On a game on graphs

TL;DR

This paper generalizes a well-known knowledge game from a line of integers to arbitrary graphs, analyzing how graph structure influences players' ability to deduce their numbers and the time needed.

Contribution

It provides a complete characterization of the game dynamics on arbitrary graphs for two players, including pre-discussion scenarios.

Findings

Complete solution for two-player game on any graph

Determines conditions for players to deduce their numbers

Analyzes the impact of graph structure on game duration

Abstract

We start with the well-known game below: Two players hold a sheet of paper to their forehead on which a positive integer is written. The numbers are consecutive and each player can only see the number of the other one. In each time step, they either say nothing or tell what number they have. Both of them will eventually figure out their number after a certain amount of time. The game is rather cooperative than competitive, and employs the notions of common knowledge and mutual knowledge. We generalize this game to arbitrary (directed and non-directed) simple graphs and try to establish for which graphs one or both of them will figure out the solution, and how long they do need to find it. We give a complete answer for the case of two players, even if they are both allowed to discuss before the start of the game.