Playing weighted Tron on Trees

TL;DR

This paper analyzes a weighted version of the Tron game on trees, demonstrating bounds on players' gains and showing that the game outcome depends on the tree structure.

Contribution

It introduces the weighted Tron game on trees and establishes bounds on players' advantages, revealing how tree structure influences the outcome.

Findings

Alice can guarantee at most 1/5 less than Bob on trees.

Existence of trees where Bob can secure at least 1/5 more than Alice.

The game outcome varies with tree structure, affecting players' strategies.

Abstract

We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, each build their own path by claiming one vertex at a time, starting with Alice. The vertices carry non-negative weights that sum up to 1 and either player tries to claim a path with larger total weight than the opponent. We show that if the graph is a tree then Alice can always ensure to get at most 1/5 less than Bob, and that there exist trees where Bob can ensure to get at least 1/5 more than Alice.