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Learning Intransitivity:

from Intransitive Geometrical

Objects to "Rhizomatic"

Intransitivity

Alexander Poddiakov

National Research University

Higher School of Economics

A B C

18-21 March 2019, Moscow

All of us know that if A>B and B>C than A>C.

This is true – but not for all types of situations.

Alongside such relations, which are called transitive,

exist other types of relations, intransitive ones.

In my presentation, I will

- explore the nature of intransitivity and introduce a

new class of intransitive objects – geometrical ones

-describe some experimental research on people’s

understanding of intransitive relations

- list 4 types of complexity of intransitive relations

- argue that the message for students should be more

multi-dimensional than “if you have violated the

transitivity axiom, you are not rational”

The transitivity axiom: if A≻B and B≻C then A≻C

where “≻” means “dominates over”, “is better than”,

“is preferable to” etc.

Many people believe that if “you have violated the

transitivity axiom, … you are not instrumentally

rational. The content of A, B, and C do not matter to

the axiom”.

(Five Minutes with K.E. Stanovich, R.F. West, and M.E.

Toplak, 2016

https://mitpress.mit.edu/blog/five-minutes-keith-e-stanovich-richard-f-west-and-maggie-e-toplak)

In contrast – we have a lot of math research on

objects in intransitive relations

(“rock-paper-scissors” relations)

2 4 9 1 6 8 3 5 7

Lets us consider 3 sets of 3 pencils of different lengths. We

compare the length of each pencil with the length of all the

other pencils.

Numbers are taken

from the magic

square presented by

Gardner (1974)

2 4 9 1 6 8 3 5 7

Red pencils beat green ones 5 out of 9 times

2 4 9 1 6 8 3 5 7

Green pencils beat blues ones 5 out of 9 times

2 4 9 1 6 8 3 5 7

Blue pencils beat red ones 5 times out of 9 times

Thus, transitivity works during

the comparison of 3 pencils – for

the relation “to be longer”…

but it does not work in more

complex situations: during

the comparison of 3 sets of 3

pencils – for the relation

“to often be longer”

Thus, transitivity works during

the comparison of 3 pencils – for

the relation “to be longer”…

but it does not work in more

complex situations: during

the comparison of 3 sets of 3

pencils – for the relation

“to often be longer”

Only arrows from winning to loosing sets

are shown

Intransitive dice

National Museum of Mathematics, USA

Purple (A): 4, 4, 4, 4, 0, 0

Yellow (B): 3, 3, 3, 3, 3, 3

Red (C): 6, 6, 2, 2, 2, 2

Green (D): 5, 5, 5, 1, 1, 1

A≻B, B≻C, C≻D, D≻A

Intransitive competition (“rock–paper–scissors”

competition) is also studied in biology and sociology

Let’s return to math - some references…

Conrey, B. et al. (2016). Intransitive dice. Mathematics Magazine, 89(2).

Gardner, M. (1974). On the paradoxical situations that arise from non-transitive

relations. Scientific American, 231(4).

Gower, T. (2017). A potential new Polymath project: intransitive dice.

https://gowers.wordpress.com/2017/04/28/a-potential-new-polymath-project-

intransitive-dice.

Grime, J. (2017). The bizarre world of non-transitive dice: Games for two or more

players. The College Mathematics Journal, 48(1).

Lebedev, A. V. (2018). [Intransitive triplets of continuous random variables]

http://www.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=21644

Pegg Jr., E. (2005). Tournament dice. http://www.mathpuzzle.com/MAA/39-

Tournament%20Dice/mathgames_07_11_05.html.

Trybuła, S. (1961). On the paradox of three random variables. Applicationes

Mathematicae, 5(4).

Yet all this research deals with numbers. I will present

geometrical and mechanical constructions in

intransitive relations. From a mathematical point of

view, it is a new class of intransitive objects. They

show deterministic (not probabilistic) intransitive

relations.

Stylized Tractors with Intransitive Towing Couplers

Which tractor should the driver choose as the leading one to

fit into to bring two tractors to a destination point?

The driver should choose A in pair A-B, B in pair B-C, and C in

pair A-C. That is, A≻B, B≻C, C≻A.

This model does not require quantitative comparisons. The

comparison of geometrical shapes is all that is necessary.

A B C

Intransitive

Double

Levers

With the same rotation force applied to the shaft,

Lever A will overpower lever B (as lever arm A is shorter

than lever arm B).

Lever B will overpower lever C.

Lever C will overpower lever A.

A A

B B C C

Levers and gears are “close relatives” in the family

of simple machines, so…

Intransitive Double Gears – with intransitive speeds

of rotation

Whatever element the first player chooses (a knob or a

gear), the second player can always choose an element

rotating faster than the element chosen by the first

player.

I have shown that Oskar van Deventer’s 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.

Moreover, in 75% of cases the

third player can not only win but

can also (!) control the

distribution of the second and

third places of the other players.

“Transitivity and intransitivity are fascinating

concepts that relate both to mathematics and to the

real world we live in”.

Teachers can “interest and engage students … as

they seek to discover which relationships are

transitive, and which are not”.

Roberts, T. S. (2004). A ham sandwich is better than

nothing: Some thoughts about transitivity. Australian

Senior Mathematics Journal, 18.

Experimental studies of people’s understanding

“which relationships are transitive,

and which are not”

Beliefs about intransitive relations are domain-

specific. The participants believe that intransitive

relations exist in some domains but not in others

(though really they are possible in the latter domains

as well).

Poddiakov, 2010, 2011; Bykova, 2018 (under

Poddiakov’s supervision)

Participants: 169 people (17-28 yrs, 121 females, 48 males)

Method

Participants were asked questions about the possibility of the

domination of A over B, B over C, and C over A for various A, B,

and C.

“Is it possible that

A beats/dominates over B,

B beats/dominates over C,

C beats/dominates over A?” in various areas.

1. There are 3 straight rigid sticks where the 1st stick is longer

than the 2nd stick, and the 2nd stick is longer than the 3rd

stick. Is it possible that the 3rd stick can be longer than the 1st

stick?

98% of answers – “say no” (right answer)

2. There are 3 objects of different mass where the 1st object is

more massive than the 2nd object, and the 2nd object is more

massive than the 3rd object. Is it possible that the 3rd object

can be more massive than the 1st object?

93% of answers – “say no” (right answer)

3. Is it possible that chess computer A regularly prevails over

chess computer B, B regularly prevails over C, but C regularly

prevails over A?

79% of answers – “say it’s possible” (right answer)

4. There are 3 boxes with 6 pencils of different lengths in each

box. We compare the length of each pencil with the length of

all the other pencils. We learn that the pencils from the 1st

box are often longer than the pencils from the 2nd box, and

the pencils from the 2nd box are often longer than the pencils

from the 3rd box.

Is it possible that the pencils from the 3rd box are often

longer than the pencils from the 1st box?

29% of answers – “say it’s possible”

(right answer)

the simpler example of intransitive

pencil sets is on the right) 2 4 9 1 6 8 3 5 7

5. There are 3 teams with 6 wrestlers in each team. During

their tournament, each wrestler of one team meets and

wrestles with each wrestler from two other teams. It is known

that the wrestlers of the 1st team beat the wrestlers of the 2nd

team more often than are beaten by them, and the wrestlers of

the 2nd team beat the wrestlers of the 3rd team more often

than are beaten by them.

Is it possible that the wrestlers of the 3rd team beat the

wrestlers of the 1st team more often than are beaten by them?

81% of answers – “it’s possible” (right answer)

!!!

Problems “3 boxes with 6 pencils in each box” and “3 teams

with 6 wrestlers in each team” are of the same logical

structure.

But, paradoxically, the most participants (81% ) believe in

intransitivity of wrestlers’ teams, and only 29% think that

intransitivity of length in pencils’ boxes is possible.

Why? We have not studied it yet. Only some hypotheses can be

formulated.

6. Is it possible that microorganisms A force out B, B force out

C, and C force out A?

91% of answers – “it’s possible” (right answer)

(Concept “intransitive competition” is widely accepted in

biology.)

7. There are 3 kinds of stylized wheeled battering rams

competing with one another: each ram tries to mark another

ram with its inserted felt-tip pen.

It is known that the 1st ram marks the 2nd ram, but avoids

being marked by the last one; and the 2nd ram marks the 3rd

ram, but avoids being marked by the last one.

Is it possible that the 3rd ram marks the 1st ram, but avoids

being marked by the last one?

86% of answers – “it’s possible”

(right answer,

a possible example is

on the right)

8. There are 3 double-gears joined in pairs, the 1st double-gear

rotates faster than the 2nd double-gear, and the 2nd double-

gear rotates faster than the 3rd double-gear.

Is it possible that the 3rd double-gear rotates faster than the

1st double-gear?

26% of answers – “say it’s possible” (right answer)

a possible example is below.

Conclusion on the experiment

Beliefs about intransitive relations are domain-

specific. The participants believe that intransitive

relations exist in some domains but not in others

(though really they are possible in the latter domains

as well).

The axiom of transitivity was applied by the

participants very selectively. In many situations it was

the correct solution.

M.S. Permogorskiy (2016, under Poddiakov’s

supervision) proposed the following original hypothesis:

Inclusion of additional information

“it is known that A>B and B>C”

into problems that deal with objectively intransitive

relations reduces the success rate (i.e., prevents

participants from drawing the right conclusion that

C>A).

A problem posed for the experimental group

There are 3 sets of 6 pencils of different lengths in each set.

Set А: 8 5 5 3 3 3

Set B: 7 7 7 2 2 2

Set C: 6 6 6 4 4 1

We have compared the length of each pencil with the length of

all the other pencils. We have learnt that pencils from Set A

are often longer than pencils from Set B, and pencils from Set

B are often longer than pencils from Set C.

Based on the data, which of the following statements is right?

- Pencils from Set A are often longer than pencils from Set C?

OR

- Pencils from Set C are often longer than pencils from Set A?

The problem posed for the control group

There are 3 sets of 6 pencils of different lengths in each set.

Set А: 8 5 5 3 3 3

Set B: 7 7 7 2 2 2

Set C: 6 6 6 4 4 1

We have compared length of each pencil with length of all the

rest pencils.

Based on the data, which of the following statements is right?

- Pencils from Set A are often longer than pencils from Set C?

OR

- Pencils from Set C are often longer than pencils from Set A?

Results

54% of correct answers - for problems with the information that

“pencils from Set A are often longer than pencils from Set B,

and pencils from Set B are often longer than pencils from Set

C”.

71% of correct answers - for problems without this information.

Conclusion on the experiment

Information such as “A>B and B>C” can lead the participants

to assume that the relationship is transitive. This can

negatively influence solving problems on objectively

intransitive relations.

Discussion

We have a rich tradition of math studies of various

intransitive objects. Yet there is no such tradition of

psychological and educational studies on

understanding-misunderstanding of the objectively

intransitive relations by people.

It seems reasonable to distinguish between four types of

situations (Poddiakov, 2010).

(1) Relations are objectively transitive and problem solvers

make correct conclusions about their transitivity.

(2) Relations are objectively transitive, but problem solvers

wrongly consider them as intransitive. Most studies are

conducted in this paradigm

(see e.g. May, 1954; Tversky, 1969; Regenwetter et al, 2011).

(3) Relations are objectively intransitive and problem solvers

make correct conclusions about their intransitivity

(e.g. intransitivity of intransitive dice, lotteries etc.)

(4) Relations are objectively intransitive, but problem solvers

wrongly consider them as transitive (e.g. because of taking the

transitivity axiom for granted).

Surprisingly, this type has been minimally studied by cognitive

and educational psychology.

Intransitive relations:

4 levels of complexity

The following classification is not exhaustive and serves to

mark some reference points.

1. Simple combinatorial intransitivity between non-interacting

objects (e.g. in intransitive dice sets, intransitive sets of sticks

etc.). Each object can be precisely described by a few

parameters (like numbers on the sides of dice). The parameters

are additive, without interactions.

2 4 9 1 6 8 3 5 7

2. Interactive intransitivity without qualitative transformations

of the objects in the intransitive relations. The information

about the objects and their interactions is complete.

3. Interactive intransitivity

with qualitative

transformations of the

objects in intransitive

relations.

Information about the

objects and their

interactions is complete.

Position A (White) ≻

Position B (Black)

Position B (Black)≻

Position C (White)

Position C (White) ≻

Position D (Black)

Position D (Black)≻

Position A (White)

Intransitive chess positions

“≻” means “is better than”

4. Interactive "rhizomatic"

(multiple, intertwining)

intransitivity in real complex

systems.

A body of biological studies is

devoted to the complex

intransitive competitions in

ecological niches. Such

competition transforms

participants. Information about

the participants, their features

and interactions is incomplete.

Pic. from: Three Minute Theory:

What is the Rhizome?

https://www.youtube.com/watch?v=gnteiRO-XfU

Conclusion

The important message for students should be more multi-

dimensional than “if you have violated the transitivity axiom,

you are not instrumentally rational, and the content of A, B,

and C do not matter”.

In complex and multi-variable situations, intransitive choices

are perfectly rational because the choice options are in

intransitive relations. That is, transitive choices in intransitive

options are a fallacy.

Various educational tools can be used to support students'

understanding of intransitive relations, and geometrical

intransitive manipulatives can be included here.

Bykova, A. (2018). [Understanding of intransitivity of superiority in various

contexts]. Diploma work. (in Russian)

Permogorskiy, M. S. (2016). [Psychological predictors of the use of the transitivity

heuristic]. PhD thesis. (in Russian) https://www.hse.ru/sci/diss/175064641

Poddiakov, A. (2006). [Intransitive character of superiority relations, and decision

making]. https://psy-journal.hse.ru/en/2006-3-3/27139364.html (in Russian)

Poddiakov, A. (2010). Intransitivity cycles, and complex problem solving.

https://www.researchgate.net/publication/237088961.

Poddiakov, A. (2017). [Non-transitivity is a fount for inventors]. English Google

translation:

https://translate.google.com/translate?sl=ru&tl=en&u=https://trv-

science.ru/2017/11/21/netranzitivnost-kladez-dlya-izobretatelej/

Poddiakov, A. (2018). Intransitive machines. https://arxiv.org/abs/1809.03869

Poddiakov, A., & Valsiner, J. (2013). Intransitivity cycles and their transformations:

How dynamically adapting systems function.

https://www.researchgate.net/publication/281288415