Szpilrajn-type theorems in economics

Andrikopoulos, Athanasios (2009): Szpilrajn-type theorems in economics.

Abstract

The Szpilrajn "constructive type" theorem on extending binary relations, or its generalizations by Dushnik and Miller [10], is one of the best known theorems in social sciences and mathematical economics. Arrow [1], Fishburn [11], Suzumura [22], Donaldson and Weymark [8] and others utilize Szpilrajn's Theorem and the Well-ordering principle to obtain more general "existence type" theorems on extending binary relations. Nevertheless, we are generally interested not only in the existence of linear extensions of a binary relation R, but in something more: the conditions of the preference sets and the properties which $R$ satisfies to be "inherited" when one passes to any member of some \textquotedblleft interesting\textquotedblright family of linear extensions of R. Moreover, in extending a preference relation $R$, the problem will often be how to incorporate some additional preference data with a minimum of disruption of the existing structure or how to extend the relation so that some desirable new condition is fulfilled. The key to addressing these kinds of problems is the szpilrajn constructive method. In this paper, we give two general "constructive type" theorems on extending binary relations, a Szpilrajn type and a Dushnik-Miller type theorem, which generalize and give a "constructive type" version of all the well known extension theorems in the literature.

• Item Type: MPRA Paper
• Language: English
• Keywords: Consistent binary consistent binary relations, extension theorems, intersection of binary relations
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D6 - Welfare Economics > D60 - General; D - Microeconomics > D0 - General > D00 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

References

Arrow K. J., (1951; second ed. 1963), Social Choice and Individuals Values, Wiley, New York.

Bade S., (2005), Nash equilibrium in games with incomplete preferences, {\it Economic Theory}, {\bf 26}, pp. 309-332.

Blackorby C., Bossert W., and Donaldson D., (1999), Rationalizable solutions to pure population problems, {\it Social Choice and Welfare}, {\bf 16}, pp. 395-407.

Bosi B., Herden G., (2006), On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller, {\it Order}, {\bf 23}, pp. 271–296.

Brightwell R., G., Scheinerman R., E., (1992), Fractional Dimension of Partial Orders, {\it Order}, pp. 139-158.

Chipman J., (1960), The foundations of utility, {\it Econometrica} 28, pp. 193-224.

Clark A. S., (1988), An extension theorem for rational choice functions, {\it Review of Economics Studies}, pp. 485-492.

Donaldson D., Weymark J.A., (1998), A quasi-ordering is the intersection of orderings, {\it Journal of Economic Theory} {\bf 78}, pp. 382-387.

Duggan J., (1999), A General Extension Theorem for Binary Relations, {\it Journal of Economic Theory} {\bf 86}, pp. 1-16.

Dushnik B., Miller E.W., (1941), Partially ordered sets, {\it American Journal of Mathematics} {\bf 63}, pp. 600-610.

Fishburn P. C., (1973), The theory of Social Choice, Princeton University Press, Princeton.

Freixas J., Puente A. M., (2001), A note about games-composition dimension, {\it Discrete Applied Mathematics} {\bf 113}, 265-273.

Hansson B., (1968), Choice structures and preference relations, {\it Synthese} {\bf 18}, pp. 443-458.

Herden J., and Pallack A., (2002), On the continuous analogue of the szpilrajn Theorem I, {\it Mathematical Social Sciences} {\bf 43}, pp. 115-134.

Miller N., (1980), A new solution set for tournament and majority voting: Further graph theoretical approaches to the theory of voting, {\it Amer. J. Polit. Sci.}, {\bf 24}, 68-96.

Nehring K., and Puppe C., (1998), Extended partial orders: a unifying structure for abstract choice theory, {\it Annals of Operations Research} {\bf 80}, 27-48.

Ok A. E., (2002), Utility Representation of an Incomplete Preference Relation, {\it Journal of Economic Theory}, {\bf 104}, 429-449

Rabinovitch I., Rival L., (1979), The rank of a distributive lattice, {\it Discrete Math.}, {\bf 25}, 275-279.

Richter M., (1966), Revealed preference theory, {\it Econometrica} {\bf 34}, pp. 635-645.

Sholomov L. A., (2000), Explicit form of neutral decision rules for basic rationality conditions, {\it Mathematical Social Sciences} {\bf 39}, pp. 81-107.

Stehr Zyklische Ordnungen M. O., (1996), Axiome und einfache Eigenschaften, University of Hamburg, Hamburg, Germany, Diplomarbeit.

Suzumura K., (1976), Remarks on the theory of collective choice, \textit{Economica} \textbf{43}, pp. 381-390.

Szpilrajn E., (1930), Sur l'extension de l'ordre partiel, {\it Fundamenta Matematicae} {\bf 16}, pp. 386-389.

Weymark J.E., (2000), A generalization of Moulin's Pareto extension theorem, {\it Mathematical Social Sciences} {\bf 39}, pp. 235-240.