keynesian uncertainty and fuzzy logic

23 Pages·2010·0.66 MB·English

IS KEYNESIAN UNCERTAINTY COMPATIBLE WITH FUZZY LOGIC FRAMEWORK?
ALEXANDRE C.GOMES DE SOUZA1
GABRIEL PORCILE 2

ABSTRACT
In the post-keynesian approach, uncertainty is understood as ‘radical uncertainty’ and cannot
be formalized by the classical probability theory. There are a great debate in how formalize
the notion of uncertainty, as conceived by Keynes in his Treatise on Probability and General
Theory. Fuzzy logic, in turns, offers a robust mathematical and computational framework of
modeling some types of uncertainties, as imprecision, ambiguity, and vagueness, that is
commonly met in subjective and incomplete information set considered by practical
decision-makers. Aligned with Dow and Gosh (2009), this paper aims to evaluate the fuzzy
logic tools in keynesian economics modeling, by presenting some detached fuzzy logic
elements of the Treatise of Probability. By considering qualitatively (high, medium, low) as
fuzzy variables its two dimensions, probability and weight, a small and exploratory fuzzy
model of keynesian uncertainty is developed.

Keywords: probability, keynesian economics models, fuzzy logic.
JEL Classification: E12, C45, C63,

1 Department of Economics, Banco Central do Brasil,. Email:[email protected]. The views expressed in this
work are those of the authors and do not necessarily reflect those from Banco Central do Brasil or its members.
2 Department of Economics, Federal University of Paraná, and Reseacher of CNPq, Brazil. Email: [email protected]
1 INTRODUCTION

Uncertainty is a central concept in economics sciences, demanding, in its several approaches,
distincts views of the world and recommendations of economic policies. One of the basics points
in discussion about this concept refers to its formal characterization, mainly around the
dichotomy of radical uncertainty and risk. The neoclassic economics theory, for example,
considers a rational and omniscient agent, having complete information of futures events and
total capacity of its processing. As a result of this view, uncertainty is understandable as risk
and formalized by the classical probability theory. On the other hand, in the post keynesian
approach, decisions are took in changeable non-ergodic economic environment. As
consequence, the future events data are not available a priori, including all the nature states of
the future and respective probabilities. In this sense, it is supposed a radical uncertainty where
the mathematical tool of classical probability is not applicable.
Fuzzy logic offers a robust mathematical and computational framework of modeling
some types of uncertainties, as imprecision, ambiguity, and vagueness, that is commonly met in
subjective and incomplete information set considered by practical decision-makers. Fuzzy
applications are very dependent on intuition, rules-of-thumb, and conventions with several
successful cases in engineering, electronics gadgets, and computational software. The main idea
of fuzzy logic is to consider not only two values true or false as the classical, aristotelic logic,
but also values between both, multi-values of degrees of truth. We can make an analogy with the
variety of gray between the black and white in the color scale. With this attribute, we are able to
handle vagueness systematically. That’s the way practical man commonly view and think when
make decisions in complex environments (MCNEILL and FREIBERGER, 1993; KLIR and
FOLGER, 1988; KLIR and YUAN, 1995, PEDRYCZ and GOMIDE, 1998).
In A Treatise on Probability, Keynes (1921) showed his first insights about uncertainty in
economics. Probability in Keynes is defined as a logical relation between two propositions sets:
a conclusion set and evidence set. Considering incomplete evidences, the decision-maker
foresees the probability of a proposition. This an enigmatic book, with a lot of concepts and
notions that until now are a little misunderstood. There are typical elements of fuzzy theory in
the Treatise, as vagueness in information, gradation approach of some concepts (rational belief
degrees, for example), critics to classical probability theory and classical logic, knowledge
expressed by an if/then structure and approximate reasoning and so on.
Gerrard (2003) argued that some post-keynesians scholars should investigate the
theoretical and empirical relevance of Keynes’s contributions to probability theory instead of
only criticize the neoclassical economics theory in respect on uncertainty formalization. In line
with this idea, Vercelli (1999) suggested researches that rebuild the Keynes work on liquidity
preference (highly related with uncertainty) that can offer more operational analytical treatment.
In other words: it’s necessary a positive agenda in this issue, as the approximations between
fuzzy logic and Keynes’s probability theory developed by Coates (2007), Chick and Dow
(2001), and Dow and Gosh (2009), and moderns approaches in decision theory that can deal
with hard uncertainty, as Simonsen and Werlang (1991), Dow and Werlang (1992a, 1992b).
This paper aims to evaluate the fuzzy logic tools in keynesian economics modeling by
presenting some fuzzy logic elements detached mainly of the Treatise of Probability, the Keynes
first work on uncertainty. By considering qualitatively (high, medium, low) as fuzzy variables its
two dimensions, probability and weight, a small and exploratory fuzzy model of keynesian
uncertainty is developed.
The structure of the paper is as follows. The next section we provide an overview of
fuzzy framework. In the following section, was presented the Keynes’s logical theory of
probability proposed in A Treatise in Probability and its mains elements were detached and we
provide some links between fuzzy logic and Keynes probability relation logic. In section 4, we
offer a small and exploratory model that formalizes probability relation in the fuzzy framework.
At the end, we conclude.

2 FUZZY FRAMEWORK

The fuzzy framework - fuzzy logic, fuzzy set, fuzzy control, and others -, is a tool of the so
called Soft Computing, a broad set of tools (neural network and genetic algorithm) that aim to
apply some aspects of human cognition and biology in computation. Specifically, fuzzy logic
considers the vagueness commonly used in our natural language and explores the human being
ability in making decisions in environments with high level of complexity and uncertainty (as
airplane piloting, medical care or weather analysis) by using imprecise and vague expressions.
For example, expressions like “next winter will be rigorous cold” and “the employment is high”
permit quickly and effectively decisions, as explained by Klir and Folger(1988, p.2)
“…simplifying complexity by making a satisfactory trade-off or compromise between the
information available to us and the amount of uncertainty we allow”.
The fuzzy approach has been used in several areas with strong results, mainly when
conventional techniques fail as in complex system control and in environments with high level
of uncertainty, commonly verified in decision-making processes. Although economics deals with
social-economics systems, with an inherent complexity and the common use of natural language,
fuzzy logic applications in economics are not common (Klir and Yuan, 1995). According Zadeh
(1982 apud Klir and Yuan, 1995, p. 451) social-economics systems are a type of system
characterized as "... a large-scale system in human judgment which, perception, and emotions
play an important role" and which aspects are the central "... pervasive fuzziness of almost ...
phenomena." Also according to Zadeh (1987, p. 24), commenting on the failure of traditional
methods of scientists, the economy has all the characteristics necessary for application of fuzzy
logic:
Take economics. …..economic systems are very complex. Second, and
more important, human psychology plays an essential role in the
behavior of such systems. And this complexity, together with human
reasoning, makes the classical mathematical approaches, based on two-
valued logic, ineffective. So, again, to approximate the way humans can
sort through large masses of data and arrive at some sort of qualitative
conclusion, it might be necessary to use fuzzy logic (ZADEH, 1987, p.
24)

Despite the great technological advances seen in recent times, many industrial and
organizational processes are still controlled manually or intuitively by an experienced
professional, as operators of industrial processes or executives of investment banks. In such
cases, simple "rules-of-thumb" are considered in their routine activities. Several limitations of
cognitive, environmental and temporal prevent him from obtaining the optimal solution by using
conventional techniques and tools. In the design of fuzzy systems is first necessary to know the
best strategies considered effective by the decision maker. The next step is to transform this
knowledge appropriately in a fuzzy system whose basic elements are linguistic variables, fuzzy
rules and inference methods presented briefly in the next section.

2.1 FUZZY SETS

The development of the fuzzy theory started in 1965 with the seminal Lofti Zadeh’s
article – Fuzzy Sets. Fuzzy sets theory is an evolution of the classic set theory because they deal
with objects that may not belong only to one category represented by a set but may belong to
more than one category (sets) differing by degrees. It makes possible mathematical
representation of special types of uncertainty differ from random uncertainty, that is adequately
treated by the Probability Theory, as vagueness , ambiguity and imprecision (ZADEH, 1965;
KLIR and YUAN, 1995).
Fuzzy sets are defined considering a set X, called universe of discourse, with a collection
of objects x. Fuzzy sets, in reality, are subsets of the broader set X. The fuzzy subsets have ill-
defined border; so, an object can be mapped (or allocated) in many subsets in the same time,
with different degrees of membership (ZADEH, 1965; KLIR E FOLGER, 1988). Formally, a
fuzzy set (more correctly subset) A in a universe of discourse X is defined by a membership
function µ which maps all objects to A. Thus,
A,

m : X (cid:190) fi(cid:190) [0,1]
A

A={(x,m )| x˛ X}
A

Function µ maps values between 0 and 1, nearer degree 0 is understood as no
A
compatibility and nearer 1, a high level of compatibility

1 if x is totally compatible with A

µ (x) = 0 if x is totally incompatible with A
AThus, the formal representation of a subset A of a universe of discourse X is defined as
0 < p < 1 if x is partially compatible with A, with degree p.
The graphical representation of a fuzzy set defined vaguely as “a real number near 5”,
using a triangular function (there are others types of functions commonly used, as trapezoidal
and Gaussian) is shown in the Graph 1 Interesting to observe that the numbers 4, 4.6, 5, are
members of this subset with membership degrees equal to 0.0, 0.5 and 1.0, respectively:

1
0,5
4,6
4,0 5,0 6,0 x
Graph 1 – Graphical Representation of Fuzzy Subset

An important concept that is used in more elaborated applications of fuzzy subsets is
linguistic variable. A linguistic variable is characterized by a quintuple (x, T(x), X, g, m) in
which x is the name of the variable, T(x) is the set of names of the linguistic values (also called
linguistic terms) X is the universe of discourse, G is a grammar, m is a semantic rule for
associate each linguistic terms their meaning in X. For example, if “employment” is defined as a
linguistic variable, than the set of terms could be, T = {low, moderate low, stable, moderate
high, high}. Each term of T(x) must be conveniently represented by a fuzzy set in the universe
of discourse, similar of the set “a real number near 5” (Graph 1).
Fuzzy relation measured the degree of association between elements of different sets. If
the degree of association is either 0 or 1, we have crisp relation, but if the degree is between 0
and 1, we have fuzzy relations (TSOUKALAS and UHRIG, 1997). Fuzzy relations are fuzzy
sets defined on Cartesian products,. We can list pairs of the Cartesian product in the plane X x Y
by the equation:

R = {((x,y), µ (x,y))}
R
The fuzzy relation can represent by different ways. Linguistically, for example, by the
statement “x is greater to y”; when, for each ordering pair (x,y) of the cartesian space, the
membership µ (x,y) must be valued accordingly. A more common way is by a membership
R
matrix of an n x m binary fuzzy relation:

m (x ,y ) m (x ,y ) ... m (x ,y )
R 1 1 R 1 2 R 1 n
 
m (x ,y ) m (x ,y ) ... m (x ,y )
 R 2 1 R 2 2 R 2 n 
 ... ... ... ... 
 
m (x ,y ) m (x ,y ) ... m (x ,y )
R m 1 R m 2 R m n

For example, matrix A represents a crisp relation and matrix B a fuzzy relation, which
can represent the statement “x is greater to y”:

1 0 1 0  1 0,3 0,9 0,8
   
0 0 1 1 0,4 0 1 0,6
   
A = B =
0 1 0 0 0,2 0,2 1 0 
   
1 1 1 0  0 0 1 0,1

It’s easy to see that element x is greater y with degree 1 but greater only with
1 1
degree 0,3 comparatively element y . The fuzzy relation notion is especially important in
2
representing diagnostic analyses between effects and plausible causes. Tsoukalas and Uhrig
(1997, p.75) offered an interesting applied exercise, that was to assume reasonable numerical
values between 0 and 1 for membership values relating members of the set S - automobile
system set – and F – automobile fault set -, i.e., complete the fuzzy relation matrix R accordingly
with the summed expression S R F:
Matrix : S R F
 s (low.gas.mileage)   1 0 0,3 0,5
1  f (bad.spark.plugs)
    1
s (excessive.vibration 0,1 1 0,8 0  
 2    f (wheel.imbalance)
 s (loud.noise) ......0,5 0,5 1 0 ..... 2 
s (hig3h.temperature)   0 0 0 1   f3(bad.muffler) 
 
 4     f (thermostat.stuck)
s (steering.instability)  0 1 0,5 0  4
5
2.2 RULES AND FUZZY LOGICAL INFERENCE

The conception of fuzzy logic has as essential idea a multivalued world distinguishing itself
frontally with the Aristotelian logic (so-called classic logic), that is bivalued (true/false;
black/white; 0/1). The traditional logic considers two basic rules of inference, Modus Ponens
and Modus Tollens. Fuzzy logic uses the same inference structure, highlighting the propositions
with If-Then and considering A and B as fuzzy sets. The inference Modus Ponen is extended to
the Generalized Modus Ponen (GMP) in the form below:

Premise 1: x is A*
Premise 2 : IF x is A THEN y is B
Conclusion: y is B*

As noted by Mendel (1985, p.359), there is a subtle difference between the Modus Ponen and
GMP. In the latter, the fuzzy sets A* and B* are not necessarily the same as the fuzzy sets A and
B considered in the second premise. This feature allows approximated logical inferences very
common in practical applications in which information is imperfect (A* and B*) and cause-
effect relationship is known (Premise 2). Tsoukalas and Uhrig (1997, p.111) explained that the
inference procedure GMP as showed above is performed analytically by composing A* with the
implication relation R(x,y):

B* = A* ° R(x, y), where ° is a composition operator, as max-min
Rules are a form of logical proposition, consisting of a collection of IF-THEN expressions
that make up the knowledge base. In practical applications, the rules are obtained from experts
(through interviews, for example) or extracted directly from numerical data in appropriate
algorithms.

Rl : IF u is F l and u is F l and .....u is F lTHEN v is Gl
1 1 2 2 p p

Where: l = 1, 2, ..M; F and G are fuzzy subsets, u and v are linguistics variables.

An example presented by Shepherd and Shi (2006, p.419) concerning the application of
fuzzy framework in the labor market follows:

IF unemployment is low THEN real wage is high
IF unemployment is high THEN real wage is low
IF unemployment is moderate THEN real wage is normal

Where: low, high, moderate and normal are terms of the fuzzy linguistic variables
unemployment and real wages.

2.3 FUZZY EXPERT SYSTEMS

Fuzzy expert system, detailed in 1, is structured by four basics elements: a rule set; a
fuzzifier; an inference ‘machine’; and a defuzzifier. The rule set represents all the tacit
knowledge obtained from an expert. The rules are defined using the terms “IF x then y”, x and y
are defined as linguistic variable. The fuzzifier transform observed data and trigger the rules
accordingly. The inference machine makes fuzzy inferences, articulating the triggered rules. The
defuzzifier, transform and map fuzzy sets in crisp sets, resulting the effective control action
(MENDEL, 1995).
Source: MENDEL (1995)

FIGURE 1 – FUZZY SYSTEMS

Nowadays, there are many software packages that help analyst to conceive fuzzy
systems, as Matlab and Mapple.

3 KEYNES’S NOTION OF PROBABILITY AND ITS FUZZY ELEMENTS

The notion of probability in Keynes (1921) is radically distinct from the frequentist
view. It’s understood as the relation between two sets of propositions: a set of conclusions a and
a body of evidence h. Based on current knowledge (evidence), the probability α is the degree of
rational belief. In Keynes words:

“Let our premises consist of any set of propositions h and our conclusion
consist of any set of propositions a, then, if a knowledge of h justifies a
rational belief in a of degree α we say that there is a probability-relation of
degree α between a and h.” (KEYNES, 1921, p.4)

Considering the notations given by Keynes (1921), the probability-relation is
commonly expressed by a = a h, but another notation suggested by Keynes (1921, p.40) is
specially interesting because explicits the relationships between a and h, as fuzzy relation does: