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.
Cite this paper
Mallick, I. (2014) Optimal Separation of Twin Convex Sets under Externalities. Advances in Pure Mathematics
, 381-390. doi: 10.4236/apm.2014.48049
 Allen, F. and Douglas, G. (2007) Understanding Financial Crises. Oxford University Press, Oxford.
 Hahn, F. (1985) Money, Growth and Stability. The MIT Press, Cambridge.
 Keynes, J.M. (1936) The General Theory of Employment, Interest and Money. Macmillan, London.
 Ray, D. (1999) Development Economics. Oxford University Press, New Delhi.
 Fudenberg, D. and Jean, T. (2005) Game Theory. Ane Books, India.
 Gale, D. (2000) Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511492310
 Osborne, M.J. and Ariel, R. (1990) Bargaining and Markets. Academic Press, Inc., San Diego.
 Coase, R.H. (1988) The Firm, the Market and the Law. The University of Chicago Press, Chicago.
 Laffont, J.-J. (1989) The Economics of Uncertainty and Information. The MIT Press, Cambridge.
 Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Microeconomic Theory. Oxford University Press, Oxford.
 Bowles, S. (2005) Microeconomics: Behavior, Institutions and Evolution. Oxford University Press, New Delhi.
 Hayami, Y. (2001) Development Economics: From the Poverty to the Wealth of Nations. Oxford University Press, Oxford. http://dx.doi.org/10.1093/0199243972.001.0001
 Azariadis, C. (1993) Intertemporal Macroeconomics. Blackwell Publishers, Cambridge.
 Blanchard, O.J. and Fischer, S. (1989) Lectures on Macroeconomics. The MIT Press, Cambridge.
 Mallick, I. (2002) Strategic Competition in Banking: Theory and Policy. Ph.D. Thesis, Jadavpur University, Kolkata.
 Dewatripont, M. and Tirole, J. (1993) The Prudential Regulation of Banks. The MIT Press, Cambridge.
 Gale, D. (1983) Money: In Disequilibrium. Cambridge University Press, Cambridge.
 Stiglitz, J.E. (1986) Economics of the Public Sector. W. W. Norton & Company, New York.
 Viscusi, W.K., Harrington, J.E. and Vernon, J.M. (2005) Economics of Regulation and Antitrust. The MIT Press, Cambridge.
 Bernard, M. (2012) Conflict, Cooperation and Coordination: Essays in Game Theory and Experimental Economics. Ph.D. Thesis, Stockholm School of Economics, Stockholm.
 Tirole, J. (1988) The Theory of Industrial Organization. The MIT Press, Cambridge.
 Hart, O. (1995) Firms, Contracts, and Financial Structure. Oxford University Press, Oxford.http://dx.doi.org/10.1093/0198288816.001.0001
 Smith, J.M. (1982) Evolution and the Theory of Games. Cambridge University Press, Cambridge.
 Edgeworth, F.Y. (1881) Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Kegan Paul, London.
 Duffie, D. (2012) Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets. 3rd Edition, Princeton University Press, Princeton.
 Shavell, S. (2004) Foundations of Economic Analysis of Law. Harvard University Press, Cambridge.
 Milgrom, P. and Roberts, J. (1992) Economics, Organization and Management. Prentice-Hall, Inc., Upper Saddle River.
 Minkowski, H. (1911) Theorie der Konvexen Korper, Insbesondere Begrundung ihres Oberflachenbegriffs. Gessamelte Abhandlungen II, Leipzig.
 Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton.
 Gale, D. (1960) The Theory of Linear Economic Models. McGraw-Hill, Inc., New York.
 Luenberger, D.G. (1969) Optimization by Vector Space Methods. Wiley, New York.
 Dixit, A.K. (1990) Optimization in Economic Theory. Oxford University Press, Oxford.
 Rudin, W. (1976) Principles of Mathematical Analysis. McGraw-Hill, Inc., London.