ELECTRONIC COMMUNICATIONS in PROBABILITY

2011

Abstract We study permutation betting markets, introduced by Chen et al.(Proceedings of the ACM Conference on Electronic Commerce, 2007). For these markets, we consider subset bettings in which each trader can bet on a subset of candidates ending up in a subset of positions.

2008

The overhand shuffle is one of the “real ” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [4] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [6] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s Lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n 3 log n) lower bound of Wilson [5] for the Rudvalis shuffle. 1

2005

In this essay we shall discuss mathematical models of card-shuffling. The basic question to answer is” how many times do you need to shuffle a deck of cards for it to become sufficiently randomized?”. This obviously depends on what we mean by shuffling and sufficiently randomized so we shall dwell quite a bit on these points too.

Journal of Combinatorial Theory, Series A, 2000

Upper and lower bounds are obtained for the number of shuffles necessary to reach the``furthest'' two hand deal starting from a given permutation of a deck of cards. The bounds are on the order of (log 2 n)Â2 and log log n, respectively.

The Annals of Applied Probability, 2010

The antique Mills Futurity slot machine has two unusual features. First, if a player loses 10 times in a row, the 10 lost coins are returned. Second, the payout distribution varies from coup to coup in a manner that is nonrandom and periodic with period 10. It follows that the machine is driven by a 100-state irreducible period-10 Markov chain. Here we evaluate the stationary distribution of the Markov chain, and this leads to a strong law of large numbers and a central limit theorem for the sequence of payouts. Following a suggestion of Pyke (2003), we address the question of whether there exists a two-armed version of this "one-armed bandit" that obeys Parrondo's paradox. More precisely, is there such a machine with the property that the casino can honestly advertise that both arms are fair, yet when players alternate arms in certain random or nonrandom ways, the casino makes money in the long run? The answer is a qualified yes. Although this "history-dependent" game is conceptually simpler than the original such games of Parrondo, Harmer, and Abbott (2000), it is nearly as complicated analytically, and open problems remain.

The Annals of Applied Probability, 2006

The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37-49] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274-325] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the (n 3 log n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77-85] for the Rudvalis shuffle.

The overhand shuffle is one of the "real" card shuffling methods in the se nse that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle (5) who showed that the mixing time with respect to variation distance is at least of order n2 and at most of order n2 log n. In this paper we use an extension of a lemma of Wilson (7) to establish a lower bound of order n2 log n, thereby showing that n2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's Lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the £(n3 log n) lower bound of Wilson (6) for the Rudvalis shuffle.

Journal of Physics Through Computation

Noise is known to disrupt a preferred act or motion. Yet, noise and broken symmetry in asymmetric potential, together, can make a directed motion possible, in fact essential in some cases. In Biology, in order to understand the motions of molecular motors in cells ratchets are imagined as useful models. Besides, there are optical ratchets and quantum ratchets considered in order to understand corresponding transport phenomena in physics. The discretized actions of ratchets can be well understood in terms of probabilistic gambling games which are of immense interest in Control theory-a powerful tool in physics, engineering, economics, biology and social sciences. In this article, we deal with Parrondo's paradox which is about a paradoxical game and gambling. Imagine two kinds of probability dependent games A and B, mediated by coin tossing. Each of the games, when played separately and repeatedly, results in losing which means the average wealth keeps on decreasing. The paradox appears when the games are played together in random or periodic sequences; the combination of two losing games results into a winning game! While the counterintuitive result is interesting in itself, the model can very well be thought of a discretized version of Brownian flashing ratchets which are employed to understand noise induced order. In our study, we examine various random combinations of losing probabilistic games in order to understand how and how far the losing combinations result in winning. Further, we devise an alternative model to study the similar paradoxical game and examine the idea of paradox in it. The work is done by computer simulations. Analytical calculations to support this work, is currently under progress. 1. Introduction: Parrondo's Paradox Game and interpretation A gambling or a probabilistic game can be depicted by coin tossing. In this Parrondo's paradox game, gambling is done with two biased coins. It is easy to understand that when we play with a

2006

The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order n^2 and at most of order n^2 n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order n^2 n, thereby showing that n^2 n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n^3 n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.

2010

The notion of gambling, traditionally applied to specified and organised games of chance, is increasingly being used to refer to a wider spectrum of behaviours and activities. In Fooled by Randomness, Taleb argues that chance plays a dominant role in many aspects of daily life, including financial markets, and that to succeed in life the role of chance must be understood in order to maximise gains and minimise losses.