INFLUENCE: a partizan scoring game on graphs

TL;DR

The paper introduces INFLUENCE, a scoring game on directed graphs, analyzes its properties, especially on paths, and provides strategies and bounds for players' scores, revealing strategic advantages and surprising outcomes.

Contribution

It formalizes the game INFLUENCE, proves it is a nonzugzwang game in Milnor's universe, and analyzes optimal strategies and score bounds on paths.

Findings

First player always scores better than second by a bounded margin of 5.

INFLUENCE is a nonzugzwang game in Milnor's universe.

Players can win almost all vertices despite initial disadvantages.

Abstract

We introduce the game INFLUENCE, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices he has taken. We prove that INFLUENCE is a nonzugzwang game, meaning that no player has interest to pass at any step of the game, and thus belongs to Milnor's universe. We study this game in the particular class of paths where black and white are alternated. We give an almost tight strategy for both players when there is one path. More precisely, we prove that the first player always gets a strictly better score than the second one, but that the difference between the score is bounded by 5. Finally, we exhibit some graphs for which the initial proportion of vertices of the color of a player is as small as possible but where this player can get almost all the vertices.