influence: A partizan scoring game on graphs

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 vertices 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 scores 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.

Incidence, a scoring positional game on graphs

“…Despite the fact that scoring games were less studied, mainly due to the difficulty to build a general framework for them, particular scoring games on graphs have still been introduced recently. One can cite the game Influence introduced by Duchêne et al in 2021 [7] which has been proven PSPACE-complete in 2022 [9], or the largest connected subgraph game, introduced by Bensmail et al, firstly as a scoring connection game [5], and then as a Maker-Breaker connection game [4]. In [19], there is a list of other particular scoring games on graphs that have been recently studied.…”

The Largest Connected Subgraph Game

“…Recently, the papers [12,13,14] have started to build a general theory around scoring games. There have also been a number of papers on different scoring games of late, such as [6,8,15,18]. In this paper, we introduce the following 2-player connection and scoring game on graphs, which, surprisingly, has not yet been studied even though it is a very natural game.…”

On a vertex-capturing game

“…For more on scoring games, we refer the reader to [9] for a survey on the topic, which also includes a formalism to deal with scoring games through scoring game notation, and the universes as mentioned above. To finish, we would like to mention a series of scoring games [2,4,6,10,12] that were recently introduced, and which feature different types of mechanisms or rules of independent interest.…”