Discrete Gambling and Stochastic Games

Maitra, A.; Sudderth, William D.

doi:10.1007/978-1-4612-4002-0

Cited by 132 publications

(149 citation statements)

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“…Clearly, as n tends to ∞, U n (f ) increases to U ∞ (f ), the maximum probability of attaining the goal with initial fortune f when there is unlimited playing time (see [11,Section 2.15] and [16,Section 3.6]). It will be shown for w ≤ 1 2 ≤ r that the supremum in (1) is attained at y = B(f ), from which it follows that the bold strategy is optimal.…”

Section: Optimality Of Bold Playmentioning

confidence: 99%

On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

Yao

2007

J. Appl. Probab.

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In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ 1 2 ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.

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Section: Optimality Of Bold Playmentioning

confidence: 99%

On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

Yao

2007

J. Appl. Probab.

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“…First, it unifies and shortens disparate proofs of positionality for the parity [CY90], limsup [MS96] and mean [Bie87,NS03] payoff function (section 4). Second, it allows us to generate a bunch of new examples of positional payoff functions (section 5).…”

Section: Introductionmentioning

confidence: 98%

“…For example, the discounted payoff [Sha53,CMH06] and the total payoff [TV87] are used to evaluate shortterm performances. Long-term performances can be computed using the meanpayoff [Gil57,dA98] or the limsup payoff [MS96] that evaluate respectively average performances and peak performances. These functions are central tools in economic modelization.…”

Section: Introductionmentioning

confidence: 99%

Pure Stationary Optimal Strategies in Markov Decision Processes

Gimbert

Stacs 2007

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Abstract. Markov decision processes (MDPs) are controllable discrete event systems with stochastic transitions. Performances of an MDP are evaluated by a payoff function. The controller of the MDP seeks to optimize those performances, using optimal strategies. There exists various ways of measuring performances, i.e. various classes of payoff functions. For example, average performances can be evaluated by a mean-payoff function, peak performances by a limsup payoff function, and the parity payoff function can be used to encode logical specifications. Surprisingly, all the MDPs equipped with mean, limsup or parity payoff functions share a common non-trivial property: they admit pure stationary optimal strategies. In this paper, we introduce the class of prefix-independent and submixing payoff functions, and we prove that any MDP equipped with such a payoff function admits pure stationary optimal strategies. This result unifies and simplifies several existing proofs. Moreover, it is a key tool for generating new examples of MDPs with pure stationary optimal strategies.

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“…See also [17] for a discussion of the optimality of bold play in some two-person games. Several authors have considered discrete versions of this problem in which the gambler's initial fortune and the amount of each bet must be integers and the gambler's goal is to attain a fortune of n. An extensive discussion of discrete gambling problems such as this can be found in [13]. Bold play remains optimal when w < 1 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Also, by (13) and the first part of the lemma, we have (12). Finally, we consider the case in which, for some j ∈ {1, .…”

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confidence: 99%

Improving on bold play when the gambler is restricted

Schweinsberg

2005

J. Appl. Probab.

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Suppose that a gambler starts with a fortune in (0, 1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume that whenever the gambler stakes an amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 − w, where w < 1 2 . Dubins and Savage showed that the optimal strategy, which they called 'bold play', is always to bet min{f, 1 − f }, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than at one time. We show that the bold strategy of always betting min{ , f, 1 − f } is not optimal if is irrational, extending a result of Heath, Pruitt, and Sudderth.

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Discrete Gambling and Stochastic Games

Cited by 132 publications

References 0 publications

On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

Pure Stationary Optimal Strategies in Markov Decision Processes

Improving on bold play when the gambler is restricted

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