We consider a situation in which members of an oligopoly have different technologies, which allow them to produce at different costs. Members may license their technology to other members. Using the Aumann-Drèze modification of the Shapley value, we compute fair prices for these licenses. We also study the problem of stability for these ``licensing coalitions.''Cooperative Game Theory, Technology Transfers, Modified Shapley Value

Fabrice Valognes

Guillermo Owen

Hélène Ferrer

English

Research Papers in Economics

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Endogenous Coalitions Formations 
Through Technology Transfers 
and Fair Prices 
Fabrice Valognes, 
Hélène Ferrer,
University of Caen Basse-Normandie, CREM-CNRS, UMR 6211
Guillermo Owen
Dept. of Mathematics, Naval Postgraduate School, Monterey, USA
April 2011 - WP 2011-09
Centre de Recherche en Économie et Management
Center for Research in Economics and ManagementENDOGENOUS COALITIONS FORMATIONS THROUGH
TECHNOLOGY TRANSFERS AND FAIR PRICES
H EL ENE FERRER, GUILLERMO OWEN*, AND FABRICE VALOGNES
Abstract. We consider a situation in which members of an oligopoly
have dierent technologies, which allow them to produce at dierent
costs. Members may license their technology to other members. Using
the Aumann-Dr eze modication of the Shapley value, we compute fair
prices for these licenses. We also study the problem of stability for these
\licensing coalitions."
1. Introduction
The objective of this paper is to provide a model of coalition formation
where rms have technologies that are perfectly replicable. Hence, within
an asymmetric oligopoly, we are interested in technology transfers between
rms. The protability of technology transfers is a two-edged sword. Faced
by a possible technology transfer between any of its competitors and result-
ing loss of its prot, a rm will try to disrupt such transfer by oering a
more attractive deal to either rm
1.
Even though endogenous coalition formation is a central theme in coop-
erative game theory, much of this literature is of limited applicability to the
study of oligopoly markets since the standard denition of the character-
istic function ignores externalities among coalitions
2. In these games, it is
implicitly assumed that the payos levels members of a coalition can attain,
are independent of the actions chosen by the players outside this coalition:
the characteristic function gives the same value irrespective of how the other
players are partitioned. By contrast, in this paper, the model explicitly de-
scribes a procedure in which individual players, when deciding to form a
coalition, consider the consequences of their actions on the behavior of the
other players.
*Corresponding author.
1See for example, La Manna [1993].
2Greenberg [1995] emphasizes that the characteristic function ignores the possibility of
externalities.
1FERRER, OWEN AND VALOGNES 2
In order to model this cooperative behavior, we shall determine the gains
each coalition can obtain by the cooperation of its members and according
to the behavioral assumption that every possible coalition of rms makes
about rms that are not in the coalition. This is done by dening two
dierent games in characteristic function form, as in Hart and Kurz [1983],
each game corresponding to a dierent assumption about how the opponents
of a particular coalition will respond. However, unlike Hart and Kurz, we do
not assume that players evaluate the payos they receive in each coalition
structure according to an extension to the Shapley Value, rst analyzed by
Owen [1977]. We rather assume that each coalition only gets its own worth,
as in Aumann and Maschler [1964], Aumann and Dr eze [1975] and Shenoy
[1979].
For our purpose, we consider here an oligopoly, producing as homoge-
neous product. Dierent producers have dierent technologies, giving rise
to dierent cost structures. The more ecient producers may (for a price)
license their technology to others. If this happens, the purchaser will then
have the same cost structure as the seller. The question is as to what price
one or another of the rms may pay for these licenses.
There are two possible models to consider here:
(1) Several rms form a coalition. They then act as though they were
a single rm, sharing the most ecient (least-cost) technology, and
pooling their prots. In other words, we have here a merger.
(2) Several rms form a coalition. In fact, however, all they do is share
technologies. Each one of them now uses the most ecient tech-
nology, but otherwise acts independently of the other rms in the
coalition. The problem is then to determine the price the several
rms must pay for the use of this technology.
While interpretation 1 above can reasonably be made, this is precisely
the sort of behavior that anti-trust laws forbid. It is in fact possible that
some mergers might be allowed, but mergers are complicated procedures,
requiring all sorts of approval (from the shareholders? from the authorities?)
and thus, even if approval is ultimately granted, the game would have to
be modied to take into account the one-time costs of the merger. We
will therefore study the situation of coalition formation where rms adopt
a competitive behavior inside the coalition.FERRER, OWEN AND VALOGNES 3
2. The Model
To simplify matters, we will assume both the demand and the cost func-
tions are linear. By changing the units if necessary, we can assume that the
demand function is given by
(1) p = A  
X
i
qi
where p is price, and qi is the amount produced by rm i.
Assume next that each rm has a linear cost structure, and that rm i has
unit costs ci. Without loss of generality, we will assume that the producers
are listed in order of eciency, with the most ecient rms listed rst, i.e.,
c1  c2  :::  cn
Then, for a Cournot-Nash equilibrium [1927, 1950], the general rule is
that (at equilibrium) rm i will produce a quantity
(2) qi =
A +
P
j cj   (n + 1)ci
n + 1
(though admittedly some care should be taken to consider the case that
some of the rms { the least ecient ones, i.e., those with highest costs {
will produce nothing at all3).
Let us suppose a coalition S forms. The members of S will then all use
the most ecient technology available to them; i.e., all will produce with
unit costs
(3) cmin = minfci j i 2 Sg
Each member of S will then produce the quantity
(4) qi =
A +
P
j cj   (n + 1)cmin
n + 1
Total production for a coalition S will then be
(5) Q(S) =
s

A +
P
j cj   (n + 1)cmin

n + 1
3The rule is that, if the qn given by equation (2) is negative, then rm n is eectively
driven out of the market. The rm will then produce 0, and the quantities given by (2)
must be recalculated, replacing n by n   1, etc.FERRER, OWEN AND VALOGNES 4
where s is the cardinality of S. We note, however, that in the summation
term,
P
j cj , each rm j's cost cj should be replaced by the lowest ck
among all k which belong to the same coalition as j does. Thus coalition
S's production, and also its prots, will depend on the behavior of rms
outside S. We therefore distinguish two special cases. Case 1 assumes that
the rms outside S all act together, and case 2 assumes that they all act
independently.
We choose to consider case 1 only: if coalition S forms, the complementary
coalition, N-S, also forms. Here we consider the minimum a coalition S can
guarantee that is we consider the worth possible scenario for the coalition S
whatever the behavior of the N-S rms.
In that case, the s rms in our coalition will all be producing with the
same unit costs, c = cmin, while the other n-s rms will all be producing
with unit costs k, given by
(6) k = minfcj j j 2 (N   S)g
i.e., the lowest unit costs among the members of N-S. Recall, now, that
they are all producing independently.
Then, by the above, each member of S would be producing
(7) qi =
A + (n   s)k   (n   s + 1)c
n + 1
while each member of N-S would produce
(8) qj =
A + sc   (s + 1)k
n + 1
(assuming of course that these quantities are non-negative). Thus total
production would be
(9) Q = sqi + (n   s)qj =
nA   (n   s)k   sc
n + 1
The price would then be
(10) p = A   Q =
A + (n   s)k + sc
n + 1
and prots, per unit, to each member rm of S would beFERRER, OWEN AND VALOGNES 5
(11) i = p   c =

A + (n   s)k   (n   s + 1)c
n + 1
2
(which turns out to be the same as qi), and so prots to a single player i
are qi
2, and total prots to the coalition S will be
(12) v(S) = sq2
i =
s[A + (n   s)k   (n   s + 1)c]
2
(n + 1)
2
We can now treat this as the characteristic function of our (semi-coope-
rative) game.
3. The Results
We consider a 4-person game, with A = 10, and ci = 0, 1, 2, and 3
respectively. If no cooperation is allowed, then we nd
q1 =
10 + 6   5(0)
5
= 3:2
Similarly, q2 = 2.2, q3 = 1.2, and q4 = 0.2. Total production is then Q
= 6.8, so p = 3.2, and the several players' prots are 10.24, 4.84, 1.44, and
0.04 respectively. We note parenthetically that rm 4 is essentially moribund
because of its high costs, and really needs some technological help if it is to
survive.
Suppose, now, that some cooperation is allowed, in the form of technology
sharing. In that case, we nd that, if the coalitions f1g and f2, 3, 4g form,
then 2, 3, and 4 all produce with unit cost 1, so
q1 =
10 + 3   5(0)
5
= 2:6
q2 =
10 + 3   5(1)
5
= 1:6
And q3 = q4 = 1.6. Total production is Q = 7.4, so p = 2.6, and prots
are 6.76, 2.56, 2.56, and 2.56. Thus v(f1g) = 6.76, v(f2, 3, 4g) = 7.68.
In a similar way, we compute v(S) for the remaining coalitions, to obtain
the characteristic function:
In each case, Q(S) denotes the production by S assuming the structure
fS, N-Sg forms.FERRER, OWEN AND VALOGNES 6
Table 1
S Q(S) p v(S) S Q(S) p v(S)
; 0 0 f2, 3g 2.8 2.4 3.92
f1g 2.6 2.6 6.76 f2, 4g 2.8 2.4 3.92
f2g 1.2 2.2 1.44 f3, 4g 1.6 2.8 1.28
f3g 0.4 2.4 0.16 f1, 2, 3g 7.5 2.5 18.75
f4g 0 2.5 0 f1, 2, 4g 7.2 2.4 17.28
f1, 2g 5.6 2.8 15.68 f1, 3, 4g 6.6 2.2 14.52
f1, 3g 4.8 2.4 11.52 f2, 3, 4g 4.8 2.6 7.68
f1, 4g 4.8 2.4 11.52 N 8 2 16
As may be seen, the function v is not superadditive. The reason is that,
for example, coalition f1, 2, 3g would nd it better to drive rm 4 out of
the market than to license to it the powerful technology owned by rm 1.
Nevertheless the function can be used for the usual game-theoretic purposes.
For example, assume the coalition f1, 2g forms. We see that, following the
Aumann-Dr eze model, value to rm 1 is (15.68+6.76-1.44)/2 = 10.5, and to
rm 2 it is (15.68-6.76+1.44)/2 = 5.18. In fact, using rm 1's technology,
they would each produce 2.8 units, and obtain a prot of 7.84. Thus, in this
case, rm 2 should pay rm 1 a licensing fee of 2.66 units for the right to
use this technology.
Suppose next that coalition f1, 2, 4g forms. In that case, the value of
the restricted game is (11, 4.55, 1.73) to 1, 2, and 4 respectively. Now if all
three rms use 1's technology, they will each produce 2.4 units and obtain a
prot of 5.76. The value can therefore be obtained if rm 2 pays 1.21 units,
and rm 4 pays 4.03 units, for the license to this technology.
For f1, 2, 3g, the restricted game has value (11.46, 5.00, 2.28). In this
case, each of the rms would produce 2.5 units, with prots of 6.25. Then
2 and 3 would pay 1.25 and 3.97 to rm 1 for the license.
Denition 1. We call \fair price", a price given by the dierence between
the Aumann-Dr eze value and the prot that each rm could obtain if each
benets from the technology transfer i.e. 8j 2 S; with j 6= i; F := 'j [v jS] 
v (S)
s
, where i is the most ecient rm of the coalition S.
Take this analysis one step further. Assume f1, 2g has formed, receiving
(as discussed above) payos (10.5, 5.18). If rm 4 were now to approach
the coalition, asking for membership, the eventual payos would be (11,FERRER, OWEN AND VALOGNES 7
4.55, 1.73). Firm 1 would be happy enough about this, but 2 would reject
the deal. Similarly, if rm 3 were to approach and ask for membership,
2 would reject the deal. Finally, each of 1 and 2 would lose if it were to
leave the coalition and try to go it alone. We conclude that the coalition
f1, 2g is stable in the sense that (a) one of the members would reject any
new prospective member, and (b) each of the current members would lose,
should it decide to leave the coalition.
This leads us to the following denition:
Denition 2. A coalition S is stable if the following hold:
(1) for any k = 2 S, either (i) there is at least one j 2 S such that
'j[vjS [ fkg] < 'j[vjS], or (ii) 'k[vjS [ fkg] < v(fkg)
(2) for each j 2 S, 'j[vjS]  v(fjg).
Example 1 (continued).
We compute the possibilities for each of the coalitions. Still following the
Aumann-Dr eze model, values to the several rms are:
Table 2
Coalition Firm 1 Firm 2 Firm 3 Firm 4
f1,2g 10.5 5.18
f1,3g 9.06 2.46
f1,4g 9.14 2.38
f2,3g 2.6 1.32
f2,4g 2.68 1.24
f3,4g 0.72 0.56
f1,2,3g 11.46 5 2.28
f1,2,4g 11 4.54 1.74
f1,3,4g 10.48 2.06 1.98
f2,3,4g 3.893 1.933 1.854
f1,2,3,4g 10.32 3.73 1.25 0.70
We note here that coalition f1, 2, 3g is also stable, as (a) its members
would lose, should 4 join the group, and (b) any one of the three members
would lose should it decide to leave and go it alone. On closer analysis, all
the three-person coalitions are stable, as are the two-person coalitions that
include rm 1. The other two-person coalitions are unstable, as is the grand
coalition N.
We turn now to consider a 5-person game, with A = 15, and ci = 0, 1, 2,
2.5 and 3 respectively. If no cooperation is allowed, then we ndFERRER, OWEN AND VALOGNES 8
q1 =
15 + (1 + 2 + 2:5 + 3)   5(0)
6
=
23:5
6
= 3:917
Similarly, q2 = 2.917, q3 = 1.917, q4 = 1.417 and q5= 0.9167. Total
production is then Q = 66.5/6 = 11.083, so p = 23.5/6 = 3.917, and the
several players' prots are 15.34, 8.507, 3.674, 2.007 and 0.8403 respectively.
We note that in this case, rm 5 is essentially moribund because of its high
costs.
We assume again that some cooperation is allowed, in the form of tech-
nology sharing. In that case, we nd that, if the coalitions f1g and f2, 3, 4,
5g form, then 2, 3, 4 and 5 all produce with unit cost 1, so
q1 =
15 + 4(1)   5(0)
6
=
19
6
= 3:167
q2 =
15 + 0   2(1)
6
=
13
6
= 2:167
And q2 = q3 = q4 = q5 = 13/6. Total production is Q = 71/6, so p =
19/6, and prots are 10.028, 4.694, 4.694, 4.694, and 4.694. Thus v(f1g) =
10.028, v(f2, 3, 4, 5g) = 18.776 .
In a similar way, we compute v(S) for the remaining coalitions, to obtain
the characteristic function:
Table 3
S Q(S) p v(S) S Q(S) p v(S)
; 0 0 f1, 2, 3g 10 3.333 33.333
f1g 3.167 3.167 10.028 f1, 2, 4g 9.5 3.167 30.083
f2g 1.667 2.667 2.778 f1, 2, 5g 9.5 3.167 30.083
f3g 0.833 2.833 0.694 f1, 3, 4g 8.5 2.833 24.083
f4g 0.417 2.917 0.174 f1, 3, 5g 8.5 2.833 24.083
f5g 0 3 0 f1, 4, 5g 8.5 2.833 24.083
f1, 2g 7 3.5 24.5 f2, 3, 4g 6 3 12
f1, 3g 6 3 18 f2, 3, 5g 6 3 12
f1, 4g 6 3 18 f2, 4, 5g 6 3 12
f1, 5g 6 3 18 f3, 4, 5g 4.5 3.5 6.75
f2, 3g 3.667 2.833 6.722 f1, 2, 3, 4g 12 3 36
f2, 4g 3.667 2.833 6.722 f1, 2, 3, 5g 11.667 2.917 34.028
f2, 5g 3.667 2.833 6.722 f1, 2, 4, 5g 11.333 2.833 32.111
f3, 4g 2.333 3.167 2.722 f1, 3, 4, 5g 10.668 2.667 28.445
f3, 5g 2.333 3.167 2.722 f2, 3, 4, 5g 8.667 3.167 18.776
f4, 5g 1.667 3.333 1.389 N 12.5 2.5 31.25FERRER, OWEN AND VALOGNES 9
Following the Aumann-Dr eze model, values to rm 1, rm 2, rm 3, rm
4 and rm 5 are:
Table 4
Coalition Firm 1 Firm 2 Firm 3 Firm 4 Firm 5
f1, 2g 15.875 8.625
f1, 3g 13.667 4.333
f1, 4g* 13.927 4.073
f1, 5g* 14.014 3.986
f2, 3g 4.403 2.319
f2, 4g 4.663 2.059
f2, 5g 4.75 1.972
f3, 4g 1.621 1.101
f3, 5g 1.708 1.014
f4, 5g 0.7815 0.6075
f1, 2, 3g* 18.718 9.454 5.162
f1, 2, 4g* 17.721 8.457 3.905
f1, 2, 5g* 17.75 8.486 3.847
f1, 3, 4g* 16.318 4.012 3.752
f1, 3, 5g* 16.347 4.041 3.694
f1, 4, 5g* 16.878 3.646 3.559
f2, 3, 4g 6.115 3.073 2.813
f2, 3, 5g* 6.144 3.102 2.755
f2, 4, 5g 6.675 2.706 2.619
f3, 4, 5g 2.897 1.97 1.883
f1, 2, 3, 4g* 19.189 8.986 4.541 3.284
f1, 2, 3, 5g* 18.710 8.507 4.062 2.748
f1, 2, 4, 5g* 18.115 7.911 3.071 3.013
f1, 3, 4, 5g* 17.810 3.830 3.433 3.375
f2, 3, 4, 5g* 7.740 3.962 3.566 3.508
N 17.260 7.190 3.106 2.115 1.579
We note here that starlit-coalitions are stable. For example, the coalition
f1, 2, 4, 5g is stable as its member would lose, should 3 join the group,
and any one of the four members would lose should it decide to leave and
go it alone. For the three-player coalitions case, we may remark that all
coalitions that include the more ecient rm are stable. Moreover, all four-
person coalitions are stable.
The question we can now ask is the following: what are the fair prices
rms 2, 4 and 5 ought to pay to rm 1 in order to be allowed to use the
most ecient technology? In fact, using rm 1's technology, they would
each produce 2.833 units, and obtain a prot of 8.03. Hence, in this case,
rm 2 should pay rm 1 a fair price that corresponds to the licensing feeFERRER, OWEN AND VALOGNES 10
of 0.12 units to be allowed to use this technology. In the same way, rm 4
should pay 4.96 units to use the technology and nally rm 5 should pay
5.02 units.
Finally, we give an existence proof for stable coalitions:
Theorem 1. Every i 2 N belongs to at least one stable coalition.
Proof. Essentially, we use a process, which starts with the singleton coalition
fig and adds new members until a stable coalition is obtained.
Let then S1 = fig, and note that 'i[vjS1] = v(fig). Thus, for this
coalition, condition (b) holds.
Now either S1 is stable, in which case our process stops, or S1 is unstable.
Since (b) holds, this means (a) does not hold. Thus there is some k 6= i for
which neither (a-i) nor (a-ii) holds; i.e., 'i[vjfi, kg  'i[vjS1] = v(fig) and
'k[vjfi, kg  v(fkg)
Let now S2 = fi, kg. Again note (b) holds. Now, if S2 is stable, the
process stops. If not, there is a rm l 6= i, k such that neither (a-i) nor (a-ii)
holds. Thus, letting S3 = fi, k, lg
'i[vjS3]  'i[vjS2]  v(fig)
'k[vjS3]  'k[vjS2]  v(fkg)
'l[vjS3]  v(flg).
We continue in this way, adding a new member to the coalition at each
step, so that Sm+1 is simply Sm plus one new member. At each step,
'k[vjSm+1]  'k[vjSm]  v(fkg) for each k 2 Sm, and 'l[vjSm+1]  v(flg)
for the new member, l, of Sm+1.
Thus condition (b) holds at each step. Eventually this process of adding
new rms must stop as there is only a nite number of rms. The nal
coalition, SM, satises condition (a) and is therefore stable. 
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Dept. of Economics & CREM-CNRS 6211, University of Caen, 19 rue C.
Bloch, F-14032 Caen Cedex, France.
E-mail address: helene.ferrer@unicaen.fr
Dept. of Mathematics, Naval Postgraduate School, Monterey, CA 93943,
USA.
E-mail address: Gowen@nps.edu
Dept. of Economics & CREM-CNRS 6211, University of Caen, 19 rue C.
Bloch, F-14032 Caen Cedex, France.
E-mail address: fabrice.valognes@unicaen.fr

Asymmetric Oligopoly and Technology Transfers&quot;,

Coalition Structures&quot;,

Cooperative Games with Coalition Structures,&quot;

Endogenous Formation of Coalitions&quot;,

Equilibrium points in n-person games&quot;.

On Coalition Formation: A Game Theoretical Approach&quot;,

Researches into the Mathematical Principles of the Theory of Wealth, translated by Nathaniel T.

The Bargaining Set for Cooperatives Games&quot;,

Values of Games With A Priori Unions&quot;

Endogenous Coalitions Formations Through Technology Transfers and Fair Prices

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Endogenous Coalitions Formations Through Technology Transfers and Fair Prices

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