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Abstract
Motivated by classical nontransitivity paradoxes, we call an
n -tuple
( x 1 , … , x n )
∈ [ 0 , 1 ] n
cyclic if there exist independent random variables
U 1 , … , U n with
P ( U i = U j )
= 0 for
i ≠ j such
that
P ( U i + 1 > U i )
= x i for
i
= 1 , … , n
− 1 and
P ( U 1 > U n )
= x n . We call
the tuple
( x 1 , … , x n )
nontransitive if it is cyclic and in addition satisfies
x i
> 1
2 for
all
i .
Let
p n
(resp. p n ∗ )
denote the probability that a randomly chosen
n -tuple
( x 1 , … , x n )
∈ [ 0 , 1 ] n is cyclic (resp. nontransitive).
We determine
p 3 and
p 3 ∗ exactly, while for
n
≥ 4 we give upper and
lower bounds for
p n
that show that
p n
converges to
1
as
n
→ ∞ .
We also determine the distribution of the smallest, middle, and largest elements in a
cyclic triple.
Keywords
nontransitivity paradoxes, probability, game theory, social
choice theory
Mathematical Subject Classification
Primary: 60C05, 91A60
Milestones
Received: 30 August 2020
Accepted: 29 November 2020
Published: 6 April 2021
Communicated by Anant Godbole