Cooperation- based Clustering for Profit- maximizing Organizational
Design
Nghia Tran, Christophe Giraud- Carrier, Kevin Seppi, and Sean Warnick, Member, IEEE
Abstract— This paper shows how the notion of value of
cooperation, a measure of the percentage of a firm’s profits
due strictly to the cooperative effects among the goods it sells,
can be used to analyze the relative economic advantage afforded
by various organizational structures. The value of cooperation
is computed from transactions data by solving a regression
problem to fit the parameters of the consumer demand function,
and then simulating the resulting profit- maximizing dynamic
system under various organizational structures. A hierarchical
agglomerative clustering algorithm can be applied to reveal the
optimal organizational substructure.
I. INTRODUCTION
Analyzing the impact of organizational structure on the
performance of profit- maximizing organizations is a difficult
task for business managers. Yet, informed design decisions
are essential to long- term profitability. While these decisions
are often ad hoc, today’s large volumes of data make
more systematic analyses possible. The key concept in such
analyses is that of the value of cooperation ( VC) experienced
by a firm, which captures the percentage of a firm’s profits
due strictly to the cooperative effects among the goods it
sells [ 1].
Such a measure provides the scientific backing for sound
organizational design decisions. For example, if a firm can
identify which of its products have strong synergies with
others, it can organize to ensure that decision makers of
related products work closely together. This may include
physically co- locating entities where interaction adds strong
value to the organization, or it may result in decentralization
when cooperation adds little value. Similarly, if a firm
identifies pieces of its business that add little cooperative
benefit to the organization as a whole, it may consider selling
off these subunits. A subunit with a healthy balance sheet
may sell for a high price without adversely affecting the
firm’s value of cooperation. On the other hand, the firm may
pursue a different strategy of retaining its decoupled subunits
but use the value of cooperation to identify an acquisition that
strongly couples their mutual benefit. Thus, divisions of a
firm that are quite independent may be cooperatively coupled
through the acquisition of another business with the right
cooperative effects. For example, a firm with two distinct
independent divisions that have no value of cooperation may
acquire another business unit that not only adds value of
cooperation with each division, but does so in a way that the
total business becomes tightly integrated. Moreover, the firm
All authors are with the Department of Computer Science, Brigham
Young University, Provo, UT 84602, USA ( phone: 801- 422- 3027; fax: 801-
422- 0169; email: { tcnghia, cgc, kseppi, sean}@ cs. byu. edu).
may identify an acquisition candidate that is struggling on its
own, and thus is inexpensive, but brings the right cooperative
effects to the organization to offset the risk of acquiring a
struggling business. The value of cooperation thus becomes
the lens through which a firm can better identify valuable
opportunities in the market environment, or costly “ baggage”
in its own organizational structure.
In this paper, we discuss a value of cooperation inspired by
the theory of industrial organization and coalition games [ 2],
[ 3], [ 4], [ 5], [ 6]. Essentially, the profit maximizing dynamics
of a given organizational structure define the value function
for a particular coalition game. Researchers in [ 7] describe
how the impact of a proposed merger can be computed by
evaluating the post- merger equilibrium prices. They consider
common functional forms of demand functions and indicate
how to conduct the merger simulation in each case. Our
value of cooperation is computed through a kind of “ reverse”
merger simulation that explores the impact of splitting the
firm into its constituent economic units to determine the
value it is realizing by unifying the objectives of these
basic units. We are then able to use this measure to drive
a hierarchical agglomerative clustering algorithm in order to
reveal the organizational substructure defined naturally from
the cooperative effects of the global product network.
II. THE VALUE OF COOPERATION
Consider a market, M, selling N products. A firm, F, is
a subset of the N products in the market, F ? 2M. This
implies that the firm controls the production and distribution
of the products assigned to it. Furthermore, we consider a
Bertrand market model, so that each firm may set the prices
of the products assigned to it. We assume that the products
of the market are partitioned between m firms ( i. e., no two
firms control the same product) and that every firm in the
market is a profit maximizing entity. Under the standard
assumption that market dynamics are stabilizing, we expect
price perturbations to re- equilibriate, which means that we
may simply consider the profits of firms at equilibrium. These
profits, in turn, define a payoff function reminiscent of those
used to define coalition games.
Let v( Fi) = pi| p= peq
be the payoff or profit of firm i at
the market equilibrium prices peq. In this way the firm may
be thought of as a coalition of ni players in an N- player
cooperative game. Each player is a one- product company that
completely manages the production, distribution, and pricing
decisions for its product. The firm, then, is a confederacy of
these one- product companies that work together to maximize
their combined profits or payoffs.
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The theory of coalition games studies the behavior of
such coalitions once the payoff function is defined for every
possible coalition. The idea is that any given coalition Fi
yields a well- defined payoff v( Fi), and then a number of
questions can be explored regarding how to distribute the
payoff among the members of the coalition, etc.
Our situation is different, however, because the payoff to a
given firm does not depend only on the products it controls,
but also on the market structure of the products outside the
firm. For example, consider a 10- product market and a three
product firm in the market. The payoff to the firm does not
depend only on the prices of the three products it controls,
but also on the prices of the other seven products. The profit-maximizing
equilibrium prices of these other seven products,
however, may be set differently depending on whether they
belong to a single firm or whether they are controlled by
seven different firms. Thus, the payoff to the three- product
firm depends on the entire market structure.
Coalition game theory addresses such situations by con-sidering
partition systems and restricted games. For our
purposes, it is sufficient to partition the N products ofMinto
m firms and assume that this structure is fixed outside of the
particular firm that we are studying. This enables us to work
with a well- defined payoff function induced by the profit-maximizing
dynamics of firms within the market without
eliminating the multiple- coalition ( i. e., multiple firm) cases
of interest.
To quantify the value of organizing a group of one-product
companies into a single firm, we need to compare
the profits the firm receives if it sets its prices as if each of its
products were independent companies with those it realizes
by fully capitalizing on cooperation between the products.
More precisely, let peq be the profit- maximizing equilibrium
prices for the given market structure. In contrast, consider the
new profit maximizing equilibrium prices achieved without
cooperation if Fi were divided into its constituent one-product
companies and each independently optimized their
prices. Let this second set of equilibrium prices serve as a
basis for comparison, or reference, and be denoted pref. We
can then define the following measure.
Definition 1: The Value of Cooperation ( VC) of a firm Fi
in marketMwith structure S = F1, F2, . . ., Fm is given by:
VCref ( Fi, S) = pi| peq - pi| pref
This VC measure captures precisely the value realized by
the firm due to cooperation within its organization. Note that
the VC measure is always non- negative since the cooperating
firm can always recover at least the non- cooperating, or
reference, profits by simply setting the prices it controls in
peq to those of pref .
As defined, the VC measure carries units of dollars and
reflects a kind of absolute dollar- value of cooperation within
the firm, thus making comparisons difficult. We, therefore,
define a “ relative” Value of Cooperation by normalizing
VCref by the equilibrium profits as follows.
Definition 2: The Relative Value of Cooperation ( RVC) of
a firm Fi in market M with structure S = F1, F2, . . . , Fm
is given by:
RVCref ( Fi, S) =
pi| peq - pi| pref
pi| peq
This RVC measure is naturally interpreted as the percent-age
of profits due to cooperation within the organization.
It is bounded between zero and one, and enables direct
comparison among firms of different sizes.
By simply replacing the equilibrium and reference prices
in the above definitions with the equilibriated profit-maximizing
prices of the market structures being compared,
one can easily use the thus modified VC and RVC to ana-lyze
the relative values of different organizational structures
within a single firm. This is a natural application of the above
framework, where the market is a firm and the firms are its
organizational divisions.
III. A SIMPLE EXAMPLE
To illustrate the use of VC and RVC, let us consider a
simple one- firm market with only two products with linear
demand given by:
Suppose that the unit production costs of each product are
c1 = c2 = 10. If we consider a firm structure where each
product is produced by an independent division, the profit
function for each division becomes:
p1( t) = q1( t) ( p1( t) - c1)
= - 3.5p21
- p1p2 + 135p1 + 10p2 - 1000 ( 2)
p2( t) = q2( t) ( p2( t) - c2)
= - 2p22
- 3p1p2 + 30p1 + 120p2 - 1000 ( 3)
Taking the partial derivatives of each profit function with
respect to the appropriate pricing variable, we get the profit-maximizing
market dynamics:
and associated equilibriated profits: p1 = 161.84 and p2 =
109.52.
Now, consider a firm structure where both products are
controlled by the same division. In this case, the firm’s profit
function becomes:
p( t) = q1( t) ( p1( t) - c1) + q2( t) ( p2( t) - c2)
= - 3.5p21
+ 165p1 - 4p1p2 + 130p2 - 2p22
- 2000
( 5)
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With this structure, the division adjusts the prices of both
products to optimize the same objective. The new dynamics
become:
These single- division dynamics drive an initial pricing
vector to the following profit- maximizing equilibrium:
and associated equilibriated profit: peq = 316.667.
The values of cooperation in this example thus become:
V C = peq-( p1+ p2) = 45.3067 and RV C = 0.1431. These
suggest that in this market, just under 15% of the profits of
the two- product firm are the direct result of its inter- division
cooperation.
IV. VC- BASED CLUSTERING
Equipped with the value of cooperation as a measure of
productive interaction between products, a firm may consider
its internal organization and cluster its products to maximize
the value of cooperation, indicating which decision makers
in the organization should work together in setting prices,
and which can work more independently. Knowing where
cooperative value resides within an organization is the critical
first step in exploiting it.
We begin by describing how one may compute a firm’s
value of cooperation from data. The algorithmic process
involves the following three steps:
1) Fit demand model from transactions data. This can
often be accomplished through standard regression
techniques. To obtain good estimates of the model
parameters, however, it is important that the data be
sufficiently “ exciting.” This can often be a challenge if
prices have remained relatively constant, or have only
been changed in very structured ways ( e. g., 20% off
everything sales).
2) Build corresponding profit- maximizing dynamic sys-tem.
This step involves constructing the firm’s profit
function for each product in its offering. Importantly,
this assumes knowledge of the total handling costs
associated with selling each product, which informa-tion
may often have to be estimated at best. Moreover,
our initial results assume that this cost structure does
not change with organization structure, i. e., although
economies of scale associated with selling more of a
product can be easily accommodated, no cost benefit
across product lines is built into the existing model. 1
3) Compute equilibrium for given organizational struc-tures.
With different assumptions about which prod-ucts
group together, the profit maximizing dynamics
1This limitation is an important focus of future work since synergistic
costs can play as important a role in total cooperation as synergistic sales.
Nevertheless, the current work demonstrating effects of synergistic sales
demonstrates the key idea underlying cooperation- based analysis.
are altered yielding a new equilibrium point for the
system. These equilibrium profits generate a new set
of equilibrium profits, and it is through a comparison
of equilibrium profits that the value of cooperation is
calculated. Note that depending on the size of the prob-lem
or the strength of nonlinearities in the dynamics,
these equilibria may often be computed analytically, or
through simulation.
A natural extension of the above algorithm would be to
automate step 3 in such way that the organizational structure
resulting in the largest value of cooperation is found. In other
words, one is interested in finding the k clusters of n products
for which the total value of cooperation is maximized for all
k from 1 to n. We propose to do this through the use of
the value of cooperation within a hierarchical agglomerative
clustering framework [ 8], [ 9].
VC- based hierarchical agglomerative clustering starts from
the reference structure, where all products are assumed to
act independently. Then, the two products which exhibit the
strongest value of cooperation are merged, and the process
is repeated, decreasing the total number of clusters by one
until all of the firm’s products are finally merged into a
single organizational structure. The hierarchical nesting adds
a natural constraint to the problem, which yields a product
hierarchy with a clear organizational interpretation.
The real novelty of our approach comes from its use
of the profit- maximizing dynamics to define the resulting
clusters. A typical approach to economic- driven clustering
may compute the same demand function from data, expand
it in a Taylor series around the market equilibrium, and
then consider the first order terms as defining a graph over
products, i. e., the product network. Various approaches to
clustering this graph might then be considered. The approach
discussed here, however, is a radical departure from such
approaches by using the demand function to characterize a
dynamic system, and then allowing this dynamic system to
define clusters over the product network.
V. EXPERIMENTS
The use of VC- based hierarchical agglomerative cluster-ing
is best highlighted through an example. In this simple
example, a firm managing 15 products is considered. The
demand function is taken to have the form:
q = Ap + B
where q ? Rn, p ? Rn, and qi, pi = 0 for i = 1, ..., n.
This linear structure may have been fit directly from data, or
it may be the result of linearizing another demand function
around a nominal set of prices.
Considering the reference structure where every product
sets its price independently to maximize its own profit, each
constituent product system has a profit function given by:
pi = qi( p)( pi - ci)
where ci ? R+ is the marginal cost of the ith product. Note
that a fixed cost could be added to the expression without
affecting the results.
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Figure 1 illustrates the matrix A in the demand function,
both in tabular and graphical form.
0 2 4 6 8 10 12 14 16
0
2
4
6
8
10
12
14
16
Fig. 1. The A Matrix of the Demand Function
The vector B in the demand function is given by:
B = [ 1130 330 330 1130 1030...
1030 1930 330 2130 1125...
2130 1930 - 150 300 330] T
and the cost vector is given by:
C = [ 110 130 130 110 110...
110 110 130 110 120...
110 110 120 120 130] T
Note that the relative strength of the own- price elasticities of
various products is visible in the strongly diagonal structure
of matrix A. In spite of this feature, Figure 2 demonstrates
that over 50% of the firm’s profits result strictly from the
cooperative effects between products.
The relative value of cooperation increases sharply as
the first few products are grouped into their respective
clusters. Once a critical clustering is achieved, however, no
improvement in the value of cooperation is observed through
subsequent centralization. This indicates that these sets of
products are fairly independent, decoupled with respect to
the market demand function. Cooperation- based clustering
identifies these groups, even when they are not apparent from
the market demand function directly.
The result of the cooperation- based clustering is shown in
Figure 3. Each row indicates a set of clusters, beginning with
15 single- product clusters and ending with a single cluster
of all the products. An interesting aspect of this clustering
is apparent in the analysis of this figure with the RVC plot.
We note that once the products have been grouped into five
0 5 10 15
0
10
20
30
40
50
60
Relative Value of Cooperation
Percetage
Number of Clusters
Fig. 2. RVC vs. Product Groupings
clusters, no more value of cooperation is derived through
further clustering. This indicates that the firm is operating
at the intersection of five rather independent markets, a fact
that is not readily apparent from inspection of the demand
function ( see Figure 1).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 11 12 15 14
1 2 3 4 5 6 7 8 9 11 12 15
1 2 3 4 5 6 7 9 11 12
1 3 4 5 6 7 12 9 11
1 11 4 5 6 7 12 9
1 4 5 6 12 9
1 4 5 6
1 4 5 6
4 6
Products
Fig. 3. CV- based Hierarchical Agglomerative Clustering of a 15 Product
Firm
VI. CONCLUSION
This paper explores the notion of value of cooperation,
which measures the percentage of a firm’s profits due strictly
to the cooperative effects among the goods it sells. The idea
is to assume profit- maximizing dynamics among firms within
the market and compare equilibrium profits in two different
scenarios. The first scenario considers the firm as it is, a
single economic entity with a unified objective and exhibiting
full cooperation among its various products. The second
scenario considers splitting the firm into its constituent
economic units and computing market equilibrium prices if
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these units were to fail to cooperate and acted completely
independently out of self interest. The difference between
the cooperative profits of the first scenario and the aggregate
profits of the independent units of the second scenario defines
what we call the value of cooperation of the firm in its current
market environment. We have illustrated the use of the value
of cooperation for product clustering in the context of optimal
organizational design.
It is instructive to contrast the value of cooperation with
other measures used to characterize cooperative games. Hart
and Mas- Colell defined a measure, called the potential, P,
that computes the expected normalized worth of the game,
i. e., the per- capita potential, P/ N, equals the average per-capita
worth ( 1/ m) i ( pi)/(| Fi|). Given a market structure,
this measure characterizes the expected profit of an average-sized
firm ( where size is measured with respect to the number
of products the firm controls) in the market, even if such a
firm does not actually exist. Moreover, the potential has been
connected to another measure, called the Shapley value, Fj ,
which yields the marginal contribution of each product in
the market. This measure characterizes how the payoff of a
coalition should be divided between members of the team. In
both cases, the potential and Shapley values do not suggest
anything about the intrinsic benefit of forming coalitions in
the first place. The value of cooperation, on the other hand,
captures the natural significance for organizing production
into multi- product firms. Nevertheless, these measures do not
yield any information about how the profit of a firm should
be efficiently invested into each of the firm’s constituent
production lines. Thus, the measures are inherently different
from the potential or Shapley value of the coalition game
that focus more on the value of a member of a coalition to
the group rather than the value of the coalition as a whole.
Future work will concretely establish the relationship
between the value of cooperation and market power and
indicate how to compute approximations to these metrics
from readily available market data.
ACKNOWLEDGMENT
We would like to thank David Sims for his thoughtful
discussions on the nature of economic systems and the
meaning of cooperative value.
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