Some online Maker-Breaker games

TL;DR

This paper studies online Maker-Breaker games where players select items in a random order, aiming to minimize or maximize the purchase cost of a subset from a family, with a focus on the implications of random permutations.

Contribution

It introduces a new model of online Maker-Breaker games involving random permutations and analyzes strategic behaviors under uncertainty.

Findings

Players' strategies depend on the random permutation of items.

The game dynamics reveal how randomness influences optimal strategies.

Results show the impact of permutation order on game outcomes.

Abstract

We consider some Maker-Breaker games of the following flavor. We have some set $V$ of items for purchase. Maker's goal is to purchase some member of a given family $\cH$ of subsets of $V$ as cheaply as possible and Breaker's goal is to make the purchase as expensive as possible. Each player has a pointer and during a player's turn their pointer moves through the items in the order of the permutation until the player decides to take one. We mostly focus on the case where the permutation is random and unknown to the players (it is revealed by the players as their pointers move).