Mathematical analysis of Soros's theory of reflexivity

0.18 MB·English

by  C. P. Kwong

Mathematical analysis of Soros’s
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a C.P. Kwong∗
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2 November 27, 2008

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n. Abstract
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f ThemathematicalmodelproposedbyGeorgeSorosforhistheory
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ofreflexivityisanalyzedundertheframeworkofdiscretedynamical
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systems. We show the importance of the notion of fixed points for

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explaining the behavior of a reflexive system governed by its cog-
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7 nitive and manipulative functions. The interrelationship between
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these two functions induces fixed points with different characteris-
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4 tics, which in turn generate various system behaviors including the
1. so-called“boomthenbust”phenomenoninSoros’stheory.
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9 Key words: Soros’s theory of reflexivity, Discrete dynamical sys-
0 tems
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JELclassifications: B41,C69
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1 Introduction
The theory of reflexivity is a theoretical construct of George Soros’s
philosophyonpricesinfinancialmarkets. Thetheorywasfirstdocu-
mentedbySorosinhisTheAlchemyofFinance(Soros,1987),expanded
∗TheChineseUniversityofHongKong,HongKong.
Address for correspondence: Department of Mechanical and Automation Engineer-
ing, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong; email: cp-
[email protected]
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in Open Society (Soros, 2000), and reiterated, using the recent credit
crisis as example, in his latest work The New Paradigm for Financial
Markets (Soros, 2008). From the outset Soros does not believe, that
themethodsofnatural sciencescan beapplied tothestudyofsocial
sciencessuchaseconomics. Simplyput,inhisownwords,thereisno
hopeto obtain Newton’slaws for finance or economics. Specifically
hedefiespricetheoriesgroundedonequilibriumascanbenoticedby
thetitleofthefirstsectionofthechapter“TheTheoryofReflexivity”
in his The Alchemy of Finance, namely “Anti-equilibrium”. His basic
argumentisthathumanbeingswillinevitablyparticipateinanypric-
ing process. Since their perspectives on price, and hence their reac-
tiveactionswhichinfluencethefutureprice,areeverchanging,there
isnoroomforequilibriumbutfluctuations. Exactlyfromthispremise
Sorosproposedhis theoryof reflexivity. Though his approach to re-
flexivity has been mainly philosophical, Soros did attempt to model
histheorymathematicallyusingthefollowingpairofequationsfrom
theverybeginning(Soros,1987):
y = f(x),
(1)
(x = φ(y).
Hecalls f thecognitivefunctioninwhich“theparticipants’percep-
tions depend on the situation” and φ the manipulative function in
which“thesituationisinfluencedbytheparticipants’perceptions”.1
The following quote from Soros (Soros, 2008, p.17) may give some
hintsonhisattitudetowardstheuseofmathematicsineconomics:
“I was not very good at math, and that led me to question the
assumptions on which the mathematical models of economists
werebased.”
Accordingtotheinventorhimselfthetheoryofreflexivityhasre-
ceivednoseriousattentionfromtheacademia sinceitsinception—it
is “totally ignored in departmentsofeconomics” (Soros, 2008, p.20).
Soros attributes this phenomenonto the imprecision in his formula-
tions,and“Asaresult,theprofessionalswhosepositionsIchallenged
coulddismissorignoremyargumentsontechnicalgroundswithout
giving them any real consideration” (Soros, 2008, p.20). We cannot
agree him more except that Soros has indeed formulated his theory
byaprecisemathematicalmodel,i.e.,Equation(1). Therealproblem
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arisesfromthemeagreattentionofthisequationbySoroshimselfor
others.ForinstanceallSoros’spublishedwritingsarenarrativeinna-
tureandneitherotherrepresentativereviewsonthetheoryofreflex-
ivityhaveaddressedthisfundamentalequation(CrossandStrachan,
1997; Bryant, 2002). Soros may be excused from the duty of per-
forming the mathematical analysis because he “was not very good
at math”. But for other mathematically inclined researchers in fi-
nance or economics, the lack of such an interest is somewhat puz-
zlingsinceEquation(1)isveryamenabletomathematicalanalysisas
we will show in this paper. The article, published by Birshtein and
Borsevici (BirshteinandBorsevici, 2002), is perhaps the only excep-
tion. There the authors make explicit the recursive nature of Equa-
tion (1) and then use graphical analysis to show that equilibria do
exist in the reflexive model, a result in contrast to Soros’s dubious-
nessonequilibriumtheory. Nevertheless,theiranalysisisincomplete
and thepresentpaper can be consideredas thecontinuation oftheir
efforts. We believe that such an analysis is indispensable to a real
understandingofthetheoryofreflexivityasformulatedbySoros,no
mattertheresultsaresupportinghisclaims,whichgobeyondfinance
toreachothersocialsciencesubjects,orotherwise. Thisisexactlythe
purposeofourstudy.
Atthisjunctionitmaybeworthwhiletoconsidertheroleofmath-
ematicalmodelinginstudyingeconomicsinordertojustifyourpresent
effort. The topic is big and controversial (Mirowski, 1986; Debreu,
1991; BeedandKane, 1991; Velupillai, 2005). However, our view is
simple (and of course, nonsingular) enough to be expressed within
a paragraph. We admit that what social scientists are trying to ex-
plore are systems of extreme complexity. Not only that the number
ofvariables canbehugebutalsocomplicatedhumanbehaviorsmay
be involved. However, it should not, in our opinion, to preclude
thepossibilityofusingsome“good”mathematicstoeffectmodeling
and analysis of these complex social systems as we do for physical
or biological systems. The mathematics so used need not be very
advanced and abstract to be effective. The works of Nobel Laure-
ates Daniel Kahneman (KahnemanandTversky, 1979) and Thomas
C.Schelling(Schelling,2006)formodelinghumanbehaviorsaretwo
typicalexamples. Itisimportanttolearnfromtheseexamples,thatin
manycasestheeffectivenessofmathematicsinanappliedsciencelies
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not in providing precise computations of variables but sufficiently
accurate qualitativedescriptionsofphenomena. Thisis particularly
truewhenwestudysystemsthatcannotbemodeledexactly.
Wedivideourpaperintoseveralsections. InSection2wesetupa
dynamicalsystemmodelforthestudyofEquation(1)anduseasim-
pleexampletoshowthebasiccharacteristicsofthismodel. InSection
3weanalyze themodelbehavior in detail. Althoughthemathemat-
ical results we shall use are well documented, we will nevertheless
supply their full descriptions for readers who are not familiar with
thesetopics. FinallyweconcludeourworkinSection4togetherwith
someadditionalcomments.
2 The System Model for Reflexivity
Equation (1) can be depicted by the following block diagram which
showsclearlythecausalrelationshipsbetweenthevariables xandy:
f
x y
φ
Figure 1: Thesystem-theoretic model ofreflexivity.
Theblockdiagramalsosuggeststheuseof“systemstheory”tostudy
reflexivity.2 For instance, in the language of systems theory, x is the
inputsignaltothecognitivefunction f andyistheoutputsignalof
thefunction(treatedasasystem).Thediagramrepresentsafeedback
systemsincetheoutputyof f isfedbacktoitsinputafteryismapped
bythemanipulative function φ. Wecan alsosaythattheoutput x of
φ is fedback toits input after x is mapped bythe cognitive function
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f. Because of this symmetry f takes the same role of φ when it is
consideredasadevicewhichmodifiesthefeedbacksignals. Foreasy
referenceinthefutureweuseS tolabelthissystem.
GivenasystemS ofFigure1withspecificcognitiveandmanipu-
lativefunctions,thevaluesof xand yarenotarbitrarybutgoverned
by the feedback connection of these two functions. Indeed they are
simply the solution(s) of Equation (1). The significance of this con-
straint on variable values, which is introduced by the feedback con-
figurationorequivalentlyEquation(1),isillustratedbythefollowing
simpleyetelucidativeexample.
Let x betheexchangerateofaparticular currency(say Eurover-
sus USD) and y be its domestic interest rate. According to the in-
terpretationofSoros,thecognitionfunction f wouldtaketheroleof
understandingthe current exchange rate x and then would form an
opinion on theinterest rate y. Themanipulative function φ reacts to
thisvalueofyandoutputsanewvalueofexchangerate. Forthesake
offurtheranalysisofthereflexivesystemS weassumetheexistence
of thesetwo functions without detailing how these functions can be
derived.3 The process continues as dictated by the feedback loop of
Figure1.
Fromthe above example it is easy to seethat thevalues of x and
y evolve with time step-by-step starting from some initial value of
x. Suppose we label this initial value as x , then we obtain subse-
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quently two sequences of numbers x ,x ,... and y ,y ,.... Clearly
1 2 1 2
the subscripts 1,2,... are used to index time. Moreover, we have
y = f(x ),x = φ(y ),y = f(x ),x = φ(y ), and so on. Thus we
0 0 1 0 1 1 2 1
can write two general expressions for the respective evolutions of x
andy(withtime)as
y = f(x ), i = 0,1,2,... (2)
i i
and
x = φ(y ), i = 0,1,2,... . (3)
i+1 i
In this way we turn S into a discrete dynamical system by “iter-
ation of functions”. Following the tradition of dynamical systems
theory we call the pair (x,y ), a point on the Euclidean plane, the
i i
stateofS andthesetofpoints{(x ,y ),(x ,y ),(x ,y ),...}the(for-
0 0 1 1 2 2
ward)orbitof(x ,y )underS. Dynamicalsystemsofthiskindhave
0 0
been extensively studied (see, for example, (HaleandKoc¸ak, 1996),
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(Holmgren,1996),and(Robinson,2004))andthisiswhyweclaimin
theIntroductionthat Equation (1)is very amenable to mathematical
analysis. Inthenextsectionwewillfrequentlyapplyexistingresults
fromdiscretedynamicalsystemstoourstudy. However,beforemov-
ingon totheformal analysis ofthedynamics ofS, wewouldliketo
use the exchange rate example to motivate our research along this
direction.
Asabove welet x be theinitial exchangerateofacurrency. The
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cognition function f takes x as input and outputs the interest rate
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y = f(x ). Suppose the manipulative function φ reacts to y in a
0 0 0
way such that x = φ(y ) = x , then we see immediately that the
1 0 0
value of x as well as the value of y will remain unchanged for all
i i
subsequent i, i.e., x = x ,y = y ,i = 1,2,.... Mathematically we
i 0 i 0
call the pair (x ,y ) the ”fixed point” of the dynamical system S,
0 0
which we shall formally define and study in the next section. The
concept of fixed point is significant in applying the theory of reflex-
ivity. This is because the existence of a fixed point (or fixed points)
impliesthatbothpartieswhodeterminetherespectivecognitiveand
manipulative functions do agree with the one-one correspondence
betweentheelementsofthisparticularpair(orpairs)of xandy. For
the exchange rate example that will mean an agreement in which a
particular exchange rate should correspond to a particular interest
rate. Thesystemthusreachesanequilibrium. Intheextremecasethe
partiesagreeon all pairs of x and y. This will happenwhen thema-
nipulative function φ is the “inverse” of the cognitive function f, or
viceversa. Itisworthtonote,beforegivingthedetails,thateventhe
initialstatedoesnotbeginatafixedpointbutinitsvicinity, thesub-
sequentstatemayapproachandsettleatthefixedpointafterseveral
iterations. Inrealtermsthepartieshavedifferentopinionsinthebe-
ginningbuteventuallyagreewitheachother. Alternativelythestate
leaves the fixed point and does not return. The dynamical behavior
ofS asinfluencedbyitsfixedpointsisourimmediatetopicofstudy.
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3 The System Behavior
3.1 Background Mathematics
Webeginwiththedefinitionofafunction,assumingthatthereaders
already knowwhatis a set. Wemuststressagain thatall themathe-
maticspresentedherearestandard(see,forexample,GaskillandNarayanaswami
(1998)),andtheirinclusionhereisforeasyreferenceonly.
Definition 1. Let A and B be sets. A relation R on A and B is a
collectionoforderedpairs(a,b) suchthata ∈ Aandb ∈ B. ♦
Definition 2. Let A and B be sets. A function f from A to B is a
relation Ron AandBsuchthatforeverya ∈ Athereisoneandonly
one pair (a,b) ∈ R. We call A the domain of the function and B its
codomain. Afunctionisalsocalledamap. ♦
f
Quite oftenwe write f: A → B or A −→ B for “a function f from
A to B,” and write a 7→ f(a) to indicate that ”a is mapped to f(a)
under f”.
Definition3. Let f: A → Bbeafunction. Therangeof f istheset
{b ∈ B thereexistsa ∈ Asuchthat f(a) = b}.
♦
(cid:12)
Notethattherang(cid:12)eof f isnotnecessarilythewholecodomainB. For
example, the function represented by y = sin(x) has the set of real
R
numbers as both its domain and codomain. However, the range
of this function is only the closed interval [−1,1]. Another example
is the function f representedby y = log(x), of which the domain is
the set {x ∈ R x > 0}. If we denote this set by R+, we may write
f: R+ → R.
(cid:12)
(cid:12)
Definition4. Afunction f: A → B isinjectiveifwhenever(a ,b) ∈
1
f and(a ,b) ∈ f,thena = a . Itiscalledsurjective,orontoifwhen-
2 1 2
ever b ∈ B, then there exists an a ∈ A such that (a,b) ∈ f. Finally,
the function is called bijective, or one-one if it is both injective and
surjective. ♦
For instance the function represented by y = sin(x) is not injective
andthefunctionrepresentedbyy = log(x)isbijective(one-one). As
a consequence, the “inverse” of this latter function exists, which is
theanti-log.
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Definition5. Let f: A → Bbeabijectivefunction. Thentheinverse
function f−1 of f is the function f−1: B → A defined by f(a) 7→ a
forevery f(a) ∈ B. ♦
f g
Let A,B, and C be setsand f,g be functions A −→ B and B −→ C.
Wemaythenwrite
f g
A −→ B −→ C
h
and introduce a function A −→ C. The function h is called the com-
positionofthefunctions f and g. Inthiscasehiscalledacomposite
function. Wewriteh = g◦ f. Foreverya ∈ A,wehave
a 7→ h(a) = (g◦ f)(a) = g(f(a)).
Notethattheorderofcompositionisimportant—f ◦gmaynoteven
makesense(for examplewhenthedomainof f doesnotincludethe
rangeofg)andifitexists,isingeneralnotidenticalto g◦ f.
Differential calculus plays a crucial role in the study of iteration
offunctions.
Definition 6. Let f: R → R be a function with domain D and let
a ∈ R. Wesaythat f hasalimit Aas xtendstoa,writtenas
lim f(x) = A,
x→a
if,forǫ > 0,thereisaδ > 0suchthatforeveryx ∈ D,
0 < |x−a| < δ implies |f(x)− A| < ǫ.
♦
Definition 7. Let f: R → R be a function with domain D. We say
that f is continuous at a point a ∈ D if, for every ǫ > 0, there is a
>
δ 0suchthat
0 < |x−a| < δ implies |f(x)− f(a)| < ǫ.
A function is said to be continuousif it is continuousat every point
in D. ♦
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Definition8. Let f: R → R beafunctionwithdomain D and a ∈ R
be a point in the interval (a−c,a+c) ⊆ D for some c > 0. We say
that f isdifferentiable ataifthefollowinglimit
f(x)− f(a)
lim ,
x→a x−a
exists. The limit, if exists, is called the derivative of f at a and is
labeled f′(a). A function is said to be differentiable if it is differen-
tiableateverypointin D. ♦
Geometrically f′(a)istheslopeofthetangentlineto f ata,whichcan
bepositiveornegative. Theabsolutevalueofthisslopeiswrittenas
|f′(a)|.
3.2 Fixed Points
We are now ready to go into the details of the dynamic behavior of
the system S. First, the idea of a “fixed-point”, which we sketch in
Section2,isformallydefined.
Definition9. Thepairofrealnumber(x¯,y¯)iscalledafixedpointof
S if
f(x¯) = y¯ and φ(y¯) = x¯.
♦
Thegraphofafunctionusuallygivesalotofinformationanditis
evenmoretruewhenweinvestigatethedynamicsof S thatconsists
oftwofunctions f andφconfiguredinthepresentway. Inparticular,
ifweplot f and φ onthesame x-y plane asin Figure2, theintersec-
tionsof f andφ,ifexist,areexactlythefixedpoints. Ifthefunctions
areinversesofeachother,thentheirgraphsoverlapatalltheirdefin-
ing points. Furthermore, the dynamics of S, i.e., the evolution of x
andywithtime,canbeeasilyvisualizedbyusingthisgraphicalrep-
resentationaswewillshowinSubsection3.3.
By the notion of composition of functions one may infer that the
dynamicsofthesystemS maybestudiedbyconsideringtheiteration
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y
φ(y)
f(x)
y¯
x
x¯
Figure 2: The ideaof afixed point.
ofthecompositefunction Γ = φ◦ f orΦ = f ◦φ. Ineithercaseonly
onesinglefunction(Γ orΦ)anditerationofonesinglevariable (x or
y) are involved. Itis indeedtrue ifwe focuson theevolutionof x or
yaloneandinthiscasethedynamical equations(2)and(3)can then
Γ
bewritten,intermsof ,as
x = Γ(x ), i = 0,1,2,..., (4)
i+1 i
Φ
or,intermsof ,as
y = Φ(y ), i = 0,1,2,... . (5)
i+1 i
Note that the orbit of x under Γ is the set {x,Γ(x),Γ2(x),...} and
the orbit of y under Φ is the set {y,Φ(y),Φ2(y),...}. However, if
Γ Φ
we consider only the iteration of or , then the interrelationship
between f and φ in determining the system behavior we have just
demonstrated,willbecomeopaque.
Definition10. Thereal number x¯ is called afixed pointofthe com-
positefunctionΓ = φ◦ f if
Γ(x¯) = φ(f(x¯)) = x¯.
Similarly, y¯isafixedpointofthecompositefunctionΦ = f ◦φif
Φ(y¯) = f(φ(y¯)) = y¯.
♦
Itiseasytoestablishthefollowingfact.
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