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Fluctuation and Noise Letters Vol. 0, No. 0 (2001) 000–000 c World Scienti?c Publishing Company

arXiv:physics/0408022v2 [physics.soc-ph] 31 Jan 2005

CAT’S DILEMMA – TRANSITIVITY VS. INTRANSITIVITY

EDWARD W. PIOTROWSKI* and MARCIN MAKOWSKI** Institute of Mathematics, University of Bialystok, Lipowa 41, PL-15424 Bialystok, Poland ep@alpha.uwb.edu.pl*, mmakowski@alpha.pl** Received (received date) Revised (revised date) Accepted (accepted date)

We study a simple example of a sequential game illustrating problems connected with making rational decisions that are universal for social sciences. The set of chooser’s optimal decisions that manifest his preferences in case of a constant strategy of the adversary (the o?ering player), is investigated. It turns out that the order imposed by the player’s rational preferences can be intransitive. The presented quantitative results imply a revision of the ”common sense” opinions stating that preferences showing intransitivity are paradoxical and undesired. Keywords : intransitivity; game theory; sequential game.

1.

Introduction

The intransitivity can occur in games with three or more strategies if the strategies A, B , C are such that A prevails over B , B prevails over C , and C prevails over A (A > B > C > A). The most known example of intransitivity is the children game ”Rock, Scissors, Paper” (R, S, P ) where R > S > P > R. The other interesting example of intransitive order is the so-called Condorcet’s paradox, known since XVIIIth century. Considerations regarding this paradox led Arrow in the XXth century to prove the theorem stating that there is no procedure of successful choice that would meet the democratic assumptions [1]. The importance of this result to mathematical political science is comparable to G¨ odel’s Incompleteness Theorem in logic [2]. It seems logical to choose an order, in a consistent way between things we like. But what we prefer often depends on how the choice is being o?ered [3,4]. This paradox was perceived by many researchers and analysts (for instance Stan Ulam described this in his book ”Adventures of a Mathematician”, some problems with intransitive options can be found in [5, 6]). On the other hand scientists have a penchant for classi?cations (rankings) on basis of linear orders and this (we think) follows from

Cat’s Dilemma – transitivity vs. intransivity

such intransitive preferences there are so suspicious for many researchers. In the paper, we present quantitative analysis of a model, which can be illustrated by the Pitts’s experiments with cats, mentioned in the Steinhaus diary [7] (Pitts noticed that a cat facing choice between ?sh, meat and milk prefers ?sh to meat, meat to milk, and milk to ?sh!). This model ?nds its re?ection in the principle of least action that controls our mental and physical processes, formulated by Ernest Mach [8] and referring to Ockham’s razor principle. Pitts’s cat, thanks to the above-mentioned food preferences, provided itself with a balanced diet. In our work, using elementary tools of linear algebra, we obtained the relationship between the optimal cat’s strategy and frequencies of appearance of food pairs. Experiments with rats con?rmed Pitts’s observations. Therefore, it is interesting to investigate whether intransitivity of preferences will provide a balanced diet also in a wider sense in more or less abstract situations involving decisions. Maybe in the class of randomized behaviors we will ?nd the more e?ective ways of nutrition? The following sections constitute an attempt at providing quantitative answer to these questions. The analysis of an elementary class of models of making optimal decision presented below permits only determined behaviors, that is such for which the agent must make the choice. Through this analysis we wish to contribute to dissemination of theoretical quantitative studies of nondeterministic algorithms of behaviors which are essential for economics and sociology – this type of analysis is not in common use. The geometrical interpretation presented in this article can turn out very helpful in understanding of various stochastic models in use. 2. Nondeterministic cat

Let us assume that a cat is o?ered three types of food (no. 1, no. 2 and no. 3), every time in pairs of two types, whereas the food portions are equally attractive regarding the calories, and each one has some unique components that are necessary for the cat’s good health. The cat knows (it is accustomed to) the frequency of occurrence of every pair of food and his strategy depends on only this frequency. Let us also assume that the cat cannot consume both o?ered types of food at the same moment, and that it will never refrain from making the choice. The eight (23 ) possible deterministic choice functions fk : fk : {(1, 0), (2, 0), (2, 1)} → {0, 1, 2}, k = 0, . . . , 7 (1)

are de?ned in Table 1. The functions f2 and f5 determine intransitive orders. The

Table 1. The table de?ning all possible choice functions fk . function fk : fk (1, 0) = fk (2, 0) = fk (2, 1) = frequency pk : f0 0 0 1 p0 f1 0 0 2 p1 f2 0 2 1 p2 f3 0 2 2 p3 f4 1 0 1 p4 f5 1 0 2 p5 f6 1 2 1 p6 f7 1 2 2 p7

parameters pk , k = 0, . . . , 7 give the frequencies of appearance of the choice function

Piotrowski and Makowski

in the nondeterministic algorithm (strategy) of the cat ( k=0 pk = 1, pk ≥ 0 for k = 0, . . . , 7). We will show the relationship between the frequency of occurrence of individual type of food in cat’s diet and the frequencies of occurrence of food pairs. Let us denote the frequency of occurrence of the pair (k, j ) as qm , where m is the number of food that does not occur in the pair (k, j ) ( 2 m=0 qm = 1). This denotation causes no uncertainty because there are only three types of food. When the choice methods fk are selected nondeterministically, with the respective intensities pk , the frequency ωm , m = 0, 1, 2, of occurrence of individual food in cat’s diet are according to Table 1. given as follows: ? food no. 0: ω0 = (p0 + p1 + p2 + p3 )q2 + (p0 + p1 + p4 + p5 )q1 , ? food no. 1: ω1 = (p4 + p5 + p6 + p7 )q2 + (p0 + p2 + p4 + p6 )q0 , ? food no. 2: ω2 = (p2 + p3 + p6 + p7 )q1 + (p1 + p3 + p5 + p7 )q0 . Three equalities above can be explained with the help of the conditional probability concept. Let us denote B3?(j +k) = {(j, k )}, P (Bj ) = qj and Cj = {j } for j, k = 0, 1, 2, j = k . The number P (Ck |Bj ) indicates the probability of choosing the food of number k , when the o?ered food pair does not contain the food of number j . Since the events of choosing di?erent pairs of food are disjoint and comprise all the space of elementary events. Hence, for each food chosen, we have the following relation:

2

7

ωk = P (Ck ) =

j =0

P (Ck |Bj )P (Bj ), k = 0, 1, 2.

(2)

By inspection of the table of the functions fk , k = 0, . . . , 7, we easily get the following relations:

7

P (C0 |B2 ) = P (

k=0 7

fk (B2 ) = 0) = p0 + p1 + p2 + p3 , fk (B1 ) = 0) = p0 + p1 + p4 + p5 ,

7

P (C0 |B1 ) = P (

k=0

P (C1 |B0 ) = P (

k=0 7

fk (B0 ) = 1) = p0 + p2 + p4 + p6 , fk (B2 ) = 1) = p4 + p5 + p6 + p7 ,

7

(3)

P (C1 |B2 ) = P (

k=0

P (C2 |B1 ) = P (

k=0 7

fk (B1 ) = 1) = p2 + p3 + p6 + p7 , 2 fk (B0 ) = 1) = p1 + p3 + p5 + p7 , 2

P (C2 |B0 ) = P (

k=0

and P (C0 |B0 ) = P (C1 |B1 ) = P (C2 |B2 ) = 0. Frequency of the least preferred food, that is the function min(ω0 , ω1 , ω2 ), determines the degree of the diet completeness. Since ω0 + ω1 + ω2 = 1, the most valuable

Cat’s Dilemma – transitivity vs. intransivity

way of choosing the food by the cat occurs for such probabilities p0 , . . . , p7 , that the function min(ω0 , ω1 , ω2 ) has the maximal value, that is for

1 . ω0 = ω1 = ω2 = 3

(4)

Any vector p = (p0 , . . . , p7 ) (or six conditional probabilities (P (C1 |B0 ), P (C2 |B0 ), P (C0 |B1 ),P (C2 |B1 ),P (C0 |B2 ), P (C1 |B2 ))), which for a ?xed triple (q0 , q1 , q2 ) ful?lls the system of equations (4) will be called an cat’s optimal strategy . Let us study this strategy in more details and subject it to geometrical analysis. For given q0 , q1 , q2 all optimal strategies are calculated. The system of equations (4) has the following matrix form: ? ? ? ?? ? 1 P (C0 |B2 ) P (C0 |B1 ) 0 q2 1 ? ? P (C1 |B2 ) 1 ?, 0 P (C1 |B0 ) ? ? q1 ? = 3 (5) 1 0 P (C2 |B1 ) P (C2 |B0 ) q0 and its solution: q2 q1 q0 = = =

1 d 1 d 1 d

P (C0 |B1 ) + P (C1 |B0 ) ? P (C0 |B1 )P (C1 |B0 ) , 3 P (C0 |B2 ) + P (C2 |B0 ) ? P (C0 |B2 )P (C2 |B0 ) , 3 P (C1 |B2 ) + P (C2 |B1 ) ? P (C1 |B2 )P (C2 |B1 ) , 3

(6)

de?nes a mapping of the three-dimensional cube [0, 1]3 in the space of parameters (P (C0 |B2 ), P (C0 |B1 ), P (C1 |B0 )) into a triangle in the space of parameters (q0 , q1 , q2 ), where d is the determinant of the matrix of parameters P (Cj |Bi ). The barycentric coordinates [9] of a point of this triangle are interpreted as the probabilities q0 , q1 and q2 . These numbers represent the heights a, b and c or the areas PQAB , PQBC and PQAC of three smaller triangles determined by the point Q (cf. Fig. 1), or the lengths of the segments formed by the edges of the triangle by cutting them with the straight lines passing through the point Q and the opposite PQBC |RB | a 1 vertex of the triangle. Hence e.g. q q2 = b = PQAC = |RA| , where the symbol |RB | represents length of the segment. The next picture (Fig. 2) presents the image of the three-dimensional cube in this simplex. It determines the area of frequency qm of appearance of individual choice alternatives between two types of food in the simplex, for which the optimal strategy exists. In order to present the range of the nonlinear representation of our interest, the authors illustrated it with the values of this representation for 10,000 randomly selected points with respect to constant probability distribution on the cube. Justi?cation of such equipartition of probability may be found in Laplace’s principle of insu?cient reason [10]. In our randomized model the a priori probability of the fact that the sum of probabilities P (Cj |Bk ) is smaller than a given number α ∈ [0, 1] equals α. The absence of optimal solutions outside the hexagon forming the shaded part of the picture (Fig. 2) is obvious, since the bright (non-dotted) part 1 1 of the picture represents the areas, for which q0 > 1 3 (or q1 > 3 , or q2 > 3 ), and the 1 total frequency of appearance of pairs (0, 1) or (0, 2) must be at least 3 in order to

Piotrowski and Makowski

C

b Q c R

a

A

B

Fig 1. The barycentric coordinates.

Fig 2. Image of the three-dimensional cube on simplex.

assure the completeness of the diet with respect of the ingredient 0 (but this cannot 1 happen because when q0 > 2 3 , then q1 + q2 = 1 ? q0 < 3 ). The system of equations (5) can be transformed into the following form: ? ? ?? ? ? 1 P (C0 |B2 ) q2 ?q1 0 3 ? q1 ? ? 1 ?? ? ? 0 q0 ? ? P (C2 |B1 ) ? = ? 3 ? q2 ? , (7) ? ?q2 1 P (C1 |B0 ) 0 q1 ?q0 ? q 0 3 which allows to write out the inverse transformation to the mapping de?ned by equations (6). By introducing the parameter λ we may write them as follows: P (C0 |B2 ) = λ λ ? 1 + 3q1 λ + 1 ? 3q2 , P (C2 |B1 ) = , P (C1 |B0 ) = . 3q2 3q1 3q0 (8)

A whole segment on the unit cube corresponds to one point of the simplex, parameterized by λ. The range of this representation should be limited to the unit cube, which gives the following conditions for the above subsequent equations: λ ∈ [0, 3q2 ], λ ∈ [1 ? 3q1 , 1], λ ∈ [3q2 ? 1, 2 ? 3q1 ]. (9)

Cat’s Dilemma – transitivity vs. intransivity

The permitted values of the parameter λ form the common part of these segments, hence it is nonempty for: max(0, 1 ? 3q1 , 3q2 ? 1) ≤ min(2 ? 3q1 , 3q2 , 1). Therefore λ ∈ [max(0, 1 ? 3q1 , 3q2 ? 1), min(2 ? 3q1 , 3q2 , 1)]. (11) (10)

It may be now noticed that for any triple of probabilities belonging to the hexagon, there exists an optimal solution within a set of parameters ((P (C0 |B2 ), P (C0 |B1 ), P (C1 |B0 ))). If we assume the equal measure for each set of frequencies of occurrence of food pairs as the triangle point, then we may state that we deal with optimal 2 strategies in 3 of all the cases (it is the ratio of area of regular hexagon inscribed into a equilateral triangle). The inverse image of the area of frequencies (q0 , q1 , q2 ) of food pairs that enable realization of the optimal strategies, which is situated on the cube of all possible strategies, is presented by four consecutive plots in Fig. 3. We present there the same con?guration observed from di?erent points of view. The segments on the ?gures correspond to single points of the frequency triangle of the individual food pairs. The greatest concentration of the segments is observed in two areas of the cube that correspond to intransitive strategies1. The bright area in the center of the cube, which may be seen in the last picture, belongs to the e?ective strategies – e?ective in the subset of frequencies of a small measure (q0 , q1 , q2 ) of the food pairs appearance. Among them, the totally incidental behavior is located, which gives consideration in equal amounts to all the mechanisms of deterministic 1 . choice pj = pk = 8 3. Example of an optimal strategy

The formulas (8) that map the triangle into a cube can be used to ?nd an optimal strategy in cases, when the probabilities (q0 , q1 , q2 ) of appearance of individual 1 1 , q1 = 1 pairs of the products are known. Let us assume that q0 = 2 3 and q2 = 6 . 1 2λ Then, according to the formulas (8), we have P (C1 |B0 ) = 3 + 3 , P (C0 |B2 ) = 1 1 2λ , P (C2 |B1 ) = λ , where λ ∈ [0, 1 2 ]. Selecting λ = 4 we have: P (C0 |B2 ) = 2 , 1 1 P (C2 |B1 ) = 4 , P (C1 |B0 ) = 2 . We may now show the solution of equations (3), 1 e.g.: p0 = 2 , p5 = p7 = 1 4 and pj = 0 for others parameters. We will obtain the following frequencies of occurrence of individual foods in the diet: ω0 = (p0 + p5 )q1 + p0 q2 = 1 1 1 + = , 4 12 3 1 1 1 = , ω1 = p0 q0 + (p5 + p7 )q2 = + 4 12 3 1 1 1 ω2 = (p5 + p7 )q0 + p7 q1 = + = . 4 12 3

(12)

The above calculations of the frequency ωj con?rm optimality of the indeterministic algorithm determined in this example.

1 See

section 4.

Piotrowski and Makowski

1

P C2 B1

P C0 B2

0 1

0 1

1

P C1 B0 P C0 B2

1 0

P C1 B0

1

0

0 0

P C2 B1

P C2 B1

0

1

10

P C0 B2

1 1

P C1 B0

0

P C1 B0

0

P C2 B1

11

0 0

P C0 B2

Fig 3. The inverse image of area of frequencies (q0 , q1 , q2 ) that enable realization of the optimal strategy, see Appendix A.

4.

Intransitive nondeterministic decisions

In the case of random selections we may talk about order relation food no. 0 < food no. 1, when from the o?ered pair (0, 1) we are willing to choose the food no. 1 more often than the food no. 0 (P (C0 |B2 ) < P (C1 |B2 )). Therefore we have two intransitive orders:

1 ? P (C0 |B2 ) < 2 , P (C2 |B1 ) < 1 2 , P (C1 |B0 ) < 1 ? P (C0 |B2 ) > 2 , P (C2 |B1 ) > 1 2 , P (C1 |B0 ) > 1 2 1 2

. .

It is interesting to see in which part of the simplex of parameters (q0 , q1 , q2 ) we may take optimal intransitive strategies. They form the six-armed star composed of two triangles, each of them corresponding to one of two possible intransitive orders 1 (Fig. 4). They dominate in the central part of triangle, near point q0 = q1 = q2 = 3 . They form darkened part of area inside the star. Optimal transitive strategies cover the same area of the simplex as all optimal strategies, however they occur less often in the center of the simplex. We illustrated this situation in the next picture (Fig. 5). In areas of high concentration of optimal transitive strategies, one of three frequencies q0 , q1 , q2 looses its signi?cance – two from three pairs of the food occur with considerable predominance. We have enough information to be able to compare the applicability range of di?erent types of optimal strategies. Let us assume the same measure of the possibility of occurrence of determined proportion of all three food pairs. This assumption means that the probability of appearance of the situation determined by a point in the triangle-domain of parameters (q0 , q1 , q2 ) does not depend on those parameters. Two thirds of strategies are optimal. There

Cat’s Dilemma – transitivity vs. intransivity

Fig 4. Optimal intransitive strategies.

Fig 5. Optimal transitive strategies.

are 33%2 of circumstances, which allow for the use of the optimal strategies that belong to the speci?ed intransitive order. There are 44% ( 4 9 ) of situations of any order that favor optimal strategies, what follows from the fact that they are measured by the surface of regular star, and its area is equal to double area of the triangle corresponding to one intransitive order reduced by the area of the hexagon 1 2 4 2 inscribed into the star. So we have: 3 +1 3 ? 9 = 9 . Appearance of the number 9 in the calculation can be easily explained by the observation that the area of the regular six-armed star is two times bigger than the area of the hexagon inscribed into it. This number (22%) is the measure of the events that favor both types of intransitive strategies. It is worth to stress that in the situation that favors optimal strategies we can always ?nd the strategy that determines the transitive order (see Fig. 5). However, we should remember that this feature concerns only the simple model of the cat’s behavior, and does not have to be true in the cases of more complicated reaction mechanisms.

2 They

are measured by the area of equilateral triangle inscribed into a regular hexagon.

Piotrowski and Makowski

5.

Conclusions

In this article, we used a stochastic variant of the principle of least action . Perhaps dissemination of usage of this principle will lead to formulation of many interesting conclusions and observations. We presented a method, which allows successful analysis of intransitive orders that still are surprisingly suspicious for many researchers. More profound analysis of this phenomenon can have importance everywhere where the problem of choice behavior is studied. For instance in economics (description of the customer preference toward products- marketing strategy) or in political science where the problem of voting exists. Analysis of intransitive orders is a serious challenge to those who seek description of our reasoning process. The quantitative observations from the previous section show that intransitivity, as the way of making the decision, can provide the diet completeness for the cat from our example. Moreover, the intransitive optimal strategies constitute the major part of all optimal strategies. Therefore, it would be wrong to prematurely acknowledge the preferences showing the intransitivity as undesired. Perhaps there are situations, when only the intransitive orders allow obtaining the optimal e?ects. The most intriguing problem that remains open, is to answer the question whether there exists a useful model of optimal behaviors, which gives the intransitive orders, and for which it would be impossible to specify the transitive optimal strategy of identical action results. Showing the impossibility of building such constructions would cause marginalization of the practical meaning of intransitive orders. On the other hand, indication of this type of models would force us to accept the intransitive ordering.

Acknowledgements The authors are grateful to prof. Zbigniew Hasiewicz and prof. Jan Sladkowski for useful discussions and comments.

Appendix A. The following mini-program written in the language Mathematica 5.0 generate four plots in Fig. 3.

Cat’s Dilemma – transitivity vs. intransivity

4 In[1] := c = N [ 3√ ]; 3 2 gencorrect := M odule[{a = c {Random[], Random[]}}, W hile[ a[[2]] > 3 , a = c {Random[], Random[]} ]; a ]; 3 setpoint := M odule[{q 1, a, x, y, v = 1, w = 1}, W hile[ 2 c v + 2 w > 1, c ; a = gencorrect; q 1 = a[[2]]; x = a[[1]] ? 2

y = a[[2]] ? 1 3 ; v = Abs[x]; w = Abs[y ] ]; {q 1, N [

√ 3 2

x?

1 2

y+ 1 3 ]} ];

setsegment := M odule[{a, q 1 = 1, q 2 = 1, l1 = 1, l2 = 0}, W hile[l1 > l2 , a = setpoint; q 1 = a[[1]]; q 2 = a[[2]]; l1 = M ax[0, 1 ? 3 q 1, 3 q 2 ? 1]; l2 = M in[2 ? 3 q 1, 3 q 2, 1] ];

1 1?l1 1?3 q2+l1 l2 1?l2 1?3 q2+l2 {{ 3lq 2 , 3 q1 , 3(1?q1?q2) }, { 3 q2 , 3 q1 , 3(1?q1?q2) }} ];

f ig [x , y , z ] := Show[Graphics3D[{GrayLevel[.0], T hickness[.002], T able[Line[setsegment], {2000}]}], V iewP oint → {x, y, z }, Axes → T rue, AxesLabel → {”P(C0 | B2 )”, ”P(C2 | B1 )”, ”P(C1 | B0 )”}, T icks → {{0, 1}, {0, 1}, {0, 1}}, BoxStyle → Dashing [{.02, .02}], AxesStyle → T hickness[.005] ]; f ig [?6, .7, .3] f ig [.6, ?3, .3] f ig [1.3, 3.4, 2] f ig [3, ?2, 3]

References

[1] K. J. Arrow, Social Choice and Individual Values , Yale University Press, New York (1951). [2] M. Gardner, Time Travel and Mathematical Bewilderments , Freeman, New York (1988). [3] A. Tversky, Elimination by aspects: A theory of choice , Psychological Review, 79 (1972) 281-299. [4] A. Tversky, Intransitivity of Preferences , Psychological Review, 76 (1969) 31-48.

Piotrowski and Makowski

[5] J. Y. Halpern, Intransitivity and Vagueness , Principles of Knowledge Representation and Reasoning, Proceedings of the Ninth International Conf., Whistler, Canada (2004); arXiv:cs/0410049. [6] E. Groes, H. J. Jacobsen, T. Tranas, Testing the Intransitivity Explanation of the Allais paradox, Theory and Decision 47 (1999), 229-245. [7] H. Steinhaus, Memoirs and Notes (in Polish), Aneks, London (1992). [8] E. Mach, The Science of Mechanics , Open Court, LaSalle, IL (1960). [9] Encyclopaedia of mathematics on cd-rom , Kluwer Academic Publishers, Dordrecht (1997). [10] P. Dupont, Laplace and the Indi?erence Principle in the ’Essai philosophique des probabilits’ , Rend. Sem. Mat. Univ. Politec. Torino, 36 (1977/78) 125-137.

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及物性分析

The

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