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On a general scheme in the theory of conflicts
On a General Scheme in the Theory of Conflicts Ferenc
Szidarovszky
Znstitute of Mathematics and Computer Science Karl Marr University of Economics Budapest IX, Dimitrov ter 8, H-1093, Hungary and Systems and Industrial
Engineering
Department
University of Arizona Tucson, Arizona 85721
Transmitted by John Casti
ABSTRACT A general model for conflict situations is presented, and balanced strategies are defined. Sufficient and necessary conditions are presented for the existence of balanced strategies. The methodology is based on the theory of linear inequalities.
1.
INTRODUCTION
Based on the fundamental work of Pawlak [I], increasing attention has been given to solution concepts for conflict situations. Zakowski [2, 31 and Wasowski [4] have introduced abstract formulations of conflict situations and studied their solutions in terms of the existence of balanced strategies. The basic model was given in [2] and [4], and this model has been generalized to an oriented total conflict situation by Zakowski [3]. In all the models the solution can be obtained as the nonnegative solution of a certain system of linear equations. The existence of such solutions has been examined by using a fundamental result of Minkowski 15, p. 1191. In this paper a general model of conflicts will be introduced, and the concept of balanced strategies will be formulated for the general case. Similarly to earlier studies, we shall verify that balanced strategies exist if and only if a certain system of linear equations has a nonnegative solution. In addition, sufficient and necessary conditions will be given by using only the
APPLIED
MATHEMATICS
AND COMPUTATION
Q Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010
34:29-37 (1989)
29
009~3003/89/$03.50
FERENC SZIDAROVSZKY
30
model parameters. We shall also demonstrate that by particular selection of the model parameters all earlier models, concepts, and methods as well can be derived as special cases. 2.
THE GENERAL
MODEL
Let X={x,,r, ,..., x,,, } denote the set of participants in the conflict situation. Assume that there are M types of conflicts. Define
&cm):x
xx
--)
[o,co),
where E(“‘)(xi, x j) is the factor by which aggression (defense) of xi against x j is more difficult than defense (aggression) of rj against xi in a conflict of type m (m = 1,2 ,..., M). Let p(“‘):X+
[O,co)
and
p:X+
[O,co),
where JL(“‘)(x~) and p(ri) give the strength of xi in a conflict of type m and the overall strength of xi, respectively.
DEFINITION. The conflict B = {X; e(l),. . . , E(~); p(I), . . . , pL(M); p} is called balanced if and only if there exist strategies X(“‘): X X X + [0, co) such that
C x(“‘)( Xi
y X
j)
<
/A’“‘(xi)
Vi,m,
(14
Vi,
(lb)
Vi,j,m,
(14
j#i
c m
1 x’““(Xi,xj) =p(xJ j#i
A(“‘)(Xi, Xi) = EyA(“lyxj,
Xi)
where ~$7) = E(“‘)(x~, ri). In this definition the relation (la) represents the fact that no participant can allocate more resources than his/her strength for aggression or defense against his/her enemies in each of the conflict types. The relation (lb) expresses that no participant can allocate more resources than his/her overall strength for aggression (defense) in all types of conflicts together, and Equation (lc) expresses that the strength allocations are proportional to the relative difficulties of aggression and deiense for each pair of participants.
31
Theory of Conflicts
Before formulating the solution methodology for finding balanced strategies some remarks are in order.
REMARK 1. Assume that for an m and a pair (i, j), E$J)EI~) # 1. Then Pqrj, Xi) = X’“‘(+ Xi) = 0. In order to prove this assertion multiply (lc) with (i, j) and with (j, i) to get
h(*)(xi, Xj)A(“‘)(Xj) x,)[l
- EI;I)&)y] = 0.
Then either X(“‘)(xi, rj) or h(m)(xj, xi) is zero, and (lc) implies that the other one is also zero. Define DC”‘) & X X X as D(m)=
((r&rj)~&j;l)&;~~#l)
(m=1,2,...,M).
Then the above discussion implies that for
P”)(Xi, Xj) = 0
(xi, xi) E D(“‘).
REMARK 2. Set M = 1, p(i) = ~1,and D (‘) = 0. Then the oriented conflict model of Zakowski [3] is obtained.
REMARK 3.
Select A4 = 1, p(l) = p again; furthermore let
D(l)=
{ X\DQ3} U
{(Xi>“j)JV(xi>
"j>
=
+
‘}a
where cp is the indicator function of conflict [q(xi, x j) = - 11, cooperation [r&x,, xi) = +l], or neutrality [cp(ri, xi) P DQI, the domain of q]. If in addition
E(!) = ‘I
0
if
(xi, xj) E D(l),
1
if
(xi,xj)
66 D(l),
then the models of Zakowski [2] and Wasowski [4] are obtained as special cases.
FERENC SZIDAROVSZKY
32
For the sake of simplicity introduce the following notation:
by@= p(*‘( Xi))
q!’ = X(*yq, Xj),
ai =
p(q).
If aim) denotes the slack variable for the relation (la), then (la), (lb), and (lc) can be summarized as
c
x’,?‘; + (y$“’ = b,(“’
Vi,m,
Pa>
Vi,
@b)
j#i
C
m
tiTj=aj
C j#i
Observe first that h’r] = 0 if (xi, rj) E Dcm), furthermore that for
A(?! = e!71)h(?! '.I 'I I.’
i > j.
(xi, xi) 6 DC”),
Therefore (2a) and (2b) can be rewritten as follows:
c
c
E!?‘j$(T! + ‘] 1,’
(x,,x,) z D'""
h’,“l! + ai”’ = bj”’
Vi,m,
(3a)
vi,
(3b)
j>i
j
(r,,rj)EDc””
&(“‘=
ci
m
where ci =
xb,!“)m
ai >, 0.
In deriving (3b) we used additionally the definition of the slack variables (y!““. I Hence we have proven the following
THEOREM 1. The nonnegative numbers x’“)(x,, x j) = h’,?) give a balanced strategy ifand only if there exist nonnegative scalars aim) such that
33
Theory of Conflicts SIT’ [i < j, (xi, xi) 4 IN”‘]
(3a) and (3b), and
and a{“‘) satisfy Equations
furthermore
X(17)=
0
if
&iT)x’i”,” if
‘I
(Xi, Xj) E II(“), (xi, xi) 4 Dcrn) and i > j.
It is easy to verify that in the above mentioned special cases REMARK. this theorem gives the main results of Zakowski [2, 31 and [4].
3.
NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF BALANCED STRATEGIES Our first result can be formulated as
THEOREM
2.
There exist balanced strategies for the conflict
situation
if
and only if
for all y,!**) (Vi, m) and zi (Vi)
satisfying the conditions Vm, (xi, xi) 4 lYm),
i < j,
Vi, m.
(54 GW
PROOF. Let A denote the matrix of coefficients of the linear equations (3a), (3b), and define
b=(b’,”
,...,
yqyp
)..., yp ,..., yy) ,..., yp&,...
b$‘,...,
bi”) ,..., b(,M’,c, ,..., c,)‘,
and
J,)‘.
FERENC SZIDAROVSZKY
34
Then the theorem of Minkowski [5, p. 1191 implies that Ax = b has nonnegative solutions if and only if bTy 2 0 for all vectors y such that ATy > 0. In our case these inequalities have the special form (4) and (5a), (5b), respectively. n
REMARK. This result has only slight practical importance, since it depends on the unknowns yi”‘) and z.*.
In the next part of this section necessary and sufficient conditions will be presented via the original model parameters only. For the sake of simplicity we assume that DC”) = 0 for all m; the general case can be studied in the same way. Observe that for each fixed m, there is at most one negative value yj”‘). Let them be denoted by ypQ)
‘t
k = 1,2 ,..., r.
’
Define &if=
{m,,m,
,..., m,}
and
I=
{i]i=i,foratleastonek}.
Note that the mk’s (k = 1,2,. . . , r) are all different, but the i, values r) may be repeated. If m 4 A, then the inequality (5a) is always satisfied, and for m E Jt’ (5a) is satisfied if and only if (k = 1,2,...,
(6) where m = mk. On defining
the relations (5b) are satisfied if and only if zi >
-
Y!*(~))
Vi.
Assume now that r; mk, i,, yi,“k), k = 1,2,. . . , r; and ~l(~(~)),i = 1,2,. . . , N, are fixed. Consider first the case when r < M, that is, &’ # { 1,2,. . . , M].
35 Note that the fixed values must satisfy the following relations:
if
iEI
20
y(m(i))
’
1
=yi;:(yiFk)) if
iEI
1 vi*
(7)
Hence the left hand side of (4) is bounded from below by
+
c
c ciy.y)),
xbjm’yjm)-
m4A
i
i
and by using the relation (7) we observe that this expression is not smaller than
where
=
c
b;““$’
-
bin’).
j#i
(Note that this lower bound can be attained by a particular selection of the y!“) t ‘s.) This expression can be rewritten as
Since the integers r, mk, i,, the positive numbers JzJ/~~“)], and the nonnegative numbers y!m(i)) I, can be selected arbitrarily, this last expression is 1 f i G+Z
FERENC SZIDAROVSZKY
nonnegative if and only if
c
A(/) + ci > 0,
a=M
(8) b,!“) - ci > 0,
where 0~ &s {1,2,..., M} is an arbitrary set, and m is arbitrary. Consider next the case where ./I = { 1,2,. . . , M }, and assume again that mk,i,,yl(kmk’,
k=1,2
,...,
M,
~j”‘(~“,
i=1,2
,... ,N,
and fixed. Note that in this case the fixed values must satisfy the following relations:
y!m(i))
’ i
=r”=ic
if
iEI,
>rnkin(~~~~)ly,(~~)I)
if
i@Z.
(Yp)
(9)
Therefore the left hand side of (4) is not greater than
where u~“‘)>,O for all m=l,2,..., iEZ
M and i 6 1. Here we assume that for
Since the indices i,, i,, . . . , i,, the absolute values jyjknlk)J,k = 1,2,. . . , M, and the nonnegative numbers ui”‘) for i P Z and m = 1,2,. . . , M are arbitrary, this expression is always nonnegative if and only if
b!“‘-ciao t
Vi,m
(10)
37
Theory of Conflicts and
771
forallsubsets
.@#.X&{l,Z J(i>
={1,2
jeJ(i)
E Jr
,...,
M),
,..., N}-i
i g {1,2 ,..., N} -i Since (11) is stronger following result:
THEOREM 3.
”
pj~&,and if
.M={1,2
,...,
M},
otherwise.
than the first inequality
There exist balanced
_
strategies
in (8), we have proved
the
if and only if the relations
(10) and (11) hold.
REMARK. In the case of the model of Zakowski [3], M = 1 and ci = 0 for all i. In that special case our conditions reduce to A(:‘) > 0 for all m, i. This condition is the same as given in Theorem 1 in Zakowski’s paper. We can similarly show that the other models and their solution criteria are also special cases of our results.
REFERENCES Z. Paw&, On conflicts, Internat. J. Man-Machine Stud. 21:127-134 (1984). W. Zakowski, Investigation of the balanced situation in theory of conflicts, Bull. Pal. Acad. Sci. Math. 32(7-8):379-382 (1984). W. Zakowski, The balanced strategy in an oriented total conflict, Bull. Pal. Acad. Sci. Math. 35(7-8):525-530 (1987). J. Wasowski, Existence of the balanced strategy in theory of conflicts, Bull. Pal. Acad. Sci. Math. 35(9-10):53-537 (1987). E. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1961.
Szidarovszky
Znstitute of Mathematics and Computer Science Karl Marr University of Economics Budapest IX, Dimitrov ter 8, H-1093, Hungary and Systems and Industrial
Engineering
Department
University of Arizona Tucson, Arizona 85721
Transmitted by John Casti
ABSTRACT A general model for conflict situations is presented, and balanced strategies are defined. Sufficient and necessary conditions are presented for the existence of balanced strategies. The methodology is based on the theory of linear inequalities.
1.
INTRODUCTION
Based on the fundamental work of Pawlak [I], increasing attention has been given to solution concepts for conflict situations. Zakowski [2, 31 and Wasowski [4] have introduced abstract formulations of conflict situations and studied their solutions in terms of the existence of balanced strategies. The basic model was given in [2] and [4], and this model has been generalized to an oriented total conflict situation by Zakowski [3]. In all the models the solution can be obtained as the nonnegative solution of a certain system of linear equations. The existence of such solutions has been examined by using a fundamental result of Minkowski 15, p. 1191. In this paper a general model of conflicts will be introduced, and the concept of balanced strategies will be formulated for the general case. Similarly to earlier studies, we shall verify that balanced strategies exist if and only if a certain system of linear equations has a nonnegative solution. In addition, sufficient and necessary conditions will be given by using only the
APPLIED
MATHEMATICS
AND COMPUTATION
Q Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010
34:29-37 (1989)
29
009~3003/89/$03.50
FERENC SZIDAROVSZKY
30
model parameters. We shall also demonstrate that by particular selection of the model parameters all earlier models, concepts, and methods as well can be derived as special cases. 2.
THE GENERAL
MODEL
Let X={x,,r, ,..., x,,, } denote the set of participants in the conflict situation. Assume that there are M types of conflicts. Define
&cm):x
xx
--)
[o,co),
where E(“‘)(xi, x j) is the factor by which aggression (defense) of xi against x j is more difficult than defense (aggression) of rj against xi in a conflict of type m (m = 1,2 ,..., M). Let p(“‘):X+
[O,co)
and
p:X+
[O,co),
where JL(“‘)(x~) and p(ri) give the strength of xi in a conflict of type m and the overall strength of xi, respectively.
DEFINITION. The conflict B = {X; e(l),. . . , E(~); p(I), . . . , pL(M); p} is called balanced if and only if there exist strategies X(“‘): X X X + [0, co) such that
C x(“‘)( Xi
y X
j)
<
/A’“‘(xi)
Vi,m,
(14
Vi,
(lb)
Vi,j,m,
(14
j#i
c m
1 x’““(Xi,xj) =p(xJ j#i
A(“‘)(Xi, Xi) = EyA(“lyxj,
Xi)
where ~$7) = E(“‘)(x~, ri). In this definition the relation (la) represents the fact that no participant can allocate more resources than his/her strength for aggression or defense against his/her enemies in each of the conflict types. The relation (lb) expresses that no participant can allocate more resources than his/her overall strength for aggression (defense) in all types of conflicts together, and Equation (lc) expresses that the strength allocations are proportional to the relative difficulties of aggression and deiense for each pair of participants.
31
Theory of Conflicts
Before formulating the solution methodology for finding balanced strategies some remarks are in order.
REMARK 1. Assume that for an m and a pair (i, j), E$J)EI~) # 1. Then Pqrj, Xi) = X’“‘(+ Xi) = 0. In order to prove this assertion multiply (lc) with (i, j) and with (j, i) to get
h(*)(xi, Xj)A(“‘)(Xj) x,)[l
- EI;I)&)y] = 0.
Then either X(“‘)(xi, rj) or h(m)(xj, xi) is zero, and (lc) implies that the other one is also zero. Define DC”‘) & X X X as D(m)=
((r&rj)~&j;l)&;~~#l)
(m=1,2,...,M).
Then the above discussion implies that for
P”)(Xi, Xj) = 0
(xi, xi) E D(“‘).
REMARK 2. Set M = 1, p(i) = ~1,and D (‘) = 0. Then the oriented conflict model of Zakowski [3] is obtained.
REMARK 3.
Select A4 = 1, p(l) = p again; furthermore let
D(l)=
{ X\DQ3} U
{(Xi>“j)JV(xi>
"j>
=
+
‘}a
where cp is the indicator function of conflict [q(xi, x j) = - 11, cooperation [r&x,, xi) = +l], or neutrality [cp(ri, xi) P DQI, the domain of q]. If in addition
E(!) = ‘I
0
if
(xi, xj) E D(l),
1
if
(xi,xj)
66 D(l),
then the models of Zakowski [2] and Wasowski [4] are obtained as special cases.
FERENC SZIDAROVSZKY
32
For the sake of simplicity introduce the following notation:
by@= p(*‘( Xi))
q!’ = X(*yq, Xj),
ai =
p(q).
If aim) denotes the slack variable for the relation (la), then (la), (lb), and (lc) can be summarized as
c
x’,?‘; + (y$“’ = b,(“’
Vi,m,
Pa>
Vi,
@b)
j#i
C
m
tiTj=aj
C j#i
Observe first that h’r] = 0 if (xi, rj) E Dcm), furthermore that for
A(?! = e!71)h(?! '.I 'I I.’
i > j.
(xi, xi) 6 DC”),
Therefore (2a) and (2b) can be rewritten as follows:
c
c
E!?‘j$(T! + ‘] 1,’
(x,,x,) z D'""
h’,“l! + ai”’ = bj”’
Vi,m,
(3a)
vi,
(3b)
j>i
j
(r,,rj)EDc””
&(“‘=
ci
m
where ci =
xb,!“)m
ai >, 0.
In deriving (3b) we used additionally the definition of the slack variables (y!““. I Hence we have proven the following
THEOREM 1. The nonnegative numbers x’“)(x,, x j) = h’,?) give a balanced strategy ifand only if there exist nonnegative scalars aim) such that
33
Theory of Conflicts SIT’ [i < j, (xi, xi) 4 IN”‘]
(3a) and (3b), and
and a{“‘) satisfy Equations
furthermore
X(17)=
0
if
&iT)x’i”,” if
‘I
(Xi, Xj) E II(“), (xi, xi) 4 Dcrn) and i > j.
It is easy to verify that in the above mentioned special cases REMARK. this theorem gives the main results of Zakowski [2, 31 and [4].
3.
NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF BALANCED STRATEGIES Our first result can be formulated as
THEOREM
2.
There exist balanced strategies for the conflict
situation
if
and only if
for all y,!**) (Vi, m) and zi (Vi)
satisfying the conditions Vm, (xi, xi) 4 lYm),
i < j,
Vi, m.
(54 GW
PROOF. Let A denote the matrix of coefficients of the linear equations (3a), (3b), and define
b=(b’,”
,...,
yqyp
)..., yp ,..., yy) ,..., yp&,...
b$‘,...,
bi”) ,..., b(,M’,c, ,..., c,)‘,
and
J,)‘.
FERENC SZIDAROVSZKY
34
Then the theorem of Minkowski [5, p. 1191 implies that Ax = b has nonnegative solutions if and only if bTy 2 0 for all vectors y such that ATy > 0. In our case these inequalities have the special form (4) and (5a), (5b), respectively. n
REMARK. This result has only slight practical importance, since it depends on the unknowns yi”‘) and z.*.
In the next part of this section necessary and sufficient conditions will be presented via the original model parameters only. For the sake of simplicity we assume that DC”) = 0 for all m; the general case can be studied in the same way. Observe that for each fixed m, there is at most one negative value yj”‘). Let them be denoted by ypQ)
‘t
k = 1,2 ,..., r.
’
Define &if=
{m,,m,
,..., m,}
and
I=
{i]i=i,foratleastonek}.
Note that the mk’s (k = 1,2,. . . , r) are all different, but the i, values r) may be repeated. If m 4 A, then the inequality (5a) is always satisfied, and for m E Jt’ (5a) is satisfied if and only if (k = 1,2,...,
(6) where m = mk. On defining
the relations (5b) are satisfied if and only if zi >
-
Y!*(~))
Vi.
Assume now that r; mk, i,, yi,“k), k = 1,2,. . . , r; and ~l(~(~)),i = 1,2,. . . , N, are fixed. Consider first the case when r < M, that is, &’ # { 1,2,. . . , M].
35 Note that the fixed values must satisfy the following relations:
if
iEI
20
y(m(i))
’
1
=yi;:(yiFk)) if
iEI
1 vi*
(7)
Hence the left hand side of (4) is bounded from below by
+
c
c ciy.y)),
xbjm’yjm)-
m4A
i
i
and by using the relation (7) we observe that this expression is not smaller than
where
=
c
b;““$’
-
bin’).
j#i
(Note that this lower bound can be attained by a particular selection of the y!“) t ‘s.) This expression can be rewritten as
Since the integers r, mk, i,, the positive numbers JzJ/~~“)], and the nonnegative numbers y!m(i)) I, can be selected arbitrarily, this last expression is 1 f i G+Z
FERENC SZIDAROVSZKY
nonnegative if and only if
c
A(/) + ci > 0,
a=M
(8) b,!“) - ci > 0,
where 0~ &s {1,2,..., M} is an arbitrary set, and m is arbitrary. Consider next the case where ./I = { 1,2,. . . , M }, and assume again that mk,i,,yl(kmk’,
k=1,2
,...,
M,
~j”‘(~“,
i=1,2
,... ,N,
and fixed. Note that in this case the fixed values must satisfy the following relations:
y!m(i))
’ i
=r”=ic
if
iEI,
>rnkin(~~~~)ly,(~~)I)
if
i@Z.
(Yp)
(9)
Therefore the left hand side of (4) is not greater than
where u~“‘)>,O for all m=l,2,..., iEZ
M and i 6 1. Here we assume that for
Since the indices i,, i,, . . . , i,, the absolute values jyjknlk)J,k = 1,2,. . . , M, and the nonnegative numbers ui”‘) for i P Z and m = 1,2,. . . , M are arbitrary, this expression is always nonnegative if and only if
b!“‘-ciao t
Vi,m
(10)
37
Theory of Conflicts and
771
forallsubsets
.@#.X&{l,Z J(i>
={1,2
jeJ(i)
E Jr
,...,
M),
,..., N}-i
i g {1,2 ,..., N} -i Since (11) is stronger following result:
THEOREM 3.
”
pj~&,and if
.M={1,2
,...,
M},
otherwise.
than the first inequality
There exist balanced
_
strategies
in (8), we have proved
the
if and only if the relations
(10) and (11) hold.
REMARK. In the case of the model of Zakowski [3], M = 1 and ci = 0 for all i. In that special case our conditions reduce to A(:‘) > 0 for all m, i. This condition is the same as given in Theorem 1 in Zakowski’s paper. We can similarly show that the other models and their solution criteria are also special cases of our results.
REFERENCES Z. Paw&, On conflicts, Internat. J. Man-Machine Stud. 21:127-134 (1984). W. Zakowski, Investigation of the balanced situation in theory of conflicts, Bull. Pal. Acad. Sci. Math. 32(7-8):379-382 (1984). W. Zakowski, The balanced strategy in an oriented total conflict, Bull. Pal. Acad. Sci. Math. 35(7-8):525-530 (1987). J. Wasowski, Existence of the balanced strategy in theory of conflicts, Bull. Pal. Acad. Sci. Math. 35(9-10):53-537 (1987). E. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1961.