Mathematical Topics on Representations of Ordered Structures and Utility Theory: Essays in Honor of Professor Ghanshyam B. Mehta (Studies in Systems, Decision and Control)

Preface
Acknowledgements
GBM, My Father
GBM, My Advisor
Contents
Contributors
General Utility Theory
Topology and Preference Relations
	1 Introduction
	2 Binary Relations
	3 Utility Functions
	4 The Order Topology
	5 The Quotient Space
	6 Utility Functions and the Quotient Space
	7 Natural Topologies
	8 Monotone and Light Mappings
	9 Continuity of the Inverse Function
	10 The Existence of Utility Functions
	11 Jordan Arcs in a Topological Space
	12 Concluding Remarks
	References
The Existence and the Non-existence of Utility Functions in Order-Theoretic, Algebraic and Topological Environments
	1 Introduction
	2 Preliminaries
	3 The Ordinal Approach
	4 The Algebraic Approach: Totally Preordered Vector Spaces
	5 The Topological Approach: Totally Preordered Topological Spaces
	6 When Order, Algebra and Topology Meet: Totally Preordered Topological Vector Spaces
	7 A Miscellany of Applications
	References
Open Questions in Utility Theory
	1 Introduction
	2 Preliminaries
		2.1 Basic Definitions
		2.2 Numerical Representations of Different Kinds of Orderings
		2.3 Order Topology
		2.4 Debreu\'s Open Gap Lemma and the Continuous Representability of Total Preorders
		2.5 The Problem of the Continuous Representability of Interval Orders and Semiorders
	3 Open Questions Related to Numerical Representability
		3.1 Open Problems About Numerical Representability of Binary Relations
		3.2 Continuous Utility Functions That Represent Different Kinds of Orderings: The State-of-Art
		3.3 Continuous Numerical Representability of Interval Orders and Semiorders
	4 The Continuous Representability Properties in General Topology
		4.1 The Continuous Representability Property (CRP)
		4.2 The Semicontinuous Representability Property (SRP)
		4.3 Yi\'s Extension Property in Topological Spaces
	5 Algebraic Representability of Different Kinds of Orderings
		5.1 Algebraic Utility
		5.2 When Algebra Meets Topology: The Continuous Algebraic Representability Property and Related Items
	6 Alternative Versions of Utility Representations
		6.1 Representations in Codomains Different from the Real Line
		6.2 Utility Through Functional Equations
		6.3 Utility Theory in the Fuzzy Setting
	7 Conclusion
	References
Particular Kinds of Preferences and Representations
Representations of Interval Orders  on Connected Separable Topological Spaces
	1 Introduction
	2 Definitions
	3 Basic Properties
	4 Weak Utility Representation Theorems
	5 Interval Order Representation Theorems
	6 Generalizations
	7 Final Remarks
	8 Proofs of Properties
	References
Searching for a Debreu\'s Open Gap Lemma for Semiorders
	1 Introduction
	2 Preliminaries
	3 Continuous SS-Representability of Semiorders
		3.1 A New Necessary Condition for the Existence of a Continuous SS-Representation
	4 The Implications of the Necessary Conditions on the SS-Representation
	5 A Continuity Approach: ε-Continuity
	6 Concluding Remarks
	References
A Note on Candeal and Induráin\'s Semiorder Separability Condition
	1 Introduction
	2 Notation and Framework
		2.1 Notation and Definitions
		2.2 Chains
		2.3 Representations with Positive Threshold
		2.4 Axioms and Previous Results
	3 Results
		3.1 Noses and Half-Noses
		3.2 Main Results
	4 Discussion
	References
Chain Representations of Nested Families of Biorders
	1 Introduction
	2 Biorders
		2.1 Notation and Framework
		2.2 Biorders
	3 Well-Gradedness and Weak Gradedness
		3.1 Well-Gradedness
		3.2 Weak Gradedness
	4 Nested Families of Biorders on Finite Sets
		4.1 Relation to the Literature
		4.2 Results
		4.3 Proofs
		4.4 Remarks and Extensions
	5 The Infinite Case
		5.1 Definitions
		5.2 Chain Representations in a Linearly Ordered Set
		5.3 Real Chain Representations
	6 Discussion
	References
A Note on Representable Group Topologies
	1 Introduction
	2 g-Natural Non-totally Disconnected Group Topologies
	3 Some Other Results on Order and Topological Algebra
	4 Open Problems
	5 Conclusion
	References
Preferences in Abstract Convex Structures
	1 Introduction
	2 Abstract Convex Structures
		2.1 Some Examples
		2.2 Lattices as Convexity Spaces
		2.3 Separation Axioms
	3 Betweenness
		3.1 Ternary Betweenness and Interval Spaces
		3.2 Betweenness in Convex Structures
	4 Convex Preferences
		4.1 Basic Definitions
		4.2 Convex Preferences in Interval Spaces
	5 Concluding Remarks
	References
Strictly Monotonic Preferences
	1 Introduction
	2 Notation, Definitions and Background
		2.1 Definitions and Notation
		2.2 Existence of Utility Representation
	3 Strictly Monotonic Preferences
	4 Strictly Monotonic Preferences on F(K)
	5 Conclusion and Further Remarks
	References
Extensions of Utility
Continuity and Continuous Multi-utility Representations of Nontotal Preorders: Some Considerations Concerning Restrictiveness
	1 Introduction
	2 Notation and Preliminary Results
		2.1 Basic Definitions Concerning Preorders and Their Continuous Representability
		2.2 Decreasing Separable Systems and Continuity  of Preorders
	3 Characterization of Continuous Representations
	4 Restrictive Results Concerning Continuous Multi-utility Representations
	5 Conclusions
	References
Jointly Continuous Multi-utility Representations of Non-total Preorders
	1 Introduction
	2 Notation and Preliminaries
	3 Utility and Multi-utility Representation on kω-Spaces
	4 Jointly Continuous Multi-utility Representations
		4.1 Properties of Submetrizable kω-Spaces
		4.2 Jointly Continuous Multi-utility Representation on Locally Compact Second Countable Spaces
		4.3 Jointly Continuous Multi-utility Representation on Submetrizable kω-Spaces
	5 Conclusions
	References
Subjective States Without the Completeness Axiom
	1 Introduction
	2 Representation of Preorders
	3 Subjective State Spaces with Incomplete Preferences
		3.1 Two Representation Theorems
		3.2 Uniqueness of the State Space
	4 Aggregation
	5 Concluding Remarks
	6 Appendix
		6.1 Preliminary Results
		6.2 Proof of Theorem 3
		6.3 Proof of Theorem 4
		6.4 Proof of Theorem 5
		6.5 Proof of Theorem 6
	References
Preference for Flexibility: A Continuous Representation in an Ordinal Setup
	1 Introduction
	2 Notation and Terminology
	3 A Basic Representation
		3.1 Proof of Lemma 1
	4 Consumption-Investment Budgeting
	5 Concluding Remarks
	References
The Arrow-Hahn Construction  in a Locally Compact Metric Space
	1 Introduction
	2 Constructive Mathematics in (Very) Brief
	3 Preference Relations
	4 The Arrow-Hahn Theorem Revisited
	5 Concluding Remarks
	References
Continuous Utility Representation  of Fuzzy Preferences
	1 Introduction
	2 Preliminaries
	3 Continuous Utility Representation
	4 Concluding Remarks
	References
Applications into Economics
Ranking Opportunity Sets Taking into Account Similarity Relations Induced by Money-Metric Utility Functions
	1 Introduction
	2 Preliminaries
	3 The Similarity Relation
	4 Ranking Opportunity Sets by Using a Simple Similarity-Based Ordering
		4.1 Mimic Property
		4.2 Axioms
		4.3 Characterization Result
	5 Other Rules for Ranking Opportunity Sets by Using the Similarity Relation
		5.1 Axioms
		5.2 Lemmas
		5.3 Characterization Results
	6 Final Remarks
	References
The Interplay Between Intergenerational Justice and Mathematical Utility Theory
	1 Introduction
	2 Preliminaries
		2.1 Preference Relations and Utility Representations
		2.2 Lexicographic Preferences and Non-representability
		2.3 Axioms for Intergenerational Justice
	3 Intergenerational Utility via Mathematical Utility Theory
		3.1 Possibility Results
		3.2 Impossibility Results
	4 Richter–Peleg Representations in Intergenerational Justice
	5 Conclusions and Future Research
	References
Comparative Risk Aversion for State-Dependent Preferences
	1 Introduction
	2 Notation
	3 Reference Sets and Risk Premiums
		3.1 Risk Premiums
	4 Comparisons of Risk and Risk Aversion
		4.1 Risk Orderings for Equal Mean Sets
		4.2 General Risk Orderings
		4.3 Risk Aversion
	5 Constant, Decreasing and Increasing Risk Aversion
		5.1 Constant Risk Aversion and Linear Risk Tolerance
		5.2 Decreasing and Increasing Absolute Risk Aversion
	6 Implications for Asset Demand
	7 Concluding Comments
	References