking of the Potter caste varies across locations in India.
Another way in which the segregated equilibrium fails to be unique is that the linear hierarchy of the
segregated equilibrium is not the only possible pattern of segregation. From the discussion in section 2.5,
there does not exist an equilibrium in which all agents reject matches with all groups other than their own
after all histories. Moreover, with two groups the segregated equilibrium is essentially the only equilibrium
that features group segregation, with the exception of knife-edge cases in which parameter values are such
that agents are just indifferent between accepting and rejecting matches with members of other groups.
However, with three or more groups there do exist other patterns of segregation that are equilibria. For
example, with three groups, if a segregated equilibrium exists then there also exists an equilibrium that is a
cycle. In this cyclical equilibrium, on the equilibrium path members of group 1 reject matches with members
of group 2, members of group 2 reject matches with members of group 3, and members of group 3 reject
matches with members of group 1, while all other matches are accepted. As in the segregated equilibrium,
in this cyclical equilibrium agents interact only with members of their own groups on the equilibrium path.
25
Other patterns are possible as well. The linear hierarchy of the segregated equilibrium seems in some sense
simpler than these other equilibria, which perhaps provides a reason to think that the segregated equilibrium
would be selected over the other equilibria. Admittedly, this is a somewhat weak justification, and it would
be interesting to try to find examples of societies with a cyclical or some other pattern of segregation, as
these other patterns also seem to be allowed by the model. However, I am not aware of any such societies.
3
Centralized and Decentralized Segregation
So far I have developed a theory of social division in which members of different groups do not interact with
each other due to a reputation effect. Two previous papers, Akerlof (1976) and Pe˛ ski and Szentes (2013)
(henceforth APS), construct closely related models. Like my model, APS feature a community of agents who
belong to payoff-irrelevant groups and who search over the community to find relationship partners. Also
like my model, APS construct equilibria in which there is a future penalty for interacting with a member of a
different group, resulting in group segregation. However, the information structure assumed by APS differs
from the information structure in my model. In APS, agents observe with whom their current relationship
partners have interacted in the past, with whom their partners’ partners have interacted in the past, and so
on to infinity. In contrast, in my model agents observe with whom their current partners have interacted
in the past, but that is all. The structure of punishments also differs between my model and APS. In APS
there is a punishment for agents who interact with members of other groups, a punishment for failure to
punish such interactions, a punishment for failure to punish failure to punish, and so on. In my model, there
is a punishment for agents who interact with members of other groups, but there is no further punishment
for agents who fail to inflict the initial stage of punishment. Because failure to punish is itself a punishable
offense in APS, it is possible to support more severe punishments in APS than in my model. In particular
Akerlof constructs an equilibrium in which both violation of the caste norms and failure to punish such a
violation lead to the same punishment, namely complete ostracism. In contrast, in my model agents who
violate the caste norms can still support a level of cooperation that is greater than zero. Agents cannot
be reduced to zero cooperation in my model because this most severe punishment is not compatible with
bilateral rationality in the absence of further punishments for failure to punish.
The differences between my model and APS suggest empirical tests that can distinguish the two models.
One difference between my model and APS is that APS does not generate a group hierarchy, and so whether
the hierarchy exists can help to distinguish my model from APS. Since evidence for the group hierarchy
was discussed in the introduction, in this section I discuss several other questions that may distinguish my
model from APS. First, is there a punishment for failure to punish failure to punish... failure to punish
26
infringements of group norms, or does the chain stop after the first step? Second, are these punishments the
most severe possible, or are they more mild? Third, do group members have the information about their
partners’ partners’... partners interactions that is necessary to implement the most severe punishments, or
is each person’s information about their partners more limited? Fourth, how does this information spread
through the population? The answers to these questions are different for different Indian castes.
An example of a caste that seems well described by APS comes from Majumdar (1958), who describes
the norms of the Chamar caste in the state of Uttar Pradesh. Majumdar emphasizes the role of the caste
panchayat in enforcing caste norms. The panchayat is a council of caste elders that meets to judge offenses
against the rules of the caste and that may also pronounce punishments. Majumdar describes a case where
the norms were violated and the response of the panchayat as follows:
“Even if a person gives food or water to an outcaste, or invites him for a smoke, without knowing the
stigma attached to the recipient of his kindness, the unwitting offender also relinquishes his membership
of the caste.... An instance of this occurred in May 1954, when K-Chamar of Bijapur village visited BChamar of Mohana. K-Chamar had been, for some reason or other expelled from his caste by the Chamar
biradari [that is, the local subcaste] of Bijapur. He came to Mohana without letting anyone know of
the disgrace, and B-Chamar as is the custom treated his guest very hospitably, and they took their
midday meals together. Soon it was known that K-Chamar was an outcaste. Consequently B-Chamar
was declared an outcaste by the Chamar caste-panchayat of Mohana.”
Four features of this account are noteworthy. First, not only do the caste rules require that K-Chamar
be punished for violating the rules of the caste, but they also require that B-Chamar be punished for failing
to punish K-Chamar. Second, both B-Chamar and K-Chamar suffer the most severe punishment possible,
complete ostracism. Third, in order for the villagers of Mohana to inflict the correct punishment on BChamar, they must know not only that B-Chamar interacted with K-Chamar, but also that K-Chamar had
previously violated the caste rules in some way. Fourth, uncoordinated gossip does not seem to be sufficient
to disseminate the information necessary to enforce this social norm. In fact, B-Chamar does not receive
the information that he needs in time. Instead, the caste panchayats of Bijapur and Mohana work actively
to gather and broadcast the necessary information, and the coordinated, purposeful intervention of these
institutions seems to be necessary to ensure that the norms are enforced.
A contrasting example of a caste that seems best described by my model comes from Hayden (1983).
Hayden describes rules in the Nandiwalla caste in the state of Maharashtra. Among the Nandiwallas a
person who has broken the caste rules is said to be eli. A person who has become eli can appear before
the caste panchayat and be reinstated in the community by paying a fine. However, while the panchayat
can coordinate the community to allow people to exit the eli state, the panchayat does not seem to play a
role in informing the community when people enter the eli state. Hayden describes the process of becoming
eli as follows: “Eli is not a status that is imposed on a person for his actions. Rather it is an automatic
27
reaction to the fact that one automatically becomes polluted by an improper act.... It does not have to be
pronounced by anyone.”
Regarding the consequences of being eli, Hayden writes:
“The [Nandiwallas] say that they ‘won’t give even fire’ to one who is eli. However, there is a certain
literal quality to this pronouncement. They won’t give him fire, but they will give him matches. They
won’t take food with him, but they will certainly drink liquor and take pa:n with him. One should not
quarrel with someone who is eli, but the latter may argue in panchayat. What seems to happen is that,
although certain specific commensal activities with other caste members are limited for one who is eli,
most aspects of his life remain unchanged. He still puts his tent in the same place in both the large
triennial encampment and in smaller camps on the road. People come to visit, and he can reciprocate.
In most ways, life goes on normally.
In addition to those mentioned above, the activities in which an ‘outcast’s’ participation is restricted
involve business, religious ceremonies, and the marriage of his children. In the first category, one should
not enter into any business arrangement with one who is eli. In the second, those who are not fully in
caste cannot participate in most group religious ceremonies. It is the third category that is potentially
the most serious. If one is in caste suspension, he cannot arrange marriages for his young children, and
other families should not honor previous arrangements by accepting his daughter or sending their own
so long as he is eli. However, if someone does honor a marriage agreement, the amount he is charged is
usually small.”
There are several key differences between the Nandiwallas as described by Hayden and the Chamars as
described by Majumdar, which suggest that my model represents Nandiwalla caste norms better than APS.
First, while the punishment for violating the caste norms among the Chamars is the most severe punishment
possible, that is, compete ostracism, the punishment for violating caste norms among the Nandiwallas is more
mild. Among the Nandiwallas some but not all forms of cooperation are withheld from violators. Second,
while there is a penalty for violating the caste norms among the Nandiwallas, the penalty for failing to punish
a violator is minimal, as represented by the “small” fine for a person who honors a marriage agreement with
an eli partner. The lack of penalty for failing to punish a violator among the Nandiwallas contrasts with the
severe penalty for failing to punish a violator among the Chamars. Third, because Nandiwalla caste members
are not punished for failing to punish other caste members who have violated the caste rules, Nandiwalla
caste members need much less information than Chamar caste members in order to follow the caste norm
successfully. As a result, Nandiwalla caste members seem to be able to acquire the information that they
need to follow the caste norm from uncoordinated gossip, and unlike the Chamar panchayat, the Nandiwalla
panchayat does not play an active role in broadcasting information about who has entered the eli state.
To summarize the discussion in this section, the central difference between my model and APS is that the
information requirements of the APS model are much more demanding than the information requirements of
my model. This difference has a concrete institutional realization. Castes described by the APS model like
the Chamars have developed a centralized institution, the panchayat, that gathers the necessary information
and broadcasts it to the population. In contrast in castes described by my model, like the Nandiwallas,
28
information diffuses through the population in a decentralized and uncoordinated way. The Chamars’ specialized information broadcasting institution allows the Chamars to inflict more severe punishments on norm
violators than the Nandiwallas. I hypothesize that these differences represent different stages in the evolution of social division. Social divisions first appear spontaneously and without centralized coordination, as
described by my model. The punishment for interacting with members of other groups is relatively mild,
and there is no punishment for failing to punish interactions with members of other groups. As divisions
harden and become more entrenched, centralized institutions develop to coordinate punishments for norm
violations, punishments become more severe, and punishments are inflicted not only for interacting with
members of other groups but also for failing to punish others’ interactions. The model of APS describes
these more formalized divisions.
4
Conclusion: When and Where Do Social Divisions Appear?
In this paper I have described a model in which social divisions appear in a population of ex ante identical
agents, as a way to support higher levels of cooperation. Social divisions are supported by a reputation
effect. It is believed that people who have interacted with members of other groups in the past will continue
to interact with members of those groups in the future, and as a result people who have interacted with
members of other groups in the past are believed to be less trustworthy. The reputation effect endogenously
creates a hierarchical relationship between groups. The informational requirements of my model are less
demanding than the requirements of other models in the literature, and so my model describes a society
in which group segregation is supported by uncoordinated gossip rather than the specialized, centralized
institutions that would be necessary to disseminate more detailed information.
The social divisions in my model differ in kind from the social divisions described in the majority of the
development literature. Most development papers on social division study social divisions defined by language
differences. For example, the commonly used ethnolinguistic fractionalization index primarily defines groups
based on language. Lazear (1999) develops a model of this kind of division, and Michalopoulos (2012) explains
when these kinds of divisions are likely to appear. In particular, Michalopoulos argues that geographical
diversity leads to linguistic diversity. In contrast, my model describes divisions between groups that speak
the same language, and my model suggests that the conditions under which these divisions are likely to
appear are quite different from the conditions that lead to linguistic diversity. By way of conclusion, I
discuss some conditions that may affect whether these “caste-like” divisions are likely to appear.
My model identifies three potentially empirically testable conditions that may affect whether a society
develops caste-like divisions. The first two conditions are relatively straightforward. First, the purpose of
29
social division is to promote cooperation in the absence of formal contract enforcement, and so societies in
which formal contract enforcement is weak and relational contracting is more important are more likely to
become divided. Second, the reputation mechanism that enforces social division requires that agents have
at least some ability to observe with whom their current relationship partners have interacted in the past.
Thus societies in which people are better able to observe and gossip about each other’s relationships are
more likely to become divided. Both of these conditions hold to a greater extent in developing countries
than developed countries, and so these conditions may help to explain why social divisions of the type that
I describe are more salient in developing countries than developed countries.
The third condition that should affect the likelihood of social division is more surprising. The central
mechanism in my model is it is believed that an agent who has interacted with a member of a different group
in the past will continue to accept matches with all members of that group in the future. In the model, this
belief is correct because all members of society are identical in all payoff-relevant respects, and so once an
agent has incurred the reputational penalty by interacting with one member of a different group, there is
no reason for the agent not to interact with all members of the group. However, if there are fundamental
differences between agents within each group, than an agent might be willing to interact with some members
of a different group but not others. In this case it cannot be inferred that an agent who has interacted with
one member of a group in the past will be willing to interact with all members of the group in the future, and
so the reputation mechanism underlying my model breaks down. Thus, somewhat paradoxically, not only is
fundamental diversity not necessary to explain the existence of caste-like divisions, but more fundamental
diversity actually reduces the likelihood that caste-like divisions will appear. This condition suggests that the
relatively greater level of fundamental diversity, for example in levels and kinds of human capital investment,
in developed countries makes caste-like divisions in developed countries harder to sustain.
As well as having implications for when social divisions are likely to appear, my model has implications
for whether social division can be affected by policy changes. Divisions based on real differences between
groups, such as divisions based on language differences, are difficult to change through policy, at least in
the short-run. However, divisions of the kind described in my model may be easier to affect through policy.
For example, policies that improve formal contract enforcement, or that increase diversity of kinds of human
capital investment, may cause social divisions to break down. Such policies may have negative consequences,
given that social division helps to promote cooperation. At the same time, social divisions also have negative
effects not captured in my model. For example, social divisions can lead to violent conflict. If these negative
effects of social division are important, then policies that break down social divisions may ultimately increase
welfare.
30
References
[1] Akerlof, George (1976), “The Economics of Caste and of the Rat Race and Other Woeful Tales,”
Quarterly Journal of Economics 90:4, 599-617.
[2] Anderson, Siwan (2011), “Caste as an Impediment to Trade,” American Economic Journal: Applied
Economics 3:1, 239-263.
[3] Bernheim, B. Douglas, and Debraj Ray (1989), “Collective dynamic consistency in repeated games,”
Games and Economic Behavior 1:4, 295-326.
[4] Chandra, Kanchan (ed.) (2012), Constructivist Theories of Ethnic Politics, Oxford: Oxford University
Press.
[5] Eeckhout, Jan (2006), “Minorities and Endogenous Segregation,” Review of Economic Studies 73:1,
31-53.
[6] Farrell, Joseph and Eric Maskin (1989), “Renegotiation in repeated games,” Games and Economic
Behavior 1:4, 327-360.
[7] Ghosh, Parikshit and Debraj Ray (1996), “Cooperation in Community Interaction Without Information
Flows,” Review of Economic Studies 63:3, 491-519.
[8] Greif, Avner (1993), “Contract Enforceability and Economic Institutions in Early Trade: The Maghribi
Traders’ Coalition,” American Economic Review 83:3, 525-548.
[9] Grimard, Franque (1997), “Household consumption smoothing through ethnic ties: evidence from Cote
d’Ivoire,” Journal of Development Economics 53:2, 391-422
[10] Hayden, Robert M (1983),“Excommunication as Everyday Event and Ultimate Sanction: The Nature
of Suspension from an Indian Caste”, Journal of Asian Studies 42:2, 291-307.
[11] Kandori, Michihiro (1992), “Social Norms and Community Enforcement,” Review of Economic Studies
59:1, 63-80.
[12] Majumdar, D.N. (1958), Caste and Communication in an Indian Village, Asia Publishing House: Bombay.
[13] Mayer, Adrian C. (1960), Caste and Kinship in Central India, Berkeley: University of California Press.
[14] Mazzocco, Maurizio and Shiv Saini (2012), “Testing Efficient Risk Sharing with Heterogeneous Risk
Preferences,” American Economic Review 102:1, 428-468.
[15] Munshi, Kaivan and Mark Rosenzweig (2006), “Traditional Institutions Meet the Modern World: Caste,
Gender, and Schooling Choice in a Globalizing Economy,” American Economic Review 96:4, 1225-1252.
[16] Munshi, Kaivan and Mark Rosenzweig (2009), “Why is Mobility in India so Low? Social Insurance,
Inequality, and Growth,” mimeo, Brown University.
[17] Okuno-Fujiwara, Masahiro and Andrew Postlewaite (1995), “Social Norms and Random Matching
Games,” Games and Economic Behavior 9:1, 79-109.
[18] Pe˛ ski, Marcin and Balazs Szentes (2013), “Spontaneous Discrimination,” American Economic Review.
31
A
Proofs
A.1
Proof of Proposition 1
Plugging equation (3) into the constraint (2) and equation (1) and rearranging yields
VBm = max v(a)
a
(12)
subject to
VBf ≤
1
[v(a) − (1 − δ)d(a)] + (1 − δ)c
δ
(13)
Recall that â was defined as the value of a that solves
max v(a) − (1 − δ)d(a).
a
Since v is strictly concave and d is strictly convex, there exists a finite value of a that maximizes â. Since
â has a maximum value, there exists V̂Bf such that the constraint (13) can be satisfied for a ≥ 0 if and only
if VBf ≤ V̂Bf , with V̂Bf defined by
V̂Bf =
1
[v(â) − (1 − δ)d(â)] + (1 − δ)c.
δ
(14)
Now, define a function φ(x) by
φ(x) = max v(a)
a
(15)
subject to
x≤
1
[v(a) − (1 − δ)d(a)] + (1 − δ)c
δ
(16)
Any fixed point of φ is a benchmark equilibrium. However, notice that φ is not well-defined for all x,
since for x > V̂Bf there is no a ≥ 0 that satisfies (16). Since v and d are continuous and differentiable, φ is
continuous and differentiable. By the envelope theorem,
∂φ
= −ψ < 1
∂x
(17)
where ψ > 0 is the Lagrange multiplier on the constraint (16). Since φ(x) is decreasing and is well-defined
for x sufficiently small, φ(V f ) has exactly one fixed point if and only if
φ(V̂Bf ) ≤ V̂Bf
(18)
Plugging in the expression for V̂Bf from (14) into (18) and rearranging yields the condition that a benchmark equilibrium exists if and only if
c≥
1
[d(â) − v(â)].
δ
32
This completes the proof.
A.2
Proof of Lemma 3
Suppose that all agents follow the segregated strategy profile and suppose that the segregated strategy profile
is an equilibrium. Let V m (g, H, g ′ , H′ ) be the value of being an agent with group and past match set (g, H)
matched to an agent with group and past match set (g ′ , H′ ), and let V u (g, H) be the value of being an
unmatched agent with group and past match set (g, H), when the distribution of agents in the pool of agents
with unfilled relationship slots is ΓS . Then we have
V m (g, H, g ′ , {H′ }) = v(āS (g, H, g ′ , H′ ))
X
V u (g, H) = −(1 − δ)c +
g′ |g′ ≥g and g′ ∈H
1 m
G − γ(g, H) u
V (g, H, g ′ , {g} ∪ {g ′ }) +
V (g, H)
G
G
Consider first the case of an agent with group and past match set (g, {g}) for some group g. We can
write V u (g, {g}) as
V u (g, {g}) = −(1 − δ)c +
G−1 u
1 m
V (g, {g}, g, {g}) +
V (g, {g})
G
G
Since the segregated strategy profile is individually incentive compatible and bilaterally rational, the level
of cooperation āS (g, {g}, g, {g}) solves
max v(a)
a
such that v(a) ≥ (1 − δ)d(a) + δV u (g, {g})
But this implies that
v(āS (g, {g}, g, {g}) = (1 − δ)d(āS (g, {g}, g, {g}) + δV u (g, {g})
which can be rewritten as
d(āS (g, {g}, g, {g}) − v(āS (g, {g}, g, {g}) = δ
1
c
G
This is just the equation that implicitly defines ā(1), so āS (g, {g}, g, {g}) = ā(1) for all g. Plugging this
into the expression for V u (g, {g}) yields that V u (g, {g}) = −(1 − δ)Gc + v(ā(1)). Moreover, the incentives
for an agent with group and past match set (g, {g} ∪ {g ′ }) for g ′ ≤ g are exactly the same as the incentives
for an agent with group and past match set (g, {g}), so V u (g, {g} ∪ {g ′ }) = ā(1) for g ′ ≤ g.
Now consider the case of an agent i with group and past match set (g, H) matched with a partner j with
group and past match set (g ′ , {g} ∪ {g ′ }), where g ′ ≥ g. āS (g, H, g ′ , {g ′ }) is individually incentive compatible
for agent j if
v(āS (g, H, {g} ∪ {g ′ })) ≥ (1 − δ)d(āS (g, H, g ′ , {g} ∪ {g ′ })) + δV u (g ′ , {g} ∪ {g ′ })
Plugging in the value for V u (g ′ , {g} ∪ {g ′}) derived above and rearranging yields that āS (g, H, g ′ , {g} ∪ {g ′})
is individually incentive compatible for agent j if āS (g, H, g ′ , {g} ∪ {g ′ }) ≤ ā(1). Therefore, āS (g, H, g ′ , {g} ∪
33
{g ′ }) is bilaterally rational individually incentive compatible for both agents if āS (g, H, g ′ , {g} ∪ {g ′ }) solves
max v(a)
(19)
such that v(a) ≥ −(1 − δ)d(a) + δV u (g, H)
(20)
and a ≤ ā(1)
(21)
a
The first constraint is the individual incentive compatibility constraint for agent i and the second constraint is the individual incentive compatibility constraint for agent j. Since this problem is the same for all
g ′ such that g ′ ≥ g, āS (g, H, g ′ , {g} ∪ {g ′ }) is the same for all g ′ such that g ′ ≥ g, which implies that we can
write
V u (g, H) = −(1 − δ)c +
γ(g, H) m
G − γ(g, H) u
V (g, H, g ′ , {g} ∪ {g ′ }) +
V (g, H)
G
G
(22)
Plugging (22) into (19) and rearranging yields that ā(g, H, g ′ , {g} ∪ {g ′ }) solves
v(āS (g, H, g ′ , {g} ∪ {g ′ }) − d(āS (g, H, g ′ , {g} ∪ {g ′ }) = δ
γ(g, H)
c
G
This is just the equation that implicitly defines ā(γ(g, H)), so āS (g, H, g ′ , {g} ∪ {g ′ }) = ā(γ(g, H)) for all
g, H, and g ′ such that g ′ ≥ g. Plugging this into the expression for V u (g, H) yields that V u (g, H) =
G
+ v(ā(γ(g, H))
−(1 − δ) γ(g,H)
Finally, consider and agent i with group and past match set (g, H) matched with a partner j with group
and past match set (g ′ , H′ ). āS (g, H, g ′ , H′ ) satisfies individual incentive compatibility for agent i if
v(āS (g, H, g ′ , H′ )) ≥ (1 − δ)d(āS (g, H, g ′ , H′ )) + δV u (g, H)
Plugging in the value for V u (g, H) derived above and rearranging yields that āS (g, H, g ′ , H′ ) is individually incentive compatible for agent i if āS (g, H, g ′ , H′ ) ≤ ā(γ(g, H)). A similar argument shows that āS (g, H, g ′ , H′ )
is individually incentive compatible for agent j if āS (g, H, g ′ , H′ ) ≤ ā(γ(g, H)). Thus āS (g, H, g ′ , H′ ) is bilaterally rational and individually incentive compatible for both agents if āS (g, H, g ′ , H′ ) solves
max v(a)
a
such that a ≤ ā(γ(g, H))
and a ≤ ā(γ(g ′ , H′ ))
Clearly the solution to this problem is āS (g, H, g ′ , H′ ) = min{ā(γ(g, H)), ā(γ(g ′ , H′ )). This completes the
proof.
A.3
Proof of Lemma 4
From the definition of ā(γ) in (6) and the fact that v(·) is convex and d(·) is concave, ā(γ) is decreasing in γ.
Since ā(γ) is decreasing in γ and since V m (γ) = v(ā(γ)), it is immediate that if γ < γ ′ then V m (γ) > V m (γ ′ ).
Since V u (G) = −(1 − δ)c + V m (G), V u (G) < V m (G). Thus it only remains to be shown that if γ < γ ′ then
V u (γ) < V u (γ ′ ).
34
Define φ(x, γ) by
φ(x, γ) = max v(a)
a
subject to the constraint
x≤
1
G
[v(a) − (1 − δ)d(a)] + (1 − δ) c
δ
γ
From the proof of proposition 1, φ(x, γ) has a fixed point for all γ such that 1 ≤ γ ≤ G if and only if
c≥
1
[d(â) − v(â)]
1−δ
(23)
Inspection of the definition of V m (γ) shows that if φ(x) has a fixed point, then V m (γ) is the fixed point
of φ(x). Thus if c satisfies the condition above, then, rearranging the constraint in the definition of φ(x, γ),
ā(γ) solves
max v(a)
a
subject to
v(a) ≥ (1 − δ)d(a) + δV u (γ)
The solution to the previous problem is decreasing in V u (γ), and ā(γ) is decreasing in γ, which implies
that V u (γ) must be increasing in γ, completing the proof.
A.4
Proof of Proposition 2
A segregated equilibrium exists if conditions 1 through 4 from section 2.6 are satisfied. From the definition
of ā(γ), if āS (g, H, g ′ , {g ′ }) = min{ā(γ(g, H)), ā(g ′ , H′ ) then condition 1 is satisfied. The argument in the
text of section 2.6 shows that condition 3 is satisfied whenever and that there exists N̄ such that for N > N̄ ,
condition 4 is satisfied. Thus it only remains to show that condition 2 is satisfied.
Suppose that all agents follow the segregated strategy profile and that the distribution of agents in the
pool of agents with unfilled relationship slots is ΓS . Let VSf (g, H) be the value that an agent with group
and past match set (g, H) expects to get from each of her future matches. Note that under the segregated
strategy profile with distribution ΓS this value is the same regardless of the group of the agent’s future
partner. Without loss of generality, suppose that γ(g, H) ≥ γ(g ′ , H′ ). Then VSm (g, H) must satisfy
max v(a)
a
subject to the constraint
VSm (g, H) ≥ (1 − δ)d(a) + δVSu (g, H)
where
VSu (g, H) = −(1 − δ)c +
G − γ(g, H) u
γ(g, H) f
VS (g, H) +
VS (g, H)
G
G
Following the same steps as in the proof of proposition 1, we can rewrite these equations as
35
VSm (g, H, g ′ , H′ ) = max v(a)
a
subject to
VSf (g, H, g ′ , H′ ) ≤
1
G
[v(a) − (1 − δ)d(a)] + (1 − δ)
c
δ
γ(g, H)
Then, continuing to follow the steps in proposition 1, we eventually get that an agent with group and
past match set (g, H) cannot profitably renegotiate to a higher level of cooperation if and only if
1
G
c ≥ [d(â) − v(â)]
γ(g, H)
δ
Since γ(g, H) ≤ G, there are no profitable renegotiate for any agent with any combination of group and
past match set if and only if
c≥
1
[d(â) − v(â)]
δ
So condition 2 for the existence of a segregated equilibrium is satisfied. Thus all of the conditions for the
existence of a segregated equilibrium are satisfied, and so a segregated equilibrium exists. Finally, lemma 3
shows that if a segregated equilibrium exists then the level of cooperation chosen must be āS (g, H, g ′ , {g ′ }) =
min{ā(γ(g, H)), ā(g ′ , H′ ). This completes the proof.
36