A note on concurrent graph sharing games

TL;DR

This paper analyzes a concurrent graph sharing game where two players select vertices with the goal of securing a minimum total weight, establishing optimal guarantees for different graph types.

Contribution

It proves that the first player can guarantee a minimum weight of 1/3 on any graph and 1/2 on trees, providing tight bounds for these scenarios.

Findings

First player can guarantee at least 1/3 weight on any connected graph.

On trees, the first player can guarantee at least 1/2 weight.

These bounds are tight, as shown by examples with cycles and trees.

Abstract

In the concurrent graph sharing game, two players, called First and Second, share the vertices of a connected graph with positive vertex-weights summing up to $1$ as follows. The game begins with First taking any vertex. In each proceeding round, the player with the smaller sum of collected weights so far chooses a non-taken vertex adjacent to a vertex which has been taken, i.e., the set of all taken vertices remains connected and one new vertex is taken in every round. (It is assumed that no two subsets of vertices have the same sum of weights.) One can imagine the players consume their taken vertex over a time proportional to its weight, before choosing a next vertex. In this note we show that First has a strategy to guarantee vertices of weight at least $1/3$ regardless of the graph and how it is weighted. This is best-possible already when the graph is a cycle. Moreover, if the graph is a tree First can guarantee vertices of weight at least $1/2$, which is clearly best-possible.