N-Person Cooperative Games
Strategic-Equilibrium
by
Gabriel J. Turbay, Ph.D.1
Keywords: Strategic-Equilibrium, Extended-Imputations, Utility-Transfer Analysis,
Linearly-Balanced Collections, Theorems of the Alternative for Matrices
ABSTRACT
Based on von Neumann and Morgenstern´s detached extended imputations relation to the
stable set solution of a cooperative game, necessary an sufficient conditions for the
structural and strategic equilibriums that characterizes the symmetric (objective) solutions
are given for all general-sum n-person cooperative games with transferable utility. The
mathematical characterization of the equilibriums for these games is accomplished in terms
of covering collections structures that are linearly-balanced, admissible utility transfers and
fundamental theorems of the alternative for matrices: The Fredholm alternative for matrices
form of the fundamental theorem of linear algebra and the Farkas lemma. The existence of
a fundamental strategic equilibrium for every game is established and shown to constitute a
von Neumann and Morgenstern un-dominated system of interrelated extended imputations.
It is shown to be not necessarily a solution but a systemic attractor where all solutions may
emerge from. Heuristic procedures to determine the fundamental equilibrium of a game and
for generating vN-M non-discriminatory solutions are given.
1
Former Research Director, Facultad de Administración UNIVERSIDAD DEL ROSARIO, Bogotá Colombia
-1-
Introduction
Except for vN-M analysis of the zero-sum and general sum 3-person
cooperative games that identifies symmetric type solutions and a possible
valuation for the players in the game, also found in the 4-person game; and the
attempt to find all solutions to a general sum n-person cooperative game by
means of the detached extended imputations, most of the existing solution
concepts and criteria appear as forced solutions that result in disconnected
versions of these symmetric solutions. When these solutions appear spread out
in eventual partitions, a fragmentation emerges of what vN-M consider an
integrated unit not to be taken in isolation as stated in “Theory of Games and
Economic Behavior”2. There, the symmetric type solutions, not individually
but as a set, not in actuality but in potentiality, are identified as the only
strategic possibilities3 for the game.
In the heuristic formulation of the stable sets or vN-M solutions, the
definition captures in terms of the domination criteria, some essential aspects
of the of the symmetric solutions stability; it does not focus however, on
those properties that would allow to characterize and establish the necessary
and sufficient conditions for the existence of such strategic equilibriums for
2
See vN-M 29.1.3 page 261
3
-2-
general n-person cooperative games. This type of characterization is the
objective hopefully to be accomplished in this paper.
Basic concepts and definitions
Let N = {1, 2,…, n} denote de set of n players and En denote the ndimensional Euclidian space. A game in characteristic function form is a pair
 = (N,v) , where v is a real-valued set function defined on the subsets of N
that satisfy the following properties:
(2.1) v() = 0
(2.2) v(S  T) ≥ v(S)+ v(T) whenever S  T =, S and T subsets of N
(2.3)
A game  = (N,v) is said to be essential if
v( N ) 
 v({i})
iN
Though essentiality is a desired property in every game, when answering the
question: If the grand coalition does not form, which coalitions are likely to
form? It may be considered that the grand coalition cannot be formed or that
the value V (N) = max [V(S)] over S N (to maintain the supper-additive
property). Somehow, as it will be shown later, it is convenient to replace the
essentiality condition above with the more general condition:
(2.4) 𝑣(𝑆) > ∑𝑖∋𝑆 𝑣({𝑖}) for every S in some covering C of N
-3-
A covering collection of N or simply a cover of N is a set
C = {C1 , . . . , Ck } of subsets of N such that ∀ 𝑗 ∈ 𝑁 ∃ 𝐶𝑖 C such that j
Ci . |C | = k is the cardinality of C . Clearly,
𝑘
⋃ 𝐶𝑖 = 𝑁
𝑖=1
A payoff vector for coalition S is any distribution vector x En where
x = (x1, , . . .,xn)t , where xj ≥ 0 if j S, xj =0 if j S
Similarly, a payoff vector x En, for a coverC of N, in the game v is a
column vector defined by
xt = (x1, x2, . . . ,xn) such that for all S C
∑ 𝑥𝑗 = 𝑣(𝑆)
𝑗∈𝑆
A payoff vector for the players in N satisfies:
 group rationality if
𝑥(𝑁) = ∑ 𝑥𝑗 = 𝑣(𝑁)
𝑗∈𝑁
 coalitional rationality if for all S N
𝑥(𝑆) = ∑ 𝑥𝑗 ≥ 𝑣(𝑆)
𝑗∈𝑆
 individual rationality if for all jN
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𝑥𝑗 ≥ 𝑣({𝑗})
An imputation for the game v is a payoff vector for N that satisfies both
individual and group rationality conditions above.
A coalition 𝑆  N is said to be efficient for a payoff vector x for S if
𝑥(𝑆) = ∑ 𝑥𝑗 = 𝑣(𝑆),
𝑗∈𝑆
An extended imputation4 is a payoff vector x for N that satisfies
individual rationality. In a 0-normalized game a payoff vector is an extended
imputation if and only if it is a non-negative vector. In the later this case the
set of all extended imputations consist of the nonnegative orthant En+
An extended imputation is said to be detached (vN-M) if
𝑥(𝑆) = ∑ 𝑥𝑗 ≥ 𝑣(𝑆) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑆  𝑁
𝑗∈𝑆
A cover C = {C1 , . . . , Ck } of N is said to be an efficient cover for x
(an extended imputation x), or a covering support for x if for all 𝐶𝑖 C ,
𝑥(𝐶𝑖 ) = ∑ 𝑥𝑗 = 𝑣(𝐶𝑖 )
𝑗∈𝐶𝑖
4
The concept of extended imputation was introduced in vN-M p.364for analytical purposes.
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An extended imputation x is said to be attainable if and only if there
exist a cover C of N that is efficient for x.
REMARK
1:
Attainable-extended-imputations
through
a
cover
C = {C1 , . . . , Ck } are payoff vectors for the covering collection C of N in
consideration. They may be thought of as the composition of k-different
payoff vectors, x(1),. . . , x(k) , for coalitions C1 , . . . , Ck respectively. So that
Ci is an efficient set for for x(i) , i =1,. . ., k. The vectors x(i) ,i=1,. . .,k are
related to each other by the fact that if Cr and Cs are two different coalitions
and their intersection is not empty, Cr Cs   then for any j Cr Cs
,
the
payoff to player j in x (r) equals the one in x (s). That is, xj(r) = xj(s) , whenever
j Cr Cs .
Extended-imputations with cover-support are not necessarily detached.
Nor, detached 5 extended–imputations must have necessarily a cover support.
Extended-imputations are not limited to the restriction imposed by
group rationality. That is, one may have.
𝑥(𝑁) = ∑ 𝑥𝑗 ≠ 𝑣(𝑁)
𝑗∈𝑁
5
Here a rational extended-imputation is equivalent to a detached extended imputation, in vN-M p.370.
However not all detached extended imputations are attainable and here we are interested in detached extended
imputations with covering support.
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An extended-imputation that satisfies individual and coalitional
rationality but x(N) < v(N) is known as a pre-imputation
Proposition 1
The set of all detached xd-imputations for the cooperative
game  = (N,v) is a convex polyhedral set
bounded bellow . Here is
referred to as the over-all rationality-set
ℛ̂ ( ) = {𝑥𝜖𝐸 𝑛 |𝑥(𝑆) = ∑
𝑗∈𝑆
Proof:
The over-all rationality-set
𝑥𝑗 ≥ 𝑣(𝑆), for all 𝑆 𝑁}.
results from the intersection of half
spaces. Hence it is a convex polyhedral set. This set is bounded bellow since
cooperative game in characteristic function form determines a unique6
“vN-M value” ||2 such that for all x in ℛ̂ ( ), | 𝑥(𝑁) ≥ 𝑣(𝑁) + | |2 .
Since we are interested in the bargaining dynamics of the game
previous to the formation of the grand coalition, for now, we will restrict our
attention to the set of extended imputations that satisfy individual and
coalitional rationality. We will refer to this set, simply, as the rationality-set,
defined by
6
See TGEB pag.
-7-
ℛ( ) = {𝑥𝜖𝐸 𝑛 |𝑥 (𝑆) = ∑
𝑗∈𝑆
𝑥𝑗 ≥ 𝑣(𝑆), for all 𝑆 ⊂ 𝑁}.
Elements of the rationality set are referred to as rational-extended imputations.
The properties of proposition 1 for the overall rationality set are
applicable to the less restricted rationality set. The former is a subset of the
later.
If an extended-imputation x is rational and has an efficient covering
support C,
then x is said to be rational attainable.
extended-imputations that have an efficient cover support, C
So, all detached
in = (N,v) are
rational-attainable.
Proposition 2
The set of all attainable extended-imputations for the
cooperative game  = (N,v) is the union of connected convex sets . This set
will be referred to, as The Bargaining - field for the game = (N,v) and
will be denoted by B ( ).
Proof: The rational set is clearly the intersection of half spaces. Hence
B ()is a convex polyhedral set. However the bargaining field is the
intersection of the hyperplanes with gradients defined by the characteristic
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vectors of the coalitions in covering collections of N . These intersections
clearly may belong or not to the rationality- set
We may restrict the negotiations to the set of rational-attainable
extended imputations, denoted by r-B (). This restriction will allow us to
see better the attractor like characteristics of the fundamental-strategicequilibrium that every cooperative games exhibits. However it eliminates too
many possibilities where bargaining processes may begin. By requiring
extended imputations to be in the rationality set all attainable extended
imputations dominated by detached –attainable extended imputations are not
considered as bargaining possibilities to begin negociation processes.
Proposition 3 r-B (  )  R (  ). The extreme points of r-B (  ),
are those of R () .
Example 1:
For the 3-person cooperative game given below, the
̅̅̅̅̅̅
bargaining field consist on the three line segments ̅̅̅̅̅̅
𝐴 𝑋° , ̅̅̅̅̅̅
𝐵 𝑋° and 𝐶
𝑋°.
The extreme points of the rationality set the for 0-normalized game here, is
given R (  ) = {x En+ | x1 +x2  10 , x1 + x3  10 and x2 +x3  6 }. Clearly
the extreme points of R (  ): A, B, C and X° are those of r-B (  ).
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3-Person general-sum game
v({i})=0, i=1,2,3 , v({1,2}) = 10, v({1,3}) = 8, v({2,3}) = 6, v({1,2,3}) = 12
x2
E (0, 12, 0)
B (8, 6, 0)
A (0, 10, 8)
x2 + x3 = 6
X o (6, 4, 2)
F(12, 0, 0)
x1 + x 2 = 10
x1
C(10, 0, 6)
D (0, 0, 12)
x3
Figure 1:
The rational-bargaining field r-B ( ) = ̅̅̅̅̅̅
𝐴 𝑋° ∪ ̅̅̅̅̅̅
𝐵 𝑋° ∪ ̅̅̅̅̅̅
𝐶 𝑋° for the general-sum
3-person cooperative game above.
The concept of rational- attainable extended-imputation is essential to
identify and characterize the structural and strategic equilibriums, and most
important the fundamental strategic equilibrium for the general-sum nperson cooperative game.
Remark 2 Any attainable payoff vector that satisfies individual and
coalitional rationality is a vector in the rationality- set. The converse is not
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necessarily. That is any vector in the rationality set is not necessarily
attainable
Utility transfer analysis7
Utility transfers are defined as vectors in En. These are expressed
basically as the difference of payoff vectors. That is for x and y any extended
imputations the utility transfer  t = (1, 2, … , n ) is defined as  = y – x
or y = x +  .
Utility transfers may not be viable or acceptable by the players
involved, so it is necessary to have a well define structure of payoff s and
transfers to understand their nature and behavior.
Proposition 4 Let x be a rational attainable xd – imputations with
cover support C . Then a extended-imputation y = x + 
also attainable through C
Proof:
, with x  y is
if and only if  (C) = 0 for all C in C .
y = x+ implies y(C) = x(C)+  (C). clearly , given
x(C) = v(C) for all C in C , Then y(C) = v(C) if and if  (C)=0 for all C in C.
7
Several concepts and results in this paper were previously developed by the author in his doctoral
dissertation. Turbay Gabriel J., RICE University (1976).
- 11 -
If  Rn, ≠0 and  (C)=0 for all C in C we say that  is a utility
transfer admissible by C
Corollary : Given x and y, x  y , two extended-imputations with
cover support C , then the utility transfer  = y - x ≠ 0, is admissible by C.
Characteristic vectors and matrices
To characterize the stability that may take place in the process of
attaining sustainable payoff claims and to find the type of utility transfers, if
any, admissible by a cover C = {C1,…,Ck }, the following simple procedure
may be followed:
The characteristic matrix of a cover C . = {C1,…,Ck } in  =(N,v), is
a 0-1 matrix Wk x n
whose rows Wi * i = 1,2,…,k, are the characteristic
vectors of the coalitions Ci C . If
Wi*.= ( wi1 , w i2 , … ,w iin ), the
elements of each row i are given by
wij = 1, if j  Ci,
and
wij =0
otherwise, i = 1,…,n .
It follows that the n-vector
J t = (1,1 ,. . . , 1) is the characteristic
vector of the set of all players N ={1,2, . . . ,n}. It is also the gradient of the
hyper-plane that contains the imputations n-1 simplex I for the game. In
general, a characteristic row-vectors of a coalition S of N is the transposed
of the gradient of the hyperplane x(S) = v(S) of attainable payoffs through S.
- 12 -
Example 1. In the 3-person zero sum game C ={{1, 2}, {2,3}} is a
covering support for both extended imputations : x = (1, 0, 1)
y = ( 2/3, 1/3, 2/3). The utility transfer  such that y = x+
and
is given by
 = (- 1/3, 1/3, - 1/3). Clearly the utility transfer  is admissible by C, ,
given that (C1 ) = 0 and (C2)= 0, for C1 = {1, 2}and C2= {2, 3}.
Once the characteristic matrix corresponding to a cover C
is
constructed , the set of admissible transfers with respect to C , if any, may be
obtained by solving for  = ( 1,  2,… , 
n
)t, the homogeneous system
W = 0 .
Player´s claims and bargaining alternatives
If an extended imputation x is supported by a cover C
the payoffs
imputed by x are realizable through the coalitions in C . Given two different
players h and k in a given coalition Cs C , we say that a coalition Ct  C is
a bargaining alternative of player h against player k if
.
h  Ct and k  Ct
Bargaining alternatives represent a strategic basis to defend a particular
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claim or stand prescribed in a given attainable extended imputation in the
context of the cover C .
In the example above, relative to x = (1, 0, 1), players 1 and 3
have no bargaining alternative
defensible
so the prescribed claims are not
however relative to y = ( 2/3, 1/3, 2/3), player 2 in coalition
{1, 2} has a bargaining alternative against player 1 and in coalition {2, 3}
he has a bargaining alternative against player 3. As long as the players
have bargaining alternatives against other coalition members that don´t,
transfers of utility may be requested by the former players against the
later ones. The admissible utility transfer structure in the example above
is  = (-, , - )t, The limit of this process relative to x is reached when the
utility transfer is equal to  = (- 1/2, 1/2, - 1/2) that gives x° = ( 1/2, 1/2,
1/2 ) at this point players 2 and 3 have bargaining alternatives that
support the corresponding claims and the pair [x° , C ]8 with C = {{1,
2}, {2,3}, {1, 3}} becomes an equilibrium for the 3-person zero-sum game.
The structure of the utility transfers, the bargaining field: the three line
segments
̅̅̅̅̅̅
𝐴 𝑥° , ̅̅̅̅̅̅
𝐵 𝑥° and ̅̅̅̅̅̅
𝐶 𝑥° , the bargaining alternatives {i, j}
versus k i, j, k =1, 2, 3 and the equilibrium x° for the game are shown in
the figure below:
8
All the stability arguments given by vN-M are applicable here see pag.
- 14 -
UTILITY TRANSFER STRUCTURE
FOR THE 3-PERSON ZERO-SUM GAME
x2
x3 = 0
(-, -, )
x1 = 0
B (1, 1, 0)
A (0, 1, 1)
x2 + x3 = 1
(, -, -)
x1
x1 + x2 = 1
Xo (. 5, . 5, . 5)
C(1, 0, 1)
(-, , -)
x2 = 0
x3
The following two propositions illustrate how the structure of
supporting coalitions for extended imputations determine the structure of the
admissible utility transfers, the bargaining alternatives for the players and
hence the stability of the claims.
Proposition 5 Let C = {C1,…,Ck } be a cover of N. If |C .| = k < n ,
then the cover C admits non-null utility transfers .
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Proof: Clearly, any system of homogeneous equations W = 0 has a
non trivial solution   0, whenever the number of equations k is less than the
number of unknowns n.
Corollary: The dimension of the admissible transfer space  (C )is the
same as the nullity of W. That is Dim[ (C )] = n-k
The rank of the characteristic matrix W of a cover C . of N depends on
the number of linearly independent rows of W.
If the characteristic vectors of a collection of subsets of N are linearly
independent we say that the corresponding coalitions are linearly independent
subsets of N
Proposition 6
Let C . = {C1,…,Cn } a covering collection of linearly
independent subsets of N . Then there exist no utility transfer   0 admissible
b: y the cover C .
Proof: Since W has full rank, that is, rank (W) = n , the corresponding
homogeneous system W = 0
has no non-trivial solution. Hence no
admissible transfers are allowed by C .
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In the example above, the collection of the three 2-person coalitions has
a corresponding characteristic matrix with full rank. This implies the
inexistence of admissible utility transfers relative to the equilibrium[x° , C ] .
Such condition is reflected in the fact that any request made by one player to
another one for a transfer in excess of the amounts prescribed by x° will not
find a bargaining alternative. The requesting player will fail to get the
additional amount from a third player because the third player would be able
to protect his actual stand using the corresponding bargaining alternative, in
which the requested player is included and may keep his actual stand but the
requesting one is excluded
Necessary and sufficient conditions for the admissibility of transfers are
given by the fundamental theorem of linear algebra in the Fredholm
alternative for matrices form. In our particular case it gives us the following:
Proposition 7. (Utility transfer admissibility theorem) Given a cover
C of N with characteristic matrix W;
Either the system
Wt = J has a solution
Or the system
W = 0 , Jt   0 has a solution
- 17 -
But not both can have a solution.
Proof: The above theorem is simply an application of the well known
fundamental theorem of linear algebra.
Corollary
If Wt = J has a solution and there exist   0 such that
W = 0, then Jt  = 0.
Proof: If Jt = t W then Jt = t W = 0, since W = 0,
Whenever the system Wt = J has a solution the corresponding cover
of N is said to be a linearly-balanced collection of subsets of N.
Proposition 8 Any linearly-balanced collection C of subsets of N is a
covering collection for N
Proof: Since t W = Jt has a solution then the elements column W*j
cannot be all zeros for otherwise t W*j = 0,. Since t W*j =1, it follows that
every j in N belongs to at least one coalition Ci in C ..
LetC .
= {C1,…,Ck } be
a linearly- balanced cover C
For all j  N , Let
D (j) = {Ci  C | j Ci} , and D
- 18 -
hl
= { Ci C | h  Cj, l Cj}.
of N.
The following proposition gives us an important property of lbalanced collections
Proposition 9 A necessary and sufficient condition for C to be l- balanced is
that for all h, l  N, h l
(∗) ∑ 𝛾𝑖 =
𝐶𝑖 ∈ 𝔇ℎ𝑙
∑ 𝛾𝑖
𝐶𝑖 ∈ 𝔇𝑙ℎ
Proof: C is l- balanced if and only if for all j  N
(𝑖) ∑ 𝛾𝑖 𝐼ℎ (𝐶𝑖 ) =
𝐶𝑖 ∈ 𝔇(𝑗)
∑ 𝛾𝑖 = 1
𝐶𝑖 ∈ 𝔇(𝑗)
. Hence for h l, h, l  N we have that
(𝑖𝑖)
∑
𝛾𝑖 𝐼ℎ (𝐶𝑖 ) = 1 =
𝐶𝑖 ∈ 𝔇(ℎ)
∑ 𝛾𝑖 𝐼𝑙 (𝐶𝑖 )
𝐶𝑖 ∈ 𝔇(𝑙)
Let I = D (h) D (l)
Both D (h)- [I  D hl ]=  and I  D hl = 
Also D (l)- [I  D lh ]and I  D lh = 
Therefore
- 19 -
(𝑖𝑖𝑖) ∑ 𝛾𝑖 𝐼ℎ (𝐶𝑖 ) +
𝐶𝑖 ∈ ℐ
∑
𝛾𝑖 𝐼𝑙 (𝐶𝑖 ) = ∑ 𝛾𝑖 𝐼ℎ (𝐶𝑖 ) +
∑
𝐶𝑖 ∈ 𝔇(ℎ𝑙)
𝐶𝑖 ∈ ℐ
𝐶𝑖 ∈ 𝔇(𝑙ℎ)
𝛾𝑗 𝐼𝑙 (𝐶𝑖 )
=0
Then ( * ) follows when we subs tract ( iii ) from (ii ). Conversely if we
can find numbers (not all zero) 𝛾1^ , … . , 𝛾𝑘^
Such that
( * ) hold true for all
j in N then (i) must also hold true therefore we must have
∑
𝛾𝑖^ 𝐼ℎ (𝐶𝑗 ) =
𝐶𝑖 ∈ 𝔇(ℎ)
Defining
∑ 𝛾𝑖^ 𝐼𝑙 (𝐶𝑖 ) = 𝑏 ( 𝑏 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝐶𝑖 ∈ 𝔇(𝑙)
𝛾𝑗 =
1
𝑏
𝛾𝑗^ , (𝑗 = 1, … . . , 𝑘) we conclude that C
must be l-
balanced with balancing vector  = (1, ……, k)T.
The following definitions discriminate deeper than those of Bondareva ( ) and
Shapley( ):
A covering collection C = {C1, . . . , Ck} of subsets on N with
characteristic matrix W, is said to be
l-balanced if   | W t  = J,  unrestricted (linearly-balanced)
n-balanced if   | W t  = J, and  i | i <0, i=1,. . . ,k (negatively balanced)
nn-balanced if    0 | W t  = J, (non-negatively-balanced)
- 20 -
Non-negatively- balanced9 coverings may be partitioned into the following
two kinds, of covering collections:
p-balanced if   > 0 | W t  = J, (positively-balanced)
w-balanced if    0 | W t  = J, and i = 0, for at least one element of  .
  Rm is referred to as the balancing vector. The elements i of ,
i=1,…,m, are the balancing coefficients of the sets in C. .
A utility transfer  0 is said to be a zero-sum utility transfer if and only if
(i) W = 0
(ii) Jt = 0
9
(admissibility)
( pareto optimality)
Here p-bal,anced corresponds to balanced and w-balanced to weakly-balanced in Shapley( )
,
- 21 -
Structural equilibrium
Let C = {C1, . . . , Ck} be covering collection of subsets on N and
X(C ) the corresponding set of rational-attainable extended imputations. Then
The pair [ X(C ), C ] constitute a structural equilibrium for the game
 = (N,v) if and only if the only utility transfers admissible by C , if any,
are zero-sum.
Now we state the fundamental structural equilibrium theorem:
Theorem I
The pair [ X(C ), C ] as above constitute a structural
equilibrium for the game  = (N,v) if and only if the collection C
is a
linearly balanced collection of subsets of N.
Proof: A direct consequence of the corollary to proposition 7 is that if
the collection C is l-balanced the only possible transfers if any must be zerosum and conversely The definition of structural equilibrium in terms of utility
transfers admissibility above implies that if the only admissible transfers by a
cover C are zero-sum then it must be l-balanced.
The following theorem
gives us a special characteristic of balanced
collections in terms of bargaining alternatives for the players:
- 22 -
Theorem II
In any structural equilibrium [ X(C ), C ], for any
extended imputation in X(C ) (of attainable claims), every player has a
bargaining alternative versus any other non-symmetric player
Proof: From proposition 9 we know
∑ 𝛾𝑖 𝐼ℎ (𝐶𝑖 ) =
𝐶𝑖 ∈ 𝔇ℎ𝑙
∑ 𝛾𝑖 𝐼𝑙 (𝐶𝑖 )
𝐶𝑖 ∈ 𝔇𝑙ℎ
Hence either the players are symmetric if the sums above equal zero or each
player has at least one bargaining alternative versus the other player.
Strategic dependence
Let h and l be different players. Relativc to a cover C of N we
say that l is strategic-dependent on h if and only if D (h)  D (l) .That is,
if any coalition in C that contains h also contains l but the converse is not
true. If D (h) =D (l)
the players are symmetric and hence strategic-
quivalent
A Strategic-dependent players have no bargaining alternatives against
the player hey depend from. Clearly,
Proposition 10 If C is l-balanced, no player can be strategic-dependent
on another player.
- 23 -
The strong structural stability exhibited10 by collections with n linearly
independent vectors motivates an analysis of the rationality set:
ℛ( ) = {𝑥𝜖𝐸 𝑛 |𝑥(𝑆) = ∑
𝑗∈𝑆
𝑥𝑗 ≥ 𝑣(𝑆), for all 𝑆 ⊂ 𝑁}
We know ℛ( ) is a convex polyhedral set. In matrix notation,
ℛ( ) = { 𝑥| 𝑉𝑥 ≥ v }, where V ( 2n- 2
x n) is
the characteristic matrix of the
proper subsets of N and v the corresponding vector of characteristic function
̂ the characteristic
values. Let 𝑥̂ an arbitrary extreme point of ℛ( ) and 𝑊
matrix of the coalitions in active at 𝑥̂ .
Proposition 10 Any 𝑥̂ extreme in ℛ( ) with its corresponding covering
support C
conform a structural-equilibrium for the game  = (N,v)
Proposition 11.
The extreme points of ℛ( ) are the basic solutions of the
system 𝑉𝑥 ≥ v and hence, there are n linearly independent subsets of n active
at any 𝑥̂ in ℛ( )
10
All vN-M stability arguments applicable to the symmetric solution to the 3-person zero-sum game are
applicable here see pp.
- 24 -
Example 2 : For the 3-person general sum game with characteristic function
given by vi = 0 if i=1, 2, 3 , v({1, 2}) = 5, v({1, 3}) = 6 , v({2, 3}) = 17,
v({1, 2, 3}) = 19, the extreme point of ℛ( ) are given by the points A, B , C,
D. Observe that A and D have a full rank n-balanced covering support while
B and C have a full rank a w-balanced cover support.
The bargaining field B ( ) here is given by the set of points
̅̅̅̅̅
conformed by the union of the segments ̅̅̅̅̅̅
𝐴𝐵 , 𝐵
𝐶 and ̅̅̅̅̅̅
𝐶𝐷.
Strategic Stability
x2
x3 = 0
D (0, 1, 1)
x1 = 0
W o =
1 0 0
WB = 1 1 0,
0 1 1
1 0 0
1
WC = 0 1 1,  = 1
1 0 1
0
 =
1
0
1
(-, -, )
1 0 1
0 1 1
(0, 11, 6) C
1 0 0  = 1
0 1 1
1
0 0 1
-1
WD = 1 0 1,  = 1
0 1 1
1
(0,-, )
(0, 8, 9) o
x1
(0, 5, 12) B
(0, ,-)
A (5, 0, 12)
x1 + x2 = 5
x3
(-, , -)
1
0
1
1
0
1
0 1 0
WA = 0 1 1 ,
1 1 0
- 25 -
-1
 = 1
1
x2 = 0
Figure 2. Strategic instability in negatively balanced Extreme points
Proposition 12
The rationality set is bounded below so that for any x in ℛ( )
𝑥(𝑁) ≥ 𝑣(𝑁) + | |2
Here, | |2 is the vN-M value11 determined by any game so that a detached
extended imputation exists with an excess e = x(N) – v(N) ≥ | |2 .
Proof: Clearly the removal of the constraint x(N) ≥ 𝑣(𝑁) wil not affect the
lower bound of the overall-rationality–set of Proposition 1 above.
To understand the relationship between n-balanced w-balance and pbalanced collections we establish the following propositions related to dual
coalitions.
Given a subset S of N, the dual of a coalition S is the set S*= N-S. If
Ws is the characteristic vector of S then the characteristic vector of S* is given
0 𝑖𝑓 𝑗𝑆
by Ws ∗ = (wj∗ ) = {
1 𝑖𝑓 𝑗 𝑆
Similarly , if D is a collection of subsets of N the dual of D is defined by
11
See theorem on page
- 26 -
D * = {D* = N-D | D D }
Proposition 13. Let C = { C1, …, Ck} be an l-balanced collection of subsets
of N with corresponding balancing vector 𝛾 = (𝛾1 , … . , 𝛾𝑘 )𝑇 , and let D be a
proper sub-collection of C such that
∑ 𝛾𝑗 ≠ 1.
𝐶𝑗 ∈𝜏
)  D * is also l-balanced with
Then the collection C ´ = (C - D
balancing vector
𝛾𝑖
𝜌
𝛾 ´ = (𝛾1 ´ , … . , 𝛾𝑘 ´)𝑡 where 𝛾𝑖´ = {
−𝛾𝑖
𝜌
Here
𝑖𝑓 𝐶𝑖 ∈ 𝒞´
𝑖𝑓 𝒞𝑖 ∈ 𝔇∗
𝜌 = 1 − ∑ 𝛾𝑗
𝐶𝑖 ∈𝔇
Proof: For any j  N let D = {C ∈ τ  j C}
Since Cj    𝐶𝑗1  * =
( - D )*  D then
∑ 𝛾𝑗´ 𝐼𝑗 (𝐶)=
𝐶𝑗 ∈𝒷
1
𝜌
∑ 𝛾𝑗 𝐼𝑖 (𝐶𝑗 ) −
𝐶𝑗 ∈[𝒞− 𝜏]
1
𝜌
- 27 -
∑ 𝛾𝑗 𝐼𝑗 (𝐶)=
𝐶𝑗 [𝜏−𝒟]∗
1
𝜌
∑ 𝛾𝑗 𝐼𝑖 (𝐶𝑗 )
𝐶𝑗∈𝒟 ∗
But i D , i  D *, hence the right most term of ( ) vanishes and we obtain
∑ 𝛾𝑗^ 𝐼𝑗 (𝐶)=
𝐶𝑗 ∈𝒷 ^
1
𝜌
𝛾𝑗 𝐼𝑖 (𝐶𝑗 ) −
[ ∑
∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 )]
𝐶𝑗 [𝜏−𝒟]∗
𝐶𝑗 ∈[𝒞−𝜏]
Adding to the expression in brackets in ( )
∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 ) − ∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 ) = 0
𝐶
𝑗∈𝒟
𝐶𝑗 ∈𝒟
And nothing that
∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 ) =
𝐶𝑗 [𝜏−𝒟]∗
1
𝜌
∑ 𝛾𝑗 𝑎𝑛𝑑 ∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 ) = ∑ 𝛾𝑗
𝐶
𝑗∈𝒟
𝐶𝑗 [𝜏−𝒟]
𝐶
𝑗∈𝒟
We obtain
1
∑ 𝛾𝑗´ 𝐼𝑗 (𝐶)=
𝐶𝑗 ∈𝒷 ^
1
𝜌
𝛾𝑗 𝐼𝑖 (𝐶𝑗 ) + ∑ 𝛾𝑗 𝐼𝑖 (𝐶𝐽 ) )
[( ∑
𝐶𝑗  𝒟
𝐶𝑗 ∈[𝒞−𝜏]
− ( ∑ 𝛾𝑗 + ∑ 𝛾𝑗 )]
𝐶𝑗 [𝜏−𝒟]
𝐶𝑗 ∈𝒟
- 28 -
1
𝜌
(1 − ∑ 𝛾𝑗 ) = = 1
𝜌
𝜌
𝐶𝑗 ∈𝜏
Which establishes that the collection C
´
is l-balanced with corresponding
balancing vector ´.
Corollary If C is p-balanced and D  C , D   , is such that
0 < ∑ 𝛾𝑖 < 1
𝑐𝑖 𝜖𝔇
Then C´´= (C - D )  D * is n-balanced with D * = {Ci  C ´| ´i <0} and
(C - D ) ={Ci  C ´| i >0}.
Proof :
A particular case occurs when D = {Ch} then C *h = C – {Ch}{Ch*} is
n-balanced, In this case  i *h > 0 for Ci  (C – {Ch}) and  h*h < 0 Ch*}.
Proposition 14 Any cover C´ ={ C1, …, Cn } of N consisting of n linearly
independent subsets of N , is linearly balanced and has a
full rank
characteristic matrix Wn x n . For any ChC the collection C´´= C - { Ch}
- 29 -
has a characteristic matrix W´n-1x
n .and
admits utility transfers  =[W-1h*]
given by the hth column of W-1. For such transfers
W  0, Wh*>0, Jt >0
Example 3 Consider the collection of subsets of N = {1, 2, 3, 4], where
C = {C1, C2, C3, C4, C5} and C1 = {1, 2, 3}, C2 = {2, 3, 4}, C3 = {3, 4, 5},
C4 = {1, 4} and C5 = {1, 2, 5}. Here, C is p- balanced with characteristic
matrix W and corresponding balancing g given by
1
0
W= 0
1
(1
1
1
0
0
1
2 −2
−1 3
4 3 −1
−2 2
[ 1 −1
1
1
1
0
0
0
1
1
1
0
1⁄4
0
1⁄4
0
1 and 𝛾 = 1⁄2 𝑊 −1 = 1/
0
1⁄4
1)
(1⁄2)
0
2 0
−2 −1
2
2 −1 −2
0
2
0
2 −1
2 ]
Considering collections of the form
C ´= [C - {Cn}]  {C’n} (n= 1, 2, 3, 4, 5) we have that proposition
12 applies, and
- 30 -
1 − 1⁄4 = 3⁄4 𝑖𝑓 𝜈 = 1, 2, 4
𝜌 = 1 − 𝛾𝜈 = {
1 − 1⁄2 = 1⁄2 𝑖𝑓 𝜈 = 3, 5
The coefficients of the balancing vector 𝛾̂
𝛾̂𝑖 =
𝑖

𝑓𝑜𝑟 𝐶𝑖  {C - {Cn}},
𝛾̂𝑖 =
−𝑖

𝑓𝑜𝑟 𝐶𝑣
All collections C =C −{𝐶𝜈 } are structurally unstable and for any n = 1, 2,
3, 4 or 5 C ´ is n-balanced. Below we give the corresponding characteristic
matrices, transfers and balancing vectors for n = 1, 2, 3, 4, 5.
Characteristic matrix
of Balancing vector
C − {𝐶𝜈 }
̅𝜀
𝑊
Characteristic matrix of
[C −{𝐶𝜈 } ]  {𝐶 ∗𝜈 }
Balancing
Vector
For n = 1
∗∗∗∗∗
00111
̅
𝑊 = 10010
11001
( 1 1 0 0 1)
−
0
̅
𝑊𝜀 = 0
0
(0)
xT = (2, -, 3, -2,-)
00011
01110
̂
𝑊 = 00110
10010
( 1 1 0 0 1)
𝐽t  = 
For n = 2
- 31 -
−1⁄3
1⁄3
𝛾̂ = 2⁄3
1⁄3
( 2⁄3 )
̂ = (1 1 1 1 1)
𝛾̂ T 𝑊
11100
∗∗∗∗∗
̅
𝑊 = 00111
10010
(1 1 0 0 1)
0
−
0
0
(0)
11100
10001
̂
𝑊 00111
10010
( 1 1 0 0 1)
x t= (-2, 3,-, 2, -)
𝐽t  = 
1⁄3
−1⁄3
̂𝛾 =
2⁄3
1⁄3
( 2⁄3 )
̂ = (1 1 1 1 1)
𝛾̂ T 𝑊
For n = 3
11100
01110
̅ = ∗∗∗∗∗
𝑊
10010
(1 1 0 0 1)
0
0
−
0
(0)
11100
01110
̂ = 11000
𝑊
10010
( 1 1 0 0 1)
𝐽 t  = 2
xT = ( 0, -2, 2, 0, 2)
1⁄2
1⁄2
̂𝛾 =
−1
1⁄2
( 1 )
̂ = (1 1 1 1 1)
𝛾̂ T 𝑊
For n = 4
11100
01110
̅
𝑊 = 00111
∗∗∗∗∗
(1 1 0 0 1)
0
0
0
−
(0)
xT = ( 2, - , - , 2 - ,)
11100
01110
̂
𝑊 = 00111
̂𝛾 =
01101
( 1 1 0 0 1)
𝐽t  = 
1⁄3
1⁄3
2⁄3
−1⁄3
( 2⁄3 )
̂ = (1 1 1 1 1)
𝛾̂ T 𝑊
For n = 5
11100
01110
̅ = 00111
𝑊
10010
( ∗∗∗∗∗ )
0
0
0
0
(−)
xT = ( 0, 2, - 2, 0, 2)
11100
01110
̂ = 00111
𝑊
10010
( 0 0 1 1 0)
𝐽 t  = 2
- 32 -
1 ⁄2
1 ⁄2
1 = ̂𝛾
1 ⁄2
( −1 )
̂ = (1 1 1 1 1)
𝛾̂ T 𝑊
The structure of the admissible utility transfers in each case v =1, 2, 3,
4, 5 is obtained from the corresponding column v of W-1 according with
proposition 14.
̂ in all cases is n-balance and only differs
We note that the Matrix 𝑊
from W in that one of the rows has been replaced by its complement (dual
coalition)
̂ above when n = 5
Had we began our transfer analysis say with 𝑊
11100
01110
̂
𝑊 = 00111 ,
10010
( 0 0 1 1 0)
1⁄2
1⁄2
1 = ̂𝛾
1⁄2
( −1 )
1 −1 0
1
0
0 2 0
0 −2
−1 1
̂
𝑊 =
1 −1 0 −1
2
2
−1
1 0
1
0
0 2
0 −2)
( 0
The type of admissible utility transfers that result when the coalition
corresponding to v = 5 (the collection with negative balancing coefficient)
removed, is given by xT = ( 0, -2, 2, 0, -2) .
- 33 -
is
We may observe that the structure of the utility transfers is identical to
the one of case v= 5 above, but in the opposite direction.
By examining the n-balanced collections in example 2 above, we note
that the coalitions with negative coefficients hinder the strategic possibilities
of both players 2 and 3 since from extreme points A and D each player can
leave the restriction that is binding him to a zero payoff and use the coalition
structure where player 1 have no bargaining alternative.
Remark 3:
Note that the admissible utility transfers that may emerge when
a coalition with negative balancing coefficient is removed are such that some
or all the players in the removed coalition will benefit from the transfer since
̂5∗  > 0. However the transfers brings an overall value loss since 𝐽𝑡  < 0 .
𝑊
That is the transfer moves from an extended imputation of attainable claims x
to another one y = x +  with corresponding value levels y(N) < x(N).
Here,
̂𝑖∗  = 0 𝑓𝑜𝑟 𝑖 ≠ 5, 𝑎𝑛𝑑 𝑊
̂5∗  > 0. 𝑇ℎ𝑢𝑠, 𝑊
̂  ≥ 0,
𝑊
𝐽𝑡  = −1 < 0
But this conditions according to the Farkas lemma occurs only when
the collection is not nn-balanced.
- 34 -
Proposition 15 (Farkas Lemma) Given a cover C
of N with characteristic
matrix W,
Either the system
Wt = J
  0 has a solution
Or the system
W  0, Jt  < 0 has a solution
But not both can have a solution.
Corollary: If there exist a utility transfer  admissible by some p-balanced
sub-collection of a nn-balanced coverC
such that W  0 then the utility
transfer must be necessarily pareto-optimal.
In brief, if a p-balanced collection F with |F |= k linearly
independent subsets, k < n, by proposition 5, it must admit utility
transfers. However, the corollary to proposition7 tells us that such
admissible transfers must be pareto-optimal. So if 𝑥̂ is an extreme
point of the rationality set, and C
for ̂𝑥
and F
C
is the corresponding l-balanced cover
, then C
coefficients non-positive.
- 35 -
must have (n-k)
of its balancing
Strategic-equilibrium
We have shown with examples that n-balanced covers are not
strategically stable because removing the coalitions will not hinder
the bargaining alternatives of players and on the contrary will give
opportunity for admissible utility transfers that will benefit at least
some of the players in the removed coalition so the players may not
be willing to stick to such coalitions that act as restraints. This
shows that an structural equilibrium may not be strategically stable.
Such condition is clearly corrected when there exist no admissible
utility transfers with W  0 and Jt  < 0.
That is, a pair
[ X(C ),C ] is an strategic-equilibrium for the
the game  =(N,v) if the cover support C for x in X(C ) is a
nn-balanced covering collection of N.
Clearly we have a strong-strategic–equilibrium if our cover has pbalanced collection of n-linearly-independent coalitions. In this case
X(C ) = {x°} This strong-equilibrium pair will be denoted simply
- 36 -
as
[x°,C ]. If the corresponding nn-balanced collection happens to
be w-balanced, the equilibrium is said to be a weak-strategic
equilibrium.
Thus, in any cooperative game we may have as many strong-strategic
equilibriums as there are p-balanced collections of subsets of N consisting of
n-linearly independent subsets of N. Clearly, Not all of these equilibriums
may satisfy individual and coalitional rationality.
If an strategic-equilibrium is in the rationality set we say that such
equilibrium is a fundamental strategic-equilibrium for the game  = ( N, v ).
Proposition 6
Any extreme point 𝑥̂ in ℛ( ) has
negatively
balanced cover support unless 𝑥̂ = x° where
Jt x°= min Jt x subject to 𝑉𝑥 ≥ v .
Proof: the dual to the above linear program is given by
max t v subject to tV = Jt , 0
The primal and dual problems are convex polyhedral sets bounded below and
above respectively. Hence both primal and dual problem have optimal
solutions12 x°, ° respectively. Further, Jt x° = °tv with ° 0 . Any other
12
See Owen G.(1982) p
- 37 -
extreme solution 𝑥̂ to the minimization problem must have 𝑥̂ (N) = x°(N)
and the corresponding balancing vector ̂ must be non negative .
As corollary to proposition 16 , we have the following theorems.
THEOREM III The pair [ X(C ),C ] constitute a strategic-equilibrium
for the game  = (N, v) if and only if the corresponding covering collection
C of N is nn--balanced .
THEOREM IV (Existence) Every cooperative game  =(N,v) has a
fundamental strategic equilibrium.
All linearly defined systems are relative invariants under strategic
equivalence.
THEOREM V (Invariance) Any equilibrium (structural or strategic)
for a game  = (N, v) is a relative invariant under strategic equivalence.
The above theorems are consequences of the LP characterization our
equilibriums . Specific proofs are given in 13
13
Turbay G.J.(1976) “On value theories for N-person cooperative games”, Doctoral Dissertation RICE UNIVERSITY.
- 38 -
Clearly any extreme point of the rationality set (basic solution) has a
l-balanced cover support. If the l-balanced cover support for 𝑥̂ happens to
be nn-balanced then 𝑥̂ must be an optimal solution to the primal and the
corresponding non-negative balanced vector
̂  0 must be a solution to the
dual problem.
Given an nn-balanced linearly independent cover C of N with
|C | = n , and let C
|C
+
+
= { Ci | i > 0 } and C 0 = { Ci | i = 0 } so that
| + |C 0 |=n.
If a feasible extreme extended imputation ̂𝑥 for the primal problem has
a nn-balanced cover C with | C | = n and the number of positive
coefficients p = | C
+
| = n, then x° = 𝑥̂ is the unique solution to the primal
problem.
If 0 < p <n then |C 0 | = n -p > 0 , the dimension of the transfer space
is (n-p) and the number of optimal extreme points is (n-p) and the equilibrium
is a weak-strategic equilibrium.
Remar k 4: The strategic-equilibrium for any cooperative game may be
obtained by using the dual simplex method for LP problems. However, as with
the covering problem in integer programming the nature of the constraint
gives high degrees of degeneracy.
- 39 -
A heuristic procedure for finding the fundamental-strategic equilibrium
of a game is to place in descending order the coalitions according to the perplayer- characteristic function value (S) =
𝑣(𝑆)
|𝑆|
and form groups of n linearly
independent nn-balanced sets with the higher possible values for (S) and
obtain the corresponding unique supported extended imputation. If we can
produce one that satisfy individual and coalitional rationality, such extended
imputation and corresponding cover support constitutea fundamental strategic
equilibrium for the game.
“Systems thinking” view of the strategic-equilibrium
The fundamental strategic equilibrium identified for cooperative games
is not necessarily a solution to the game. It may be thought of as a
fundamental attractor from which all solutions to the game may emerge.
The following views can be traced to vN-M heuristic “systems
thinking” views in the TGEB:
 In relation to the symmetric solution of the 3-person zero sum
game:
- 40 -
(1) Solutions are not simply outcomes, but a system of
interrelated outcomes; none of them, by itself, is to be
considered a solution.
(2) The outcomes in the solution are not events but “possible
events”, so solutions are conditional interrelated sets of
outcomes.
 In relation to the discriminatory solutions:
(3) These solutions were never thought to exist in advance,
they simply emerge as sets that satisfy the stability
conditions : “… these situations arose even in the
extremely simple framework of the zero-sum….”
 Regarding composition and decomposition of extended
imputations:
(4) brings out the possibility of “viewing as one” two separate
occurrences”
The above four basic points are reflected in our approach mainly as follows:
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 Rational-attainable extended-imputations with their
corresponding covering support represent conditional systems.
 Through one such extended-imputation we are viewing as “one”
different possible occurrences
 The possible occurrences are related among them by the fact that
they represent bargaining –alternatives: That is, mutually
exclusive possibilities for the players that may be realizable.
 In each interrelated alternative the players can obtain the
bargaining claim prescribed by the extended imputation in
consideration.
Example 4: Consider a 4–person game with characteristic function given by
200 if |S| = 4
v(S) =
120 if |S| =3 and player 1 is not in S
120 if |S| =2 and player 1 is in S
0
otherwise
The extended-imputation x° = (80, 40, 40, 40) t is rational attainable
extended imputation with p-balanced cover support given by
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C = { {1, 2}, {1, 3}, {1, 4}, {2, 3, 4}}. And corresponding balancing vector
° = (1/3, 1/3, 1/3, 2/3) t . hence the pair [x°, C ] is a fundamental-strong
equilibrium the game .
It represents the following conditional system of interrelated possible
occurrences:
(80, 40, 0, 0) if
{1,2}
occurs
(80, 0, 40, 0) if
{1,3}
occurs
(80, 0, 0, 40) if
{1,4}
occurs
( 0, 40, 40, 40) if
{2, 3, 4}
occurs
The fundamental strategic-equilibrium is a pair [x°, C ] that all0w us
to “view as one”, one extended imputation supported by one cover structure
that describes four possible occurrences. The extended imputation x° is
actually the composition of four payoff vectors (extended imputations) for the
four corresponding possible coalitional eventualities.
Strategic-equilibrium and vN-M non discriminatory solutions
Continuing with the example above, if the players in the equilibrium
cover form syndicates, they could realize binding agreements so as to bargain
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with the excluded players for is incremental value contribution if the grand
coalition is to form. Thus,
If players 1 and 2 form a syndicate [1, 2] they may possibly agree:
(1) To form a syndicate so as to secure the characteristic function value to the
coalition v{1, 2} = 120
(2) To split the 120 as prescribed by x°= (80, 40, 40, 40)
t
and use the
corresponding “syndicate dividends” as disagreement payoffs for coalition
{1,2} d = (80, 40, 0, 0)
(3) To split with coalition {3, 4} their incremental contribution ( if the grand
coalition is to form ) e34= v(N) - v({1, 2}). So that a syndicate external
division rate  must be established with the excluded coalition {3,4} in the
negotiation of e34.
(4) To split the proceeds of the syndicate negotiations according to a
prescribed syndicate internal division rate.
The possible outcomes that may occur if the syndicate [1, 2]forms and
the corresponding binding agreements are accepted by the players, are given
by the following conditional system described by a generic row vector of
imputations :
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If [3, 4] form a counter-syndicate with internal splitting rate 
x1
80 +   e34
With
x2
x3
40 (1-) e34
x4
(1-) e34
(1-)(1-)e34
0< , ,  <1
If players 3 and 4 negotiate separately we obtain:
x1
80 +  (1- (3 +4 )) e34
With
x2
40+(1-) (1-(3 +4 ))e34
x3
x4
3 e34
4 e34
0< , , 3, 4 <1
We could continue the same procedure for all possible syndicates.
The above constructive description of obtaining possible outcomes14,
when applied to the general sum 3-person cooperative game, the possible
outcomes description emerge as
vN-M non-discriminatory “objective”
solutions.
This is the subject of forthcoming papers entitled “Rethinking the solutions to the general-sum 3-person
cooperative game” and “ The Stronger Player Paradox”
14
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References
Bondareva O. N
Farkas
Fredholm
Hichmann and Hirsh
Owen G
Shapley L.
Turbay G. J.
Von Neuman J. and O. Motgenstern
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