A Greedy Chip-firing Game

TL;DR

The paper introduces the hunger game, a deterministic analogue of Markov chains, which accurately mimics their behavior and exhibits concentration properties similar to stochastic simulations, with potential periodicity in rational cases.

Contribution

It presents the hunger game as a new deterministic model that replicates Markov chain behaviors and analyzes its convergence and periodicity properties.

Findings

Hunger game concentrates around stationary distribution with discrepancy ~ N^{-1}

Exhibits concentration for hitting measures and times with discrepancy ~ N^{-1}

In rational cases, the game is eventually periodic with a tiling structure in configuration space

Abstract

We introduce a deterministic analogue of Markov chains that we call the hunger game. Like rotor-routing, the hunger game deterministically mimics the behavior of both recurrent Markov chains and absorbing Markov chains. In the case of recurrent Markov chains with finitely many states, hunger game simulation concentrates around the stationary distribution with discrepancy falling off like $N^{-1}$, where $N$ is the number of simulation steps; in the case of absorbing Markov chains with finitely many states, hunger game simulation also exhibits concentration for hitting measures and expected hitting times with discrepancy falling off like $N^{-1}$ rather than $N^{-1/2}$. When transition probabilities in a finite Markov chain are rational, the game is eventually periodic; the period seems to be the same for all initial configurations and the basin of attraction appears to tile the configuration space (the set of hunger vectors) by translation, but we have not proved this.