Abstract
We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process.
Highlights
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In the case a trapping action is chosen, the gambler wins if the last success epoch is isolated by the trap from the other success epochs
Motivation to study this game stems from connections to the best-choice problems with random arrivals [1,2,3,4,5,6,7,8,9] and the random records model [10,11]
Summary
In the case a trapping action is chosen, the gambler wins if the last success epoch is isolated by the trap from the other success epochs Motivation to study this game stems from connections to the best-choice problems with random arrivals [1,2,3,4,5,6,7,8,9] and the random records model [10,11]. Similar results have been obtained for the best-choice problem with some other pacing processes [1,4,7,9] In this context, trapping can be employed to test optimality of the myopic strategy, which fails if in some situations the action bygone outperforms but a trapping action is better still.
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