A min-max random game on a graph that is not a tree

TL;DR

This paper investigates a modified two-player game on a non-tree graph where outcomes depend on move sequences and move counts, revealing a sharp threshold phenomenon despite increased complexity and dependence issues.

Contribution

It extends classical threshold results to a more complex game on non-tree graphs, addressing the challenges of outcome dependence and loss of independence.

Findings

Existence of a sharp threshold in parameter p for the modified game

Dependence of outcomes on move sequences and counts

Complexity due to non-tree graph structure

Abstract

We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the case where all different game histories lead to different outcomes, a classical result due to Pearl (1980) says that in the limit when the number of moves is large, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. We are interested in a modification of this game where the outcome is determined by the exact sequence of moves played by the first player and by the number of times the second player has played each of the two possible moves. We show that also in this case, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. Since in the modified game, different game histories can lead to the same outcome, the graph associated with the game is no longer a tree which means independence is lost. As a result, the analysis becomes more complicated and open problems remain.