# Utility representation theorems for Debreu separable preorders

Published on May 1, 2012in Journal of Mathematical Economics0.634
· DOI :10.1016/j.jmateco.2012.02.005
Gerhard Herden11
Estimated H-index: 11
,
Estimated H-index: 5
(Central Economics and Mathematics Institute)
Abstract
We prove the existence of arbitrary (resp., semicontinuous, continuous) utility representations for arbitrary (resp., semicontinuous, continuous) preorders satisfying some weakened Debreu order separability conditions. In this way we widely generalize a classical result for total preorders that essentially is due to Debreu.
• References (16)
• Citations (7)
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References16
#1Itzhak Gilboa (TAU: Tel Aviv University)H-Index: 39
#2Fabio Maccheroni (Bocconi University)H-Index: 29
Last. David Schmeidler (OSU: Ohio State University)H-Index: 47
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A decision maker (DM) is characterized by two binary relations. The first reflects choices that are rational in an "objective" sense: the DM can convince others that she is right in making them. The second relation models choices that are rational in a "subjective" sense: the DM cannot be convinced that she is wrong in making them. In the context of decision under uncertainty, we propose axioms that the two notions of rationality might satisfy. These axioms allow a joint representation by a sing...
#1Gianni Bosi (UniTS: University of Trieste)H-Index: 12
#2Gerhard HerdenH-Index: 11
The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial orde...
#1Gerhard HerdenH-Index: 11
#2Andreas PallackH-Index: 4
Abstract One of the best known theorems in order theory, mathematical logic, computer sciences and mathematical social sciences is the Szpilrajn Theorem which states that every partial order can be refined to a linear order. Since in mathematical social sciences one frequently is interested in continuous linear orders or preorders, in this paper the continuous analogue of the Szpilrajn Theorem will be discussed.
#1José Carlos R. Alcantud (University of Salamanca)H-Index: 19
#2Carlos Rodríguez-Palmero (University of Valladolid)H-Index: 7
Abstract In this paper we characterize the existence of semicontinous weak utilities for acyclic binary relations. We shall reobtain directly from that result sufficient conditions that are available in the literature.
#1Gianni BosiH-Index: 1
We present a separation theorem in pairwise normally preordered bitopological spaces, which slightly generalizes both a well known separation theorem by Nachbin in normally preordered topological spaces, and a separation theorem by Kelly in pairwise normal topological spaces. Based on this result, we give necessary and su‐cient conditions for the existence of semicontinuous order-preserving functions on such spaces. Further, we discuss the existence of upper semicontinuous order-preserving funct...
Abstract This is the first of two papers on (continuous) utility functions. While the second paper is mainly dedicated to the extension problem on (continuous) utility functions this paper presents general existence theorems on (continuous) utility functions on arbitrary preordered (topological) spaces. At first generalizations of the classical Birkhoff representation theorem are proved. Then as the main result of this paper a very general theorem which presents necessary and sufficient conditio...
#1V. L. LevinH-Index: 1
#1Gerard DebreuH-Index: 1
#1Jaap Van BrakelH-Index: 1
#1Andrew Mas-Colell (University of California, Berkeley)H-Index: 1
Abstract The Walrasian equilibrium existence theorem is reproved without the assumptions of complete of transitive preferences.
Cited By7
#1Paolo Bevilacqua (UniTS: University of Trieste)H-Index: 7
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Last. Magalì E. Zuanon (University of Brescia)H-Index: 1
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#1Paolo BevilacquaH-Index: 7
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Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space.
#1Gianni Bosi (UniTS: University of Trieste)H-Index: 12
#2Magalì ZuanonH-Index: 4
Abstract We introduce the concept of quasi upper semicontinuity of a not necessarily total preorder on a topological space and we prove that there exists a maximal element for a preorder on a compact topological space provided that it is quasi upper semicontinuous. In this way, we generalize many classical and well known results in the literature. We compare the concept of quasi upper semicontinuity with the other semicontinuity concepts to arrive at the conclusion that our definition can be vie...
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#2Kevin Munger (NYU: New York University)H-Index: 5
Part I. Basics: 1. The analysis of politics 2. Becoming a group: the constitution 3. Choosing in groups: an intuitive presentation 4. The formal analytics of choosing in groups Part II. Spatial Theory: 5. Politics as spatial competition 6. Two dimensions: elusive equilibrium Part III. Extensions: Collective Choice, Uncertainty, and Collective Action: 7. The collective-choice problem: impossibility 8. Uncertainty 9. Voting as a collective-action problem Solutions to selected problems.
#1Alfio Giarlotta (University of Catania)H-Index: 11
A NaP-preference (necessary and possible preference) on a set A is a pair $${\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}$$ of binary relations on A such that its necessary component $${\succsim^{^{_N}} \!\!}$$ is a partial preorder, its possible component $${\succsim^{^{_P}} \!\!}$$ is a completion of $${\succsim^{^{_N}} \!\!}$$, and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-...
#1Gianni Bosi (UniTS: University of Trieste)H-Index: 12
#2Magalì ZuanonH-Index: 4
Given an interval order on a topological space, we characterize its representability by means of a pair of upper semicontinuous real-valued functions. This characterization is only based on separability and continuity conditions related to both the interval order and one of its two traces. As a corollary, we obtain the classical Rader’s theorem concerning the existence of an upper semicontinuous representation for an upper semicontinuous total preorder on a second countable topological space.
#2Gianni BosiH-Index: 12
Last. Magalì ZuanonH-Index: 4
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We introduce a new kind of representation of a not necessarily total preorder, called strong multi-utility representation, according to which not only the preorder itself but also its strict part is fully represented by a family of multi-objective functions. The representability by means of semicontinuous or continuous multi-objective functions is discussed, as well as the relation between the existence of a strong multi-utility representation and the existence of a Richter-Peleg utility functio...
#1Alfio Giarlotta (University of Catania)H-Index: 11
#2Salvatore Greco (University of Catania)H-Index: 56
A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–...