Perpetuities in Fair Leader Election Algorithms

Abstract

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the firstorder Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

A Binomial Splitting Process in Connection with Corner Parking Problems

“…Unlike g and g 2 , which is O(1) for large z, g V is exponentially small for large z. When p = 1 2 , we see that (14) has the closed form solutioñ…”

“…It also arose in a leader election algorithm, in which a truncated binomial number of contestants were advanced at each stage during a contest. Namely, X n is the number of rounds taken until the election comes to an end; see [13] and [14] for a broad framework for these types of problems.…”

“…In this paper we prove that the random variable X n has both a logarithmic mean and a bounded variance for large n. Also, the distributions do not approach a fixed limit law due to the fluctuations present in the distributions. For a similar context, see [12], [13], [14] and the references therein.…”

From coin tossing to rock-paper-scissors and beyond: a log-exp gap theorem for selecting a leader

“…See also [17], [27] for recurrences similar to (2.3). Alternatively, instead of considering only one round, one may also wait for a random number of times T n until a WOD set is reached.…”