Sharp thresholds for half-random games I

TL;DR

This paper investigates the threshold bias in biased Maker-Breaker positional games where one player plays randomly and the other plays optimally, revealing that randomness significantly tilts the advantage towards the random player.

Contribution

It determines the sharp threshold bias for classical graph games in the half-random setting with a clever Breaker, extending understanding of probabilistic intuition in positional games.

Findings

Threshold bias is significantly tilted towards the random player.

The probabilistic intuition still applies to the connectivity bottleneck.

Random play against an optimal opponent is disadvantageous for the random player.

Abstract

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph.