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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 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.

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 Wilcoxon–Mann–Whitney

test for stratified samples related to Efron's paradox dice have been revealed [Thangavelu,

Brunner, 2007].

In an adjacent ―intransitive‖ area—not 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 areas—geometry and mechanics—a 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 one—2 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

level—that of mathematical combinatorics. The second co-author has invented the first example

of meta-intransitivity—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 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. Lebedev’s 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 numbers—from set B, and blue

numbers from set C.

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In the second set, we take red numbers from set B, green numbers—from set C, and blue

numbers from set A.

In the third set, we take red numbers from set C, green numbers—from 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, 8—meta-

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 (m≥0).

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 = 4・3m,

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 positions—for a set with meta-intransitivity

level equal to 1, 36 positions—for 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 = Z・3m.

Conclusion

Thus, the formerly known intransitive sets have been significantly expanded by a new

type—meta-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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