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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2310.08935 |
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| _version_ | 1866909229722370048 |
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| author | Liang, Huaijin Chen, Zengjing |
| author_facet | Liang, Huaijin Chen, Zengjing |
| contents | The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated ran-domly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., ABABB if r = 2 and s = 3). They established the Parrondo effect if r + s divides J, and conjectured it in four other situations, including the case J = 2 with r >= 1 and s >= 1. We prove the conjecture in the latter case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08935 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture about Parrondo's paradox for two-armed slot machines Liang, Huaijin Chen, Zengjing Probability Computer Science and Game Theory 60J10, 60F05 The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated ran-domly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., ABABB if r = 2 and s = 3). They established the Parrondo effect if r + s divides J, and conjectured it in four other situations, including the case J = 2 with r >= 1 and s >= 1. We prove the conjecture in the latter case. |
| title | Proof of a conjecture about Parrondo's paradox for two-armed slot machines |
| topic | Probability Computer Science and Game Theory 60J10, 60F05 |
| url | https://arxiv.org/abs/2310.08935 |