Excel: Conditional probabilities of winning a tieb

问题:

I have been reading the following paper (http://strategicgames.com.au/PhD.pdf - page 21-22) and I am attempting to reproduce table 2.5 and 2.3 on page 23 in Excel.

Formulas for the calculation:

My Excel output:

I used the following steps to calculate the values.

1. Firstly I calculated the value 0.52 using the 3rd formula above - 0.62*(1-0.60)/(0.62*(1-0.60)+(1-0.62)*0.60)
2. Secondly I calculate the value 0.80 using the 1st formula above - =($E$1*C8)+($G$1*D7) and then copy and paste this formula into the other cells.

Where am I going wrong calculating the table values?

I should be receiving the following answer based on the paper's table.

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Excel implementation using solution below

回答1:

If you are going to use those formulas, then as the paper says, you need to use two tables, P_A and P_B.

Your calculation assumed that player A is on serve until someone reaches 7 or the score is (6,6). This incorrectly gives a big advantage to player A.

You can use a single table that is neither P_A nor P_B in the paper, but instead an interlaced version that I will call P_C that gives the probability that player A wins at a given score assuming that player A serves first, and thus at scores (a,b) where a+b mod 4 is 0 or 3.

P_C(a,b) = p_A * P_C(a+1,b) + (1-p_A) * P_C(a,b+1) when a+b mod 4 is 0 or 3.
P_C(a,b) = (1-p_B) * P_C(a+1,b) + p_B * P_C(a,b+1) when a+b mod 4 is 1 or 2.
P_C(6,6) = p_A * (1-p_B)/(p_A * (1-p_B) + p_B * (1-p_A)).
P_C(7,_) = 1.
P_C(_,7) = 0.