## The Coin Tossing Game With Each Player Starting With Different Amounts of Money

Each player can have between £0 and £5.

If A has £0, then he cannot play, so will have $0 for ever after.

If A has £1, then after 1 toss he will have either £0 or £2, each with probability 1/2, and £1, £3, £4 or £5 with probability 0..

If A has £2, then after 1 toss he will have either £1 or £3, each with probability 1/2, and £0, £2, £4 or £5 each with probability 0..

If A has £4, then after 1 toss he will have either £3 or £5, each with probability 1/2, and £0, £1, £2 or £4 with probability 0.

If A has £5, then player B has £0, so cannot play and player A has £5 for ever after.

We can construct the table:

A has\A has After 1 Toss | £0 | £1 | £2 | £3 | £4 | £5 |

£0 | 1 | 0 | 0 | 0 | 0 | 0 |

£1 | 1/2 | 0 | 1/2 | 0 | 0 | 0 |

£2 | 0 | 1/2 | 0 | 1/2 | 0 | 0 |

£3 | 0 | 0 | 1/2 | 0 | 1/2 | 0 |

£4 | 0 | 0 | 0 | 1/2 | 0 | 1/2 |

£5 | 0 | 0 | 0 | 0 | 0 | 1 |

\[P= \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 & 0 & 0 \\ 0 & 0 & 1/2 & 0 & 1/2 & 0 \\ 0 & 0 & 0 & 1/2 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right)\]

.An absorbing state of a Markov chain when the probability of that state in the transition matrix is 1. Rewrite the table as

A has\A has After 1 Toss | £0 | £5 | £1 | £2 | £3 | £4 |

£0 | 1 | 0 | 0 | 0 | 0 | 0 |

£5 | 0 | 1 | 0 | 0 | 0 | 0 |

£1 | 1/2 | 0 | 0 | 1/2 | 0 | 0 |

£2 | 0 | 0 | 1/2 | 0 | 1/2 | 0 |

£3 | 0 | 0 | 0 | 1/2 | 0 | 1/2 |

£4 | 0 | 1/2 | 0 | 0 | 1/2 | 0 |

\[\left( \begin{array}{cc} P & O \\ S & Q \end{array} \right)\]

where\[P=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right), \; O= \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right), S= \left( \begin{array}{cc} 1/2 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1/2 \end{array} \right), \; Q= \left( \begin{array}{cccc} 0 & 1/2 & 0 & 0 \\ 1/2 & 0 & 1/2 & 0 \\ 0 & 1/2 & 0 & 1/2 \\ 0 & 0 & 1/2 & 0 \end{array} \right)\]

\[(I-Q)^{-1}= \left( \begin{array}{cccc} 1.6 & 1.2 & 0.8 & 0/4 \\ 1.2 & 2.4 & 1.6 & 0.8 \\ 0.8 & 1.6 & 2.4 & 1.2 \\ 0.4 & 0.8 & 1.2 & 1.6 \end{array} \right)\]

If a player starts with £2 the expected length of the game is (from row 2)1.2+2.4+1.6+0.8=6. If a player starts with £3 the expected length of the game is (from row 3) 0.8+1.6+2.4+1.2=6.