Extension of Monotonic Functions and Representation of Preferences

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

[1]  Roee Teper,et al.  Learning the Krepsian state: Exploration through consumption , 2016, Games Econ. Behav..

[2]  M. Richter.  Revealed Preference Theory , 1966 .

[3]  R. Aumann.  UTILITY THEORY WITHOUT THE COMPLETENESS AXIOM , 1962 .

[4]  Larry E. Jones,et al.  A Competitive Model of Commodity Differentiation , 1984 .

[5]  G. Bosi,et al.  Normally preordered spaces and continuous multi-utilities , 2016 .

[6]  Christopher P. Chambers,et al.  On Multiple Discount Rates , 2018 .

[7]  Efe A. Ok,et al.  Coalitional Expected Multi‐Utility Theory , 2019, Econometrica.

[8]  Barton L. Lipman,et al.  REPRESENTING PREFERENCES WITH A UNIQUE SUBJECTIVE STATE SPACE , 2001 .

[9]  Hiroki Nishimura.  The transitive core: inference of welfare from nontransitive preference relations , 2018 .

[10]  Continuous extension of preferences , 1993 .

[11]  Efe A. Ok,et al.  Utility representation of an incomplete and nontransitive preference relation , 2016, J. Econ. Theory.

[12]  Andreu Mas-Colell,et al.  The Price Equilibrium Existence Problem in Topological Vector Lattice s , 1986 .

[13]  K. Basu,et al.  Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian , 2003 .

[14]  Igor Kopylov.  Canonical utility functions and continuous preference extensions , 2016 .

[15]  Efe A. Ok,et al.  On the multi-utility representation of preference relations ✩ , 2011 .

[16]  Efe A. Ok,et al.  A Comprehensive Approach to Revealed Preference Theory , 2016 .

[17]  T. Bewley.  A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities , 1991 .

[18]  Paulo Klinger Monteiro,et al.  Some results on the existence of utility functions on path connected spaces , 1987 .

[19]  Strictly monotonic preferences on continuum of goods commodity spaces , 2009 .

[20]  M. Marinacci,et al.  A Smooth Model of Decision Making Under Ambiguity , 2003 .

[21]  D. Schmeidler,et al.  A More Robust Definition of Subjective Probability , 1992 .

[22]  G. Herden.  On a lifting theorem of Nachbin , 1990 .

[23]  Andreu Mas-Colell,et al.  A model of equilibrium with differentiated commodities , 1975 .

[24]  G. Mehta.  Topological Ordered Spaces and Utility Functions , 1977 .

[25]  S. Greco,et al.  Rational preference and rationalizable choice , 2018, Economic Theory.

[26]  S. Greco,et al.  Necessary and possible preference structures , 2013 .

[27]  I. Gilboa,et al.  Objective and Subjective Rationality in a Multiple Prior Model , 2010 .

[28]  G. Debreu.  Mathematical Economics: Representation of a preference ordering by a numerical function , 1983 .

[29]  Nicholas C. Yannelis,et al.  The Core of an Economy Without Ordered Preferences , 1991 .

[30]  G. Mehta,et al.  Some general theorems on the existence of order-preserving functions , 1988 .

[31]  B. Peleg.  UTILITY FUNCTIONS FOR PARTIALLY ORDERED TOPOLOGICAL SPACES. , 1970 .

[32]  Rosa L. Matzkin,et al.  Testing strictly concave rationality , 1991 .

[33]  Geoffroy de Clippel.  On the redundancy of the implicit welfarist axiom in bargaining theory , 2015, J. Econ. Theory.

[34]  E. Minguzzi,et al.  Normally Preordered Spaces and Utilities , 2011, Order.

[35]  Pawel Dziewulski,et al.  Revealed time preference , 2018, Games Econ. Behav..

[36]  S. Afriat.  THE CONSTRUCTION OF UTILITY FUNCTIONS FROM EXPENDITURE DATA , 1967 .

[37]  Aloisio Araujo,et al.  Lack of Pareto Optimal Allocations in Economies with Infinitely Many Commodities: The Need for Impatience , 1985 .

[38]  Some lifting theorems for continuous utility functions , 1989 .

[39]  T. Bewley.  Existence of equilibria in economies with infinitely many commodities , 1972 .

[40]  Donald J. Brown,et al.  Myopic Economic Agents , 1981 .

[41]  Philip J. Reny,et al.  A Characterization of Rationalizable Consumer Behavior , 2014 .

[42]  Uzi Segal,et al.  The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach , 1987 .

[43]  Juan Carlos Candeal,et al.  The existence of utility functions for weakly continuous preferences on a Banach space , 2006, Math. Soc. Sci..

[44]  Peter P. Wakker,et al.  Continuity of preference relations for separable topologies , 1988 .

[45]  John Duggan,et al.  A General Extension Theorem for Binary Relations , 1999 .

[46]  S. Cerreia-Vioglio.  Objective rationality and uncertainty averse preferences , 2016 .

[47]  H. Varian.  The Nonparametric Approach to Demand Analysis , 1982 .

[48]  E. Prescott,et al.  Stochastic Monotonicity and Stationary Distributions for Dynamic Economies , 1992 .

[49]  Faruk Gul,et al.  Temptation and Self‐Control , 1999 .

[50]  D. Schmeidler.  A Condition for the Completeness of Partial Preference Relations , 1971 .