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Various objects in intransitive relations of superiority (intransitive sets of sticks, dice, machines, and chess positions) are described. In meta-intransitive systems, each system has its own intransitive internal cycle of superiority between its components, but all the systems themselves form an intransitive cycle (meta-cycle) too. Meta-intransitive dice (meta-dice)—three sets of dice in each of which dice form their own intransitive cycles of superiority, and relations between the sets are intransitive too—are described. Meta-intransitive machines (levers) are shown. The formerly known intransitive sets can be considered as intransitive sets with zero-level of meta-intransitivity. A way to build multi-level meta-intransitive sets based on nested Condorcet-like compositions is introduced. Meta-intransitivity of various objects can become a new interesting sub-area of logics and mathematics of intransitive relations.
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Meta-Intransitivity: Meta-Dice, Levers and Other Opportunities
ALEXANDER PODDIAKOV
National Research University Higher School of Economics
apoddiakov@gmail.com
ALEXEY V. LEBEDEV
Lomonosov Moscow State University
avlebed@yandex.ru
Abstract
Various objects in intransitive relations of superiority (intransitive sets of sticks, dice, machines,
and chess positions) are described. In meta-intransitive systems, each system has its own
intransitive internal cycle of superiority between its components, but all the systems themselves
form an intransitive cycle (meta-cycle) too. Meta-intransitive dice (meta-dice)three sets of
dice in each of which dice form their own intransitive cycles of superiority, and relations
between the sets are intransitive tooare described. Meta-intransitive machines (levers) are
shown. The formerly known intransitive sets can be considered as intransitive sets with zero-
level of meta-intransitivity. A way to build multi-level meta-intransitive sets based on nested
Condorcet-like compositions is introduced. Meta-intransitivity of various objects can become a
new interesting sub-area of logics and mathematics of intransitive relations.
Objects in intransitive relations of superiority (dominance)
Many math lovers know about paradoxical intransitive
1
dice: die A beats die B more often than
is beaten by it; B beats C more often than is beaten by it; but C beats A more often than is beaten
by it like in the game of rock-paper-scissors. ―With any of these sets of dice you can operate a
betting game so contrary to intuition that experienced gamblers will find it almost impossible to
comprehend even after they have completely analyzed it‖ [Gardner, 2004, p. 286].
Stochastic intransitivity of dice wins can be better comprehended via a simpler
deterministic structure of tournaments of sticks (e.g., pencils) of different lengths. If one
organizes a tournament between ―teams‖ of pencils and compares the length of each pencil from
each set with the length of each pencil from the other sets, s/he can see that pencils of Set A are
more often longer than pencils of Set B, which are more often longer than pencils of Set C, and
the latter tend to be longer than those of Set A (see Figure 1). In other words, the results of the
tournament between such pencil teams will be intransitive: Team A beats B, B beats C, and C
beats A. Thus, the elementary relation ―to be longer than‖ is transitive, but the more complex
relation ―to be longer more often than‖ can be intransitive.
1
A note on terminology: in the math literature, the terms ―intransitive‖ and ―non-transitive‖
(e.g., ―intransitive dice‖ and ―non-transitive dice‖) are used as synonyms in spite of some
difference between the logical terms ―intransitive relation‖ and ―non-transitive relation‖. In this
article in cases of description of our own objects we will use the term ―intransitive‖ as explicitly
related to the concept of intransitive cycles.
2
Figure 1 An example of intransitive sets of
sticks (e.g., pencils).
Numbers to define the pencils’ lengths are
taken from the magic square presented by
Gardner 0
In the last 50 years various sets of intransitive dice have been invented:
- Efron dice described by Gardner [2004];
- dice with numbers π, e, ϕ [Beardon, 1999/2018]0;
- Double Whammy dice such that if each player rolls two identical dice (instead of a
single die) then the direction of ―beating‖ gets reversed (for the single dice set, the direction is
A>B>C>A, but for the double dice set, the direction is AA<BB<CC<AA) [Grime, 2017];
- O. van Deventer’s dice for a three player game (―here two opponents may each choose a
die from the set of seven, and yet there will be a third die with a better chance of beating each of
them‖) [Ibid.];
- dice for a four player game [Pegg, 2005] etc.
Many studies of intransitive dice sets have been conducted for the last years 0Akin, 2019;
Buhler et al., 2018; Conrey et al., 2016; Grime, 2017; Hązła et al., 2020; Hulko, Whitmeyer,
2019; Polymath, 2017]0. In mathematical statistics, problems with WilcoxonMannWhitney
test for stratified samples related to Efron's paradox dice have been revealed [Thangavelu,
Brunner, 2007].
In an adjacent ―intransitive‖ areanot of discrete but continuous random variables, the
nontransitivity problem of the stochastic precedence relation for three independent random
variables with distributions from a given class of continuous distributions has been studied
[Lebedev, 2019].
In distant areasgeometry and mechanicsa class of intransitive simple machines
(double levers, gears, wedges, inclined planes etc.) of such geometrical shapes that these objects
interact with one another like in the rock-paper-scissors game has been invented [Poddiakov,
2018] (Figures 2, 3).
Figure 2 Poddiakov’s Intransitive
Gears.
Gear A rotates faster than B in pair A-
B, B rotates faster than C in pair B-C,
and C rotates faster than A in pair A-C.
Figure 3 Van Deventer’s Non-Transitive Gears-and-Ratchets
based on Poddiakov’s Intransitive Gears
https://i.materialise.com/forum/t/non-transitive-gears-by-
oskar/1167.
Whichever of the three knobs you turn, one of its neighbors
will rotate 2 times faster and the other one2 times slower.
See also [Van Deventer, 2019]
3
In game with van Deventer’s Non-Transitive Gears-and-Ratchets, whatever element the
first player chooses, the second player can always choose an element rotating faster than the
element chosen by the first player. Moreover, the Non-Transitive Gears-and-Ratchets can be
used for a three player game. If the first two players choose an element (a knob or a gear) each,
the third player always chooses an element and the direction of its rotation so that it will rotate
faster than the elements chosen by other players.
In logical board games such as chess or checkers, the concept of intransitivity can be
applied to build intransitive positions [Poddiakov, 2017] (Figures 4, 5). In chess, the number of
such intransitive chains is extremely high (on a par with the number of all possible positions on
the chess board) and their length can vary greatly, from a four-position chain involving only a
king and a pawn for White and Black to astronomically long chains with a number of pieces on
both sides [Filatov, 2017].
Figure 4 Intransitive chess positions by
Poddiakov [2017].
White moves first in all the positions (―
means ―is better than‖). Position A for White is
preferable to Position B for Black (i.e., when
offered a choice, one should choose A),
Position B for Black is preferable to Position C
for White, which is preferable to Position D
(Black) but the latter is preferable to Position
A (White).
Figure 5 Intransitive checkers positions by S.
Zhurakhovsky.
4
Pointing to intransitive chess positions is a shortest proof of the fact well-known to
programmers working on algorithms for chess computers: any fixed quantitative estimate of
winningness (preferability) of a position of White without taking a position of Black into account
is impossible, and any fixed quantitative estimate of winningness (preferability) of a position of
Black without taking a position of White into account is impossible too. If positions of White
and Black can form intransitive cycles, only context-dependent comparative estimates of
winningness of the positions are possible (like in intransitive dice sets for which any fixed
quantitative estimates of winningness are impossible too) [Poddiakov, 2017].
How rare are intransitive chess and checkers positions? (Cf. [Conrey et al., 2016;
Polymath, 2017]. Are intransitive positions possible in other strategic games on a marked board
(e.g., in Go)? What are the minimum size of a board and the minimum level of complexity of
game interactions between pieces (characters) of the game to make intransitive positions
possible? We do not know it yet. (We speak namely about intransitivity of positions, not
intransitivity of game strategies or individual chess players’ win-losses relations [West, Hankin,
2008].
Meta-intransitive systems
These various intransitive systems (dice, machines, chess and checkers positions) led the first co-
author to introducing the concept of meta-intransitivity [Poddiakov, 2021].
All the formerly known intransitive dice sets can be compared with one another and
linearly ordered by intransitivity: for example, from sets of dice whose intransitivity is weak (the
dice beat one another with a probability slightly higher than 50%) to sets of dice whose
intransitivity is strong (probability of a win is closer to 75% which is the proven limit). So, the
intransitive dice sets can be ordered transitively. The same concerns intransitive machines: e.g.,
one can construct intransitive gears sets with different gear ratios and, respectively, different
intransitivity (from weak intransitivity for small ratios to strong one for large ratios). Intransitive
chess positions can be ordered transitively too (e.g. via quantity of moves necessary for win
from positions in which a large number of moves is necessary for win to positions in which a
small number is necessary).
In contrast, in meta-intransitive systems, each system has its own intransitive internal
cycle of superiority (dominance) between its components but all the systems themselves form an
intransitive cycle (meta-cycle) too. The analogies which come to mind are complex biological,
social, bio-social, technological ―rhizomatic‖ (multiple, intertwining) systems, self-supporting
and self-developing ones [Poddiakov, 2019; Poddiakov, Valsiner, 2013].
Yet meta-intransitivity can be found at a much simpler and in a sense more fundamental
levelthat of mathematical combinatorics. The second co-author has invented the first example
of meta-intransitivitymeta-intransitive dice (meta-dice): three sets of dice in each of which
dice form their own intransitive cycles of superiority, and relations between the sets are
intransitive as well.
Meta-intransitive dice (meta-dice)
Lebedev’s meta-dice are built as a Condorcet-like composition including 3 sets of intransitive
dice. A short description of the Condorcet paradox which can be understood by secondary school
students is presented by Beardon [1999/2018]:
―In the <…> Voting Paradox there are 3 candidates for election. The voters have to rank
them in order of preference. Consider the case where 3 voters cast the following votes: ABC,
BCA and CAB:
A beats B by 2 choices to 1
B beats C by 2 choices to 1
but A cannot be the preferred candidate because A loses to C, again by 2 choices to 1.
This is an example of intransitivity‖.
5
A body of work has been devoted to the voting paradox and its meaning for logic, social
sciences, and practices (see a review and analysis in [Gehrlein, 2006]. It is important for us that
elements in the Condorcet paradox are ordered in the way described above:
ABC
BCA
CAB
One can see that the first element of any line goes to the last position in the next line and
moves all the other elements one position to the left without changing the sequence. This
principle has been used to make the meta-intransitive dice.
Set 1
Set 2
Set 3
Die 1
3
2
1
2
1
3
1
3
2
3
2
1
2
1
3
1
3
2
3
4
5
4
5
3
5
3
4
3
4
5
4
5
3
5
3
4
3
9
5
9
5
3
5
3
9
6
9
6
9
6
6
6
6
9
Die 2
2
1
3
1
3
2
3
2
1
2
1
3
1
3
2
3
2
1
2
6
4
6
4
2
4
2
6
5
6
4
6
4
5
4
5
6
5
8
4
8
4
5
4
5
8
5
8
9
8
9
5
9
5
8
Die 3
1
3
2
3
2
1
2
1
3
4
3
3
3
3
4
3
4
3
4
5
3
5
3
4
3
4
5
4
5
3
5
3
4
3
4
5
4
7
7
7
7
4
7
4
7
4
7
9
7
9
4
9
4
7
Table. Lebedevs meta-dice.
Each face of each die has not one, but 3 numbers: a red number, a green number, and a blue
number. After rolling a die, a player receives a red ―coin‖ if her/his red number is bigger than her
or his rival’s red number, a green ―coin‖ if her/his green number is bigger than her or his rival’s
green number, and a blue ―coin‖ if her/his blue number is bigger than her or his rival’s blue
number. (These cases are not mutually exclusive: a player can get not one but two or three coins
of different colors simultaneously.)
One die is considered superior over another die if the percentage of its wins (defined as
all the coins received by the die), is bigger than the percentage of the other die’s wins.
Winningness of a set of dice is counted for each color and so is described by 3 numbers.
A set of dice is considered superior over another set of dice if winningness of at least two
of its colors is higher than winningness of the other set.
Now let us take 3 intransitive dice sets of a usual (not meta-intransitive) type but with
different winningness. Let us order the sets from the most winning one (marked as set A) to the
least winning one (marked as set C) and build new sets accordingly to the Condorcet principle.
Namely, we do the following.
In the first set, we take red numbers from set A, green numbersfrom set B, and blue
numbers from set C.
6
In the second set, we take red numbers from set B, green numbersfrom set C, and blue
numbers from set A.
In the third set, we take red numbers from set C, green numbersfrom set A, and blue
numbers from set B.
Thus, each set is intransitive, and the set of these sets is intransitive too (in the sense of
introduced relations of superiority for the dice and the sets).
One can see that Condorcet-like compositions are used here at two levels:
- the level of interactions of the dice inside the sets;
- the level of interactions between the sets of dice.
Meta-intransitive levers
The same method of ―nested‖ Condorcet-like compositions (we use term ―nested‖
analogically with nested loops in programming) can be applied to make meta-intransitive
machines. In Figure 6, simply intransitive double levers are shown, and in Figures 7, 8meta-
intransitive ones.
Figure 6 Poddiakov’s intransitive double levers [Poddiakov, 2018].
With the same rotation force applied to the shaft, Lever A will overpower Lever B as A’s lever
arm (the perpendicular distance from the fulcrum to the line of action of the effort) is shorter
than B’s lever arm. Lever B will overpower Lever C and Lever C will overpower Lever A.
Figure 7 Poddiakov’s meta-intransitive levers.
The red, green and blue levers form internal intransitive cycles inside sets X, Y, and Z.
The yellow, pink, and white levers form intransitive cycles between sets X, Y, and Z.
7
Figure 8 Use of the meta-
intransitive levers: a an
interaction inside a set, b an
interaction between sets.
In each of sets X, Y, and Z, red levers overpower green levers, green levers overpower blue
levers, and blue levers overpower red levers. In interactions between the sets, yellow levers
overpower pink levers, pink levers overpower white levers, and white levers overpower yellow
levers. In contrast with the meta-dice, in meta-intransitive levers, structure of relations between
wins and losses is deterministic, and the levers win one another with probability 100%. If one
wishes to win, s/he has to make intransitive choices of the levers like intransitive choices of
intransitive dice.
Are “nested” Condorcet-like compositions a universal way to create meta-
intransitivity of various meta-levels?
Building of multi-level nested Condorcet-like compositions seems a way to build meta-
intransitive sets with various numbers of meta-levels.
Yet a problem of universality of the way of meta-intransitive systems constructing
through Condorcet-like compositions is open. For example, it is hard to imagine its immediate
application to build meta-intransitive chess positions.
This non-universality is explicated in differences between formulas describing
quantitative relations between the initial zero level of meta-intransitivity (i.e. without it) and
higher levels.
The minimum number of elements (dice) required to build a meta-intransitive dice set
with a given meta-intransitivity level is the following:
Nmin = 3m+1,
where Nmin - the minimum number of elements (dice) required to build a meta-
intransitive dice set, m the level of meta-intransitivity (m0).
For example, at least 3 dice are required to build a usual intransitive dice set (with meta-
intransitivity level equal to 0), at least 9 dice are required to build a meta-intransitive dice set of
the 1st level (with meta-intransitivity level equal to 1), 27 dice are required to build a meta-
intransitive dice set of the 2nd level (with meta-intransitivity level equal to 2) etc.
The same concerns meta-intransitive machines.
The minimum number of positions of White and Black in chess or checkers required to
build a meta-intransitive set with a given level of meta-intransitivity is the following:
Nmin = 43m,
where Nmin - the minimum number of positions of White and Black (totally) required to
build a meta-intransitive set, m the level of meta-intransitivity.
8
For example, at least 4 positions (2 of White and 2 of Black) are required to build their
intransitive set of zero-meta-intransitivity, at least 12 positionsfor a set with meta-intransitivity
level equal to 1, 36 positionsfor a set with meta-intransitivity level equal to 2 etc.
In general, the minimum number of components required to build an m-meta-intransitive
set for which the minimum number of components at zero-meta-intransitivity level is Z can be
expressed in the following way: Nmin = Z3m.
Conclusion
Thus, the formerly known intransitive sets have been significantly expanded by a new
typemeta-intransitive ones. Two examples from different domains:
- combinatorics of numbers in meta-intransitive dice; and
- combinatorics of geometrical shapes in meta-intransitive machines
have been shown.
The formerly known intransitive sets can be considered as ones of zero-level of meta-
intransitivity. A way to build multi-level meta-intransitive sets has been introduced.
On one hand, meta-intransitivity of various objects is joy for lovers of math problems. On
the other hand, it can become a new interesting sub-area of logics and mathematics of
intransitive relations.
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Grime, J. (2017). The bizarre world of nontransitive dice: games for two or more players. The College Mathematics Journal. 48(1): 2-9. doi.org/10.4169/college.math.j.48.1.2. Hązła, J., Mossel, E., Ross, N. (2020). The probability of intransitivity in dice and close elections. Probab. Theory Relat. Fields. 178: 951-1009. doi.org/10.1007/s00440-020-00994-7
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