Solving the Three-Player-Game

TL;DR

This paper analyzes a three-player game involving sequential rounds, calculating the expected number of rounds until all win-lose relationships are observed, illustrating concepts in probability and dynamic programming.

Contribution

It provides an exact solution to the expected number of rounds in a three-player game, combining probability theory and dynamic programming techniques.

Findings

Expected number of rounds calculated

Illustrates application of probability and dynamic programming

Serves as educational example in probability textbooks

Abstract

In this paper we solve the three-player-game question. A three-player-game consists of a series of rounds. There are altogether three players. Two players participate in each round, at the end of the round the loser quits and the third player enters the ring and another round starts. The game terminates if all six win-lose relationships appear. During each round, two players win with equal probability. One is asked to calculate the expectation of the number of rounds. It turns out to be an exemplary question that involves probabiltiy theory and dynamic programming. It can serve as an instance or exercise in the chapter of conditional expectation of any elementary or advanced textbook on probability.