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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.04153 |
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| _version_ | 1866910592892141568 |
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| author | Ramsey, David |
| author_facet | Ramsey, David |
| contents | This article considers a problem arising from a two-player game based on the classical secretary problem. First, Player 1 selects one object from a sequence as in the secretary problem. All of the other objects are then presented to Player 2 in the same order as in the original sequence. The goal of both players is to select the best object. The optimal response of Player 2 is adapted to the optimal strategy in the secretary problem. This means that when Player 2 observes an object that is the best seen so far, it can be inferred whether Player 1 selected one of the earlier objects in the original sequence. However, Player 2 cannot compare the current object with the one selected by Player 1. Hence, this game defines an auxiliary problem in which Player 2 has incomplete information on the relative rank of an object. It is shown that the optimal strategy of Player 2 is based on both the number of objects to have appeared and the probability that the current object is better than the object chosen by Player 1 (if Player 1 chose an earlier object in the sequence). However, this probability is dependent on the previously observed objects. A lower bound on the optimal expected reward in the auxiliary problem is defined by limiting the memory of Player 2. An upper bound is derived by giving Player 2 additional information at appropriate times. The methods used illustrate approaches that can be used to approximate the optimal reward in a stopping problem when there is incomplete information on the ranks of objects and/or the optimal strategy is history dependent, as in the Robbins' problem |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Stackelberg Game based on the Secretary Problem: Optimal Response is History Dependent Ramsey, David Computer Science and Game Theory Optimization and Control 60G40, 91A20, 91A27, 49N30, 05A05 G.3; G.2.0 This article considers a problem arising from a two-player game based on the classical secretary problem. First, Player 1 selects one object from a sequence as in the secretary problem. All of the other objects are then presented to Player 2 in the same order as in the original sequence. The goal of both players is to select the best object. The optimal response of Player 2 is adapted to the optimal strategy in the secretary problem. This means that when Player 2 observes an object that is the best seen so far, it can be inferred whether Player 1 selected one of the earlier objects in the original sequence. However, Player 2 cannot compare the current object with the one selected by Player 1. Hence, this game defines an auxiliary problem in which Player 2 has incomplete information on the relative rank of an object. It is shown that the optimal strategy of Player 2 is based on both the number of objects to have appeared and the probability that the current object is better than the object chosen by Player 1 (if Player 1 chose an earlier object in the sequence). However, this probability is dependent on the previously observed objects. A lower bound on the optimal expected reward in the auxiliary problem is defined by limiting the memory of Player 2. An upper bound is derived by giving Player 2 additional information at appropriate times. The methods used illustrate approaches that can be used to approximate the optimal reward in a stopping problem when there is incomplete information on the ranks of objects and/or the optimal strategy is history dependent, as in the Robbins' problem |
| title | A Stackelberg Game based on the Secretary Problem: Optimal Response is History Dependent |
| topic | Computer Science and Game Theory Optimization and Control 60G40, 91A20, 91A27, 49N30, 05A05 G.3; G.2.0 |
| url | https://arxiv.org/abs/2409.04153 |