A Treatise on Sucker's Bets

By Shalosh B. Ekhad and Doron Zeilberger

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(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 non-transitive 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) Efron-type 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) Efron-type 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) Efron-type 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 (tie-less) three-deck 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 (tie-less) four-deck 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) three-deck 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) three-deck 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) three-deck 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 tri-variate 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 Ekhad-Zeilberger tri-variate distribution
1/4*exp(-1/2*x^2-1/2*y^2-1/2*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 sucker-bets statistics,

the input file generates the output file.

• If you want to see a computer-generated 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 Ekhad-Zeilberger tri-variate distribution
1/4*exp(-1/2*x^2-1/2*y^2-1/2*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

• If you want to see the the pure linear recurrence operator in, a3, annihilating the mixed moment (a1,a2,2*a3) of the tri-variate distribution whose joint prob. density function is
1/4*exp(-1/2*x^2-1/2*y^2-1/2*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 the pure linear recurrence operator in, a3, annihilating the mixed moment (a1,a2,2*a3+1) of the tri-variate distribution whose joint prob. density function is
1/4*exp(-1/2*x^2-1/2*y^2-1/2*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.

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 tri-variate distribution whose joint prob. density function is

1/4*exp(-1/2*x^2-1/2*y^2-1/2*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^2-1/2*y^2-1/2*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.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger