A MultiComputational Exploration of Some Games of Pure Chance
by Thotsaporn Aek Thanatipanonda and Doron Zeilberger
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.tex
Posted: Sept. 25, 2019.
[Published in J. of Symbolic Computation 104 (2021), 3868.]
In the spirit of "multiculturalism", we use four kinds of computations: simulation, numeric, symbolic, and "conceptual"
to explore some "games of pure chance" inspired by children board games like "Snakes and Ladders" (aka as "Chutes and Ladders")
and "gambler's ruin with unlimited credit". Even more interesting than the many computergenerated actual results
described in this paper and its website extension, is
our broadminded, ecunemical approach, not favoring, a priori, any one of the above four kinds of computation, but
showing that, a posteriori, symbolic computation is the most important one, since (except for simulation, that is very inaccurate)
numerics can be made more efficient with the help of symbolics (in the "downward" direction),
and, (in the "upward" direction)
the mere existence of certain symboliccomputational algorithms imply interesting "qualitative" results, that
certain numbers are always rational, or always algebraic, and certain sequences are always polynomial, or
Crecursive, or algebraic, or holonomic.
This article is accompanied by four Maple packages, and numerous input and output files, that
readers can use as templates for their own investigations.
Added Feb. 24, 2020: The editor in charge of this submission, Manuel Kauers, used three referees.
Two of them understood our paper, but one of them completely misunderstood both the content and the
"big picture". Here is our
response .
Added June 11, 2020: Here is a proof of Prop. 17
The problem can be thought as starting at m >=0,
find the expected number of rounds until reaching
location 0 or less (two steps back or one step
forward with 1/2 chance each).
Denote this number by g(m).
We see that g(m) satisfies the recurrence relation
g(m) = 1+1/2*g(m2)+1/2*g(m+1) for m >=1
or equavalently
g(m) = 2*g(m1)g(m3)2, m >=2.
With the initial conditions g(1)=g(0)=0 and
g(1)=1+sqrt(5) (that was already proved in chapter 3
using generating function), we have that
g(m) and f(m)
where f(m) :=
2*m+(42*phi)+2*(fibonacci(m+2)*phifibonacci(m+3)),
satisfy the same initial conditions
and recurrence relation.
Hence g(m)=f(m) for all m>=1.
Maple packages
Sample Input and Output Files for SnakesAndLadders.txt

If you want to see the list of length 99 whose ith entry is the rational function in t
whose coefficient of t^{n} (in its Maclaurin expansion) is the probability of a Solitaire
"Chutes and Ladders" (American version, manufactured by Winning Moves Games) ending after n moves, lasting if you are currently at position i
The input gives you
the output.

If you want to see the list of length 99 whose ith entry is the rational function in t
whose coefficient of t^{n} (in its Maclaurin expansion) is the probability of a Solitaire
"Traditional Snakes and Ladders" (manufactured by Cardinal Industries) ending after n moves, if you are currently at position i
The input gives you
the output.

If you want to see a computergenerated article about a very simple "Snakes and Ladders games"
where there are neither snakes nor ladders,
and at each move you can advance either one or two moves (both with probability 1/2)
and you finish as soon as you reach 30 or 31
The input gives you
the output.

If you want to see a computergenerated article about a very simple "Snakes and Ladders games"
where there are 50 squares, and at each move you advance 1 with probability 1/3 and 2 with probability 2/3
where there is one chute (snake) from position 6 to 3, and one ladder from position 8 to 15,
The input gives you
the output.

If you want to see a computergenerated article about the actual Chutes and Ladders game
manufactured by Winning Moves Games
The input gives you
the output.

If you want to see a computergenerated article about the actual Snakes and Ladders game
manufactured by Cardinal Industries
The input gives you
the output.
Sample Input and Output Files for PosPileGames.txt

If you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he has at least n dollars, and each donor donates one dollar with probability 1/2 and two dollars with probability 1/2
The input gives you
the output.

More generally, if you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he has at least n dollars, and each donor donates one dollar with probability p and two dollars with probability 1p, for
arbitrary p (between 0 and 1)
The input gives you
the output.

If you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he has at least n dollars, and each donor donates 1, 2, 3, 4, 5, 6 dollars each with probability 1/6
The input gives you
the output.

If you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he has at least n cents, and each donor donates a penny, a nickel, a dime, or a quarter, each with probability 1/4
The input gives you
the output.

More generally, if you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he has at least n cents, and each donor donates a penny with probability p1, a nickel with probability p5,
a dime with probability p10, and a quarter with probability 1p1p5p10, in terms of the symbols p1,p5,p10,
The input gives you
the output.

More generally, if you want to see an article about the random variable (expectation, variance, etc.) "number of donors" that a beggar needs until
he gets n cents, if each donor gives 1,2, ..., k, cents, each with probability 1/k, for general k (as well, as of course, n)
[These are, up to exponentially small terms, polynomials in n but rational functions in k]
The input gives you
the output.

If you want to see an article about the sequence a(n), the probability of the first player winning
if each, independently take 1 or 2 tokens each with probability 1/2, and the first player
to gather n chips is declared the winner
The input gives you
the output.
[Note that the linear recurrence is of order 4.]

More generally,
if you want to see an article about the sequence a(n), the probability of the first player winning
if each, independently take 1 or 2 tokens each with probability p and 1p respectively, and the first player
to gather n chips is declared the winner
The input gives you
the output.
[Note that the linear recurrence is of order 4.]

If you want to see an article about the sequence a(n), the probability of the first player winning
if each, independently take 1 or 2 or 3 tokens each with probability 1/3 each, and the first player
to gather n chips is declared the winner
The input gives you
the output.
[Note that the linear recurrence is of order 10.]

If you want to see an article about the sequence a(n), the probability of the first player winning
if each, independently take 1 or 2 or 4 tokens each with probability 1/2,1/4,1/4 respectively each, and the first player
to gather n chips is declared the winner
The input gives you
the output.
[Note that the linear recurrence is of order 20.]
Sample Input and Output Files for GenPileGames.txt

If you want to see a computergenerated article about the random variable (expectation, variance, etc.)
"number of cointosses needed until you accumulate n dollars",
if at each coin toss you win a dollar with probability p and lose a dollar with probability 1p
(where p>1/2, to make sure that you achieve that goal with probability 1)
The input gives you
the output.

More generally, if you want to see a computergenerated article about the random variable (expectation, variance, etc.)
"number of cointosses needed until you accumulate n dollars",if at each coin toss you win a dollar with probability p
and lose k dollars with probability 1p
(where p>k/(k+1), to make sure that you achieve that goal with probability 1)
The input gives you
the output.

If you want to see a computergenerated article about the random variable (expectation, variance, etc.)
"number of cointosses needed until you accumulate ONE dollar",
if at each coin toss you LOSE 1 dollar with probability p and WIN 2 dollars with probability 1p
(where p<2/3, to make sure that you achieve that goal with probability 1). You get algebraic expression in terms of the symbol p.
The input gives you
the output.

Consider a game where players take turns tossing a fair coin and at each step either move left or right one unit
(each with probability 1/2), and whoever reaches the location m first is declared the winner.
What can you say about the probability of the first player winning the game?
It is (1+f(m))/2, where f(m) satisfies a certain secondorder (inhomogeneous) linear recurrence equation that enables
(found using the amazing Zeilberger algorithm), that enables
you to compute many terms efficiently once you know, thanks to Maple, that
f(1)=1Pi/4, f(2)=5 16/Pi
The input gives you
the output.

Consider a game where players take turns and at each turn move one unit to right with probability p and move k units to the left
with probability 1p
and whoever reaches the location m first is declared the winner, for various k and various NUMERIC p (>=k/(k+1))
What can you say about the probability of the first player of winning the game?
Using the amazing Zeilberger algorithm you can find a linear recurrence equation that enables
you to compute many terms efficiently once you computed the few initial conditions.
The input gives you
the output.

Consider a game where players take turns and at each turn move one unit to right with probability p and move k units to the left
with probability 1p
and whoever reaches the location m first is declared the winner, where p is SYMBOLIC and assumed that (>k/(k+1)),
so the game is guaranteed to terminate.
What can you say about the probability of the first player of winning the game?
Using the amazing Zeilberger algorithm you can find a linear recurrence equation that enables
you to compute many terms efficiently once you computed the few initial conditions.

If you want to see 4 computergenerated articles demonstrating the numerical procedures NS and ProbAheadN
The input gives you
the output.
Sample Input and Output Files for VGPileGames.txt