2nd mini-conference

“Rationality, Behavior, Experiment”

September 1-3, 2010; Moscow

Intransitivity cyclesand 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, "intransitivity-oriented" researchers underline that in cases of choices between interacting objects with multi-dimensional attributes, transitivity may be violated and, respectively, intransitivity of the choices can be reasonable (Anand, 1993; Bar-Hillel & 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).

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.

Die А:7, 7, 7, 7, 1, 1

Die В:6, 6, 5, 5, 4, 4

Die D:8, 8, 8, 2, 2, 2

An example by Ainley (1978)

Die С:9, 9, 3, 3, 3, 3

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 А

tool acting on non-sensitive zone

sensitive zone

Object В

sensitive zone

tools acting on non-sensitive zone

Object С

non-sensitive zone

sensitive zone

tools acting on

Competitive contexts

Let us consider combative systems with differentiated subsystems of attack, defense and vulnerable parts, (Poddiakov, 2000, 2006).

Weapon А

gun non-vulnerable part

vulnerable part

Weapon В

vulnerable part

gun non-vulnerable part

Weapon С

non-vulnerable part

vulnerable part

gun

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 double-gear 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).

ResultsThere were not significant differences between groups before demonstration of the intransitive objects.

% of correct answers

% of wrong

answers

% of answers difficult to

answer"

"towers" group

68 28 4

"double-gears" group

74 25 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

% of wrong answers

% of answers difficult to

answer"

before after before after before after

"towers" group

68 74 28 22 4 4

"double-gears" group

74 84.4 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 In.obs Iobs / In.obs

"towers" group

4 16 1 : 4

"gears" group

31 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.