A Treatise on Sucker's Bets
By Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )
Written: Oct. 27, 2017
In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with
a set of four dice, let's call them A,B,C,D, whose faces are marked with
[0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5]
respectively, where die A beats die B, die B beats die C, die C beats die D,
but, surprise surprise, die D beats die A! This was an amazing demonstration
that "being more likely to win" is not a transitive relation.
But that was only one example, and of course, instead of dice, we can use
decks of cards, where they are called (by Martin Gardner, who popularized this
way back in 1970) , "sucker's bets".
Can you find all such examples, with a specified number of decks, and
sizes? If you have a computer algebra system (in our case Maple), you sure
can!
Not only that, we can figure out how likely such sucker bets are, and derive, fully automatically,
statistical information!
Maple packages

SuckerBets.txt,
a Maple package to generate sucker bets (alias sets of nontransitive dice , alias Efron dice)

SuckerBetsAnalysis.txt,
a Maple package to analyze the Efron distribution for three decks

MutliAlmkvistZeilberger.txt,
a Maple package that accompanied this article,
by Moa Apagodu and Doron Zeilberger, that is needed here.

SMAZ.txt,
another Maple package that accompanied this article,
by Moa Apagodu and Doron Zeilberger, that is needed here.
Sample Input and Output Files for the Maple package SuckerBets.txt
If you want to see the unique (up to trivial cyclic symmetry) Efrontype set of four dice, where there are no ties,and only
1,2,3,4,5,6 are used
the input file generates the
output file.
If you want to see all the 38 different (up to trivial cyclic symmetry) Efrontype sets of four dice, where there are no ties,and only
1,2,3,4,5,6,7 are used,
that contains the original Efron set popularized by Martin Gardner
the input file generates the
output file.
If you want to see all the 755 different (up to trivial cyclic symmetry) Efrontype sets of four dice, where there are no ties,and only
1,2,3,4,5,6,7,8 are used,
the input file generates the
output file.
If you want to see ALL the (tieless) threedeck sucker bets with the sizes of the decks ranging from 3 to 5, and with
the number of different denominations that show up ranging from the smallest possible one all the way to 4 more
the input file generates the
output file.
If you want to see ALL the (tieless) fourdeck sucker bets with the sizes of the decks ranging from 3 to 4, and with
the number of different denominations that show up ranging from the smallest possible one all the way to 2 more
the input file generates the
output file.
If you want to see ALL the five (up to trivial cyclic symmetry) threedeck sucker bets with each deck with three cards, with all different
denominations (labeled 1 through 9)
the input file generates the
output file.
If you want to see ALL the thirteen (up to trivial cyclic symmetry) threedeck sucker bets with each deck with four cards, with all different
denominations (labeled 1 through 12)
the input file generates the
output file.
If you want to see ALL the 1732 (up to trivial cyclic symmetry) threedeck sucker bets with each deck with five cards, with all different
denominations (labeled 1 through 15)
the input file generates the
output file.
Sample Input and Output Files for the Maple package SuckerBetsAnalysis.txt

If you want to see the first 12 terms of the sequence "number of sets of 3 decks of cards, with n cards each"
that are sucker bets (equivalently, the number of words in the alphabet {1,2,3} with n occurrences of each letter
such that

The number of times 1 is ahead of 2 exceeds the number of times 2 is ahead of 1

The number of times 2 is ahead of 3 exceeds the number of times 3 is ahead of 2

The number of times 3 is ahead of 1 exceeds the number of times 1 is ahead of 3
the input file generates the
output file.

If you want to see the enumerating list up to trivial cyclic rotations (i.e. divided by 3) and the probabilities
the input file generates the
output file.

If you want to see explicit (polynomial) expressions, in n, for the mixed moments (i,j,k) for
0 ≤ i,j,k ≤ 5 for the Sucker Bets trivariate distribution with 3 decks of cards, each with n cards
the input file generates the
output file.

If you want to see explicit (polynomial) expressions, in c, for the mixed moments (i,j,k) for
0 ≤ i,j,k ≤ 16 for the EkhadZeilberger trivariate distribution
1/4*exp(1/2*x^21/2*y^21/2*z^2(x*y+x*z+y*z)*c)*2^(1/2)/Pi^(3/2)/(c+1)*(c^2+1)*(1+2*c)^(1/2)
the input file generates the
output file.

If you want to see the limit of the above as c goes to 1 (from below), conjecturally
(and rigorously proved up to all mixed moments (i,j,k) with i,j,k ≤ 5)
the
limiting scaled distribution of the suckerbets statistics,
the input file generates the
output file.

If you want to see a computergenerated article about the moments of the Sucker Bets statistics
and its relation to
(and rigorously proved up to all mixed moments (i,j,k) with i,j,k ≤ 5)
the
the EkhadZeilberger trivariate distribution
1/4*exp(1/2*x^21/2*y^21/2*z^2(x*y+x*z+y*z)*c)*2^(1/2)/Pi^(3/2)/(c+1)*(c^2+1)*(1+2*c)^(1/2)
the input file generates the
output file.

If you want to see the above in Maple format (so that you can use it in your own work)
the input file generates the
output file.
Input and Output Files Needed for this project from the Maple package MultiAlmkvistZeilberger.txt
Input and Output Files Needed for this project from the Maple package SMAZ.txt

If you want to see the the pure linear recurrence operator in, n, annihilating the
mixed moment (2n,2n,2n) of the trivariate distribution whose joint prob. density function is
1/4*exp(1/2*x^21/2*y^21/2*z^2(x*y+x*z+y*z)*c)*2^(1/2)/Pi^(3/2)/(c+1)*(c^2+1)*(1+2*c)^(1/2)
In other words, a recurrence satisfied by the sequence
Int(1/4*exp(1/2*x^21/2*y^21/2*z^2(x*y+x*z+y*z)*c)*2^(1/2)/Pi^(3/2)/(c+1)*(c^2+1)*(1+2*c)^(1/2)*x^(2*n)*y^(2*n)*z^(2*n),
x=infinity..infinity,y=infinity..infinity,z=infinity..infinity);
the input file generates the
output file.
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