Geoscience Reference
In-Depth Information
General conclusions and references follow; but before proceeding to the core of
the exposition, two main points have to be cleared.
The first one is about the heavy formal content of this paper; let it be said that
such formalizing can lead to very practical insights; we refer for this aspect to the
paper by de Koster and Paelinck (
1985
), where a conflict situation is studied by
laying bare its weakest link. We love to mention here that, having had once the
privilege to comment on one of Walter's expositions in the field, we could show the
isomorphism of two approaches.
Second, that we choose a novel approach—hypergraph theory—explains that
few (recent) reference could be quoted, together with the fact that another field of
investigation—to wit spatial econometrics—took up much of the author's attention.
14.2 Negotiation Space
14.2.1 Linear Case
To start with, the following assumptions are put forward:
A
1
: two agents, A and B, negotiate about two divisible issues represented on the
0-1 line;
A
2
: the agents are each commanding one of the issues;
A
3
: they are assumed to handle a linear preference function, which can include
externalities (namely the result obtained by the opponent);
A
4
: constraints in negotiation space are also linear.
The following representation will be explored (Fig.
14.1
):
Figure
14.1
represents the starting positions and also two constraints on the
willingness of concession; for instant agent
A
is willing to move to point
at a
constant substitution rate
a
, and analogously for
B
(constraint substitution rate
b
),
α
and
ʲ
being respectively right and down from the (0,1) and (1,0) points.
A
has a
linear preference function with a substitution rate strictly larger than
a
,
B
one
strictly smaller than
b,
those substitution rates possibly including externalities as
defined previously. The feasible region is the polytope (simplex, later polyhedron)
to the north-east of the relevant area.
The problem has however no feasible solution as logical constraints:
α
x
A
þ
x
B
¼
1
ð
14
:
1a
Þ
y
A
þ
y
B
¼
1
ð
14
:
1b
Þ
prevail, only
exchange
of all or part of the content of the litigious issues being
possible.
Constraints (
14.1a
) and (
14.1b
) imply graphically that the solutions should lie on
circle(s) (segments) centered around the unweighted center of gravity of the
relevant area,
C
, so retention of a non-zero part of the initial “endowments”
(
α
and
ʲ
) precludes final agreement.
