Explicit Expressions for Moments of the Duration of a 3-Player Gambler's Ruin

By Shalosh B. Ekhad and Doron Zeilberger

.pdf    .tex

Written: Sept. 15, 2023

Dedicated to Jean-Paul Delahaye, The French analog of Martin Gardner

Using experimental mathematics and symbolic computation, we derive many moments for the duration of a 3-player (fair) gamblerâ€™s ruin.

# Maple package

• GR3moms.txt, a Maple package to investigate the 3-Player (fair) Gambler's Ruin

## Sample Input and Output files for the Maple package GR3moms.txt

• If you want a computer-generated article with explicit expressions (as polynomials in A and B) of the first 20 moments of the duration of a fair 2-player gambler's ruin with intial capitals A and B, then

the input file produces the output file .

• If you a TERSE version of the above

the input file produces the output file .

• If you the first 20 (i) binomial (ii) straight (iii) about-the-mean (iv) scaled, and some limiting information, for 2-player gambler's ruin with TIES

the input file produces the output file .

• If you want a computer-generated article with explicit expressions (as polynomials in A, B, and C) of the first 10 moments of the duration of a fair 3-player gambler's ruin with intial capitals A, B, and C, then

the input file produces the output file .

• If you want the list of the first 20 moments about the mean of the duration of 2-player gambler's ruin with initial capitals A,B

the input file produces the output file .

• If you want the list of the first 40 moments about the mean of the duration of 2-player gambler's ruin with initial capitals A,B

the input file produces the output file .

It also gives you the special case A=B, and the limiting scaled moments of the latter.

• If you want the list of the first 10 moments about the mean of the duration of 3-player gambler's ruin with initial capitals A,B,C

the input file produces the output file .

[Note: this is contained in the file below, but since the latter file is so large, we keep it]

• If you want the list of the first 20 moments about the mean of the duration of 3-player gambler's ruin with initial capitals A,B,C

the input file produces the output file .

Warning: large file.

• Consider a 2-player gambler's ruin with equal initial capitals, A, and A, If you want a list of 20 numbers whose first entry is the expected duration divided by A2, whose second entry is the LIMIT of the variance of the duration divided by A4, as A goes to infinity, while the remaining entries are the Limits, as A goes to infinity, of the third through the 20th scaled moments:

the input file produces the output file .

• Consider a 3-player gambler's ruin with equal initial capitals, A, A, and A, If you want a list of 10 numbers whose first entry is the expected duration divided by A2, whose second entry is the LIMIT of the variance of the duration divided by A4, as A goes to infinity, while the remaining entries are the Limits, as A goes to infinity, of the third through the 15th scaled moments:

the input file produces the output file .