A Discrete-Time Homing Problem with Two Optimizers
A stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Games |
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| Online Access: | https://www.mdpi.com/2073-4336/14/6/68 |
| _version_ | 1797381015514120192 |
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| author | Mario Lefebvre |
| author_facet | Mario Lefebvre |
| author_sort | Mario Lefebvre |
| collection | DOAJ |
| description | A stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>X</mi><mi>n</mi></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula> in the continuation region <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula>, and the second player seeks to maximize the survival time in <i>C</i>. The process starts at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mi>x</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and the game ends the first time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>n</mi></msub><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula>. An exact expression is derived for the value function, from which the optimal solution is obtained, and particular problems are solved explicitly. |
| first_indexed | 2024-03-08T20:45:17Z |
| format | Article |
| id | doaj.art-f87f087a6e78442fa30e92dbc1676902 |
| institution | Directory Open Access Journal |
| issn | 2073-4336 |
| language | English |
| last_indexed | 2024-03-08T20:45:17Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Games |
| spelling | doaj.art-f87f087a6e78442fa30e92dbc16769022023-12-22T14:10:25ZengMDPI AGGames2073-43362023-10-011466810.3390/g14060068A Discrete-Time Homing Problem with Two OptimizersMario Lefebvre0Department of Mathematics and Industrial Engineering, Polytechnique Montréal, 2500, Chemin de Polytechnique, Montréal, QC H3T 1J4, CanadaA stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>X</mi><mi>n</mi></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula> in the continuation region <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula>, and the second player seeks to maximize the survival time in <i>C</i>. The process starts at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mi>x</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and the game ends the first time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>n</mi></msub><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula>. An exact expression is derived for the value function, from which the optimal solution is obtained, and particular problems are solved explicitly.https://www.mdpi.com/2073-4336/14/6/68random walkfirst-passage timehoming problemdifference gamedynamic programmingdifference equation |
| spellingShingle | Mario Lefebvre A Discrete-Time Homing Problem with Two Optimizers Games random walk first-passage time homing problem difference game dynamic programming difference equation |
| title | A Discrete-Time Homing Problem with Two Optimizers |
| title_full | A Discrete-Time Homing Problem with Two Optimizers |
| title_fullStr | A Discrete-Time Homing Problem with Two Optimizers |
| title_full_unstemmed | A Discrete-Time Homing Problem with Two Optimizers |
| title_short | A Discrete-Time Homing Problem with Two Optimizers |
| title_sort | discrete time homing problem with two optimizers |
| topic | random walk first-passage time homing problem difference game dynamic programming difference equation |
| url | https://www.mdpi.com/2073-4336/14/6/68 |
| work_keys_str_mv | AT mariolefebvre adiscretetimehomingproblemwithtwooptimizers AT mariolefebvre discretetimehomingproblemwithtwooptimizers |