TITLE:
Optimal Separation of Twin Convex Sets under Externalities
AUTHORS:
Indrajit Mallick
KEYWORDS:
Convex Set, Separation, Externality
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.4 No.8,
August
22,
2014
ABSTRACT: This paper studies the outcomes of independent and interdependent
pair-wise contests between economic agents subject to an optimal external
decision problem for each pair. The external decision maker like the government
or regulator is faced with the problem of how to devise rules and regulations
regarding contests. In this paper, a decision problem is faced under negative
and positive externalities. A pair of entities is represented by disjoint
convex sets in a small area in a neighborhood. I assume that each entity
imposes an equal externality on the other (and the other only) and thus they can
be considered to be twins. Among the group of twins in any neighborhood, there
is a set of twin pairs such that, for each pair in the set, each twin can
impose a strictly negative externality on the other (and the other only), and
this is a potential welfare loss which concerns the decision maker. A
separating hyper-plane can block the negative externalities between any pair of twins given convexity. However, this
can be costly if positive externality from the neighborhood is also
blocked by the separation technology. Thus, this paper compares the pair-wise
utility from separation to that of non-separation. A simple representation of
the decision problem is developed with respect to a single and isolated
neighborhood. A complete characterization of the decision problem is obtained
with a large number of pair-wise intersecting neighborhoods.