Cat's Dilemma - transitivity vs. intransitivity

2.1 MB·

by  Edward W. Piotrowski

Fluctuation andNoiseLetters
Vol.0,No.0(2001)000–000
(cid:13)c WorldScientificPublishingCompany
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CAT’S DILEMMA – TRANSITIVITY VS. INTRANSITIVITY
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p EDWARDW.PIOTROWSKI*andMARCINMAKOWSKI**
- Institute of Mathematics, Universityof Bial ystok,
c Lipowa 41, PL-15424 Bial ystok, Poland
o [email protected]*, [email protected]**
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c Received(receiveddate)
i Revised(reviseddate)
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Accepted (accepted date)
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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
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adversary(theofferingplayer),isinvestigated. Itturnsoutthattheorderimposedbythe
v
player’srationalpreferencescanbeintransitive. Thepresentedquantitativeresultsimply
2
arevisionofthe”commonsense”opinionsstatingthatpreferencesshowingintransitivity
2
areparadoxical andundesired.
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8 Keywords: intransitivity;gametheory;sequential game.
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4
1. Introduction
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/ 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
mathematicalpoliticalscienceiscomparabletoG¨odel’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 illus-
trated 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 prin-
ciple 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 de-
cisions. 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 parame-
ters (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 prob-
abilities 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 strat-
egy 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 of-
ten 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
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(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.