A Relational Theory of Status Hierarchy
Hiroki Takikawa1
1
Frontier Research Institute for Interdisciplinary Sciences(FRIS), International Advanced Research
and Education Organization, Tohoku University 27-1 Kawauchi, Aoba-ku, SENDAI 980-8576
∗
Corresponding author (takikawa@sal.tohoku.ac.jp, tel: +81-22-795-3147)
Abstract
The emergence of status hierarchy, which is defined as a social order that arranges individuals in
society from top to bottom, is one of the main themes in sociology. It has been widely observed
among young adolescents, task groups, gang groups, scientist groups, and firms in markets. Despite
the empirical importance of this phenomenon, the complete theoretical understanding on it has been
lacking. In recent years, R. Gould proposed a path breaking theory of status hierarchy. By using
game theory and social network theory, he showed that status hierarchy can be considered as a sort
of equilibrium when players assign attachments to all other players according to their qualities. The
major difficulty in Gould model, however, is the unwarranted assumption of the limitless resources
such as time and emotional costs which players have to pay in executing their attachment strategies.
Here we extend Gould model to be theoretically more coherent and empirically more valid: to
incorporate multidimensional choices for taking resource constraints into consideration. Unlike in
Gould model, in our model, a player chooses just one allocation strategy as a multidimensional
choice; that is, a player here has to decide attachment levels for all the other players at the same
time. Based on these assumptions, we obtain the following results.
(i) An individual status is totally determined by the individual ’s relative quality in the social
system. This observation can be contrasted with that of Gould model in which an individual status
is dyadically determined.
(ii) Our model is a genuine generalization of Gould model; that is, the results of Gould model can
1
be obtained under the specific conditions in our model.
(iii) Social influence can work in both of directions, that is, it can reinforce the existing inequality
or reduce the inequality depending on some parameters.
These results dramatically differ from Gould model because statuses are completely relationally
determined in our model; in other words, statuses fully depend on the configuration of the strategies
and attributes of all players in the whole system. Our model can be considered as a kind of network
formation model and has a wide applicability.
2
1
Introduction
How does the differentiation of social status emerge is one of the central questions in sociology. The
differentiation of status often take a hierarchical form which is called status hierarchy(Gould 2002;
Chase and Lindquist 2009). Many researchers study status hierarchy in a wide variety of social
fields such as gang groups(Whyte 1943), task groups(Bales 2001; Ridgeway and Berger 1986), young
adolescents(Savin-Williams 1987), scientist groups(Blau 1974) and even firms in markets(Podolny
1993; White 2002). In status hierarchy, each individual occupies the different statuses which are
organized hierarchically from top to bottom. While the individual taking the higher status usually
has the larger amount of resources or powers and enjoys a set of privileges , he or she also has to
undertake a set of obligations accompanying his or her high status. Status hierarchy is considered
to fulfill various collective functions such as leadership, coordination, the division of labor, and the
effective allocation of resources. However it also implies inequality among individuals, therefore
individuals with the lower statuses often suffer from various malady such as depression.
Each status has been considered, in a sociological tradition, to be generated by the act of deference
by other people(Ridgeway 1984). For example, in task groups, some members are more talkative or
active, then they are given deference by other members and come to occupy leadership positions. In
scientist groups, scientists tend to give respect to scholars who have accomplished excellent works,
then those scholars come to be more prominent and occupy the higher positions in the prestige
hierarchy in the academic communities. In sum, a status can be regarded as a kind of rewards for
contributions to a given society or groups which is made possible by underlying qualities possessed
by the persons who receive the deference. Therefore we will call this aspect the quality consideration.
If the quality consideration is all that can explain status differentiation, it might be considered
that a status might be a mere reflection of underlying natural ability or quality possessed by a person.
In other words, social status hierarchy might be considered to perfectly correspond to underlying
natural quality hierarchy.
However we sociologist also have viewed the emergence of status differentiation as a consequence
3
of the social process(Ridgeway and Erickson 2000). That implies that the mechanism determining
the form of status hierarchy can not be reduced completely to natural differences, and that social
forces do affect the form of status hierarchy. Sociologist I.Chase picks up two hypotheses in his
prominent paper(Chase 2002); one states that status hierarchy or dominance order is predetermined
by individual characteristics, which is called the ”prior attributes” hypothesis; the other claims that
social structure plays critical role in the formation of hierarchy, which is called ”social dynamics”
hypothesis. And then he examines which hypotheses are valid using animal experiments. His experimental results show that social dynamics hypothesis is more appropriate even in the formation of
dominance order among fishes.
What kind of social forces works behind the emergence of status hierarchy? While a wide variety
of candidates has been proposed by researchers, here we will mention only two among them. One
is related learning under uncertainty or social influence, and the other is concerning social norms of
reciprocity. Social influence is influence that comes form the other persons surrounding an individual,
say, i, and that affects the formation of i’s subjective judgement about the underlying quality of a
person who is a candidate of a recipient of i’s deference. The fact that certain person’s subjective
judgement are influenced by the judgement of other persons surrounding him or her is well established
as seen in the famous social psychological experiments(Asch 1956; Milgram 1974). Hence, Since
an individual status is (at least partially) determined by the perceived underlying quality of that
individual, the resulting status is affected by social influence because the judgement of qualities are
influenced by other persons’ judgement.
Social norms of reciprocity is an another factor having effects on the formation of status hierarchy. This perspective was stressed by Gould theory of status hierarchy which we will concentrate on
later(Gould 2002). Basically, the reciprocity consideration means the follows; when ego gives some
amount of respect(attachment) to alter, ego expects that alter should return the same amount of
respect(attachment) to alter; if this expectation of ego is not fulfilled, ego feels discomfort or unhappiness. This norm of reciprocity is considered to work as a counter balance force against the formation
4
of a ”meritcratic hierarchy” that the pure quality consideration might yield because it prevents the
most attractive persons from monopolizing all the other peoples attachement(This point will be fully
explained by using a formal model later).
To wrap up the above arguments, the formation of status hierarchy involves three distinct factors;
individual quality, social norms of reciprocity and social influence. The question is: how do these
factors shape the particular form of status hierarchy and what exact mechanisms work. Here we have
to go beyond informal arguments and construct a formal theory of status hierarchy. Unfortunately
there are few formal models on the formation of status hierarchy despite its importance. In these
circumstance, however, Gould theory of status hierarchy come to draw much attention recently. To
name a few, Lynn, Podolny and Tao(2009) build their theory of social construction of status based
on Gould model. Whitemeyer and Wittek (2010) also examine inequalities in network structures
using the mathematical model inspired by Gould model. Review articles on this topic devote a
considerable space for reviewing this theory(Chase and Lindquist 2009; Kadushin 2012). The formal
model proposed by Gould is so simple and tractable and at the same time provides the powerful
explanation of the emergence of status hierarchy by incorporating above mentioned three factors.
Nevertheless we consider Gould model has a critical limitation so that it cannot provide the firm
basis of analyzing status hierarchy. Specifically the model sets the unacceptable assumption of the
limitless resources such as time and emotional costs. The Goal of this paper is to extend Gould
model to be theoretically more coherent and empirically more valid; to incorporate multidimensional
choices for taking resource constraints into consideration.
Outline of our paper is as follows. Section 2 gives a critical examination of Gould model and
propose the alternative idea incorporating multidimensional choices. Section 3 and 4 formulate our
idea as a basic model and then show basic results. this will establish a relational theory of status
hierarchy. Section 5 shows our model can be interpreted as a generalization of Gould model. In
section 6 we will incorporate social influence mechanism into our model and then show that social
influence can either reinforce the existing inequality or reduce the inequality depending on some
5
parameters. Section 7 is conclusion.
2
A critical examination of Gould model
Gould’s theory of status hierarchy beautifully integrates the perspective of game theory and that of
network theory, thus combining the individualistic accounts (cf. Homans 1961; Coleman 1990) with
the structural accounts (cf. White 1970; Sørensen 1996) in theoretical tradition of status hierarchy.
His strategy is starting from the assumptions concerning individuals and then deriving structural
outcomes as a consequence of rational choices and interactions of individuals. We firstly describe
the assumption of individual behaviors in his model, and then explain how those behaviors yield the
structural outcomes.
There are two basic assumptions of individual behaviors in his theory. The one is that people
have a desire to attach other people with the attractive qualities. In the utility terms, as people give
attachment to the others with the higher qualities, they gain more utilities from these attachment
behaviors. Needless to say, this utility assumption derives from the above mentioned fact that people
tend to give deference to the others with high qualities. The another behavioral assumption is related
to the social norms of reciprocity. That means, people prefer reciprocal give-and-take in terms of
respect and attachment. if an individual does not receive the adequate amount of attachment from
the person to which he or she has given attachment, then he or she would feel uncomfortable from
this occurrence. More specifically, the difference between the amount of attachment that ego gives
alter and the amount of attachment that ego receives from alter affects ego’s utility; if the difference
is negative, ego gains negative utility and vice versa.
Here a rational person confronts a sort of trade off. Individual i were more satisfied if he or she
gives attachment to other’s with the higher qualities, all other things being equal. However if the
other person’s quality is much higher than i’s quality, he or she cannot expect to receive the sufficient
amount of reciprocal attachment form this person, and then would feel uncomfortable. Therefore i’s
strategic choice is determined by the balance between the quality consideration and the reciprocity
6
consideration.
Furthermore, this involves a kind of game situation, a situation in which each person’s outcome
depends on not only his or her own behavior but also other persons behavior(Easley and Kleinberg
2010). Here social norms of reciprocity imply the interdependency because the utility from the
reciprocity consideration depends on alter’s behavior of returning respect.
How can we derive the structural implication from these behavioral principles? Game theory
can be applied to these situations. Game theory predicts that when players act rationally, stable
states eventually come to appear in which all player involved have no incentive to deviate form the
strategies which they currently adopt, which are called Nash equilibria. This equilibrium concept
refers to a macro structural state. In the case which we are considering, the question is whether a
particular Nash equilibrium that the theory predicts would correspond to a macro state of status
hierarchy. And Gould theoretical contribution consists in his demonstrating that status hierarchy
can be modeled as a kind of Nash equilibrium based on the above behavioral assumptions. In order
to explain about this point, we have to take a look at the way to represent hierarchy in terms of
network theory.
Status hierarchy can be represented in terms of social network theory. Specifically, a weighted
directed graph is useful for depicting a form of hierarchy.
[Figure 1]
Here nodes represent players involved whereas directed edges denote the attachment of the players
whose directions represent the ”flow” of attachment. The weights given to nodes mean the amount of
the attachment. Thus we can define individual ranks in the hierarchy as the choice statuses: the sum
of the received attachment. It implies that, as players receive the more attachment from others, they
come to occupy the higher statuses. This idea connects the individual behaviors with the form of the
social structure because the configurations of the edges, that means, the structure of a given network,
7
results from the choice behaviors of all individuals. For this reason, in this framework, Game theory
and the concept of Nash equilibrium predicting the final distribution of individual choice behaviors
can be applied to the formation of status hierarchy.
As explained above, Gould model gives us a tremendous framework for analyzing the emergence
of status hierarchy. However this does not mean that his model is free from limitations. Especially
we consider that the serious limitation of Gould model consists in the dyadic model of tie formation.
We devote the remaining parts of this section to explaining this limitation.
The dyadic assumption which resides in Gould model is related to the attachment conferring
behavior. The dyadic model means here that the level of the attachment to one alter is determined
independently from the level of the attachment to other alters. Hence, the amount of the attachment
devoted to alter A poses no constraints to the amount of attachment devoted to alter B. In terms of
resources, there are no resource constraints such as time or emotional costs when people give attachments to others. However, this contradicts Gould’s own statement like ”......quality judgments are
informative to observers in part because they are costly to make. Positive comments, demonstrations
of attention, or expressions of interest reflect approval, thereby influencing opinion, if everyone knows
that they are not made lightly; and they will not be made lightly if those making them understand
them as forms of deference”(Gould 2002: 1149). In addition, this also lacks the empirical adequacy.
In sociological traditions, the decision as to whom and to what extent people devote their resources
in the face of constraints is regarded as one of the central problems confronted by the people in the
interpersonal environment. For example, the famous formulation of structural hole by Burt incorporates the idea of resource constraints(Burt 2004). Looking at the more mundane situation, if we
had almost infinite time and emotional energies, our distress about friend choice would be dissolved
in the most parts.
Theoretically speaking, as we observed before, this assumption of limitless resources implies or, we
should say, is required by the dyadic model. Once we take resources constraint into the consideration,
we cannot hold the dyadic assumption any more. For, if there are resource constraints, each decision
8
concerning the attachment level to each other person cannot be independent. If a person decides
to increase the amount of attachment to one person, then this implies he or she has to decrease
the amount of attachment to some other person at the same time. More generally, if we admit the
resource constraints, we cannot hold any more that there are strategies for every other players as
mutually independent assignments of attachment. Instead we should think there is only one strategy
as to how the total amount of attachment is allocated optimally for every other players. That is
why we call our model multidimensional choice model. Furthermore you may notice that a status
hierarchy game here change form a two player game to a multi-player game. We also call this a
relational theory of status hierarchy in contrast to original dyadic theory because here, as we will
see, individual actions and hence individual statuses are determined by not each dyadic pair and their
simple accumulation but the whole configuration of the all players behaviors in the social system.
Here is the illustration of the comparison between the dyadic model and the relational model.
[Figure 2]
Based on the above ideas, we will investigate the following question using a mathematical model.
1. Is there any status hierarchy equilibria with multidimensional choices which sociologically make
sense?
2. If so, what kind of status hierarchy emerge?
3. In what respects is our relational model superior to the original dyadic model?
3
Basic model
In this section we will explicate our model, which is an extension of Gould model in a relational way.
Let N describe all players in a closed social system. In this model, each player can reach any other
players, that is, give/receive attachment to/from any others. Each player has one sort of attribute
9
which is called quality. quality might mean a talent, ability, a good character and any attribute that
attracts other players. We denote individual j’s quality as qj .
Each player’s strategy is concerning attachment assignments. Our formulation of attachment
differs from Gould’s formulation. In Gould model, attachment strategies for every other players
are determined independently whereas, in our model, a player has to determine one attachment
allocation strategy at one time. In other words, ai1 , ai2 , ...ain represents an attachment weight of
∑
player 1, 2, ...n as a partner. Because aij is an weight, j6=i aij = 1. Note that ai is a vector not
scalar. Thus, A strategy in our model is multidimensional, which is completely different from the
original formulation.
Concerning utility assumption, we largely follow the Gould formulation. As explained before,
utility sources consist in two parts: one is related to the quality consideration and the other is
related to the reciprocity consideration. In the first part, player i’s utility comes from player j’s
quality which player i’s attaches to. Furthermore we assume i’utility linearly increases as j’s quality
increases and its amount is multiplied by attachment weight aik . In the second part, i’s utility
depends on the gap between i’s attachment to j and j’s attachment to i. Because player i feels
uncomfortable when j has not reciprocated i sufficiently, the sign of the gap, aij − aji , is negative.
Furthermore we add two trick according to Gould’s formulation. One is that we multiply the gap,
aij − aji , by aij . That means, as the attachment weight of player j as a partner is higher, then the
gap between aij and aji matters more. The other trick is that those quantities are multiplied by the
symmetry parameter s which represents the weight of symmetry considerations or the strength of
the reciprocity norms.
Taking the above all into the consideration, the utility function of player i is as follows:
ui =
∑
aij qj
−
s
j6=i
| {z }
utility from the quality consideration
|
∑
j6=i
aij (aij − aji )
{z
.
(1)
}
(dis)utility from the reciprocity consideration
Hence our rational player i tries to maximize this utility function choosing the certain attach10
ment allocation strategy, a∗ = (a11 , ...a1j , ..., a1n ). In other words, The problem here is the utility
maximization problem under the resource constraint: that is,
max ui subject to
∑
aij = 1.
(2)
j6=i
4
Results of the basic model
Under the assumption of the existence of the interior solution, we obtain the following solution.
a∗ = (a∗i1 , ..., a∗ij , ..., a∗in )
= (..., a∗ij =
(3)
∑
(n − 2)qj − k6=i,j qk
1
+
, ...)
n−1
(2n − 3)s
(4)
Furthermore, we obtain a status ranking score which can be calculated as the summation of
received attachments:
∑
(n − 2){(n − 1)qj −
aki = 1 +
a.i =
(2n − 3)s
k6=i
∑
k6=i qk }
(5)
These formulas appear a little complicated at a first glance. However, these have clear sociological
interpretations; more specifically, these capture relational aspects of social status.
In order to make it concrete, we explain the points by using numerical example. Set the values
of the parameters in the following way: N = 4, s = 6, q1 = 1, q2 = 2, q3 = 3, q4 = 4. Then using the
formula(4), Nash solution can be calculated and Nash equilibrium be represented in the following
adjacency matrix form.



0 a∗12 a∗13 a∗14
0 7 10 13

a∗21 0 a∗23 a∗24 

 = 1 5 0 11 14
 ∗
∗
∗
a31 a32 0 a34  30 6 9 0 15
a∗41 a∗42 a∗43 0
7 10 13 0

Focusing on the attachment from 1 to 2, it can be rewritten in the following:
11
2’s relative quality
a∗12
z
}|
{
1 q2 − q3 + q2 − q4
=
+
3
30
1 2−3+2−4
+
=
3
30
(6)
(7)
The first term in the right-hand side can be interpreted as a base line, which would come to be
the attachment weight of player 2 as a partner if player 1’s total attachment were allocated equally
to every other players. The second term represents ”player 2’s relative quality” because it includes
the quality difference between player 2 and player 3 and the difference between player 2 and player
4. Therefore we could say that the amount of attachment received by player 2 is determined by
not only player 2’s absolute quality but also every other player’s qualities in this social system; in
other words player 2’s status is relationally determined. That is why we would like to call our theory
a relational theory of status hierarchy. Additionally, because the denominator of the second term
includes the symmetry parameter s, we could argue along with the original Gould theory that the
quality consideration is tempered by the reciprocity consideration. To put it simply, as the reciprocity
consideration is stronger, the difference between each player’s received attachment become smaller,
and then the total inequality become more reduced.
The comparison of our model with Gould model helps us to understand the relational theoretical
aspects of our theory.
The Nash Solution in Gould model is:
a∗ij =
qi + 2qj
3s
(8)
It can be observed that each strategy is only dyadically determined in Gould model because the
solution contains only quality i and j who directly interact with each other. On the other hand, as
we saw before, our formula contains everyone else’s quality except for i, that means, i’s status is not
determined dyadically but determined by every other persons’ qualities in the whole system. The
implication is that, in the relational theory the qualities of players, say, k and l, who do not directly
12
involve the dyadic relation between i and j do affect the allocation of attachment for j by i, while in
Gould model, the attachment level from i to j is completely independent of who exists in the social
system except for directly interacting persons i and j.
[Figure 3]
5
model II as a generalization of Gould model
In this section, we will show that our model with a minor modification can be a genuine generalization
of Gould model. To put it differently, Gould model is able to cover only rather specific situations
according to our theory.
The idea of how to modify our model is simple: all we have to do is to allow self-attachment in
our model. We further assume that self-attachment brings no utilities so that it can be interpreted as
withholding attachment to the alters. This withdrawal happens when one cannot accrue the positive
utility from the alters by giving any more attachment to the alters. One would withhold attachment
to the alters and devote the remaining attachment weight to oneself as far as self attachment is
allowed. However, this means that there are no resource constraints in the substantive sense because
the total attachment is not wiped out here.
[Figure 4]
We will call the model allowing self-attachment the model II( whereas our first model is called
model I when the distinction is needed). It can be easily shown that model II has a Nash solution as
the same as that of Gould model except that attachment here is interpreted as a weight, not a raw
quantity. The important thing is that this holds only when the model II has the interior solution.
13
When the model II does not have the interior solution, thus being introduced resource constraints,
this setting can be appropriately analyzed by model I which can cover the situations with resource
constraints. That means, our model is a generalization of Gould model.
What parameters determine the condition in that each model can be applied? A quick look at
the solutions reveals that the key parameter is the symmetry parameter s. When the reciprocity
consideration is too high, each player prefers self-attachment to giving attachment to the alters. In
this social condition, original Gould model holds because the condition is satisfied that the interior
solution exists in the model II. When the strength of the reciprocity consideration is moderate,
each player prefers conferring attachment to the outer alters, thus having to consider an attachment
allocation for every other players. In this case, the model I can be applied. When the symmetry
parameter is too low, the condition that the interior solution exists in the model I no longer holds,
and then the solution comes to be the corner solution in which some player gives no attachment to
some other players.
[Figure 5]
It can be observed here that Gould model can be applied only rather special ”asocial” case in
which people withdraw the part of their attachment and devote it to themselves. Our model naturally
includes this case but can be extended to the more general case in which the allocation of attachment
matters. Therefore, we could conclude that our model has more general applicability than Gould
model.
6
Social influence model
So far we have considered the basic model which does not incorporate social influence. However,
one of the goal of Gould model is to explicate the effect of social influence on the formulation of
status hierarchy, as we saw in the first and second section. In this section, we will incorporate
14
social influence into our model and then show that Gould’s claim that social influence amplifies the
inequality in the hierarchy does not always hold.
As in Gould social influence model, we replace the objective quality parameter qi with the perceived quality Qi which is formulated as follows:
Qi =
(1 − ω)qi
| {z }
+ αω
the exogenous quality
|
∑
k6=j
akj (α = q̄)
{z
(9)
}
the endogenous quality
Perceived quality is divided into two parts: the exogenous quality and the endogenous quality.
The exogenous quality means the objective i’s quality correctly perceived by other actors while the
endogenous quality reflects other person’s evaluations which a player trying to assess i’s quality draws
on as a reference. According to this formula, as more other people give more attachment to i, one
gives more positive evaluations on i’s quality. ω represents the weights on the endogenous quality
compared to the exogenous quality. Larger ω means the greater effects of social influence in the
system. What is different from Gould model here is the existence of parameter α. This functions as
a standardizing parameter. We have to correct the balance between q and a by multiplying a by α
because a here means attachment weights, not a raw quantities as in Gould model.
We can calculate the solution of the game with social influence in the exactly same way as before.
Here we will show only the case with N = 4.
∑
1 (1 − ω)(2qj − k6=i,j qk )
aij = +
3
5s − 5αω
(10)
Obviously this is more simplified than the original formula seen below.
a∗ij =
P
sω(3s+2ω) k Qk
]
(s+ω)[3s−(3n−5)ω]
(n − 2)ω 2
(1 − ω)[(2s + ω)Qi + sQi +
3s2 − (2n − 7)sω −
(11)
Our formula implies that whether social influence amplifies the differences depends on how
strongly reciprocity works. More specifically, if s < α, social influence amplifies the natural differences between persons, while if s > α, social influence does reduce the inequality in the resulting
15
hierarchy compared with the condition without social influence. That makes sense sociologically
because the reinforced perception caused by social influence would force players to withdraw attachment more than before if the reciprocity consideration is strong enough compared to the quality
consideration. Therefore it can be concluded that our model is able to capture the reality of social
influence more than the original mode.
7
Conclusion
We’d like to conclude our paper with two remarks. First we proposed a relational theory of status
hierarchy. This theory states that Each person’s status is relationally determined, that is, a social
status is not only determined by not only his attribute but also all other’s behaviors and their
attributes in the social system or social ’field’. Secondly, we generalized and refined the original
Gould’s findings. Here we showed the original Gould formula is a special case of our model which
includes self-attachment. In addition to this, we replicated and refined the proposition on social
influence mechanism.
We hope we have shown sufficiently that our model is theoretically more coherent and attractive
than Gould model and we believe it turns out to be empirically plausible and useful. Concrete
empirical applications would be our future task.
16
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18
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