Limit theorems for Parrondo's paradox

关于双臂老虎机Parrondo悖论的一个猜想的证明

Proof of a conjecture about Parrondo's paradox for two-armed slot machines

本研究针对Ethier和Lee提出的J=2且r≥1、s≥1时带Futurity奖励的双臂老虎机存在Parrondo效应的猜想，完成了该猜想的证明，推进了Parrondo悖论在老虎机模型中的相关研究。

关于Parrondo对策的随机游动和扩散

On Random Walks and Diffusions Related to Parrondo's Games

G. Harmer和D. Abbott研究了与博弈相关的随机游动的行为。R. Parrondo。这些游戏说明了一个明显的悖论，即失败游戏的随机和确定性混合物可能会产生获胜游戏。本文利用整数模$m$的加法群上的经典循环随机游动，以一种直接的方式导出了一类包含Parrondo对策的一般对策的强律极限.然后，我们考虑的问题时，随机混合物的公平游戏相关的这些散步可能会导致赢得比赛。虽然这些问题的背景是基本的，但仍然存在悬而未决的问题。这些散步的结构的扩展到一类移位扩散，导致的事实是，两个公平的移位扩散的随机混合物可能是短暂的$+\infty$。

Combining losing games into a winning game

Parrondo's paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec's theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

悖论博弈中的机会反转

Reversals of chance in paradoxical games

我们提出了两个新的矛盾的特点，当他们结合起来的集体游戏。除了再现所谓的Parrondo效应，即从两个公平游戏的交替中获得获胜游戏之外，还出现了一种新的效应，即，当改变游戏之间的混合概率时，存在电流反转。我们提出了一个详细的研究，通过一个离散时间马尔可夫链分析，获得分析表达式的平稳概率为有限数量的球员。我们还提供了一些定性的见解，这种新的电流反转效果。

Parrondo games with two-dimensional spatial dependence

Parrondo games with one-dimensional spatial dependence were introduced by Toral and extended to the two-dimensional setting by Mihailović and Rajković. $MN$ players are arranged in an $M\times N$ array. There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or nonrandom pattern of games $A$ and $B$. Of interest is $μ_B$ (or $μ_C$), the mean profit per turn at equilibrium to the set of $MN$ players playing game $B$ (or game $C$). Game $A$ is fair, so if $μ_B\le0$ and $μ_C>0$, then we say the Parrondo effect is p…

Strategy-Driven Limit Theorems Associated Bandit Problems

Motivated by the study of asymptotic behaviour of the bandit problems, we obtain several strategy-driven limit theorems including the law of large numbers, the large deviation principle, and the central limit theorem. Different from the classical limit theorems, we develop sampling strategy-driven limit theorems that generate the maximum or minimum average reward. The law of large numbers identifies all possible limits that are achievable under various strategies. The large deviation principle provides the maximum decay probabilities for deviations from the limiting domain. To describe the flu…

A model for an epidemic with contact tracing and cluster isolation, and a detection paradox

We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation. This enables us to apply general limit theorems for super-critical Crump-Mode-Jagers branching processes. Notably, we compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection; and we explain the reasons behind this seeming paradox.