First to reach $n$ game

TL;DR

This paper analyzes a two-player game where the winner of each round is determined by drawing balls from an urn under three different regimes, revealing how the game's properties vary across these regimes.

Contribution

It introduces a novel analysis of a multi-regime urn-based game, exploring the probabilistic properties of players' net profits in each regime.

Findings

Properties of net profits differ drastically across regimes

Different regimes exhibit distinct probabilistic behaviors

The analysis provides insights into urn-based stochastic game dynamics

Abstract

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and type 2). At each round, we randomly pick a ball from the urn, and its type determines which of the two players wins. We study the game under three regimes. In the first and the third regimes, a ball is taken without replacement, whilst in the second regime, it is returned to the urn with one more ball of the same colour. We study the properties of the random variables equal to the properly defined overall net profits of the players, and the results are drastically different in all three regimes.