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Fluctuation andNoiseLetters Vol.0,No.0(2001)000–000 (cid:13)c WorldScientificPublishingCompany 5 0 0 2 n a J 1 CAT’S DILEMMA – TRANSITIVITY VS. INTRANSITIVITY 3 ] h p EDWARDW.PIOTROWSKI*andMARCINMAKOWSKI** - Institute of Mathematics, Universityof Bial ystok, c Lipowa 41, PL-15424 Bial ystok, Poland o [email protected]*, [email protected]** s . s c Received(receiveddate) i Revised(reviseddate) s Accepted (accepted date) y h p 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 2 adversary(theofferingplayer),isinvestigated. Itturnsoutthattheorderimposedbythe v player’srationalpreferencescanbeintransitive. Thepresentedquantitativeresultsimply 2 arevisionofthe”commonsense”opinionsstatingthatpreferencesshowingintransitivity 2 areparadoxical andundesired. 0 8 Keywords: intransitivity;gametheory;sequential game. 0 4 1. Introduction 0 / The intransitivitycanoccuringameswith threeor morestrategiesif the strategies s c A, B, C are such that A prevails over B, B prevails over C, and C prevails over A i (A>B >C >A). The most known example of intransitivity is the children game s y ”Rock, Scissors, Paper” (R,S,P) where R > S > P > R. The other interesting h example of intransitive order is the so-called Condorcet’s paradox, known since p XVIIIth century. Considerations regarding this paradox led Arrow in the XXth : v century toprovethe theoremstatingthatthere is noprocedureofsuccessfulchoice Xi that would meet the democratic assumptions [1]. The importance of this result to mathematicalpoliticalscienceiscomparabletoGödel’sIncompletenessTheoremin r a logic [2]. Itseems logicalto chooseanorder,ina consistentwaybetweenthings we like. But whatwepreferoftendependsonhowthechoiceisbeingoffered[3,4]. Thisparadox was perceivedby many researchersand analysts (for instance Stan Ulam described this inhisbook”AdventuresofaMathematician”,someproblemswithintransitive options can be found in [5,6]). On the other hand scientists have a penchant for classifications (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 fish, meat and milk prefers fish to meat, meat to milk, and milk to fish!). This model finds its reflection in the principle of least action that controlsour mental andphysicalprocesses,formulatedby 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 confirmed 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 find the more effective waysof nutrition? The followingsections constitute anattempt at providingquan- titative answerto these questions. The analysisofanelementary classofmodels of making optimal decision presented below permits only determined behaviors, that is such for which the agent must make the choice. Throughthisanalysiswewishtocontributetodisseminationoftheoreticalquan- titative studies of nondeterministic algorithms of behaviors which are essential for economicsandsociology–thistypeofanalysisisnotincommonuse. Thegeometri- calinterpretationpresentedinthisarticlecanturnoutveryhelpfulinunderstanding of various stochastic models in use. 2. Nondeterministic cat Let us assume that a cat is offered 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 regardingthecalories,andeachonehassomeuniquecomponentsthatarenecessary 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 offered types of food at the samemoment,andthatitwillneverrefrainfrommakingthechoice. Theeight(23) possible deterministic choice functions f : k f :{(1,0),(2,0),(2,1)}→{0,1,2}, k=0,...,7 (1) k are defined in Table 1. The functions f and f determine intransitive orders. The 2 5 Table1. Thetabledefiningallpossiblechoicefunctionsfk. functionfk: f0 f1 f2 f3 f4 f5 f6 f7 fk(1,0)= 0 0 0 0 1 1 1 1 fk(2,0)= 0 0 2 2 0 0 2 2 fk(2,1)= 1 2 1 2 1 2 1 2 frequencypk: p0 p1 p2 p3 p4 p5 p6 p7 parametersp ,k=0,...,7givethefrequenciesofappearanceofthechoicefunction k Piotrowski and Makowski in the nondeterministic algorithm (strategy) of the cat ( 7 p = 1, p ≥ 0 for k=0 k k k =0,...,7). P 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 q , where m is the m number of food that does not occur in the pair (k,j) ( 2 q = 1). This deno- m=0 m tation causes no uncertainty because there are only thrPee types of food. When the choice methods f are selected nondeterministically, with the respective intensities k p , the frequency ω , m=0,1,2, of occurrence of individual food in cat’s diet are k m according to Table 1. given as follows: • food no. 0: ω =(p +p +p +p )q +(p +p +p +p )q , 0 0 1 2 3 2 0 1 4 5 1 • food no. 1: ω =(p +p +p +p )q +(p +p +p +p )q , 1 4 5 6 7 2 0 2 4 6 0 • food no. 2: ω =(p +p +p +p )q +(p +p +p +p )q . 2 2 3 6 7 1 1 3 5 7 0 Threeequalitiesabovecanbeexplainedwiththehelpoftheconditionalprobability concept. Let us denote B = {(j,k)}, P(B ) = q and C = {j} for j,k = 3 (j+k) j j j 0,1,2, j 6=k. The number P−(C |B ) indicates the probability of choosing the food k j of number k, when the offered food pair does not contain the food of number j. Since the events of choosing different pairs of food are disjoint and comprise all the space of elementary events. Hence, for each food chosen, we have the following relation: 2 ω =P(C )= P(C |B )P(B ), k =0,1,2. (2) k k k j j Xj=0 Byinspectionofthetableofthefunctionsf ,k=0,...,7,weeasilygetthefollowing k relations: 7 P(C |B )=P( f (B )=0)=p +p +p +p , 0 2 k 2 0 1 2 3 Xk=0 7 P(C |B )=P( f (B )=0)=p +p +p +p , 0 1 k 1 0 1 4 5 Xk=0 7 P(C |B )=P( f (B )=1)=p +p +p +p , (3) 1 0 k 0 0 2 4 6 kY=0 7 P(C |B )=P( f (B )=1)=p +p +p +p , 1 2 k 2 4 5 6 7 kY=0 7 f (B ) P(C |B )=P( k 1 =1)=p +p +p +p , 2 1 2 3 6 7 2 kY=0 7 f (B ) P(C |B )=P( k 0 =1)=p +p +p +p , 2 0 1 3 5 7 2 kY=0 and P(C |B )=P(C |B )=P(C |B )=0. 0 0 1 1 2 2 Frequency of the least preferred food, that is the function min(ω ,ω ,ω ), deter- 0 1 2 minesthedegreeofthedietcompleteness. Sinceω +ω +ω =1,themostvaluable 0 1 2 Cat’s Dilemma – transitivity vs. intransivity way of choosing the food by the cat occurs for such probabilities p ,...,p , that 0 7 the function min(ω ,ω ,ω ) has the maximal value, that is for 0 1 2 ω =ω =ω = 1. (4) 0 1 2 3 Any vector p~=(p ,...,p ) (or six conditional probabilities (P(C |B ), P(C |B ), 0 7 1 0 2 0 P(C |B ),P(C |B ),P(C |B ),P(C |B ))),whichforafixedtriple(q ,q ,q )fulfills 0 1 2 1 0 2 1 2 0 1 2 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 q , q , q all optimal strategies are calculated. The system of equations 0 1 2 (4) has the following matrix form: P(C |B ) P(C |B ) 0 q 1 0 2 0 1 2  P(C1|B2) 0 P(C1|B0)  q1 = 31 1 , (5) 0 P(C |B ) P(C |B ) q 1 2 1 2 0 0      and its solution: P(C |B )+P(C |B ) q = 1 0 1 1 0 −P(C |B )P(C |B ) , 2 d(cid:18) 3 0 1 1 0(cid:19) P(C |B )+P(C |B ) q = 1 0 2 2 0 −P(C |B )P(C |B ) , (6) 1 d(cid:18) 3 0 2 2 0(cid:19) P(C |B )+P(C |B ) q = 1 1 2 2 1 −P(C |B )P(C |B ) , 0 d(cid:18) 3 1 2 2 1(cid:19) defines a mapping of the three-dimensional cube [0,1]3 in the space of parameters (P(C |B ),P(C |B ),P(C |B )) into a triangle in the space of parameters 0 2 0 1 1 0 (q ,q ,q ), where d is the determinant of the matrix of parameters P(C |B ). The 0 1 2 j i barycentric coordinates [9] of a point of this triangle are interpreted as the probabilities q , q and q . These numbers represent the heights a, b and c or the 0 1 2 areasP , P and P of three smaller trianglesdetermined by the point Q QAB QBC QAC (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 vertex of the triangle. Hence e.g. q1 = a = PQBC = |RB|, where the symbol |RB| represents length of the segment. q2 b PQAC |RA| The next picture (Fig. 2) presents the image of the three-dimensional cube in this simplex. It determines the area of frequency q of appearance of individual choice m 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. Justification of such equipartition of probability may be found in Laplace’s principleofinsufficientreason[10]. Inourrandomizedmodeltheaprioriprobability of the fact that the sum of probabilities P(C |B ) is smaller than a given number j k α ∈ [0,1] equals α. The absence of optimal solutions outside the hexagon forming theshadedpartofthepicture(Fig.2)isobvious,sincethebright(non-dotted)part ofthe picturerepresentsthe areas,forwhichq > 1 (orq > 1, orq > 1), andthe 0 3 1 3 2 3 total frequency of appearance of pairs (0,1) or (0,2) must be at least 1 in order to 3 Piotrowski and Makowski C a b Q c A R B Fig1. Thebarycentriccoordinates. Fig2. Imageofthethree-dimensionalcubeonsimplex. assurethecompletenessofthedietwithrespectoftheingredient0(butthiscannot happen because when q > 2, then q +q =1−q < 1). 0 3 1 2 0 3 The system of equations (5) can be transformed into the following form: q −q 0 P(C |B ) 1 −q 2 1 0 2 3 1  −q 0 q  P(C |B )  =  1 −q , (7) 2 0 2 1 3 2  0 q1 −q0  P(C1|B0)   31 −q0  which allows to write out the inverse transformation to the mapping defined by equations (6). By introducing the parameter λ we may write them as follows: λ λ−1+3q λ+1−3q P(C |B )= , P(C |B )= 1 , P(C |B )= 2 . (8) 0 2 3q 2 1 3q 1 0 3q 2 1 0 A whole segmenton the unit cube correspondsto one point of the simplex, param- eterized 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∈[0,3q ], λ∈[1−3q ,1], λ∈[3q −1,2−3q ]. (9) 2 1 2 1 Cat’s Dilemma – transitivity vs. intransivity The permitted valuesofthe parameterλ formthe commonpartofthese segments, hence it is nonempty for: max(0,1−3q ,3q −1) ≤ min(2−3q ,3q ,1). (10) 1 2 1 2 Therefore λ∈[max(0,1−3q ,3q −1),min(2−3q ,3q ,1)]. (11) 1 2 1 2 Itmaybe nownoticedthatforanytripleofprobabilitiesbelongingto the hexagon, there exists an optimal solution within a set of parameters ((P(C |B ),P(C |B ), 0 2 0 1 P(C |B ))). Ifweassumetheequalmeasureforeachsetoffrequenciesofoccurrence 1 0 of food pairs as the triangle point, then we may state that we deal with optimal strategies in 2 of all the cases (it is the ratio of area of regular hexagon inscribed 3 into a equilateral triangle). The inverse image of the area of frequencies (q ,q ,q ) 0 1 2 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. Wepresenttherethesameconfigurationobservedfromdifferentpointsofview. The segments on the figures correspond to single points of the frequency triangle of the individualfoodpairs. Thegreatestconcentrationofthesegmentsisobservedintwo areasofthe cube thatcorrespondtointransitivestrategies1. The brightareainthe center of the cube, which may be seen in the last picture, belongs to the effective strategies – effective in the subset of frequencies of a small measure (q ,q ,q ) of 0 1 2 the food pairs appearance. Among them, the totally incidental behavioris located, which gives consideration in equal amounts to all the mechanisms of deterministic choice p =p = 1. j k 8 3. Example of an optimal strategy The formulas (8) that map the triangle into a cube can be used to find an optimal strategy in cases, when the probabilities (q ,q ,q ) of appearance of individual 0 1 2 pairs of the products are known. Let us assume that q = 1, q = 1 and q = 1. 0 2 1 3 2 6 Then, according to the formulas (8), we have P(C |B ) = 1 + 2λ, P(C |B ) = 1 0 3 3 0 2 2λ, P(C |B ) = λ, where λ∈[0,1]. Selecting λ= 1 we have: P(C |B ) = 1, 2 1 2 4 0 2 2 P(C |B ) = 1, P(C |B ) = 1. We may now show the solution of equations (3), 2 1 4 1 0 2 e.g.: p = 1, p = p = 1 and p = 0 for others parameters. We will obtain the 0 2 5 7 4 j following frequencies of occurrence of individual foods in the diet: 1 1 1 ω =(p +p )q +p q = + = , 0 0 5 1 0 2 4 12 3 1 1 1 ω =p q +(p +p )q = + = , (12) 1 0 0 5 7 2 4 12 3 1 1 1 ω =(p +p )q +p q = + = . 2 5 7 0 7 1 4 12 3 Theabovecalculationsofthefrequencyω confirmoptimalityoftheindeterministic j algorithm determined in this example. 1Seesection4. Piotrowski and Makowski PHC2ÈB1L PHHC0ÈÈB2LL 1 0 00 11 11 1 PHC1ÈB0L PHC1ÈB0L 1 PHC0ÈÈB2LL1100 00 0 0 PHC2ÈB1L PPHHCC22ÈÈBB11LL 110 00 PHC0ÈB2L 1 11 PHC1ÈB0L PHC1ÈB0L 0 PHC2ÈB1L 000 00 111 PPHHCC0ÈÈBB2LL Fig 3. The inverse image of area of frequencies (q0,q1,q2) that enable realization of the optimal strategy, seeAppendixA. 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 offeredpair (0,1) we are willing to choose the food no. 1 more often than the food no. 0 (P(C |B ) < P(C |B )). Therefore we have two 0 2 1 2 intransitive orders: • P(C |B )< 1, P(C |B )< 1, P(C |B )< 1. 0 2 2 2 1 2 1 0 2 • P(C |B )> 1, P(C |B )> 1, P(C |B )> 1. 0 2 2 2 1 2 1 0 2 Itisinterestingtoseeinwhichpartofthesimplexofparameters(q ,q ,q )wemay 0 1 2 take optimal intransitive strategies. They form the six-armed star composed of two triangles,eachof them correspondingto one of two possible intransitive orders (Fig.4). Theydominateinthecentralpartoftriangle,nearpointq =q =q = 1. 0 1 2 3 Theyformdarkenedpartofareainsidethestar. Optimaltransitivestrategiescover 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). Inareasofhighconcentrationofoptimaltransitivestrategies,oneofthree frequencies q , q , q looses its significance– two from three pairs ofthe food occur 0 1 2 with considerable predominance. We have enough information to be able to com- pare the applicability range of different 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 (q ,q ,q ) 0 1 2 does not depend on those parameters. Two thirds of strategies are optimal. There Cat’s Dilemma – transitivity vs. intransivity Fig4. Optimalintransitivestrategies. Fig5. Optimaltransitivestrategies. are 33%2 of circumstances, which allow for the use of the optimal strategies that belong to the specified intransitive order. There are 44% (4) of situations of any 9 order that favor optimal strategies, what follows from the fact that they are mea- sured by the surface of regular star, and its area is equal to double area of the triangle correspondingto one intransitiveorderreducedby the areaofthe hexagon inscribed into the star. So we have: 1 + 1 − 2 = 4. Appearance of the number 2 3 3 9 9 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 alwaysfindthe strategythat determines the transitiveorder(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. 2Theyaremeasuredbytheareaofequilateraltriangleinscribedintoaregularhexagon. Piotrowski and Makowski 5. Conclusions In this article,we useda stochasticvariantofthe principle of least action. Perhaps disseminationof usageof this principle will leadto formulationofmany interesting conclusions and observations. We presenteda method, which allowssuccessful analysis of intransitive ordersthat still are surprisingly suspicious for many researchers. More profound analysis of this phenomenoncan have importance everywherewhere the problemof choice be- haviorisstudied. Forinstanceineconomics(descriptionofthecustomerpreference toward products- marketing strategy) or in political science where the problem of votingexists. Analysisofintransitiveordersisaseriouschallengetothosewhoseek description of our reasoning process. Thequantitativeobservationsfromtheprevioussectionshowthatintransitivity, 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 acknowledgethe preferencesshowingthe intransitivity as undesired. Perhapsthere aresituations,whenonlytheintransitiveordersallowobtainingtheoptimaleffects. The most intriguing problem that remains open, is to answer the question whether thereexistsausefulmodelofoptimalbehaviors,whichgivestheintransitiveorders, and for which it would be impossible to specify the transitive optimal strategy of identical action results. Showing the impossibility of building such constructions wouldcausemarginalizationofthe practicalmeaningofintransitiveorders. Onthe otherhand,indicationofthistypeofmodelswouldforceustoaccepttheintransitive ordering. Acknowledgements The authors are grateful to prof. Zbigniew Hasiewicz and prof. Jan S ladkowski for useful discussions and comments. Appendix A. Thefollowingmini-programwritteninthelanguageMathematica 5.0 generatefour plots in Fig. 3. Cat’s Dilemma – transitivity vs. intransivity In[1]:= c=N[ 4 ]; 3√3 gencorrect:=Module[{a=c{Random[],Random[]}},While[a[[2]]> 2, 3 a=c{Random[],Random[]}]; a]; setpoint:=Module[{q1,a,x,y,v=1,w=1},While[2v+ 3w >1, c 2 a=gencorrect; q1=a[[2]]; x=a[[1]]− c; 2 y =a[[2]]− 1; v =Abs[x]; w =Abs[y]]; 3 {q1,N[√3x− 1y+ 1]}]; 2 2 3 setsegment:=Module[{a,q1=1,q2=1,l1=1,l2=0}, While[l1>l2,a=setpoint;q1=a[[1]];q2=a[[2]]; l1=Max[0,1−3q1,3q2−1]; l2=Min[2−3q1,3q2,1]]; {{3lq12,13−ql11,31(1−3qq12+ql12)},{3lq22,13−ql12,31(1−3qq12+ql22)}}]; − − − − fig[x ,y ,z ]:=Show[Graphics3D[{GrayLevel[.0],Thickness[.002], Table[Line[setsegment],{2000}]}], ViewPoint→{x,y,z}, Axes→True, AxesLabel→{”P(C |B )”,”P(C |B )”,”P(C |B )”}, 0 2 2 1 1 0 Ticks→{{0,1},{0,1},{0,1}}, BoxStyle→Dashing[{.02,.02}], AxesStyle→Thickness[.005]]; fig[−6,.7,.3] fig[.6,−3,.3] fig[1.3,3.4,2] fig[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.