the role of topology
TRANSCRIPT
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THE ROLE OF TOPOLOGY
IN DECISION THEORY
J. C. Candeal1
1Universidad de Zaragoza
XVII Encuentro de Topologıa. Zaragoza, Noviembre 26-27,2010
J. C. Candeal Topology and Decision Theory
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 5: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/5.jpg)
Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 6: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/6.jpg)
Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 7: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/7.jpg)
Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 8: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/8.jpg)
Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 9: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/9.jpg)
Outline
1 IntroductionMathematics and EconomicsDecision Theory
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 10: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/10.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background
The marginalist period: 1838-1947
Cournot (theory of the firm and single market equilibrium, 1838),Walras (theory of the consumer and general equilibrium, 1874),Edgeworth (exchange economy and contract curve, 1881),Marshall (demand theory, 1890), Pareto (general equilibrium andoptimal resource allocation, 1896), Hicks (stability of equilibriumand barganing, 1946),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 11: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/11.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background
The marginalist period: 1838-1947
Cournot (theory of the firm and single market equilibrium, 1838),Walras (theory of the consumer and general equilibrium, 1874),Edgeworth (exchange economy and contract curve, 1881),Marshall (demand theory, 1890), Pareto (general equilibrium andoptimal resource allocation, 1896), Hicks (stability of equilibriumand barganing, 1946),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 12: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/12.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background (continued)
The set-theoretic/linear models period: 1848-1960
Arrow (social choice theory, 1951), Arrow-Debreu (generalequilibrium, 1954), McKenzie (general equilibrium, 1954), vonNeumann and Morgenstern (game theory, 1947), Nash (gametheory, 1950), Dantzig (linear programming, 1949),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 13: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/13.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background (continued)
The set-theoretic/linear models period: 1848-1960
Arrow (social choice theory, 1951), Arrow-Debreu (generalequilibrium, 1954), McKenzie (general equilibrium, 1954), vonNeumann and Morgenstern (game theory, 1947), Nash (gametheory, 1950), Dantzig (linear programming, 1949),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 14: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/14.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background (continued)
The period of integration: 1961-1980
Debreu (regular economies, 1970), Aumann (large economies,1964), Sen (social choice, 1970), Koopmans (optimal growththeory, 1965), Smale (global analysis, 1976),...,etc.
The period of spread: 1981-present
Decision theory (foundations, utility theory, risk, uncertainty,....,social choice), extensions of the Arrow-Debreu model (incompletemarkets, financial markets, infinite dimensional spaces,...),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 15: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/15.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background (continued)
The period of integration: 1961-1980
Debreu (regular economies, 1970), Aumann (large economies,1964), Sen (social choice, 1970), Koopmans (optimal growththeory, 1965), Smale (global analysis, 1976),...,etc.
The period of spread: 1981-present
Decision theory (foundations, utility theory, risk, uncertainty,....,social choice), extensions of the Arrow-Debreu model (incompletemarkets, financial markets, infinite dimensional spaces,...),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 16: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/16.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Historical background (continued)
The period of integration: 1961-1980
Debreu (regular economies, 1970), Aumann (large economies,1964), Sen (social choice, 1970), Koopmans (optimal growththeory, 1965), Smale (global analysis, 1976),...,etc.
The period of spread: 1981-present
Decision theory (foundations, utility theory, risk, uncertainty,....,social choice), extensions of the Arrow-Debreu model (incompletemarkets, financial markets, infinite dimensional spaces,...),...,etc.
J. C. Candeal Topology and Decision Theory
![Page 17: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/17.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Forewords
Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).
Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).
Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....
Social choice theory: Is it possible to aggregate individualpreferences into a social one ?
J. C. Candeal Topology and Decision Theory
![Page 18: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/18.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Forewords
Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).
Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).
Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....
Social choice theory: Is it possible to aggregate individualpreferences into a social one ?
J. C. Candeal Topology and Decision Theory
![Page 19: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/19.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Forewords
Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).
Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).
Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....
Social choice theory: Is it possible to aggregate individualpreferences into a social one ?
J. C. Candeal Topology and Decision Theory
![Page 20: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/20.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Forewords
Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).
Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).
Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....
Social choice theory: Is it possible to aggregate individualpreferences into a social one ?
J. C. Candeal Topology and Decision Theory
![Page 21: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/21.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Forewords
Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).
Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).
Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....
Social choice theory: Is it possible to aggregate individualpreferences into a social one ?
J. C. Candeal Topology and Decision Theory
![Page 22: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/22.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Objectives
Goals:
Presenting in a unified framework both classical and recentresults in utility theory.
Introducing to social choice theory: combinatorial andtopological models.
J. C. Candeal Topology and Decision Theory
![Page 23: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/23.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Objectives
Goals:
Presenting in a unified framework both classical and recentresults in utility theory.
Introducing to social choice theory: combinatorial andtopological models.
J. C. Candeal Topology and Decision Theory
![Page 24: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/24.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Mathematics and EconomicsDecision Theory
Objectives
Goals:
Presenting in a unified framework both classical and recentresults in utility theory.
Introducing to social choice theory: combinatorial andtopological models.
J. C. Candeal Topology and Decision Theory
![Page 25: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/25.jpg)
Outline
1 Introduction
2 Utility Theory: The Ordinal ApproachThe Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
![Page 26: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/26.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem
Theorem (Cantor, 1895)
Let (X ,-) be a totally ordered set that is unbordered, dense, anddenumerable. Then there exists an order monomorphism from Xonto Q.
Let X be, P = {- ⊆ X × X ; - transitive and total} andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 27: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/27.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem
Theorem (Cantor, 1895)
Let (X ,-) be a totally ordered set that is unbordered, dense, anddenumerable. Then there exists an order monomorphism from Xonto Q.
Let X be, P = {- ⊆ X × X ; - transitive and total} andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 28: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/28.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem
Theorem (Cantor, 1895)
Let (X ,-) be a totally ordered set that is unbordered, dense, anddenumerable. Then there exists an order monomorphism from Xonto Q.
Let X be, P = {- ⊆ X × X ; - transitive and total} andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 29: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/29.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem
Theorem (Cantor, 1895)
Let (X ,-) be a totally ordered set that is unbordered, dense, anddenumerable. Then there exists an order monomorphism from Xonto Q.
Let X be, P = {- ⊆ X × X ; - transitive and total} andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 30: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/30.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem (continued)
Notation. PR = {- ∈ P ;- is representable}
Definition
- ∈ P is perfectly separable if ∃ D ⊆ X , D countable in such away that for every x , y ∈ X , with (y , x) /∈ -, there is d ∈ D suchthat (x , d), (d , y) ∈ -.
J. C. Candeal Topology and Decision Theory
![Page 31: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/31.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem (continued)
Notation. PR = {- ∈ P ;- is representable}
Definition
- ∈ P is perfectly separable if ∃ D ⊆ X , D countable in such away that for every x , y ∈ X , with (y , x) /∈ -, there is d ∈ D suchthat (x , d), (d , y) ∈ -.
J. C. Candeal Topology and Decision Theory
![Page 32: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/32.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
The representation problem (continued)
Notation. PR = {- ∈ P ;- is representable}
Definition
- ∈ P is perfectly separable if ∃ D ⊆ X , D countable in such away that for every x , y ∈ X , with (y , x) /∈ -, there is d ∈ D suchthat (x , d), (d , y) ∈ -.
J. C. Candeal Topology and Decision Theory
![Page 33: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/33.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Characterizing representability
Theorem (Milgram, 1941 and Birkhoff, 1948)
Let -∈ P. Then - ∈ PR iff - is perfectly separable.
Notation. For a given -∈ P, τ- denotes the order topology on X .
Theorem
Let -∈ P. Then - ∈ PR iff τ- is second countable.
J. C. Candeal Topology and Decision Theory
![Page 34: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/34.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Characterizing representability
Theorem (Milgram, 1941 and Birkhoff, 1948)
Let -∈ P. Then - ∈ PR iff - is perfectly separable.
Notation. For a given -∈ P, τ- denotes the order topology on X .
Theorem
Let -∈ P. Then - ∈ PR iff τ- is second countable.
J. C. Candeal Topology and Decision Theory
![Page 35: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/35.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Characterizing representability
Theorem (Milgram, 1941 and Birkhoff, 1948)
Let -∈ P. Then - ∈ PR iff - is perfectly separable.
Notation. For a given -∈ P, τ- denotes the order topology on X .
Theorem
Let -∈ P. Then - ∈ PR iff τ- is second countable.
J. C. Candeal Topology and Decision Theory
![Page 36: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/36.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Characterizing representability
Theorem (Milgram, 1941 and Birkhoff, 1948)
Let -∈ P. Then - ∈ PR iff - is perfectly separable.
Notation. For a given -∈ P, τ- denotes the order topology on X .
Theorem
Let -∈ P. Then - ∈ PR iff τ- is second countable.
J. C. Candeal Topology and Decision Theory
![Page 37: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/37.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Set-up
Notation. Pa = {-∈ P; - is antisymmetric }.
Two canonical examples:
1. Lexicographic plane:
(R2,-L) :
(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.
2. The first uncountable ordinal:
([0, ω1),-):
W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .
J. C. Candeal Topology and Decision Theory
![Page 38: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/38.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Set-up
Notation. Pa = {-∈ P; - is antisymmetric }.
Two canonical examples:
1. Lexicographic plane:
(R2,-L) :
(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.
2. The first uncountable ordinal:
([0, ω1),-):
W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .
J. C. Candeal Topology and Decision Theory
![Page 39: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/39.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Set-up
Notation. Pa = {-∈ P; - is antisymmetric }.
Two canonical examples:
1. Lexicographic plane:
(R2,-L) :
(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.
2. The first uncountable ordinal:
([0, ω1),-):
W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .
J. C. Candeal Topology and Decision Theory
![Page 40: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/40.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Set-up
Notation. Pa = {-∈ P; - is antisymmetric }.
Two canonical examples:
1. Lexicographic plane:
(R2,-L) :
(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.
2. The first uncountable ordinal:
([0, ω1),-):
W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Set-up
Notation. Pa = {-∈ P; - is antisymmetric }.
Two canonical examples:
1. Lexicographic plane:
(R2,-L) :
(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.
2. The first uncountable ordinal:
([0, ω1),-):
W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Definitions and questions
Definition.
A chain (X ,-) is said to be:(i) planar (or, a Debreu chain) if it contains a subchain that isorder isomorphic to a non-representable subset of the lexicographicplane (R2,-L).
(ii) long if it, or its dual, contains a subchain which is orderisomorphic to [0, ω1).
Questions:
(1) Are there any other orders, different from the above two, thatwill cause problems ?
(2) How many such other orders can we find ?
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Definitions and questions
Definition.
A chain (X ,-) is said to be:(i) planar (or, a Debreu chain) if it contains a subchain that isorder isomorphic to a non-representable subset of the lexicographicplane (R2,-L).
(ii) long if it, or its dual, contains a subchain which is orderisomorphic to [0, ω1).
Questions:
(1) Are there any other orders, different from the above two, thatwill cause problems ?
(2) How many such other orders can we find ?
J. C. Candeal Topology and Decision Theory
![Page 44: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/44.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Definitions and questions
Definition.
A chain (X ,-) is said to be:(i) planar (or, a Debreu chain) if it contains a subchain that isorder isomorphic to a non-representable subset of the lexicographicplane (R2,-L).
(ii) long if it, or its dual, contains a subchain which is orderisomorphic to [0, ω1).
Questions:
(1) Are there any other orders, different from the above two, thatwill cause problems ?
(2) How many such other orders can we find ?
J. C. Candeal Topology and Decision Theory
![Page 45: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/45.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (1):
Definition.
An uncountable short (i.e., no long) chain that does not containany uncountable representable subchain is called an Aronszajnchain.
Theorem (Beardon et al., 2002)
(i) There exists an Aronszajn chain.(ii) Every Aronszajn chain is order isomorphic to a subchain of(Qω1
00(−1, 1),-L).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (1):
Definition.
An uncountable short (i.e., no long) chain that does not containany uncountable representable subchain is called an Aronszajnchain.
Theorem (Beardon et al., 2002)
(i) There exists an Aronszajn chain.(ii) Every Aronszajn chain is order isomorphic to a subchain of(Qω1
00(−1, 1),-L).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (1):
Definition.
An uncountable short (i.e., no long) chain that does not containany uncountable representable subchain is called an Aronszajnchain.
Theorem (Beardon et al., 2002)
(i) There exists an Aronszajn chain.(ii) Every Aronszajn chain is order isomorphic to a subchain of(Qω1
00(−1, 1),-L).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (2):
Theorem (Beardon et al., 2002)
Let (X ,-) be an arbitrary chain. Then the following assertions areequivalent:(i) - /∈ Pa
R .(ii) (X ,-) is long chain, a Debreu chain or an Aronszajn chain.
Remark.
A Souslin chain contains a copy of an Aronszajn chain.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (2):
Theorem (Beardon et al., 2002)
Let (X ,-) be an arbitrary chain. Then the following assertions areequivalent:(i) - /∈ Pa
R .(ii) (X ,-) is long chain, a Debreu chain or an Aronszajn chain.
Remark.
A Souslin chain contains a copy of an Aronszajn chain.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem
Solving question (2):
Theorem (Beardon et al., 2002)
Let (X ,-) be an arbitrary chain. Then the following assertions areequivalent:(i) - /∈ Pa
R .(ii) (X ,-) is long chain, a Debreu chain or an Aronszajn chain.
Remark.
A Souslin chain contains a copy of an Aronszajn chain.
J. C. Candeal Topology and Decision Theory
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological ApproachThe Continuous Representation ProblemCharacterizing PCR
The Continuous Representability PropertyCharacterizing CRP
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
The continuous representation problem
Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.
PC = {- ∈ P ;- is closed in X × X}
PCR = {- ∈ P ;- admits a continuous utility function}
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
The continuous representation problem
Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.
PC = {- ∈ P ;- is closed in X × X}
PCR = {- ∈ P ;- admits a continuous utility function}
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
The continuous representation problem
Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.
PC = {- ∈ P ;- is closed in X × X}
PCR = {- ∈ P ;- admits a continuous utility function}
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
The continuous representation problem
Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.
PC = {- ∈ P ;- is closed in X × X}
PCR = {- ∈ P ;- admits a continuous utility function}
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
The continuous representation problem
Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.
PC = {- ∈ P ;- is closed in X × X}PCR = {- ∈ P ;- admits a continuous utility function}
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing PCR
Theorem
PCR = PC⋂
PR .
Corollary (Debreu’s open-gap lemma)
If τ ≡ τ-, for some -∈ P, then -∈ PCR iff -∈ PR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing PCR
Theorem
PCR = PC⋂
PR .
Corollary (Debreu’s open-gap lemma)
If τ ≡ τ-, for some -∈ P, then -∈ PCR iff -∈ PR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing PCR
Theorem
PCR = PC⋂
PR .
Corollary (Debreu’s open-gap lemma)
If τ ≡ τ-, for some -∈ P, then -∈ PCR iff -∈ PR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Two cornerstone results
Theorem (Eilenberg, 1941)
Let (X , τ) be a connected and separable topological space. Thenany continuous total preorder defined on X has a continuous utilityfunction.
Theorem (Debreu, 1964)
Let (X , τ) be a second countable topological space. Then anycontinuous total preorder defined on X has a continuous utilityfunction.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Two cornerstone results
Theorem (Eilenberg, 1941)
Let (X , τ) be a connected and separable topological space. Thenany continuous total preorder defined on X has a continuous utilityfunction.
Theorem (Debreu, 1964)
Let (X , τ) be a second countable topological space. Then anycontinuous total preorder defined on X has a continuous utilityfunction.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Two cornerstone results
Theorem (Eilenberg, 1941)
Let (X , τ) be a connected and separable topological space. Thenany continuous total preorder defined on X has a continuous utilityfunction.
Theorem (Debreu, 1964)
Let (X , τ) be a second countable topological space. Then anycontinuous total preorder defined on X has a continuous utilityfunction.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
CRP: Definition
Definition
Let (X , τ) be a topological space. Then τ satisfies the ContinuousRepresentability Property (CRP) if PC = PCR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
CRP: Definition
Definition
Let (X , τ) be a topological space. Then τ satisfies the ContinuousRepresentability Property (CRP) if PC = PCR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).(iv) If τ is locally connected and separable (Candeal et al., 2004).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Sufficient conditions for CRP
Theorem
Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).(iv) If τ is locally connected and separable (Candeal et al., 2004).(v) If τ is separably connected and satisfies CCC. In particular, thisimplies that the weak topology of a Banach space has CRP(Campion et al., 2006).
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Some characterizations of CRP
Theorem
Let (X , τ) be a metric space. Then τ satisfies CRP iff it is secondcountable (≡separable).
Theorem
Let (X ,-) be a totally preordered space. Then τ- satisfies CRP iffit is second countable.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Some characterizations of CRP
Theorem
Let (X , τ) be a metric space. Then τ satisfies CRP iff it is secondcountable (≡separable).
Theorem
Let (X ,-) be a totally preordered space. Then τ- satisfies CRP iffit is second countable.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Some characterizations of CRP
Theorem
Let (X , τ) be a metric space. Then τ satisfies CRP iff it is secondcountable (≡separable).
Theorem
Let (X ,-) be a totally preordered space. Then τ- satisfies CRP iffit is second countable.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Some characterizations of CRP (continued)
Theorem
Let (V ,+, ·R, τ) be a locally convex topological real vector space.Then τ satisfies CRP iff it satisfies CCC.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Some characterizations of CRP (continued)
Theorem
Let (V ,+, ·R, τ) be a locally convex topological real vector space.Then τ satisfies CRP iff it satisfies CCC.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing CRP
Definition
A topology τ on X is said to be (totally) preorderable if τ ≡ τ-,for some -∈ P.
Theorem (Herden and Pallack, 2000 and Campion et al., 2002)
Let (X , τ) be a topological space. The topology τ satisfies CRP ifand only if all its preorderable subtopologies are second countable.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing CRP
Definition
A topology τ on X is said to be (totally) preorderable if τ ≡ τ-,for some -∈ P.
Theorem (Herden and Pallack, 2000 and Campion et al., 2002)
Let (X , τ) be a topological space. The topology τ satisfies CRP ifand only if all its preorderable subtopologies are second countable.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP
Characterizing CRP
Definition
A topology τ on X is said to be (totally) preorderable if τ ≡ τ-,for some -∈ P.
Theorem (Herden and Pallack, 2000 and Campion et al., 2002)
Let (X , τ) be a topological space. The topology τ satisfies CRP ifand only if all its preorderable subtopologies are second countable.
J. C. Candeal Topology and Decision Theory
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic ApproachThe Case Of GroupsThe Case Of Real Vector Spaces
5 Social Choice Theory
6 Appendix
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The additive representation problem
Let (X ,+) be a group, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+) be a group. A total preorder -∈ P is said to betranslation-invariant if x - y =⇒ x + z - y + z andz + x - z + y , ∀x , y , z ∈ X .
J. C. Candeal Topology and Decision Theory
![Page 82: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/82.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The additive representation problem
Let (X ,+) be a group, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+) be a group. A total preorder -∈ P is said to betranslation-invariant if x - y =⇒ x + z - y + z andz + x - z + y , ∀x , y , z ∈ X .
J. C. Candeal Topology and Decision Theory
![Page 83: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/83.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The additive representation problem
Let (X ,+) be a group, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+) be a group. A total preorder -∈ P is said to betranslation-invariant if x - y =⇒ x + z - y + z andz + x - z + y , ∀x , y , z ∈ X .
J. C. Candeal Topology and Decision Theory
![Page 84: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/84.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The additive representation problem
Let (X ,+) be a group, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+) be a group. A total preorder -∈ P is said to betranslation-invariant if x - y =⇒ x + z - y + z andz + x - z + y , ∀x , y , z ∈ X .
J. C. Candeal Topology and Decision Theory
![Page 85: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/85.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PAR
Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.Remark. PAR PR
⋂PA.
Theorem
Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:..
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PAR
Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.
Remark. PAR PR⋂
PA.
Theorem
Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:..
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PAR
Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.Remark. PAR PR
⋂PA.
Theorem
Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:(i) -∈ PAR ..
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PAR
Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.Remark. PAR PR
⋂PA.
Theorem
Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:(i) -∈ PAR .(ii) -∈ PA and there is a base of the origin for τ-, say Ve , suchthat, for each B ∈ Ve ,
⋃n∈N nB = X .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous additive representation problem
Let (X ,+) be a group, τ a topology on X , P the space ofpreferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.PCAR = {- ∈ P ;- admits a continuous and additiveutility function}.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous additive representation problem
Let (X ,+) be a group, τ a topology on X , P the space ofpreferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.PCAR = {- ∈ P ;- admits a continuous and additiveutility function}.
J. C. Candeal Topology and Decision Theory
![Page 91: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/91.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous additive representation problem
Let (X ,+) be a group, τ a topology on X , P the space ofpreferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.PCAR = {- ∈ P ;- admits a continuous and additiveutility function}.
J. C. Candeal Topology and Decision Theory
![Page 92: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/92.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous additive representation problem
Let (X ,+) be a group, τ a topology on X , P the space ofpreferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Notation.PCAR = {- ∈ P ;- admits a continuous and additiveutility function}.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCAR
Theorem (De Miguel et al., 1998)
PCAR = PC⋂
PAR .
Corollary (algebraic version of the open-gap lemma for groups)
If τ ≡ τ-, for some -∈ P, then -∈ PCAR iff -∈ PAR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCAR
Theorem (De Miguel et al., 1998)
PCAR = PC⋂
PAR .
Corollary (algebraic version of the open-gap lemma for groups)
If τ ≡ τ-, for some -∈ P, then -∈ PCAR iff -∈ PAR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCAR
Theorem (De Miguel et al., 1998)
PCAR = PC⋂
PAR .
Corollary (algebraic version of the open-gap lemma for groups)
If τ ≡ τ-, for some -∈ P, then -∈ PCAR iff -∈ PAR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+) be a group and τ a topology on X . Then τ satisfies theContinuous Algebraic Representability Property (CARP) ifPCAR = PC
⋂PA.
Theorem
Let (X ,+) be a group and τ a topology on X . Then the followingassertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PA, there is an absorbing basis of the origin
for τ-.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+) be a group and τ a topology on X . Then τ satisfies theContinuous Algebraic Representability Property (CARP) ifPCAR = PC
⋂PA.
Theorem
Let (X ,+) be a group and τ a topology on X . Then the followingassertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PA, there is an absorbing basis of the origin
for τ-.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+) be a group and τ a topology on X . Then τ satisfies theContinuous Algebraic Representability Property (CARP) ifPCAR = PC
⋂PA.
Theorem
Let (X ,+) be a group and τ a topology on X . Then the followingassertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PA, there is an absorbing basis of the origin
for τ-.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
A sufficient condition for CARP
Corollary
Let (X ,+) be a group and τ a connected topology on X . Then τsatisfies CARP.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The linear representation problem
Let (X ,+, ·R) be a real vector space, P the space of preferencesand S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+, ·R) be a real vector space. A total preorder -∈ P is saidto be homothetic if x - y =⇒ λx - λy , ∀ x , y ∈ X , λ ≥ 0.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The linear representation problem
Let (X ,+, ·R) be a real vector space, P the space of preferencesand S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+, ·R) be a real vector space. A total preorder -∈ P is saidto be homothetic if x - y =⇒ λx - λy , ∀ x , y ∈ X , λ ≥ 0.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The linear representation problem
Let (X ,+, ·R) be a real vector space, P the space of preferencesand S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+, ·R) be a real vector space. A total preorder -∈ P is saidto be homothetic if x - y =⇒ λx - λy , ∀ x , y ∈ X , λ ≥ 0.
J. C. Candeal Topology and Decision Theory
![Page 103: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/103.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The linear representation problem
Let (X ,+, ·R) be a real vector space, P the space of preferencesand S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
Definition
Let (X ,+, ·R) be a real vector space. A total preorder -∈ P is saidto be homothetic if x - y =⇒ λx - λy , ∀ x , y ∈ X , λ ≥ 0.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PLR
Notation. PH = {- ∈ P ;- is homothetic}.PL = PA
⋂PH .
PLR = {- ∈ P ;- admits a linear utility function}.
Theorem
Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PLR
Notation. PH = {- ∈ P ;- is homothetic}.PL = PA
⋂PH .
PLR = {- ∈ P ;- admits a linear utility function}.
Theorem
Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:
J. C. Candeal Topology and Decision Theory
![Page 106: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/106.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PLR
Notation. PH = {- ∈ P ;- is homothetic}.PL = PA
⋂PH .
PLR = {- ∈ P ;- admits a linear utility function}.
Theorem
Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PLR
Notation. PH = {- ∈ P ;- is homothetic}.PL = PA
⋂PH .
PLR = {- ∈ P ;- admits a linear utility function}.
Theorem
Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .(ii) -∈ PL and (X ,+, ·R, τ-) is a topological real vector space.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PLR
Notation. PH = {- ∈ P ;- is homothetic}.PL = PA
⋂PH .
PLR = {- ∈ P ;- admits a linear utility function}.
Theorem
Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .(ii) -∈ PL and (X ,+, ·R, τ-) is a topological real vector space.
Consequence. PLR = PL⋂
PR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous linear representation problem
Let (X ,+, ·R) be a real vector space, τ a topology on X , P thespace of preferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 110: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/110.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous linear representation problem
Let (X ,+, ·R) be a real vector space, τ a topology on X , P thespace of preferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 111: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/111.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous linear representation problem
Let (X ,+, ·R) be a real vector space, τ a topology on X , P thespace of preferences and S = {(a, b) ∈ R2; a ≤ b}
Question:
- ∈ P, ∃ u : X −→ R, u continuous and linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?
J. C. Candeal Topology and Decision Theory
![Page 112: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/112.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCLR
Notation. Let (X ,+, ·R) be a real vector space equipped with atopology τ . PCLR = {- ∈ P ;- admits a linear and continuousutility function}.
Theorem
PCLR = PC⋂
PLR .
Corollary (algebraic version of the open-gap lemma for real vectorspaces)
If τ ≡ τ-, for some -∈ P, then -∈ PCLR iff -∈ PLR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCLR
Notation. Let (X ,+, ·R) be a real vector space equipped with atopology τ . PCLR = {- ∈ P ;- admits a linear and continuousutility function}.
Theorem
PCLR = PC⋂
PLR .
Corollary (algebraic version of the open-gap lemma for real vectorspaces)
If τ ≡ τ-, for some -∈ P, then -∈ PCLR iff -∈ PLR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCLR
Notation. Let (X ,+, ·R) be a real vector space equipped with atopology τ . PCLR = {- ∈ P ;- admits a linear and continuousutility function}.
Theorem
PCLR = PC⋂
PLR .
Corollary (algebraic version of the open-gap lemma for real vectorspaces)
If τ ≡ τ-, for some -∈ P, then -∈ PCLR iff -∈ PLR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
Characterizing PCLR
Notation. Let (X ,+, ·R) be a real vector space equipped with atopology τ . PCLR = {- ∈ P ;- admits a linear and continuousutility function}.
Theorem
PCLR = PC⋂
PLR .
Corollary (algebraic version of the open-gap lemma for real vectorspaces)
If τ ≡ τ-, for some -∈ P, then -∈ PCLR iff -∈ PLR .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
An important consequence
Corollary
The only topological totally ordered real vector space is R or itsdual.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then τ satisfies the Continuous Algebraic RepresentabilityProperty (CARP) if PCLR = PC
⋂PL.
Theorem
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then the following assertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PL, (X ,+, ·R, τ-) is a topological real
vector space.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then τ satisfies the Continuous Algebraic RepresentabilityProperty (CARP) if PCLR = PC
⋂PL.
Theorem
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then the following assertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PL, (X ,+, ·R, τ-) is a topological real
vector space.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
The continuous algebraic representability property
Definition
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then τ satisfies the Continuous Algebraic RepresentabilityProperty (CARP) if PCLR = PC
⋂PL.
Theorem
Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then the following assertions are equivalent:(i) τ has CARP(ii) For each -∈ PC
⋂PL, (X ,+, ·R, τ-) is a topological real
vector space.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
A sufficient condition for CARP
Corollary
Let (X ,+, ·R) be a real vector space equipped with a connectedtopology τ . Then τ satisfies CARP. In particular, if (X ,+, ·R, τ) isa topological real vector space, then τ satisfies CARP.
Remark
The analogous of the last statement of the corollary is not longerfor groups.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
A sufficient condition for CARP
Corollary
Let (X ,+, ·R) be a real vector space equipped with a connectedtopology τ . Then τ satisfies CARP. In particular, if (X ,+, ·R, τ) isa topological real vector space, then τ satisfies CARP.
Remark
The analogous of the last statement of the corollary is not longerfor groups.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
The Case Of GroupsThe Case Of Real Vector Spaces
A sufficient condition for CARP
Corollary
Let (X ,+, ·R) be a real vector space equipped with a connectedtopology τ . Then τ satisfies CARP. In particular, if (X ,+, ·R, τ) isa topological real vector space, then τ satisfies CARP.
Remark
The analogous of the last statement of the corollary is not longerfor groups.
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice TheoryArrovian FrameworkTopological AggregationCharacterizating social welfare functionals
6 Appendix
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model
Definition
A social welfare functional F : Pn −→ P satisfies:
Pareto condition if, for any pair of alternatives x , y ∈ X andany preference profile (-j) ∈ Pn, we have that x -j y for allj ∈ N implies xF (-j)y , and also that x ≺j y for all j impliesxF (-j)sy .
the independence of irrelevant alternatives (IIA) condition if,for any pair of alternatives x , y ∈ X and any pair of preferenceprofiles (-j), (-
′j) ∈ Pn with the property that, for every j ,
x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -
′j x , we have that
xF (-j)y ⇐⇒ xF (-′j)y and yF (-j)x ⇐⇒ yF (-
′j)x .
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model
Definition
A social welfare functional F : Pn −→ P satisfies:
Pareto condition if, for any pair of alternatives x , y ∈ X andany preference profile (-j) ∈ Pn, we have that x -j y for allj ∈ N implies xF (-j)y , and also that x ≺j y for all j impliesxF (-j)sy .
the independence of irrelevant alternatives (IIA) condition if,for any pair of alternatives x , y ∈ X and any pair of preferenceprofiles (-j), (-
′j) ∈ Pn with the property that, for every j ,
x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -
′j x , we have that
xF (-j)y ⇐⇒ xF (-′j)y and yF (-j)x ⇐⇒ yF (-
′j)x .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model
Definition
A social welfare functional F : Pn −→ P satisfies:
Pareto condition if, for any pair of alternatives x , y ∈ X andany preference profile (-j) ∈ Pn, we have that x -j y for allj ∈ N implies xF (-j)y , and also that x ≺j y for all j impliesxF (-j)sy .
the independence of irrelevant alternatives (IIA) condition if,for any pair of alternatives x , y ∈ X and any pair of preferenceprofiles (-j), (-
′j) ∈ Pn with the property that, for every j ,
x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -
′j x , we have that
xF (-j)y ⇐⇒ xF (-′j)y and yF (-j)x ⇐⇒ yF (-
′j)x .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model (continued)
Theorem (Arrow, 1951)
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that for every profile(-j) and every x , y ∈ X it holds that x ≺i y ⇒ xF (-j)sy .
Remark
Suppose that X contains at least three elements and letF : Pn
a → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that F (-j) =-i , forevery profile (-j) ∈ Pn
a .
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model (continued)
Theorem (Arrow, 1951)
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that for every profile(-j) and every x , y ∈ X it holds that x ≺i y ⇒ xF (-j)sy .
Remark
Suppose that X contains at least three elements and letF : Pn
a → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that F (-j) =-i , forevery profile (-j) ∈ Pn
a .
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Arrow’s model (continued)
Theorem (Arrow, 1951)
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that for every profile(-j) and every x , y ∈ X it holds that x ≺i y ⇒ xF (-j)sy .
Remark
Suppose that X contains at least three elements and letF : Pn
a → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that F (-j) =-i , forevery profile (-j) ∈ Pn
a .
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Sketch of proof
Definition
A coalition A ⊆ N = {1, . . . , n} is said to be decisive if for any pairx , y ∈ X , any profile (-j) ∈ Pn such that x ≺j y , for every j ∈ A,then xF (-j)sy .
Fact (Fishburn, 1970 and Hansson, 1972)
Under the hypotheses of Arrow’s theorem the set of decisivecoalitions is an ultrafilter on N.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Sketch of proof
Definition
A coalition A ⊆ N = {1, . . . , n} is said to be decisive if for any pairx , y ∈ X , any profile (-j) ∈ Pn such that x ≺j y , for every j ∈ A,then xF (-j)sy .
Fact (Fishburn, 1970 and Hansson, 1972)
Under the hypotheses of Arrow’s theorem the set of decisivecoalitions is an ultrafilter on N.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Sketch of proof
Definition
A coalition A ⊆ N = {1, . . . , n} is said to be decisive if for any pairx , y ∈ X , any profile (-j) ∈ Pn such that x ≺j y , for every j ∈ A,then xF (-j)sy .
Fact (Fishburn, 1970 and Hansson, 1972)
Under the hypotheses of Arrow’s theorem the set of decisivecoalitions is an ultrafilter on N.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Chichilnisky’s model
Definition
Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:
anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).
unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.
continuity, i.e., if F is continuous
.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Chichilnisky’s model
Definition
Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:
anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).
unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.
continuity, i.e., if F is continuous
.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Chichilnisky’s model
Definition
Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:
anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).
unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.
continuity, i.e., if F is continuous
.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Chichilnisky’s model
Definition
Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:
anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).
unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.
continuity, i.e., if F is continuous
.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Chichilnisky’s model
Definition
Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:
anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).
unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.
continuity, i.e., if F is continuous.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n = 2. Then there is no social choice rule.
Sketch of proof (Candeal and Indurain, 1994)
If such a social choice rule would exist then the boundary of theMobius strip would be a continuous deformation retract of thewhole strip. But this is not possible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n = 2. Then there is no social choice rule.
Sketch of proof (Candeal and Indurain, 1994)
If such a social choice rule would exist then the boundary of theMobius strip would be a continuous deformation retract of thewhole strip. But this is not possible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n = 2. Then there is no social choice rule.
Sketch of proof (Candeal and Indurain, 1994)
If such a social choice rule would exist then the boundary of theMobius strip would be a continuous deformation retract of thewhole strip. But this is not possible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation(continued)
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n ≥ 2. Then there is no social choice rule.
Sketch of proof
Based on Algebraic Topology techniques. If such a ruleF : (S1)n → S1 does exist then it can be shown that thefundamental group of S1, π1(S
1), is Abelian and any element isdivisible by n. But this leads to a contradiction since π1(S
1) = Z.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation(continued)
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n ≥ 2. Then there is no social choice rule.
Sketch of proof
Based on Algebraic Topology techniques. If such a ruleF : (S1)n → S1 does exist then it can be shown that thefundamental group of S1, π1(S
1), is Abelian and any element isdivisible by n. But this leads to a contradiction since π1(S
1) = Z.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Some impossibility results in topological aggregation(continued)
Theorem
Suppose that P = S1, linear preferences over the closed disk B2,and let n ≥ 2. Then there is no social choice rule.
Sketch of proof
Based on Algebraic Topology techniques. If such a ruleF : (S1)n → S1 does exist then it can be shown that thefundamental group of S1, π1(S
1), is Abelian and any element isdivisible by n. But this leads to a contradiction since π1(S
1) = Z.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox
Important facts
(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.
(2) Contractible spaces have the property that all theirhomopoty groups are trivial.
(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox
Important facts
(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.
(2) Contractible spaces have the property that all theirhomopoty groups are trivial.
(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox
Important facts
(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.
(2) Contractible spaces have the property that all theirhomopoty groups are trivial.
(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox
Important facts
(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.
(2) Contractible spaces have the property that all theirhomopoty groups are trivial.
(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox
Important facts
(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.
(2) Contractible spaces have the property that all theirhomopoty groups are trivial.
(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Theorem
Suppose that P is connected and homotopic to a polyhedron. If asocial choice rule exists for any number of individuals, then P iscontractible.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Theorem
Suppose that P is connected and homotopic to a polyhedron. If asocial choice rule exists for any number of individuals, then P iscontractible.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Question
When contractibility is a sufficient condition ?
Theorem
Suppose that P is connected and homeomorphic to a polyhedron.Then, a social choice rule exists for any number of individuals iff Pis contractible.
Theorem (Chichilnisky and Heal, 1983)
Suppose that P is a connected parafinite CW-complex. Then asocial choice rule exists for any number of individuals iff P iscontractible.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Question
When contractibility is a sufficient condition ?
Theorem
Suppose that P is connected and homeomorphic to a polyhedron.Then, a social choice rule exists for any number of individuals iff Pis contractible.
Theorem (Chichilnisky and Heal, 1983)
Suppose that P is a connected parafinite CW-complex. Then asocial choice rule exists for any number of individuals iff P iscontractible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Question
When contractibility is a sufficient condition ?
Theorem
Suppose that P is connected and homeomorphic to a polyhedron.Then, a social choice rule exists for any number of individuals iff Pis contractible.
Theorem (Chichilnisky and Heal, 1983)
Suppose that P is a connected parafinite CW-complex. Then asocial choice rule exists for any number of individuals iff P iscontractible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
The resolution of the social choice paradox (continued)
Question
When contractibility is a sufficient condition ?
Theorem
Suppose that P is connected and homeomorphic to a polyhedron.Then, a social choice rule exists for any number of individuals iff Pis contractible.
Theorem (Chichilnisky and Heal, 1983)
Suppose that P is a connected parafinite CW-complex. Then asocial choice rule exists for any number of individuals iff P iscontractible.
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Further developments
(1) Dropping anonymity and replacing unanimity by Paretocondition while keeping continuity leads to the existence of aunique manipulator on spheres.
(2) Futher remarks and generalizations of Chichilnisky andHeal results can be given by Horvath (2001), Eckmann (2004),Weinberger (2004), Ardanza-Trevijano et al. (2007),. . . , etc.
(3) Arrow’s theorem can be formulated and proved using thetopological model. This fascinating link was discovered byBaryshnikov (1993) !
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Further developments
(1) Dropping anonymity and replacing unanimity by Paretocondition while keeping continuity leads to the existence of aunique manipulator on spheres.
(2) Futher remarks and generalizations of Chichilnisky andHeal results can be given by Horvath (2001), Eckmann (2004),Weinberger (2004), Ardanza-Trevijano et al. (2007),. . . , etc.
(3) Arrow’s theorem can be formulated and proved using thetopological model. This fascinating link was discovered byBaryshnikov (1993) !
J. C. Candeal Topology and Decision Theory
![Page 157: THE ROLE OF TOPOLOGY](https://reader031.vdocuments.us/reader031/viewer/2022021106/6205845aceabe73e7c11ff1c/html5/thumbnails/157.jpg)
IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Further developments
(1) Dropping anonymity and replacing unanimity by Paretocondition while keeping continuity leads to the existence of aunique manipulator on spheres.
(2) Futher remarks and generalizations of Chichilnisky andHeal results can be given by Horvath (2001), Eckmann (2004),Weinberger (2004), Ardanza-Trevijano et al. (2007),. . . , etc.
(3) Arrow’s theorem can be formulated and proved using thetopological model. This fascinating link was discovered byBaryshnikov (1993) !
J. C. Candeal Topology and Decision Theory
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
Economics does not live by Topology alone
Definition
A social welfare functional F : Pn −→ P is said to be a two-sidedm serial dictatorship if there are a natural numberm ∈ N = {1, . . . , n}, an injection π of N(m) = {1, . . . ,m} into Nand a partition {A,B} of π(N(m)) such that, for any pair ofalternatives x , y ∈ X and any preference profile (-j) ∈ Pn, wehave that xF (-j)y if and only if x ∼π(k) y for all k ∈ N(m) orelse, there is i ∈ N(m) so that x ∼π(k) y for all k < j andx ≺π(i) y whenever i ∈ A, or y ≺π(i) x whenever i ∈ B.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
A characterization result in the Arrovian model
Theorem
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional. Then the followingassertions are equivalent:(i) F satisfies PI and IIA,(ii) F is either trivial, or there is m ∈ N such that F is a two-sidedm serial dictatorship.
Proof
Based on theory of ordered algebraic systems.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
A characterization result in the Arrovian model
Theorem
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional. Then the followingassertions are equivalent:(i) F satisfies PI and IIA,(ii) F is either trivial, or there is m ∈ N such that F is a two-sidedm serial dictatorship.
Proof
Based on theory of ordered algebraic systems.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals
A characterization result in the Arrovian model
Theorem
Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional. Then the followingassertions are equivalent:(i) F satisfies PI and IIA,(ii) F is either trivial, or there is m ∈ N such that F is a two-sidedm serial dictatorship.
Proof
Based on theory of ordered algebraic systems.
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Outline
1 Introduction
2 Utility Theory: The Ordinal Approach
3 Utility Theory: The Topological Approach
4 Utility Theory: The Algebraic Approach
5 Social Choice Theory
6 Appendix
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 1
Suppose that τ satisfies CCC and let - be a continuous totalpreorder defined on V . Observe that it is sufficient to show that itis representable. Since (V , τ) is a separably connected topologicalspace, by a result of Campion et al. (2006), there is a continuousorder-preserving function u : V → L, where L denotes the long line.Notice that u(V ) is an interval of L. Furthermore, there is anordinal α0 ∈ ω1 that bounds u(V ). Indeed, otherwise u(V )exhausts L and then by considering(Vα)α∈ω1 = (u−1(α, α + 1))α∈ω1 , we obtain an uncountable familyof pairwise disjoint non-empty open subset of V , which contradictsCCC. So, there is a countable ordinal that bounds u(V ) andtherefore u(V ) can be identified with a subset of the real line. Thismeans that - is representable.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 1 (continued)
For the converse suppose that, by way of contradiction, (V , τ)does not satisfy CCC. Let us then show that there is a continuoustotal preorder defined on V which is not representable. This willimply that (V , τ) does not satisfy CRP. To that end, let (pβ)β∈I
be a family of seminorms that define the topology τ on V and let(Uα)α∈ω1 be an uncountable family of pairwise disjoint non-emptyopen subset of V . Without loss of generality we can assume that,for each α, there is εα > 0 such that Uα can be chosen of the formUα = {v ∈ V ; there are vα ∈ V , βα ∈ I and ; pβα(v − vα) < εα}.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 1 (continued)
Indeed, we can single out an uncountable family of pairwisedisjoint non-empty basis open subset of V , say, (Vα)α∈ω1 . Recallthat each Vα is of the form Vα = {v ∈ V ; there are vα ∈ V ,β1, . . . , βn ∈ I and ε1, . . . , εn > 0 ; pβj
(v − vα) < εj ; j = 1, . . . , n}.Then, by considering for each α ∈ ω1, εα =min{εj ; j = 1, . . . , n}we can assume each Uα to be of the form described above.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 1 (continued)
For each α ∈ ω1, consider the (non-trivial) real interval [0, εα].Notice that [0, εα] ⊂ R is order-isomorphic to [0, α] ⊂ L. Denotethis order isomorphism by φα. Define now the function u : V → Las follows:
u(v) ={φα(εα − pα(v − vα)), v ∈ Uα
0 , v ∈ V \⋃
α∈ω1Uα
It remains to prove that u so defined is a continuous function sincethen by considering the relation on V defined as: v -u w iffu(v) ≤ u(w), (v ,w ∈ V ), it is straightforward to see that -u is anon-representable continuous total preorder.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 1 (continued)
To show the continuity of u let b ∈ L. Then it is sufficient to seethat u−1(L(b)) and u−1(G (b)) are open subsets of V . If b = 0then u−1(L(b)) = u−1(∅) = ∅ and u−1(G (b)) =
⋃α∈ω1
Uα both ofwhich are, obviously, open subsets of V .Now, if b ∈ L \ {0} then for each α ∈ ω1 such that b < α, let usdefine the subsets Aα = {v ∈ Uα; pα(v − vα) > εα − φ−1
α (b)} andBα = {v ∈ Uα; pα(v − vα) < εα − φ−1
α (b)}. Notice that both Aα
and Bα are non-empty open subsets of V . Then,u−1(L(b)) = (
⋃b<α Aα)
⋃(V \
⋃α∈ω1
Uα) and
u−1(G (b)) =⋃
b<α Bα, whence open subsets of V too. Thisconcludes the proof.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 2
The “only if”part is obvious. For the converse, let u : G → R be anadditive order-preserving function for -. Let us show that u is alsocontinuous. To that end, recall that if S ⊆ R is a subgroup of theusual additive group of the reals, then S = {0}, or there is somea ∈ (0,∞); S = aZ = {az ; z ∈ Z}, or S is dense in R (see, e.g.Choquet (1966, p.56)).As in the proof of Theorem 4.1 it is sufficient to show that, forevery b ∈ R, u−1(L(b)) and u−1(G (b)) are open subsets of G .Consider the subgroup of the reals S = u(G ) ⊆ R. According tothe above result, three cases need to be distinguished. Case (i): IfS = {0} then the results is obvious because u is constant.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
Proof 2 (continued)
Case (ii): If S = aZ, then there are uniques = u(g), s ′ = u(g ′) ∈ S such that b ∈ [s, s ′]. Thus,u−1(L(b)) = L(g ′) and u−1(G (b)) = G (g), which are open subsetsof G since u is order.preserving and - is continuous.Case (iii): If S is a dense subgroup of the reals thenu−1(L(b)) =
⋃{g∈G ;u(g)<b} L(g) and
u−1(G (b)) =⋃{g∈G ;b<u(g)} G (g), which are open subsets of G
since u is order.preserving and - is continuous. This ends theproof.
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
That’s all
THANKS
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IntroductionUtility Theory: The Ordinal Approach
Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach
Social Choice TheoryAppendix
That’s all
THANKS
J. C. Candeal Topology and Decision Theory