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First to reach $n$ game

Stanislav Volkov, Magnus Wiktorsson

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.

First to reach $n$ game

Abstract

We consider a game with two players, consisting of a number of rounds, where the first player to win 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.

Paper Structure

This paper contains 7 sections, 7 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Constant model
  3. Martingale approach
  4. Main results
  5. Pólya's model
  6. Anti-OK Corral model
  7. Remaining proofs

Key Result

Theorem 2.1

Let $E_{n,p}=\mathbb{E} Z_{n,p}$ and $z=pq$. Then where $C_j=\frac{(2j)!}{j!(j+1)!}$ are Catalan numbers.

Figures (1)

  • Figure 1: A path of the process in case $\tau_Y<\tau_X$.

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 8 more