Updating Utility Functions on Preordered Sets
We consider the problem of extending a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt&q...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/22/4688 |
| _version_ | 1797458526342217728 |
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| author | Pavel Chebotarev |
| author_facet | Pavel Chebotarev |
| author_sort | Pavel Chebotarev |
| collection | DOAJ |
| description | We consider the problem of extending a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> defined on a subset <i> P </i> of an arbitrary set <i> X </i> to <i> X </i> strictly monotonically with respect to a preorder ≽ defined on <i> X </i>, without imposing continuity constraints. We show that whenever ≽ has a utility representation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is extendable if and only if it is gap-safe increasing. This property means that whenever <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>≻</mo></mrow></semantics></math></inline-formula><i><b>x</b></i>, the infimum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the upper contour of <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>′</mo></msup></semantics></math></inline-formula> exceeds the supremum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the lower contour of <i><b>x</b></i>, where <i><b>x</b></i>, <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>∈</mo><mover accent="true"><mi>X</mi><mo>˜</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo>˜</mo></mover></semantics></math></inline-formula> is <i> X </i> completed with two absolute ≽-extrema and, moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is weakly increasing. The completion of <i> X </i> makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation <i> u </i> of ≽, it provides an extension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> that coincides with <i> u </i> on a region of <i> X </i> that includes the set of <i> P</i>-neutral elements of <i> X </i>. An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when <i> P </i> is a Pareto set. |
| first_indexed | 2024-03-09T16:38:24Z |
| format | Article |
| id | doaj.art-9db3405981af4cffbe20185ccd8d3c0c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T16:38:24Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9db3405981af4cffbe20185ccd8d3c0c2023-11-24T14:54:29ZengMDPI AGMathematics2227-73902023-11-011122468810.3390/math11224688Updating Utility Functions on Preordered SetsPavel Chebotarev0Technion–Israel Institute of Technology, Haifa 3200003, IsraelWe consider the problem of extending a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> defined on a subset <i> P </i> of an arbitrary set <i> X </i> to <i> X </i> strictly monotonically with respect to a preorder ≽ defined on <i> X </i>, without imposing continuity constraints. We show that whenever ≽ has a utility representation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is extendable if and only if it is gap-safe increasing. This property means that whenever <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>≻</mo></mrow></semantics></math></inline-formula><i><b>x</b></i>, the infimum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mspace width="-0.89996pt"></mspace><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the upper contour of <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>′</mo></msup></semantics></math></inline-formula> exceeds the supremum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> on the lower contour of <i><b>x</b></i>, where <i><b>x</b></i>, <i><b>x</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mo>∈</mo><mover accent="true"><mi>X</mi><mo>˜</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo>˜</mo></mover></semantics></math></inline-formula> is <i> X </i> completed with two absolute ≽-extrema and, moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> is weakly increasing. The completion of <i> X </i> makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation <i> u </i> of ≽, it provides an extension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><msubsup><mrow></mrow><mrow><mi>P</mi></mrow><mrow></mrow></msubsup></mrow></semantics></math></inline-formula> that coincides with <i> u </i> on a region of <i> X </i> that includes the set of <i> P</i>-neutral elements of <i> X </i>. An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when <i> P </i> is a Pareto set.https://www.mdpi.com/2227-7390/11/22/4688extension of utility functionsmonotonicityutility representation of a preorderlifting theorems |
| spellingShingle | Pavel Chebotarev Updating Utility Functions on Preordered Sets Mathematics extension of utility functions monotonicity utility representation of a preorder lifting theorems |
| title | Updating Utility Functions on Preordered Sets |
| title_full | Updating Utility Functions on Preordered Sets |
| title_fullStr | Updating Utility Functions on Preordered Sets |
| title_full_unstemmed | Updating Utility Functions on Preordered Sets |
| title_short | Updating Utility Functions on Preordered Sets |
| title_sort | updating utility functions on preordered sets |
| topic | extension of utility functions monotonicity utility representation of a preorder lifting theorems |
| url | https://www.mdpi.com/2227-7390/11/22/4688 |
| work_keys_str_mv | AT pavelchebotarev updatingutilityfunctionsonpreorderedsets |