chan - uncertainty in economics and the applications of fuzzy logic in contract laws

8/12/2019 Chan - Uncertainty in Economics and the Applications of Fuzzy Logic in Contract Laws

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TitleUncertainty in economics and the application of fuzzylogic in contract laws

Author(s) Chan, Wing-kin, Louis; sl8Pe

Citation

Issue Date 2003

URL http://hdl.handle.net/10722/56123

RightsThe author retains all proprietary rights, (such as patentrights) and the right to use in future works.


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U n c e r t a i n t y in E c o n o m i c s a n d t h e

A p p l i ca t i o n o f Fu zzy Lo g i c i n

C o n t r a c t L a w s

L o u i s C h a n W i n g K i n

August 18, 2003

A B S T R A C T

In this paper, I attempt to fuzzify the judicial process by realizing the fact

that some legal concepts are fuzzy in nature, meaning that they are not

som ethin g with clear-cut boun dary or definition. Under the currec t legal

system in force in Hong Kong - the common law system, judges, plaintiffs

and defendants process like defuzzifiers who take all the relevant fuzzy legal

concepts into consideration so as to come out with some legal decisions.

By utilizing the doctrine of precedent or

'stare decisis',

they can project the

expected duration and the result of litigation. I will show also the lawmakers'

intention in minimizing the possible fuzziness which is positively related to

the costs of information and how fuzziness can be compared and ranked

under different cases. Finally, I will conclude by showing that default rules

in contract laws are serving exactly the purpose of minimizing transaction

costs in contract formation by reducing the fuzziness in the judicial process.


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D E C L A R A T I O N

I declare that this thesis represents my own work, except where due ac

knowledge is made, and that it has not been previously included in a thesis,

dissertation or report submitted to this University or to any other institution

for a degree, diploma or other qualifications.

N a m e : C h a n W i n g K i n

U n iv ers i t y N o : 1 9 9 8 0 1 0 1 1 8

S ig na t ure :

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A C K N O W L E D G E M E N T S

First and foremost, I would like to thank Dr. Joe Chan, to whom I own my

biggest intellectual debt. This thesis could not have been completed without

his many insightful suggestions and criticisms and his unfailing support and

encouragem ent. He introduced me to many numerical techniques, got me

started on the empirical studies, and made useful comments on various parts

of the thesis.

I am also grateful to Mr. Alex Shing for kindly agreeing to act as an editor

for my thesis and generously sharing his time and academic experience and

human capital. Discussions with him, either formally or informally, inspired

me a lot. Any remaining errors are exclusively mine.

Special thanks are due to Mr. Steven Ho, Mr. Ivan Tse, Ms. Ng Ha Fung

for their useful and interesting comments.

Last but not least, I am thankful to my family and Ms. Eriko Chan for their

spir itual support.

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I n t r o d u c t i o n

Economics and law always work together, no matter from the delineation

and delimitation of property rights which forms the basis of any kinds of

excha nge, or to the litigation of breaches of con tracts in cou rts. Eco nom ists

assist lawmakers in justifying proposed rules and pursuits of efficiency and

welfare m axim ization . Lawm akers, on the other hand , help econo mists in

explaining and predicting human behaviour by forming the rules of the game

for different economic agents to play with.

However, the reality is too complex that many legal concepts cannot be

defined in an exact manner, or in other words, they are fuzzy in nature. In

this paper, I will attempt to fuzzify the judicial process by realizing the fact

that some legal concepts are indeed, fuzzy in nature, meaning that they are

not something with clear-cut boundary or definition. And see how this could

help the judges, plaintiffs and defendants decide their legal actions.

This article has 4 sections. Section 1, reviews briefly the development of the

analysis of economic behaviour under uncertainty. Section 2, introduces the

fuzzy set theory and its basic operations, more importantly, the differences

between the fuzzy set theory and the traditional classical set theory. Section

3, discusses the doc trine of preceden ts and the way it assists judg es, plaintiffs

and defendants in making decisions of resolving legal disputes. An applica

tion of the fuzzy set theory to the contract law will be presented through a

hypothetical example in which the concept of a 'valid' contract is re-defined

via the fuzzy set theory. Section 4, illustrates the functions of default rules

in reducing fuzziness and transaction cost in contract formation.

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1.

U n cer t a in t y in t he h i s t o ry o f eco n o m ic t ho u g ht

The concept of probability has been widely used in economics. It is the fun

damental building block in the mainstream analysis of economic behaviour

under uncertainty. Despite its importance and widespread use in economics,

the concept of probability retains an essential vagueness. Economists make

much use of the powerful logical calculus of probability but, beyond the ax

iomatic definition, there often lies considerable ambiguity as to what prob

ability represents empirically. Prob ability may be interp reted as a relative

frequency or as a degree of belief or something in between. Without a clear

empirical interpretation of probability it is unclear how economic theories

using the prob ability c alculus can be given any exp lanato ry significance. In

this section, I will briefly review the development of uncertainty in economic

thought .

T h e E x p e c t e d U t i l i t y H y p o t h e s i s

The expected utility hypothesis originates from Daniel Bernoulli 's (1738) so

lution to the St. Petersburg Paradox

2

in 1713 by Nicholas Bernou lli. Th e

Paradox challenges the old idea that people value random ventures according

to its expe cted retu rn. Th e Para dox is like th at : a fair coin will be tossed

until a head appears; if the first head appears on the nth toss, then the payoff

is 2

n

. The paradox is that the expected retun is infinite

References: M. Blaug 'Economic theory inretrospect ,5th ed., Cambridge University

Press, (1997) and 'The History ofEconomicThought Website',Retrieved 23 March

2003,

from http://homepage.newschool.edu/het.

2

Economists including Frank Knight and John Maynard Keynes utilized as illustration

this Paradox in their analysis of risk and uncertainty.

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http://homepage.newschool.edu/hethttp://homepage.newschool.edu/hethttp://homepage.newschool.edu/het


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Bernoulli 's solution involved two revolutionary ideas: the diminishing marginal

utility and the expected utility. He assumed diminishing marginal utility such

that people's utility from wealth, u(w) is not linearly related to wealth (w)

but rather increases at a decreasing rate, that is u'(w) > 0 and u (w) R is a utility function

over outcomes.


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A x i o m a t i z a t i o n o f t h e E x p e c t e d U t i l i t y H y p o t h e s i s

In 1944, John von Neumann and Oskar Morgenstern attempted to axiomatize

this expected utility hypothesis in terms of agent's preferences over lotteries

4

references. They started from the valuation of a compound lottery which is

a lottery that yields another lottery as prizes.

Let den ote an outcom e belonging to a set of outcom es, denote

the proba bility of outcom e occuring

Moreover, as the set of proba bility mea sures on X.

A particular lottery Z with two possible outcomes: where 50% probability, it

yields a ticket for another lottery A, while with another 50% probability, it

yields a ticket for a different lottery B, can be represented by

where A and B are the lotteries which serve as outcomes of the lottery, Z, the

agent is playing. In general, a compound lottery is a set of K simple lotteries

that are compounded by probabilit ies

so th at it indica tes lottery pk with probab ility .

Thus the compound lottery Z is of the form:

and it can even be reduced to a simple lottery

which can be interpreted as the probability Xi^X occuring.

They further agrued that if an agent has preferences defined over these lot

teries, the n the re exists a utility function th at assigns a util

ity to every lotte ry th at represents these preferences, and in othe r

4

Reference: 'Theory of Gam es and Econom ic Behaviour', Princeton Universi ty Press.

(1944)

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words, they argued that agents have preferences over lotteries but not over

the outcomes. However, if lotteries are merely distributions, it might not be

reasonable to say that an agent would 'prefer ' one particular distr ibution to

another

After the axiomatization of the expected utility theory, von Neumann and

Morgenstern in 1953 showed that it is possible to characterize agents' pref-

erences among risky alternatives under a set of specific axioms governing the

preference of the agents.

Suppose there is a lottery that pays x1 with probability p1 and x

2

with

prob ability 1 - P1. Let u(x) be the utility function and set the p rice of good

to $1. von Neumann and Morgenstern claimed that an agent's preference

among such a lottery can be expressed as the expected utility of consumption

th at the lottery yields, th at is, . And , in general,

this expected utility of the lottery E[u(x)] is not equal to the utility of the

expected value of the lottery u(E[x]). More precisely, if an agent perfers the

certain outcome of the expected value to the lottery, u(E[x]) > E[u(x)], then

he is said to be risk-averse. If an agent perfers the lo ttery to th e certain

outcome of the expected value, u(E[x]) < E[u(x)], then he is said to be risk-

loving. And if an agent is indifferent between the lotte ry and th e ce rtain

outcome of the expected value, u(E[x]) = E[u(x)], then he is said to be risk-

neutral .

In fact, von Neumann and Morgenstern have followed the classical view that

randomness and probabilit ies 'exist ' inherently in Nature since in their anal

ysis probabilities are assumed to be 'objective' or exogenously given by "Na-

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ture"

and therefore cannot be altered by the actions of the agen ts. Underlying

the classical view of unc ertain ty is the 'principle of non-sufficient rea son'

5

',

which states that, equal probabilities must be assigned to each of serveral

outcomes if there is an absence of positive ground for assigning unequal ones.

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Keynes has presented his criticism on this classical notion in chapter 4 of this 'Treatise

on Probability' whe re he called it the 'principle of indifference.'

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T h e R e l a t i v e F r e q u e n c y A p p r o a c h

Due to the deficiencies in the above classical view, Richard von Mises has

developed in 1928, ano ther para digm , namely, the 'relative frequency'. Th e

relative frequentists claim that if an event occurs k times in n identical and

independent trials, provided that the number of trails is arbitrarily large,

then k/n will represent as the 'objective' probability of that event

6

. How

ever, this relative frequentist approach fails to address events which are 'high

degree unique' such as the outcome of the World War III, for example.

The R ev ea led Be l i e f A ppro a ch

Many statisticians and philosophers have long argued that randomness is

not an objectively measurable phenonomenon but rather a 'knowledge' phe

nomena, therefore probabilities are indeed as 'epistemological' issue. By this

view, a coin toss is not necessarily characterized by randomness: if one knew

the shape and weight of the coin, the strength of the tosser, the atmospheric

cond ition of the environm ent in which the coin is tossed, e t c . , one could

predict if it would be heads or tails with certainty. But, as this information

is com mo nly unavailable or is too expensive to acquire, it is convenient to as

sume it is a random event and attach probabilities to heads and tails. In this

sense, probabilities are actually a measure of the lack of knowledge about the

conditions which might affect the outcome of an event, and in other words,

6

It should be noted that this view is, in some sense, closely related to the "law of large

numbers" set out by Jacob Bernoull i in 1713. It states that i f an event occurs k t imes in n

identical and indep enden t t rials, provided th at th e number of t rai ls is arbi trari ly large, th en

k/n would be arbi trari ly close to the 'object ive ' probabil i ty which exists independently, of

that event .

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a measure of our beliefs about the event.

Frank P. Ramsey (1926) disagreed on this 'objective' view of probability and

claimed that it is the personal belief that governs probabilities but not the

'know ledge '. However, this "sub jectivist" a pproach makes it impossible to

build a predictive theory of behaviour under uncertainty since if assigned

probabilities are assumed to be subjective, the randomness itself will also

be a subjective phenomeno n. In the 1926's paper, he attem pte d to derive

a theory of behav iour under uncertain ty with subjective probabilities. Th e

basic idea is that subjective probabilities can be inferred from observation

of agents' actions in a random venture in which the agents share the same

knowledge. For instance, suppose an agent faces a gamble with two possible

outcomes, x and y, where x>y, and suppose further that he faces a choice

between two lotteries p and q defined over x and y. In the outset, we do not

know what p and q are composed of, but if an agent chooses p over q, then

we can deduce that he must believe that p assigns a greater probability to

x relative to y, or vice versa.

7

This 'revealed belief approach, however, has

been criticised on its empiricial applicability since a belief cannot be known

in advanc e un less we observe the choice of the agents. Some econom ists such

as B. O. Koopman pointed out that subjective probabilities are not necessar

ily revealed through choice, and even it is the case, they are usually revealed

through intervals rather than some single numerical measures. This view of

probability is sometimes called the 'intuitionists' which held that probabili

ties are directly derived from intuition prior to any experiment.

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It is similar to the idea of the "revealed preference" in modern day microeconomics.

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T h e S t a t e - P r e f e r e n c e A p p r o a c h

Kenneth J. Arrow and Gerard Debreu held this 'intuitionist' view and devel

oped the "state-preference" approach which has since become the dominant

method of incorporating uncertainty in general equilibrium models where

payoffs are not merely money but actual bundles of goods.

The main idea of the "state-preference" approach is that it reduces choices

under uncertainty to a conventional choice problem by altering the commod

ity stru ctu re. Th e state-preference approach assumed th at preferences are

formed over bundles of state-contingent commodities. The basic proposition

of the state-preference approach is that commodities can be differentiated by

their "state s of na tu re" . It means th at two identical goods will be trea ted

differently and command different prices if they exist under different states.

More precisely, with a set (S) of mutully exclusive "state of nature (s;)",

where S;S V i= l. ..n , one can index every comm odity by the stat e of na ture

when it is delivered and thus form a set of "state-contingent" markets.

This approach has been extensively applied to insurance industry since the

insurance contract is highly 'state-contingent' in the sense that it pays the

insured inde mn ities when a insured event occurs. Th e simplest model is

a two-state model with a fixed insurance premium per dollar of coverage,

7 .

Th e set of sta tes is where H is a sta te when an accident

hap pens , and NH is a state when no accidents happens. Let endowed income

be where U H is the wealth when an accident happens and

the otherw ise, moreover, such th at the agent will suffer a

loss when an accident happe ns. Assuming there exists a s tate-indepen dent

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utility function over payoffs as follow:

this gives the expected utility of the agent with the initial endowment, where

and denote the subjective probability tha t an accident will hap pen

and the otherwise respectively.

A "fair" insuranc e contra ct can then be formed as where

denotes the insurance premium when no accident happen s and denotes the

indemnity net of the premium when an accident happens. As a result, if an

agent purcha ses the insurance con tract, then her expec ted utility

will be as follows:

It is worth noting th at , bo th are not con stants , but variables and

that depend on the agent's choice of the amount of insurance coverage.

Assuming th at is the insurance premium per dollar

of coverage. Th e idem nity net of prem ium , will be equal to

that is the amount of coverage minus the insurance premium.

The expected profit of the insurance company will be:

Und er perfect com petitio n, this expec ted profit, IT will be equal to zero.

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After rearranging the terms:

(1)

the ratio of insurance payment to the net indemnity can be shown to be equal

to the subjective odds of the accident.

By sub stitutin g equation (1) becomes:

and it implies th at In other words, it implies th at the insuranc e

premium per dollar of coverage is equal to the subjective probability of the

accident.

The maximization problem facing the agent will become as follows:

(3)

whe re the ob jective of the age nt is to choose the am oun t of insurance coverage

C, given the fixed insurance premium per dollar of coverage, 7.

The first order condition of this maximization:

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2)


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The solution of this maximization problem can be rearranged to:

given th at the insuranc e is 'fair ', th at is equa tion (5) can be reduced

further to:

6)

it implies

This means that an agent facing a fair insurance scheme will choose to be

fully insured against the accident such that the entire loss from the accident

will be recovered. However, this result ap plies only for limiting cases in which

the condition for fair insurance, 7 = 7r

s

, holds. Whenever this condition fails

to hold, the result will be different, and the agent might not choose to be fully

insured against the accident, but rather depending on her atti tude towards

risk.

The development of the theory of choice under uncertainty after the Second

World War was a success story resting 'on solid axiomatic foundations ...

[with] important breakthroughs in the analytic of risk, risk aversion and

their application to economic issues'

8

. It reflects the mainstream view that

the concept of 'uncertainty' has been closedly related to that of probabilistic

risk.

8

Journalof EconomicPerspectives,Vol. 1, (1987) p.121

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(5)


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In fact, Adam Smith was the first classical economist to use the word un

certainty

interchan geab ly with proba bility. In his discussion of wage

dif-

ferentials in 'the liberal professions' such as lawyers, Smith (1937: pp.106)

indicates that by uncertain he meant the 'probability or improbability of

success'. Alfred Marshall (1961: pp.l35n), on the other hand, agreed that

given the law of diminishing utility, 'gambling is, in the long run, a sure way

to lose utility' for the marginal utility of gaining 100 pounds was less than

the m arginal util i ty of losing pound s. There was nothing uncertain abo ut

the long-run ou tcom e of gamb ling - only probabilistic risk. In the case of

an equal probability of gain or loss, Marshall indicated that the probabilistic

outcomes could be compared to 'certain ' expectations. Therefore, Marshall

did not assign a major role to uncertainty in his analysis.

Beginning in the twentieth century, however, non-mainstream economists in

cluding Frank Knight and John Maynard Keynes and the Post Keynesians,

based their analysis on an explicit distinction between the concept of uncer

tainty and probabilistic risk.

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certainty, is that in the former the distribution of the outcome

in a group of instances is known (either through calculation a

priori or from statistics of past experience, while in the case of

uncertainty this is not true, the reason being in general that it

is impossible to form a group of instances, because the situation

dealt with is in a high degree unique.

In other words, Knight believed that the element of risk in a venture can be

estimated by utilizing objective probabilities, whereas uncertainty cannot be

objectively determined, but only inferred from personal (subjective) experi

ences,

observed (vague) outcomes and imperfect (approximate) knowledge,

with varying degrees of ambiguity and subjectivity.

As a result, since business decisions, to a large exte nt, hinge on vague and in

complete knowledge of all existing information while machines requires only

exactness and objectivity; the functions of gathering economic resources and

assuming risk can then be assigned to a mechanical device, while the de

cision making under uncertainty to an entrepreneur. The entreprenuer not

only surpasses a calculating machine in analyzing incomplete data, but fur

ther excels over machines in predicting future outcomes involving the firm

and the market. Without possessing exact knowledge, the entrepreneur tr ies

to consider each relevant and possible future element along with their ef

fects upon each other. Sometimes, the entrepreneur may need to rely on her

'feeling' or 'intuition'. This is an extension and result of the entrepreneur's

knowledge, expertise, individual characteristics, and the social and psycho

logical factors which directly or indirectly impact the entrepreneur's envi

ronment. These factors form the basis of the decision making process under

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uncertainty. Thus, the entrepreneur can be regarded as an organizer of the

production factors and as the predictor and bearer of risks and uncertainties

associated with her business venture.

This inclusion of uncertainty as a fundamental part of the entrepreneurial

decision making process was also formalized by Frank Knight (1921):

The adventurer has an opinion as to the outcome, within more

or less narrow limits. If he is inclined to make the venture, this

opinion is either in expectation of a certain definite gain or a belief

in the real probability of a larger one. Outside the limits of the

anticipation any other result becomes more and more improbable

in his mind as the amount thought of diverges either eay.

Knight associated risk with either frequentist (statistical) or Bayesian prob

abilities. Un certain ty was associated only with unique events. To Knight,

an uncertain future is the basis of the existence of business profits.

For Keynes, on the other hand, uncertainty involves situations where decision

makers believe that no relevant probabilities exist today that can be used as

a basis for scientifically predic ting future events . As Keyn es ind ica ted , by

uncertainty he did

not mean merely to distinguish what is known for certain from

what is only probable. The game of roulette is not subject, in this

sense to uncert ainty ... Th e sense in which I am using the term s is

t h a t. .. there is no scientific basis on which to form any calculable

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ratio na l to hold in a proposition on the b asis of the available evidence. To

Keynes, probabilities are a property of the beliefs which agents hold about

the world. In con trast to the relative frequency ap proach in which proba

bilities are viewed as long-run relative frequencies that are a property of the

world itself. Moreover, unlike the Bayesians, Keynes treated proabilities as

objective, rather than subjective. More importantly, Keynes permits human

'freewill ' to cre ate future outcom es or future state s of the world in his analysis

which is truly a revolutionary way of modelling the entrepreneurial market

system which is logically inconsistent with the axioms underlying classical

theories in which the future is exogenously chosen by the 'Nature'.

To Keynes, uncertainty arises when there is more than one hypothesis enter

ing into expectation. And he argued that economic expectations are 'objec

t ive ' in the sense that given the same knowledge, different individuals will

have the same belief about a proposition

10

; and 'subjective' in the sense that

different individuals might have different information sets

(knowledge)

which

authorize them to entertain different beliefs about a proposition.

11

"In the sense important to logic, probability is not subjective. A

proposition is not probable because we think it so. When once the

facts are given which determine our knowledge, what is probable

or improbable in those circumstances has been fixed objectively,

and is indep ende ntly of our opinion." (TP , p.4)

1 0

It is sometimes called the "Harsanyi Doctrine" or "common prior" assumption.

It is different from the rational expectation hypothesis in which both the agents and

the government authorities know the (same) underlying economic system.

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In fact, it is very similar to the idea of fuzzy logic

12

in modern decision the

ory. T he origin al idea of fuzzy logic is at tri bu te d to Lotfi Zad eh in 1965.

The main idea lies in the concept of fuzzy sets which are defined by specific

me mb ership functions. Let X denote a universal set, and \XA the m embership

function by which th e fuzzy set A is defined. St ate d in can onica l form:

The sum of the membership grades is not necessarily one. The membership

function does not describe a probability (random) distribution. It describes a

possibility (non-random, subjective) distribution. An occurrence is possible

does not mean it is probable, however, if an element is impossible then it is

also improbable.

Those different hypothesis can be seen as different elements belonging to

different crisp sets, through assigning each of the hypothesis with different

membership values, by the fuzzy set theory, a newly-defined fuzzy set is then

formed to take all of them into consideration. An entrepren eur who ente rtain s

various hypothesis over the expectation on next year's interest rate, can form

a fuzzy set describing for example, 'approximately 10 per cent', so as to

take into account all possible situations authorized by his own knowledge

or inform ation. Obviously, this fuzzy set contains a subjective evaluation

on how the interest rate will change which are based on the entrepreneur's

own knowledge or information. Bu t, by so doing, one can formalize tho se

exp ectatio ns in a more realistic ma nner. Undoubtedly, Keynes recognizes

12

More on fuzzy logic and fuzzy sets theory will be discussed in the next section.

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the idea of fuzzy thinking in human decision-making process.

In choosing between alternative actions, Keynes argued that the decision

maker must take into consideration not only the probabilit ies attached to

each of the different possible outcomes but also the 'goodness' of each out

come. The mathematical expectation of an action is defined as follows. Let

A represent the degree of 'goodness' of some action. The probability, p, is

the rational degree of belief that the degree of'goodness', A , will be attached

if the action is chosen. The m athem atical expec tation,

E,

of the action is

defined as:

E = pA

by which an agent will choose an action to maximize this mathematical

expectat ion.

Though Keynes agreed that the mathematical expectation had recognized

both the probability and 'goodness' of an action, he was not fully satisfied

with this form of exp ecta tion. He presented four specific criticisms a gains t

the mathematical expectat ion.

Keynes's f irst cr it icism on the mathematcial expectation is that i t assumes

that the 'goodness' of each outcome is numerically measurable and arith

metically ad ditive. His second crit icism on mathem atical e xpecta tion is i ts

requirement that probabilit ies are numberically measurable, in TP, however

these num erical proba bilities are considered as jus t a small subset of all possi

ble probabilities. Keynes argued that numerical probabilities had been given

undue attention only because of their potential for mathematical manipu-

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may either increase or decrease, according to the new knowledge strengthens

the unfavourable or the favourable evidence; but something seems to have

increased in either case, - we have a more substantial basis upon which to

rest our conclusion. One may express this by saying tha t an accession of

new evidence increases the weightof an argument. New evidence will some

times decrease the probability of an argument, but it will always increase its

weight.' (TP , p.77) Whenever the number of completing hypothesis enter

ing into expectation formation is reduced, the weight is increased, and by

Keynes's definition, uncertainty is also then reduced.

His final criticism is th at mathem atical expectation does not take any accoun t

of the 'risk' a ttached to the choice of any action. Keynes's concept of risk,

R, is defined as follows:

R = p(A - E)

= p(l - p)A

= pqA

= qE

where q= 1-p. Keynes defined risk, R = qE. Keynes interpreted the mathe

matical expectation, E,as measuring the net immediate sacrifice required in

the hope of gaining the payoff A . In other words,E is the maximum amount

that a risk-neutral agent would be willing to pay for a gamble, (A ,0;p, 1-p),

that is payoff A with probability, p, and zero payoff with 1-p, i.e. payoff

A with probability, p, and zero payoff with 1-p. Given that q represents

the probability that the sacrifice is made in vain, it follows that Keynes de

fined risk as the mathematical expectation of the loss attached to the action.

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Keynes's notion of risk is a recognition that an agent makes choice not only

depends on expected gain but also expected loss. Most essentially, Keynes

argue d th a t high risk acts as a dete rrent to action. As the risk associate d

with an action increases, the desirability of that action falls,

ceteris paribus.

Keynes illustrated his argument for the importance of risk with reference to

the St. Petersburg paradox (TP, p.349-352): in which Peter engages to pay

Paul one shillings if a head appears at the first toss of a coin, two shillings if

it does not ap pe ar until the second, and in general, shillings if no head

appears under the rth toss. Mathematically, the value of Paul 's expectation

is if th e num ber of tosses is not in any case to exceed

n

in all,

and if this restriction is removed. It follows th at , Pa ul should

pay shillings in th e first case, and an infinite sum in th e second . N oth ing ,

it is said, could be more paradoxical, and no sane Paul would engage on

these terms even with an honest Peter . The ma them atical expe ctation of

this game is infinite yet it is reasonable that Paul is only willingg to pay a

small stake to play the game.

In recent decades, mainstream economics has associated uncertainty either

with situations where decision-makers possess information regarding their

explict (objective) probabilities or agents form Bayesian subjective probabil

ities.

Most New Classical and New Keynesian mo dels assume th e existence of

objective probability distribution functions that represent an external reality.

Th e 'N at ur e' will dete rmin e the future sta te of the world th at w ill exist. It

follows that, therefore, that society cannot alter this external reality. Agent's

have no 'free will' to alter the ir long-run econom ic future . (Lawso n, 1988)

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T h e S u b j e c t i v e P r o b a b i l i t y T h e o r i e s : T h e R a t i o n a l E x p e c t a t i o n s

H y p o t h e s i s

Subjective probability theories imply that although an objective probability

reality exists, decision-makers toda y do not possess sufficient me ntal capa city

to 'know' it. Agents can order an exhaustive list of all possible outcomes by

subjective probabilities such that the sum of all these probabilities equals to

one. In the short ru n, subjective probab ilities can be a type of knowledge

th at need not ma tch the external reality th at is presumed t o exist. In the

long run, however, subjective probabilities tend to coverge with objective

probabilities that are a property of an external and unamendable reality. In

the long run, rational agents will make optimal choices. This viewpoint of

probability forms the basis of the rational expectations hypothesis (REH).

The rational expectations hypothesis has been playing an important rule in

mo dern economic litera ture. Th e rationa l expe ctation hypo thesis specifies

that both the agents and the government authorities know the 'same' under

lying economic system. It is a technical principle of model co nstruction which

assures nothing more than consistency between an endogenous mechanism

of expectations formation and general equilibrium.

John Muth's hypothesis of rational expectations is a technical

model-building principle, not a distinct, comprehensive macroe-

conomic theory. Recent research utilizing this principle has rein

forced many of the policy recommendations of Milton Friedman

and other postwar monetarists but has contributed few, if any,

orginal policy proposals. My own research has been concerned al-

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most exclusively with the attempt to discover a useful theoretical

explanation of business cycles.

13

Robert Lucas and other proponents of rational expectations hypothesis be

lieve that the most important and reasonable justification for rational ex

pectations is that it is the only hypothesis of expectations formation which

is compatible with the principles of general equilibrium, as it aspires to be

rigorously based on the maximization of utility and profits. It proves indis

pensable to extend these principles to the process of expectation formation,

assuming that information, which is a scarce resource, is used in an efficient

way.

The argument works, but all we can say is that economics agents will not

consciously commit ex ante errors of prediction. Similarly, it is und oub t

edly correct to assert that if economic agents realize

ex post

that they have

committed errors of prediction they will try to correct them, but is not for

certain that the learning process must rapidly converge towards an equilib

rium. The equilibrium identified by the hypothesis of rational expectations

should therefore be considered as a tran sitory equilibrium only. Yet, this

point of view is not compatible with the equilibrium method used by Lu

cas who utilizes the substantive version of rational expectation which implies

that the 'environment' remains rigidly beyond the reach of any action of con

trol or transformation on the part of the economic agents. The environment,

in fact, is defined as the whole complex of variables over which economic

agents have no control, but which influences their decisions. An exception is

1 3

Lucas, R.E. , Jr and Sargent , T.J. , eds. , 1981,Rational Expectations and Econom etric

Practice, p. 1-2

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made only for the authorities who are endowed with the power to modify the

rules of economic policy, which obviously constitute an essential part of the

environment. Therefore a rational agent never makes systematic mistakes

14

,

not only ex ante but also ex post.

15

Nev ertheless, since information is scarce and can be acqu ired only at a cost, it

would be important to know the specific nature of the cost function in order

to see whether rational expectation emerges as the solution of a problem of

con straine d ma xim ization. Th is result should be considered very unlikely,

however. Moreover, we are not sure if economic agents man age to avoid

systematic

ex post

errors, that depends on the quality and quantity of the

existing information, and on procedures for handling that information

16

.

14

Obviously, Keynes's theory of liquidity preference is inconsistent with the rational

expectations hypothesise, since the underlying substantive rationality refuses to attribute

any economic value to strategic learning.

15

This implies tha t a ration al agent has no economic incentives to avoid syste matic

mistakes: strategic learning, which aims to avoid systematic mistakes in order to discover

a strategy more profitable than that adopted so far, would be deprived of any economic

value and would become unintelligible.

16

Frydman and Phelps, eds., 1983

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2 . The F uzzy Se t Theo ry

The term

fuzzy

in th e sense used in thi s pape r was first intro du ced in 1962

by Zadeh

17

in a paper concerned with the transition from circuit theory to

system theory in which he called for a "mathematics of fuzzy or cloudy quan

tities which are not describable in term s of probab ility distribu tion s". Th is

revolutionary paper was followed three years later by technical exposition of

just such a mathematics now termed the theory offuzzy sets.

18

Much of the decision-making in the real world takes places in

an environment in which the goals, the constraints and the con

sequen ces of possible actions are not known precisely. To deal

quantitatively with imprecision, we usually employ the concepts

and techniques of probability theory and, more particularly, the

tools provided by decision theory, control theory and information

theory. In so doing, we are tacitly accepting the premise that im

precision - whatever its nature - can be equated with randomness.

This, in our view, is a questionable assumption. Specifically, our

contention is that there is a need for differentiation between ran

domness and fuzziness, with the latte r being a major source of

impre cision in ma ny decision processes. By fuzziness, we me an

a type of imprecision which is associated with

fuzzy sets,

that

is , classes in which there is no sharp transition from member

ship to nonm em bersh ip. For exam ple, the class of green objects

is a fuzzy set. So are the classes of objects characterized by such

17

L. A. Zadeh, From Circuit Theory to System Theory", Proceedingsof the Institute of

Radio E ngineers50 (1962) 856-865.

18

L.A. Zadeh, Fuzzy Sets",Information and Control8 (1979) 509-534.

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commonly used adjectives as large, small, significant, important,

serious, simple, acc urate , app roxim ate, etc. Actually, in sha rp

contrast to the notion of a class or a set in mathematics, most of

the classes in the real world do not have crisp boundaries which

separate those objects which belong to a class from those which

do not. In this connection, i t is imp ortan t to note tha t, in the

discourse between humans, fuzzy statements uch as "John is

sev

eral inches taller than Jim," x is much larger than y," "the stock

market has suffered a sharp decline convey information despite

the imprecision of the italicized words

19

.

It is worth emphasizing that it is not a statement implying that probability

theory itself is wrong - it suggests only that there are forms of uncertainty

where the probability theory may give an inappropriate representation. The

point is that in the decision process under uncertainty, certain forms of im

precision occur that are intrinsic to the problem and for which the probability

calculus is inad equ ate. Bellman a nd Zadeh give a concise abstra ct classifi

cation of these forms of imprecision in terms of "classes in which there is no

sharp transition from membership to nonmembership".

Gaines

20

in his 1981's paper has given an example of a planning situation

where the role of imprecise statements as very accurate representations of

information is apparent:

1 9

R. E. Bellman and L. A. Zadeh "Decision-making in a Fuzzy Environment" Manage

ment Science 17 B141-B142 (1970)

2 0

B . R. Gaines, "Logical Foundations for Database Systems", in: E.H. Mamdani and B.

R. eds. , Fuzzy reasoning and Its Appications pp.289-308. Academic Press, Londo n. (1981)

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unwarranted precision can itself be highly misleading since ac

tions may be taken based on it - "we will deliver 7 parcels each

weighing 15.2 Kilograms at the rear entrance of building 6A on

15th February at 9.03 p.m.", "we will deliver some heavy equip

ment to your site Saturday evening", and "see you with the goods

over the weekend", may each refer to the same event but are

clearly not interchangeable, i.e., each conveys an exact meaning

th at (presum ably) prop erly represents what is to occur. If we

prefer the precision of the first statement it is not for its own

sake but because the tighter tolerances it implies on the actual

situation allow us to plan ahead with greater accuracy and less

use of resources. However, if the th ird sta tem ent really represen ts

all that can be said it would be ridiculous to replace it with ei

ther of the previous ones. It would be equally ridiculous to say

nothing. However, even the least precise of the three statements

does provide a basis for plann ing and ac tion. A key aspe ct of

executive action is planning under uncertainty and normal lan

guage provides a means for imprecision to be clearly and exactly

expressed (Gaines [2, p.303])

It is a very precise representation of the situations where people operate and

make decisions in the real world. Any decision theory must also be able to

represent adequately so as to explain and predict agents' behaviour under

real world setting.

Again it is worth noting that this statement is totally independent of any

requirement for precision in the development of science - it does not in itself

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oppose to the attempts of modelling any decision process as precisely as

possible. As Karl Pop per ha s noted in the contex t of the philosophy of

science:

both precision and certainty are false ideals. They are impos

sible to attain, and therefore dangerously misleading if they are

uncritically accepted as guides. The quest for precision is analo

gous to the quest for certainty, and both should be abandoned. I

do not suggest, of course, that an increase in the precision of say,

a prediction, or even a formulation, may not sometimes be highly

desirable. W ha t I do suggest is th at it is always undes irable to

make an effort to increase precision for its own sake especially

linguistic precision - since this usually leads to lack of clarity, and

to a waste of time and effort on preliminaries which often turn

to be useless, because they are bypassed by the real advance of

the subje ct: one should never try to be more precise tha n the

problem situation demands (Popper 7, p.

17).

Most of the applications of fuzzy set theory in decision analysis arises through

the interpretation of some forms of imprecise statement as placing a

'possi

bilistic restriction'

on the class of events which satisfy t ha t state m en t. Th is

restriction is then represented through a set with graded membership such

that any event has a 'degree of membership' in the set defining the ex ten t

to which it is consistent w ith the possibilistic restriction. It represents the

human decision processes more closely than the classical set theory in which

only black-or-white logic is allowed.

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Opponents to the fuzzy set theory, however argue that the imprecision in

volved is not essential and does not require formal represe ntation . Moreover,

they suspect the need for fuzzy set theory in decision analysis since they

believed that we already have tools to deal with uncertainty and impreci

sion, namely, the proba bility calculus. However, exam ples can be given to

show that the conventional interpretation of probability theory in terms of

likelihoods or frequencies is not app ropria te to th e kinds of imprecision exem

plified above. The term

green

defines a fuzzy set of objects not because the

colour of any one of them varies each time it is examined but because there is

reasonable doubt about whether a borderline case belongs to the set or not.

The parcel delivery example above gives three definitions of the nature of the

event which are increasingly fuzzy only in allowing the deliverer greater and

gre ate r freedom in the class of actio ns which satisfy h is definition. W ha t is

defined is not thep robability of an event occurring but the range of 'possible'

events that may occur.

There appears a need for differentiation between randomness and fuzziness,

with the latter being a major source of imprecision in many decision pro

cesses.

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2 . 1 R a ndo m ness V ersus F uzz ines s

Ran dom ness, has to do with uncertainty concerning memb ership or nonmem-

bership of an object in a non-fuzzy set. It has to do with sets in which there

is a sharp transition from membership to nonmembership, where the grades

of membership can takes on, zero or unity, two values only.

Fuzziness, on the othe r han d, is a type of imprecision which is associated with

fuzzy sets. It has to do with sets in which there is no sharp transition from

mem bership to nonmem bership, but have grades of mem bership interm ediate

between these two extreme situations. For example, the class ofgreen objects

is a fuzzy set. So are the classes of objects characterized by such commonly

used adjectives as large, small, substantial, significant, important, serious,

simple, accurate, approximate, etc. In fact,, in sharp contrast to the notion

of a class or a set in mat hem atic s, m ost of the classes in the real world do not

have crisp boundaries which separate those objects which belong to a class

from those which do not.

To illustrate the difference, the fuzzy statement "Investing in Company A

will give you and your family a handsome reward", is imprecise by virtue

of the fuzziness of the term s "ha ndsom e rewa rd". On ther other han d, th e

sta tem en t "The probab ility tha t Com pany is ope rating at a profit is 0.9" is

a measure of the uncertainty concerning the membership of Company A in

the non-fuzzy set of companies which are operating at a profit.

In fact, the mathematical techniques for dealing with fuzziness are quite dif-

ferent from those of classical probability theory. They are simpler in many

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aspects because of the fact that the notion of probability measure in prob-

abilty theory corresponds to the simpler notion of membership function in

the th eory of fuzziness. Furth erm ore, the corresponde nts of a + b and ab ,

where a and b are real numbers, become the simpler operation Max(a, b)

and Min(a, b). For this reason, even in those cases in which fuzziness in a

decision process can be simulated by a probabilistic model, it is generally

advantageous to deal with it through the techniques provided by the fuzzy

set theory rather than through the employment of the conceptual framework

of probability theory.

2 .2 Bas ic Fuzzy Set Theory and i t s Operat ions

Conversation contains many vague words from everyday gossip as "The

weather is

hot

to an economist's statement that "The economic perfor

mance of Hong Kong will become better in the coming years." Fuzzy sets

were proposed to deal with such vague words and expressions. Fuzzy sets can

handle such vague concepts as "a set of good stud ents " an d "people living

close to the poverty line," which are unable to be expressed by conventional

set theory. Th e words "good" and "close" give ambiguou s ideas. The se

vague expressions are not allowed in conventional set theory and one has to

define terms exactly like "the set of students whose G.P.A.s are higher than

3.8,"

or "the set of people whose average monthly family incomes are lower

th an $4,000." A calculation of a stu de nt's G.P.A. will show if the stud ent

belongs to the former set; and a calculation of a family's monthly income

will show if th at family belongs to the la tter set, for exam ple. These con

ventional sets, which are defined exactly, are called "crisp sets" in fuzzy set

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theory. The question raised above involves the representation of soft and im

precise information . Zade h

21

introduced the concept of a fuzzy set in order

to quantitatively represent such information.

In the following sub-sections, I will first present the crisp set theory and

the n I will introdu ce the fuzzy set theory and its various ope ration s. Th e

applications of the fuzzy set theory to the economics of (contract) law will

be discussed in the next section.

2 1

L . A. Zadeh, 'Fuzzy Sets ' , Information and Control 8 (1965), p.338-353

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2 . 2 . 1 C r i sp Se t s a nd C ha ra ct er i s t i c F unct io ns

Let X denote a universal set and X^ denote a characteristic function by which

the crisp set A is defined. Th e cha racte ristic function X can be defined by

a mapping, stated in canonical form:

It indica tes th at if the eleme nt x belongs to A, X is 1, and if it does not

belong to A, X^ is 0.

In crisp set theory, union, intersection, and complement are defined as follows.

Cr i s p Se t s

Union of crisp sets A and

Intersection of crisp sets A and

Com plem ent of crisp sets A and

The natural operations on sets, such as the union and intersection, are also

readily defined and White

2 2

has shown that the definitions used by Zadeh

23

2 2

R. B. W hite "The Consistency of the Axiom of Comp rehension in the Infinite-valued

Predicate Logic of Lukasiewicz", Journal of Philosophical Log ic 8 pp.509-534 (1979)

2 3

L .

A. Zadeh, 'Fuzzy Sets ' , Information and Control 8 (1965), p.338-353

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led to a set theory in which the axiom of comprehension, that every predi

cat e defines a set, is consistent and d oes not lead to the paradox es in classical

set theo ry. Indee d, at a former level the re are close links betwe en classical

probability theory and fuzzy set theory.

2 . 2 . 2 F uzzy Se t s a nd Mem bersh ip F unct io ns

Let X den ote a universal set and deno te a me mb ership function by which

th e fuzzy set A is defined. Th e me m bersh ip function can be defined by a

mapping, stated in canonical form:

Th e value of for the fuzzy set A is called the m em bers hip value or the

grade of me mb ership of The mem bership value represents the degree

of x belonging to the fuzzy set A.

It indicate s the sum of the me mb ership grades is not necessarily one. Th e

membership function does not describe a probability (random) distr ibution.

It describes a possibility (non-random , subjective) distributio n. An occur

rence is possible does not mean it is probable, however, if an element is

impossible then it is also improbable.

In fuzzy set theory, union, intersection, and complement are defined as fol

lows.

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Fuzzy Sets

Union of fuzzy sets A and

Inters ectio n of fuzzy sets A and

Co mp lem ent of fuzzy sets A and

Union of fuzzy sets A and B: union AuB of fuzzy sets A and B is a fuzzy set

denned by the membership function:

Intersection of fuzzy sets A and B: intersection AnB of fuzzy sets A and B

is a fuzzy set defined by the membership function:

Co m plem ent of fuzzy set A: com plem ent of fuzzy set A is a fuzzy set

defined by the membership function:

The value of the characteristic functions for crisp sets defined above was ei

ther 0 or 1 but the m emb ership value of a fuzzy set can be an arb itrary real

value between 0 and 1 as indicated by the mem bership function above. Th e

closer the value of to 1, the higher the grade of me mb ersh ip of the

elem ent x in fuzzy set A. If the eleme nt x com pletely belong s to

th e fuzzy set A. If , the eleme nt x does not belon g to A at all.

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2 . 2 . 4 The use of

For a fuzzy set A we can define the following

Strong

Weak

T he is a significant conc ept in fuzzy set theory , the are chosen

and assigned arb itrarily by the decision maker. Such assignm ents are typ

ically selected from the membership grades of the elements in set A. But,

the decision-maker can in fact, choose any real number from zero to one as

an In othe r words, the decision maker may decide th at all eleme nts

with membership grades less than or equal to any number between 0 and 1

are insignificant. In other words, some of the may not be in the set A.

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2 . 2 . 5 Ex t ens io n P r inc ip l e

Extend mapping / :

X >

Y to relate fuzzy set A on X to fuzzy set B on Y:

7)

The extension principle simple extends operations from a fuzzy set A to f(A)

as follows:

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in the structure, while other lower courts such as Magistrates' Court binds

no other court. Most cases of authority, therefore, will originate from either

the Court of the Final Appeal or the Hong Kong Court of Appeal.

Under the common law legal system, the main development of contract law

has been thro ugh th e process of precede nt. Its principles are established

by case law. With the doctrine of precedent, judges found the pre-existing

principle and applied it to the new facts brought before him. And, in theory,

a jud ge can not c reate new law bu t must ap ply old law to new facts. W henever

a new pro blem of law for which there is no pre-existing custom ary or com mon

law principle comes before the courts to be decided upon, the judge makes

a ruling which must subsequently be followed by all other judges. In other

words, a decision made by a court is binding on other courts in later cases

where the facts are similar. As a consequence, the common law with those

precedents gradually became predictable and could be applied to new cases

with a degree of certainty.

Therefore, the precedents and cases form the basis by which the plaintiff

and th e defendant can predict the result of their litigation. On the oth er

hand, these precedents enable the judges, plaintiffs and defendants to make

judgment in a more objective manner facing the subjective and highly unique

evidences in different cases. In fact, "judge-made" law is a major source of

law, by quantity.

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3 .1 Ap pl ic at io n in Law: A Via /id C ontra ct Red ef ined Via Fuzz y

Set s

A simple 'valid' contract can be broadly defined as an agreement between

two or more parties that is binding in law. This means that the agreement

generates rights and obligations that may be enforced in the courts. There are

many ways in which the essential structure of a contract can be analyzed.

One of the most common is to see a contract as consisting of three basic

elements: (1) offer and acceptance, (2) consideration and (3) intention to be

legally bound.

The three basic elements in the formation of a valid simple contract.

(1 . ) Of fer and A cce pta nc e

By offer and acceptance, it means the contracting parties must have reached

agre em ent with each othe r. An offer may be defined as a sta tem en t of willing

ness to contract on specified terms made with the intention that, if accepted,

it shall become a binding co ntract. An offer may be mad e in writing, by

words or implied from conduct. It may be addressed to one particular per

son, a group of persons, or the world at large, as in an offer of a reward.

On the other hand, acceptance may be defined as an unconditional assent,

communicated by the offeree to the offeror, to all terms of the offer, made

with the intention of accepting, whether an acceptance has in fact occureed

is ascertained from the behaviour of the contracting parties, including any

correspondence that has passed between them.

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An offer is a statement of the terms by which the offeror is prepared to

be bou nd . Th e pa rty ma king the offer is referred to as th e offeror. Th e

pa rty to who m it is m ad e is called the offeree. If an offer is acc epte d the n

the agree men t exists. However, disputes may sometimes arise due to the

disagreement on whether a statement made by the offeror an offer at all.

Some statements are not offers, though they may appear so. Such statements

can not be accepted so as to form valid contrac ts. The mo st common of

such statem ents are invitations to treat. An invitation to treat is made

at a preliminary stage and consists of one party, the invitor, extending an

invita tion t o ano ther p arty, the invitee, to make an offer. This occurs, for

example at an auction, where auctionerr invites the audience to bid for the

goods on sale. His invitation is the invitation to treat only, but not an offer.

Each b id is an offer. An offer is acc epte d by th e auc tion ee r by the fall of

his ham me r: s6 Sale of Goo ds Ordinance (Cap 26). W here an auction is

advertised as being "without reserve", the auctioneer can withdraw any item

before the auction is held as held in the case,

Harris v Nickerson [1873].

However, once the bidding begins, an auctioneer who refuses to sell to the

highest bidder will be liable to pay damages as shown by the precedent,

Warlow v Harrison.

Moreover, a reque st for bids or ten de rs will not

be an offer unless it is coupled with the promise to accept the highest bid

as determined in a Hong Kong case, Lobley Co Ltd v Tsang Yuk Kie

[1997].

Moreover, where goods are displayed in a self service store on the shelves, or

in a shop window, the display is an invitation to treat, not an offer to sell.

W hen the c ustom er picks up the com mod ity off the shelf in a self service store

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and takes it to the cashier, it is indeed the customer who is making the offer

as established in the case,

Pharm aceutical Society of Great Britain v

Boots Cash Chem ists (Southern) Ltd [1953]. Besides, the position is

the same if the goods are in the window on display. This was determined in

the case, Fisher v Bell [1960], where the defendant was charged with the

offence of offering for sale a flick knife, Lord Parker C. J. stated that " the

display of an article with a price on it in a shop window is an invitation to

treat, but not an offer by the shop owner." The defendant who had displayed

such a knife in his shop, was acq uitted . Th e ma in consequences of this are

that under the law of contract, shops are not bound to sell goods at the price

indicated and a customer cannot demand to buy a particular item on display.

On the other hand, an acceptance can be defined as an unconditional assent,

communciated by the offeree to the offeror, to all terms of the offer, made

with the intention of accepting. By unconditional, it means that the offeree

must accept the terms proposed by the offeror unconditionally or without

introducing any new terms which the offeror has not had the opportunity

to consider. Th e introdu ction of new term s is referred to as a "counter

offer" and its effect in law is to bring the original offer to an end. This was

established in the case Hyde v Wrench [1840], in which the defenda nt

offered to sell a farm to the plaintiff for 1,000. In reply, the plaintiff offered

950.

This was rejected by the defendant. Later, the plaintiff purported to

accept the original offer of 1,000. It was held by the court that there was no

contract; the counter-offer of 950 had impliedly rejected the original offer

which was no longer capable of acceptance.

Moreover, whether an acceptance has in fact occurred is ascertained from the

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behaviour of the parties, including any correspondence that has passed be

tween th em . As a general rule, acceptance will not b e effective unless comm u

nicated to the offeror by the offeree or by someone with his or her authority.

An unco mm unica ted m ental assent will not suffice. In

The Leonidas D,

C. A. [1985], Goff L. J. said th at it is "ax iom atic th at acc epta nce of an offer

cannot be inferred from silence save in the most exceptional circumstances."

The communication of acceptance must be actually received by the offeror,

and where the means of communication are instantaneous (oral, telephone,

telex, fax), the contract will come into being when and where acceptance is

received.

It is worth noting that there are two exceptions to the rule that acceptance

must be communicated. The first one concerns unilateral contracts. In the

case of unilateral contract, one party promises to do something for another if

that other does a particular task. But there is no obligation to do that task.

This was seen in

Carlill v Carbolic Sm oke Ball Co ,

where the company

offered 100 to anyone who used the smoke ball and subsequently caught

influenza. The re was no obligation on anyone to buy and use the smoke

ball. However, if a person did so, like Mrs Carlill, the contract was complete

on the use of the ball and th e catching of the infection. At this poin t, th e

unilateral offer is accepted and the contract is complete, the company is still

bound though it will not know of this at the time.

The other exception concerns acceptance made through the post. Where the

post is the appropriate means of communication between the parties, unless

the parties have agreed otherwise, the letter containing the offer is effective

when th e offeree receives it. And a letter of revoc ation is effective w hen it

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is received, because the revocation of an offer must be communicated to the

offeree before a cce ptan ce is m ad e by th e offeree. W hile a let ter of acc epta nce

is effective as soon as it is validly pos ted, or put into th e ha nd s of a post-office

employee who may officially take letters for posting.

As illustrated in Byrne v Van Tienhoven [1880]. Here the defenda nts,

in Cardiff posted an offer to the plaintiffs in New York on 1st October. On

8th October, however, the defendants had changed their minds, and they

posted a letter of revocation. Meanwhile, the plaintifs had received the offer,

had accepted by telegram on 11th October, and sent a letter confirming

acce ptanc e on 20th Octob er. Th e letter of revocation did not arrive until

25th Oc tobe r. Th e court held th at a letter of accep tance was valid when

posted, while a letter of revocation was only valid when it was received.

Thus ,

the contract had been formed, probably on 11th October, but if not

by the telegram, certainly by the letter of 20th October because both were

sent before the revocation arrived.

In contr act law, even if the lette r of acceptanc e goes astray in the p ost a nd the

offeror is not to ld, he will still be bou nd in the con trac t. Th is was establish ed

in

Household Fire Insurance v Grant [1879],

where Gra nt applied for

shares in the plantiff 's company, and a letter of allotment was posted to

G ran t bu t was never received. W hen the compa ny went into liquidation ,

Grant was asked to contribute the outstanding amount on his shares to the

company's assets. The court held that Grant was a shareholder, the contract

was made when the allotment letter was posted.

However, like many of the rules to be found in contract law, the parties can

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avoid this rule if the y wish. Mo re precisely, the offerer ca n indic ate in his

offer, that an acceptance will not be valid until he actually "receives notice

of it in writing" as in the case Holwe ll Sec urities v Hug hes [1974]-

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( 2 . ) C o ns idera t io n

Consideration is an essential element in the formation of contract. A consid

eration may be defined as consisting a detriment to the promisee or a benefit

to the prom isor: " ... som e right, inter est, profit or benefit accru ing to the

one party, or some forbearance, detriment, loss or responsibility given, suf-

fered or undertaken by the other." The worlds "benefit" and "detriment" do

not refer to whether or not the bargain is an advantageous one. Indeed, it

means the contracting parties must have provided valuable consideration to

each other.

Consideration is called "executory" where there is an exchange of promises

to perform acts in the future, for instance, a bilateral contract for the supply

of goods whereby A promises to deliver goods to B at a future date and

B promises to pay on delivery. Alternatively consideration is referred to as

"executed" where one party performs an act in fulfilment of a promise made

by the other, for example, the unilateral contract where A offers a reward to

anyone who provides certain information.

However, past consideration, unlike executory or executed consideration, is

not a valid conside ration. Con sideration is said to be past when it consists

of some service or benefit previously rendered to the promisor. W he the r a

consideration is past is a matter of fact.

In Re McArdle, C. A.

[1951], a

woman carried out work to a house jointly owned by members of her family.

After the work had been completed, her relatives signed a document promis

ing to pay her for the work. It was held that she could not recover the sum

prom ised as her considera tion was pas t. Here the promise to pay is made

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could expect extra paym ent. In

Stilk v Myrick [1809]

where Stilk, a

seam an, signed up for a voyage from London to the B altic and back. During

the voyage two seamen de serted. Th e maste r of the ship agreed to divide

their pay between the remaining members of the crew if they would work

the ship back to London without the two deserters being replaced. On their

return the master refused to pay Stilk and the other seamen. It was held by

the court that Stilk and other seamen had not provided any consideration

for the master's promise. They merely agreed to do what they were already

bound to do.

On the other hand, in

Hartley v Ponsonby [1857],

the

plaintiff

an able

seaman, signed up with the master a ship for a voyage from London to Aus

trali a and back. The re was a crew of 36, but 17 of them deserted on arrival

in Australia. The master agreed to pay the plaintiff 40 if he helped to sail

the ship to Bombay with the remaining crew. It was held, however, that the

plaintiff 's original contract was terminated because it was dangerous to said

the ship with a crew of 19 seamen. Therefore, the plaintiff had entered into

a new contract and had to provide good consideration for hte mater's promise.

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(3 . ) Intent ion to be lega l ly bound

Further to offer, acceptance and consideration, for a contract to be valid, the

parties must intend to make the sort of bargain which is legally binding -

Intention to be legally bound. The law expects a business person to intend

his business arrang em ents. Business people who do not wish to enter into

a binding contract must record in writing a specific rebuttal of the legal

contract - for example, by using clear words such as 'this agreement merely

records the parties' wishes and is not binding in law', or 'this is not intended

to create a legal contract and has no legal effect on the parties. '

In order to assist the courts in deciding whether or not an intention to be

legally bound exists, two presumptions are made in the law of contract.

First, in social and domestic agreements there is no intention to be legally

bound. As illustrated in

Balfour v Balfour [1919],

where the defendant

was a civil servant stationed in what is now Sri Lanka. He and his wife came

to England on leave. When it was time to return, he left his wife in England

for the good of her health. They agreed tha t he would pay her 30 per

m onth while they were ap art . Late r the wife divorced him and he stopped

pay ing. She sued him, unsuccessfully. However, in an oth er case

Merritt v

Merritt [1970], where an agreemen t entered into by a husba nd and wife,

after they had separated, was held to be binding. It is worth noting that it

is jus t a presu mp tion but not a rule.

In Wu Chiu-kuen v Chu Shui-ching [1992], theplaintiff Mr Wu was

employed as an attendant at a mahjong "school". The defendant, a patron,

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Mr Chu, gave some money to the plaintiff who bought five Mark Six tickets

with th e money an d gave them to the defendant. One of the tickets won a

prize. The plaintiff alleged that there was an agreement to share the win

nings. There was a lack of evidence as to the terms of any agreement such as

how many tickets to buy, how to buy and how to share. The plaintiff gave

evidence that he counted the bills given to him only when he arrived at the

Jockey Club where he decided to contribute an equal amount of money and

chose how to buy the tickets. The court found in favour of the plaintiff and

held that arrangements of this sort are very often informal and even loose

at times. What is important is that the persons involved have acted on the

informal arrangements and conducted themselves in such a way that it is

clear from all the circumstances that they have agreed and intended to buy

the tickets togeth er and sh are the winnings, if any, togethe r. In my view,

unless the p arties ' arrange men ts coupled with their conduct pu rsuant to such

arrangements are so uncertain that a reasonable man cannot conclude that

they have agreed and intended to buy and share the tickets together, I think

the court should give effect to such an agreement.

Second, in commercial agreements the presumption is that the parties have

the inte ntion to be legally boun d. As in

Edw ards v Skyways [1964],

where the defendant, an airline agreed with British Airline Pilots Association

to pay "ex

gratia

payments "approximating to" an easily calculated sum to

pilots made redundant, the plaintiff a pilot, sued for such a payment under

the agreement. The defendant claimed that there was no intention to create

legal relations and that the promise was too vague. The court held that it was

a commercial agreement. Therefore, the presumption was that there was an

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intention to create legal relations. The onus was on the defendant to rebut

the pre sum ption which they had not done. Th e use of words

ex gratia

merely meant that the defendant did not admit any pre-existing liability,

rather than there was no legally binding agreement.

After explaining the meaning of the three basic elements in a valid contract,

the next step is to show how such a contract can be redefined via fuzzy sets

by utilizing these basic elements.

Let's denote OA*, C* and I* be the three fuzzy sets representing the offer

and acceptance, consideration and intention to be legally bound respectively.

For simplicity, I will assume the legal intention be measured by the number of

correspondence between the contracting parties alone. The possibili ty that

the contracting parties will indicate their intention, for example, by making

reliance expenditure though common is not considered here.

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F uzzy Se t s

D e s c r i p t i o n

1.

Of fer and acceptance

2 .

C o ns idera t io n ( C * )

3 . In t en t io n t o be l eg a l ly

bo und ( I* )

*) is a fuzzy set whose mem bersh ip

grades represent the degree of per

ceived clarity of the offer and accep

tance compared to the legal stan

dard. Let XjGZ (Z is the universal

set of contracts) where i refers to a

particular contract.

is a fuzzy set whose membership

grades represent the perceived

suf-

ficiency of the consideration com

pared to the legal stan dard . Let

X(EZ, where i refers to a particu

lar contract.

is a fuzzy set whose membership

grades represent the degree of per

ceived intention to be legally bound

compared to the legal stan dard . For

simplicity, I will assume the legal in

tention be measured by the number

of correspondence between the con

tracting parties alone.

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Now suppose we have six contracts, their corresponding membership grades

of the three fuzzy sets A*, B* and C* are assumed as follows:

Via OA*, the offer and acceptance of all the contracts, with the exception of

Contract No.5 which is considered to be containing no offer and acceptance,

are similar to the legal standard.

Via C*, the consideration of Con tract No.2 and Co ntract No. 4 are considered

to be largely insufficient compared with the legal standard, while Contract

No.3 specified a value of consideration that is completely consistent with the

legal standard.

Via I*, I assumed that for parties who have been communicating with each

other for more than 2 times, demonstrate an intention to be legally bound

that is perfectly consistent with the legal standard.

Th e set f2* consists of the sets OA *, C* and I*. Sem antically, a valid c on trac t

is composed of an offer and acceptance, consideration and an intention to be

legally bound.

Moreover, we can obtain the membership grades of the fuzzy set

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'valid' Contract by the extension principle introduced in Section 2.25 above.

The solution is obtained by joining the membership grades of the offer and

acceptance with that of the consideration and the intention to be legally

boun d. We then obtain the maximum mem bership grade amo ng all the

me mb ership grades to obtain each con tract 's mem bership grade in We

proceed by utilizing the membership grades of OA*, C* and I*.

Each of the individual tr iplets represents the mapping of a contract in OA*

onto an element contained in both C* and I*. The first element of each triplet

represents the m em bership grade of a particular contract in OA*. T he second

element is the membership grade of the contract in C* and the third element

represents the membership grade of the contract in I*.

The minimum membership grade of each triplet is selected via the intersec

tion of the m em bersh ip grades in the trip let. If any of the m em bersh ip gra des

contained in the triplet is zero, the membership grade of their intersection

will also be zero. In other words, a contract must exist in each of the OA*,

C* and I*, for it to be in the new fuzzy set

Each element in OA* has six minimal membership grades. By applying the

max-min principle, the highest membership grade is then selected as the

membership grades of each contract in the new fuzzy set

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Contract No.l 's membership grade is

Contract No.2's membership grade is



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There fore, = 0.6 / Co ntra ct No .l , 0.2 / Co ntrac t No.2 , 0.6 / Co ntra ct

No.3 , 0.1 / C on trac t No.4 , 0.0 / C ontr act No.5.

Dele ting Co ntra ct No.5 from since its me mb ership grades is zero, the

becomes

=

0.6/Contract No.l

, 0.2/

'Contract No.2,

0.6/Contract No.3

,

0.1/Contract No.4

(11)

T he fuzzy set is said to be describe d by

(12)

The membership grades indicate the degree of inclusion and the level of

conjectural unce rtainty attr ib uta ble to each contract. Thu s the judge can

expect Contract No.l and No.3 exhibit a relatively strong compliance with

the legal sta nd ar d. W hile he can expect Co ntrac t No.2 and No.4 display a

weak compliance with the legal standard.

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In addition, other fuzzy variables can be added to the above example, for

instance, the ' legal ' capacity, LC* of the contracting parties, where LC* can

be a fuzzy number indicating, for example, the age of the parties.

Each type of contract can be defined uniquely in a manner similar to the

above . Most impo rtantl y, the possibility distribu tion gene rated by the set

can be used to describe the degree of membership of a particular contract's

inclusion in the set. Th e judg e can evaluate the possibility of a pa rticu lar

contract being a valid one by projecting the contract into the set

The judge might evaluate the fu

chan - uncertainty in economics and the applications of fuzzy logic in contract laws

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