Leader election using random walks

TL;DR

This paper generalizes classical leader election by using integer-valued random walks to determine remaining players, analyzing asymptotic behavior and connections to branching processes and stable point processes.

Contribution

It introduces a novel random walk-based leader election model and characterizes its asymptotics and stability properties, linking to branching processes and stochastic fixed point equations.

Findings

Asymptotic behavior of remaining players' positions

Distribution of total rounds until all players leave

Characterization of stable point processes under thinning

Abstract

In the classical leader election procedure all players toss coins independently and those who get tails leave the game, while those who get heads move to the next round where the procedure is repeated. We investigate a generalizion of this procedure in which the labels (positions) of the players who remain in the game are determined using an integer-valued random walk. We study the asymptotics of some relevant quantities for this model such as: the positions of the persons who remained after $n$ rounds; the total number of rounds until all the persons among $1,2,\ldots,M$ leave the game; and the number of players among $1,2,\ldots,M$ who survived the first $n$ rounds. Our results lead to some interesting connection with Galton-Watson branching processes and with the solutions of certain stochastic-fixed point equations arising in the context of the stability of point processes under thinning. We describe the set of solutions to these equations and thus provide a characterization of one-dimensional point processes that are stable with respect to thinning by integer-valued random walks.