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Presentation by Alexander Poddiakov

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2nd mini-conference
“Rationality, Behavior, Experiment”
September 1-3, 2010; Moscow
Intransitivity cycles
and complex problem solving
Alexander Poddiakov
State University—Higher School of Economics
E-mail: apoddiakov@hse.ru
If A is superior to B, and B is superior to C, does
it necessitate that A is superior to C? What
happens if superiority relations form a cycle, an
intransitive loop?
The world is both transitive and intransitive
(containing both transitive and intransitive
relations), and answers to these questions may
vary in different domains and situations.
In general it has been shown in many studies
that in some types of situations transitivity of
superiority relations is kept (e.g., in cases of
comparison of uni-dimensional non-interacting
objects—if rod A is longer than rod B, and B is
longer than C, it signifies that A is longer than
C), and also it has been shown that in some
other, more complex types of situations
transitivity is violated.
Distinguishing between these types can be an
important and difficult problem.
Yet perhaps not less important scientific and
teaching problem is that the issue of
transitivity/intransitivity polarizes some groups
of researchers (authors of handbooks, teachers,
etc.) seeming to be belonged to different
paradigms, in Kuhn's (1962) terms, or having
different creeds concerning
transitivity/intransitivity, in Fishburn's terms.
Analyzing relation between some researchers'
adherence to transitivity and others' rejection of
transitivity, Fishburn introduces analogy with
Euclidian and non-Euclidian geometries: "an
analogous rejection of non-Euclidean geometry
in physics would have kept the familiar and
simpler Newtonian mechanics in place, but that
was not to be" (Ibid., p. 117).
Namely, some scientists prove that transitivity
of superiority (of relation "is better than", etc.)
all things considered is not really violated in
their opponents' examples of intransitivity
(Chan, 2003), or cannot be violated in principle
(Ivin, 1998).
This extremely "transitivity-oriented" part of
researchers use the term "axiom of transitivity"
as a universally accepted principle and a
normative rule for correct decision making
without any discussions of possible
intransitivity of superiority relations (Zinoviev,
1972), as a key condition for rational actions
(Kozielecki, 1979).
Respectively, it leads to their opponents'
criticisms of the "axiom of transitivity", proofs
of its non-universality, and conclusions that
"rational theory choice does not obey the
transitivity axiom" (Baumann, 2005, p. 238),
"there is a genuine violation of transitivity" in
many situations (Temkin, 1999, p. 780), etc.
Transitions between "transitivity-" and
"intransitivity-oriented" approaches are
possible: Anand formally shows that "any
'intransitive' behaviour can be given a transitive
description", and "any 'transitive' behaviour can
be given an intransitive description" (Anand,
1993, p. 344-345). Yet it looks as if the
possibility of this universal formal transition
does not satisfy and reconcile groups of the
researchers with one another.
In real life, one can distinguish between 4 types of
situations.
1. superiority relations are objectively transitive, and a
person considers them transitive in the right way, in
correspondence with the "axiom of transitivity"
2. superiority relations are objectively transitive, but a
person considers them intransitive, making different
mistakes (these facts are studied in Tversky's
approach)
3. superiority relations are objectively intransitive, and a
person considers them intransitive in the right way
4. superiority relations are objectively intransitive, but a
person considers them transitive in a wrong way, e.g.,
in correspondence with the "axiom of transitivity".
Two last types of situations are rarely studied in
psychology, but they are very interesting. I think
that it is better to study them using "complex
problem-solving" approach rather than
"decision-making" approach. "Complex
problem-solving" approach deals with humans'
understanding of multi-variable situations with
lots of interacting factors, in contrast with
"decision-making" approach which often
considers choices between a small number of
options non-interacting with one another. Of
course, boundaries between approaches are
conventional, and theories working with
complex problem solving like with decision
making are possible (Huber, 1995).
Yet, at least in some theories of decision
making, impossibility of interactions is
introduced as an axiom (Kozielecki, 1979), and
I think that it leads to fatal effects on
understanding of transitivity/intransitivity in
these theories. In contrast with it, "intransitivityoriented" researchers underline that in cases of
choices between interacting objects with multidimensional attributes, transitivity may be
violated and, respectively, intransitivity of the
choices can be reasonable (Anand, 1993; BarHillel & Margalit, 1988; Fishburn, 1991;
Roberts, 2004; Temkin, 1996).
Examples:
Stochastic models
A most famous stochastic model of
intransitivity are intransitive dice, often called
Efronian dice. In sets of dice with specially
constructed numbers on their faces, the 1st die
wins the 2nd die (rolls a higher number) more
often than losses, the 2nd die wins the 3rd die
more often than losses, etc., but the last die wins
the 1st die more often than losses (Deshpande,
2000; Roberts, 2004).
An example by Ainley (1978)
Die Рђ:
7, 7, 7, 7, 1, 1
Die Р’:
6, 6, 5, 5, 4, 4
Die РЎ:
9, 9, 3, 3, 3, 3
Die D:
8, 8, 8, 2, 2, 2
One can see that die A beats die B two times more often
than is beaten by die B; die B beats die C two times more
often than is beaten; die C beats die D two times more often
than is beaten; but die D beats die A two times more often
than is beaten by die A. Thus, relation "to roll a higher
number more often" is not transitive.
Existence of intransitive dice signifies
ambivalence of classical "money pump"
reasoning: paradoxically, one may buy more
advantageous objects (the next die) in
intransitive way again and again, pumping
money into a bank.
Deterministic intransitive relations:
"competitive" and "cooperative" models
Not only stochastic, but also exactly deterministic
intransitive relations—both cooperative and
competitive—between objects of well-differentiated
structures are possible (Poddiakov, 2000, 2006).
Object Рђ
tool acting on п‚ѕп‚®
non-sensitive zone
sensitive zone
Object Р’
Object РЎ
sensitive zone
non-sensitive zone
tools acting on п‚ѕп‚® sensitive zone
non-sensitive zone
tools acting on п‚ѕп‚®
Competitive contexts
Let us consider combative systems with differentiated
subsystems of attack, defense and vulnerable parts,
(Poddiakov, 2000, 2006).
Weapon Рђ
Weapon Р’
Weapon РЎ
non-vulnerable part
gun п‚ѕп‚ѕп‚ѕп‚ѕп‚®
vulnerable part
vulnerable part
non-vulnerable part gun п‚ѕп‚ѕп‚ѕп‚®
vulnerable part
non-vulnerable part gun п‚ѕп‚ѕп‚ѕп‚ѕп‚®
This model works in sports, interindividual
conflicts, etc. Also it can explain combative
relations between species, social groups, etc.,
described in many domains. For example,
Journal Nature has published a series of articles
with words Rock, Paper, Scissors in titles about
combative relations between species (Kerr et al.,
2002; Kirkup & Riley, 2004; Reichenbach et al.,
2007). For example, it has been shown that in
agar culture, Phallus impudicus replaces
Megacollybia platyphylla, M. platyphylla
replaces Psathyrella hydrophilum, but P.
hydrophilum replaces P. impudicus (Boddy,
2000).
Cooperative contexts
Suppose there are 3 physicians—A, B, C
Physician A is a specialist in treating organs X, has
healthy organs Y, and suffers from disease of organs Z.
Physician B is a specialist in treating organs Y, has
healthy organs Z, and suffers from disease of organs X.
Physician C is a specialist in treating organs Z, has
healthy organs X, and suffers from disease of organs Y.
It is evident that physician A should dominate—in expertise and
power in doctor/patient relationship--over B, B should dominate
over C, and C should dominate over A. An analogical intransitive
loop of domination is in models A psychotherapist for a
psychotherapist, and A teacher for a teacher, etc.
All these examples show that in many kinds of
situations rational choices must be intransitive,
and keeping transitivity of choices is irrational.
Respectively, a cornerstone of rationality is
understanding of not only transitivity, but also
intransitivity.
Thus, investigations of cognitive development
related to this understanding seem necessary. Yet
really we have a lot of experiments, in which
participants are provoked to make wrong
intransitive choices (between candidates for some
jobs, between lotteries, etc.) in situations requiring
transitive inferences, and few experiments in
which intransitive choices of participants are
absolutely reasonable.
A possible starting point for such investigations
can be the following pilot study (Poddiakov,
2010, in press).
An aim of the experiment:
to investigate influence of people's observation
of triads of objects being in intransitive relations
of superiority, on changes of their beliefs about
possibility/impossibility of existence of other
"intransitive" objects in different domains.
Method
Participants were asked 8 questions.
1. There are 3 straight rigid rods such that the 1st
rod is longer than the 2nd rod, and the 2nd rod is
longer than the 3rd rod. Can the 3rd rod be
longer than the 1st rod?
2. There are 3 objects of different mass such that
the 1st object is more massive than the 2nd
object, and the 2nd object is more massive than
the 3rd object. Can the 3rd object be more
massive than the 1st object
3. Is it possible that computer chess A regularly
prevails on computer chess B, B regularly
prevails on C, but B regularly prevails on C?
4. There are 3 teams with 6 wrestlers in each
team. While their tournament, each wrestler of
one team meets and wrestles with each wrestler
from two other teams. It is known that wrestlers
of the 1st team beat wrestlers of the 2nd team
more often than are beaten by them, and
wrestlers of the 2nd team beat wrestlers of the
3rd team more often than are beaten by them.
Is it possible that wrestlers of the 3rd team beat
wrestlers of the 1st team more often than are
beaten by them?
5. There are 3 boxes with 6 pencils of different
length in each box. We compare length of each
pencil with length of all the rest pencils. We
learn that pencils from the 1st box more often are
longer than pencils from the 2nd box, and
pencils from the 2nd box more often are longer
than pencils from the 3rd box. Is it possible that
pencils from the 3rd box more often are longer
than pencils from the 1st box?
6. Is it possible that microorganisms A replace B,
B replace C, and C replace A?
7. There are 3 kinds of weapons--mobile assault
towers competing with one another: each tower
tries to mark another tower with its color marker.
In is known that the towers are such that the 1st
tower marks the 2nd tower, but avoids marking
by the last one; and the 2nd tower marks the 3rd
tower, but avoids marking by the last one.
Is it possible that the 3rd tower marks the 1st
tower, but avoids marking by the last one?
8. There are 3 double-gears such that, in case of
joints in pairs, the 1st double-gear rotates faster
than the 2nd double-gear, and the 2nd doublegear rotates faster than the 3rd double-gear.
Is it possible that the 3rd double-gear rotates
faster than the 1st double-gear?
Options for answers:
"it is possible"
"it is impossible“
"I find it difficult to answer"
After that, the participants were shown specially
designed triads of mechanical devices interacting
with one another in intransitive deterministic
way.
In the 1st experimental group, participants were
shown plastic models of "Intransitive" Mobile
Assault Towers.
In the 2nd experimental group, participants were
shown three "intransitive" double-gears such
that, in case of joints in pairs, double-gear A
rotated faster than B, B rotated faster than C, but
C rotated faster than A.
After the demonstration the participants of both
groups were asked to answer the same 8
questions again, either confirming or changing
their previous answers.
Participants: 89 graduate students of 17-21 yrs
(40 in group 1, 49 in group 2).
Results
There were not significant differences between
groups before demonstration of the intransitive
objects.
"towers"
group
"doublegears"
group
% of
correct
answers
68
74
% of
wrong
answers
28
25
% of answers
difficult to
answer"
4
1
In both groups:
- most participants (from 94% to 97%) correctly
answered that intransitive straight rigid rods and
intransitive massive objects are impossible.
- most participants (90% in group 1 and 88% in
group 2) believed in existence of intransitive
fighting towers.
- most participants (65% in group 1 and 67% in
group 2) do not believe in existence of
intransitive gears.
After demonstration of the intransitive objects, there
were significant positive changes of number of correct
answers in group 1 (p<0,01) and group 2 (p<0,05).
% of correct
answers
"towers"
group
"doublegears"
group
% of answers
difficult to
answer"
before after before after before after
68
74
28
22
4
4
74
84.4
% of wrong
answers
25
15.3
1
0.3
Yet patterns of the corrected answers were
different in cases of the towers and the gears.
Let us consider increase of number of correct
answers in both groups.
At first sight, the "towers" group looks a bit
worse: increase of correct answers is equal to
6%, in contrast with 10% in "gears" group.
Yet let us consider relation between correct
answers:
- about the observed objects (i.e., the towers in
"towers" group, or the gears in "gears" group),
and
- about the rest objects, which were not
presented for observation.
Iobs - increase of number of correct answers
about objects presented for observation
I n.obs - increase of number of correct answers
about other objects, which were not presented
for observation.
Iobs
"towers" 4
group
In.obs
16
Iobs / In.obs
1:4
"gears" 31
group
8
3.9 : 1
It signifies a paradoxical situation.
In "towers" group, changes of beliefs about other
objects were stimulated by observation of the
objects, about which the participants anyway had
right beliefs before the demonstration. (Almost
all participants knew the right answer to the
question about intransitive towers--"it is
possible".) But observation of the real towers
caused stronger effect on beliefs about many
other objects, i.e., caused effect of positive
transfer.
In contrast, observation of the real intransitive
gears, in which most participants did not believe
before the demonstration, caused large increase
of correct answers about the gears themselves,
but much lesser increase of correct answers
about other objects. Effect of positive transfer
was much weaker here than this effect in
"towers" group.
Thus, the models of intransitive towers provide
opportunities for right generalizations in much
larger measure than the models of intransitive
gears.
Conclusions
1. Domain-specificity of beliefs about
intransitivity.
Beliefs about intransitive relations of superiority
were domain-specific: the participants thought
that objects, being in intransitive relations, are
possible in some domains and impossible in
other domains (though really they are possible in
the latter domains as well). Axiom of transitivity
was applied by the participants very
selectively—if applied at all. In many situations
it was the right solution.
2. Influence of different objects on beliefs about
intransitivity.
Demonstration of different "intransitive" objects
caused different effects. The results show that
design of models of "intransitive" objects,
observation and exploration of which have positive
effects on beliefs about transitivity/intransitivity is
possible and advisable. Some of the models provide
more opportunities for right generalizations (have
more "heuristic power") than the others. Both types
of models can be used for investigation of
understanding of transitivity/intransitivity.
3. Speculative conclusion.
A program of investigation of historical and
ontogenetic development of humans' understanding
of transitivity/intransitivity as fundamental
properties of the world is necessary. Dynamics of
setting and solving (or declaring as unsolvable)
different simple and complex problems of
transitivity/intransitivity in socio- and ontogenesis
is an integral part of cognitive development as a
whole.
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