Resolving the Ellsberg Paradox by Assuming That People Evaluate Repetitive Sampling

Schneeweiss: Resolving the Ellsberg Paradox by Assuming that People Evaluate Repetitive Sampling

Sonderforschungsbereich 386, Paper 153 (1999)

Online unter: http://epub.ub.uni-muenchen.de/

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Resolving the Ellsb erg Paradox by

Assuming that People Evaluate

Rep etitive Sampling

Hans Schneeweiss

Ellsb erg designed a decision exp eriment where most p eople violated

the axioms of rational choice He asked p eople to bet on the outcome of

certain random events with known and with unknown probabilities They

usually preferred to b et on events with known probabilities It is shown that

this b ehavior is reasonable and in accordance with the axioms of rational

decision making if it is assumed that p eople consider b ets on events that are

rep eatedly sampled instead of just sampled once

Key words Ellsb ergs paradox rational decision making Sure Thing Princi

ple sub jective probabilities

Intro duction

Ellsb erg designed an exp eriment where p eople had to decide b etween

b ets on risky lotteries with known probabilities or on uncertain events where

probabilities were not known They usually preferred to b et on the lottery

thereby blatantly violating the rationality axioms of Bayesian decision theory

sometimes also referred to as Sub jective Exp ected Utility SEU theory This

b ehavior has therefore b een called a paradox The Ellsb erg Paradox has

since b ecome the paradigm for a new concept in decision theory ambiguity

Probability statements can be more or less ambiguous dep ending on how

strong an individual b elieves in the assertion of these probabilities

Otherwise rational p eople seem to express a preference for unambiguous

probabilities ie for probabilities that are ob jective and completely known

to them They shrink from ambiguous probabilities ie probabilities that

are only of a sub jective kind or that are ob jective but only vaguely known A

typical example would b e an urn I with red and black balls of an unknown

prop ortion out of which one ball is to b e drawn If indeed nothing is known

ab out the prop ortion of red and black balls in the urn then one might be

indierent of whether to bet on Red or on Black just as with an urn I I

where the red and the black balls are in equal numb er and where this prop or

tion is known So in b oth cases the indierence b etween b etting on Red or

on Black can b e regarded as an expression of assigning equal probabilities to

b oth colors In the second case urn I I however the probabilityofdrawing

a red ball say is ob jectively given as whereas in the rst case urn I the

ob jective probability of the same event is unknown and it is only a sub jective

probability which can be asserted as b eing It turns out that although

in b oth cases most p eople are indierent when confronted with a choice of

b etting on Red or on Black they typically prefer to have the ball drawn from

the urn I I with known prop ortion regardless of whether they b et on Red

or on Black This b ehavior is well do cumented bynumerous exp eriments It

was rst made public by Ellsb erg and has since been known as the

Ellsb erg paradox It is regarded to b e paradoxical b ecause in b oth urns the

sub jective probabilities for Black and Red are equal and therefore and yet

BlackI Black I I and at the same time Red I Red I I Here means is

strictly preferred toand Black I denotes the bet on the eventofablack

ball b eing drawn if the ball is drawn from urn I

Ellsb erg also designed another exp eriment with only one urn but with

balls of three dierent colors where the nature of the paradox can b e studied

more closely We shall analyze this situation in the next section

The ndings of Ellsb erg have b een veried in a large numb er of similar

b etting exp eriments and many suggestions have b een prop osed for under

standing the apparent paradox One can of course simply ignore the prob

lem and discard the observed b ehavior as b eing irrational and not worth any

further study But as the observed b ehavior in the Ellsb erg exp eriment is

rather p ersistent and therefore can hardly b e dismissed on the basis of b eing

irrational an explanation for it is called for in particular as this kind of

b ehavior is probably prevalent in many practical decisions eg in economics

or in business see eg Sarin and Web er

The central trait of the observed b ehavior seems to b e that most p eople

shrink from uncertain events the ob jective probabilities of which are unknown

or only vaguely known Betting on an event with known ob jective probabil

ity like Red I I is preferred to b etting on an event with the same sub jective

probability which however is not substantiated byanob jective probability

and is therefore ambiguous like Red I Ambiguity is a quality attached to

probability assertions A p erson may assign a probability to some event but

maybe more or less certain ab out the value of this probability It is ques

tionable whether a measure of ambiguity adequate for all kinds of uncertain

circumstances can be found but as a concept to describ e situations as in

Ellsb ergs exp erimentitisworth studying

Attempts have b een made to mo del the b ehavior of the ma jorityofpeople

in Ellsb ergs exp eriment and to study the conditions under whichambiguity

is p erceived by individuals in decisions under uncertainty For a recentsur

vey see Camerer and Web er see also Kepp e and Web er and

Eisenb erger and Web er A famous axiomatic approach that results in

a sub jective exp ected utility theory with probabilities replaced by capacities

has b een prop osed by Gilb oa see also Schmeidler For a recent

further development of this approach where ob jective probabilities are in

corp orated in the theory see Eichb erger and Kelsey For an empirical

test see Mangelsdor and Web er Recently a dierent criterion for de

cision making in the face of uncertaintygoverned byinterval probabilities was

prop osed byWeichselb erger and Augustin Schneewei

tried to explain Ellsb ergs paradoxbyemb edding Ellsb ergs exp eriments in

a game theoretic framework

Here a dierent approachischosen I shall argue that the typical b ehavior

of p eople in the Ellsb erg exp eriment can be explained by assuming that

they consider sub consciously the act of drawing a ball from an urn as a

rep etitive act despite the fact that they are told the ball will b e drawn only

once In evaluating the p ossible gains and losses from participating in a

lottery people imagine the lottery to be played several times and consider

the average amount they might gain or lose For a lottery or urn with

known probabilities of gains or losses this average amount is rather certain

due to the lawoflargenumb ers But if the probabilities are unknown the

result of rep eated lottery draws will also b e unknown no matter howmany

rep etitions are considered When confronted with a choice between urn I

with unknown probability and urn I I with known probability of drawing

a red ball a p erson might assign the same sub jective probability to Red

for b oth urns as long as one draw is considered But if that p erson p erhaps

only sub consciously imagines rep eated draws from the urn she cho oses and

if she is risk averse then she will cho ose urn I I b ecause it is with this urn only

that the average gain of rep eated draws will b e rather certain whereas the

uncertainty of gains from rep eated draws out of urn I will remain uncertain

The pap er will analyze the distribution of gains under rep eated draws

in Ellsb ergs exp eriment and will show that risk averse p eople will always

prefer the less ambiguous situation in complete accordance with the axioms

of rational choice and thus in accordance with Bayesian decision theory In

the next section the Ellsb erg paradox is reviewed in a setting somewhat

dierent from what was describ ed ab ove Section gives the main argument

howtoevaluate the result of rep eated draws and why less ambiguous events

are preferred to more ambiguous ones even if their sub jective probabilities

do not dier Section contains some concluding remarks

The Ellsb erg paradox

The Ellsb erg exp eriment or rather one of two suggested exp eriments con

sists in b ets on the outcome of a single draw from an urn whichcontains

red balls and blackoryellow balls in an unknown prop ortion A p erson

is given a choice to b et on the outcome of the draw to b e Red or to b e Black

and another choice to b et on whether the outcome will b e Red or Yellowor

whether it will be BlackorYellow In each case the p erson wins Euro if

the color he b ets up on do es indeed show up otherwise nothing is gained or

lost This decision situation is depicted in the following diagram Table

which shows the payo function dep ending on the color of the ball and on the

b etting act chosen When asked to cho ose b etween b ets R or B most p eople

Table Payos in Ellb ergs exp eriment

Numb er of balls

Bet on Symbol Red Black Yellow

Red R

Black B

RedorYellow R Y

BlackorYellow B Y

decide for R When the same p eople are then asked to cho ose b etween R Y

or B Y they typically decide for B Y Only few p eople would cho ose

R Y Some p eople decide for B in the rst decision problem and for R Y

in the second problem

Let us discuss the choice of the rst group the ma jority of p eople The

arguments for the last group are completely analogous First one might

think that since nothing is known ab out the prop ortion of blackandyellow

balls a p erson should assume owing to the principle of insucient reason

that Black and Yellow are equally likely to show up Not that the p erson

thinks both colors to be in equal prop ortion in the urn he only bases his

decision on the implicit assumption that the prop ortion is Perhaps a

b etter description of this attitude would b e to say that the decision is made

as if the prop ortion of black and yellow balls were equal or more technically

the sub jective probabilities of Black and Yellow are equal for this person

For given that nothing is known ab out the way the balls were put into the

urn why should there b e a higher sub jective probability for a blackballto

b e drawn than for a yellow ball If someone follows this argument then he

will b e indierentbetween R and B in the rst problem and b etween R Y

and B Y in the second problem In symb ols R B and R Y B Y

and P R Y P B Y P being a b ecause P R P B

sub jective probability This however is not what is observed as the actual

b ehavior of most p eople

One can argue against this reasoning that it assumes at the outset the ex

istence of sub jective probabilities This assumption is indeed what Bayesian

decision analysis is based up on There are strong arguments for making this

assumption For a recent information based argument see Ferschl

Moreover one can prove the existence of sub jective probabilities for anyra

tional decision maker from certain axioms of consistent rational choices

between acts with uncertain outcome Savage These axioms do not

necessarily imply that P B P Y This follows only if in addition to the

axioms of rational choice the principle of insucient reason is taken as an

extra assumption But even without this additional principle the observed

b ehavior of individuals in the Ellsb erg exp eriment stands in contrast to the

Bayesian decision rule

Supp ose the decision maker as a Bayesian attaches sub jective not neces

sarily equal probabilities to the events of drawing a ball with a sp ecic color

and supp ose he has a NeumannMorgenstern utility function u where we

can take without loss of generality u and u Then the exp ected

sub jective utilities of the four acts of the Ellsb erg exp eriment are just the

sub jective probabilities of the events b etted up on Thus if R is preferred to

B R B then P R PB which implies P R P Y PB P Y

and this again implies R Y B Y These preferences which follownat

urally from a Bayesian decision framework are however contradicted by the

observed b ehavior of p eople that instead prefer B Y to R Y The same

contradiction arises for those p eople the minority for which B R and

R Y B Y only the very few with R B and R Y B Y have

preferences in accordance with Bayesian decision theory

The conclusion of the Ellsb erg exp eriment then is that most p eople con

trary to what Bayesians say do not base their decision on sub jective proba

bilities This is seen as a paradox Against this one may argue that it is a

paradoxonlyto Bayesians that is only to those that believeinsubjective

probabilities But the paradox go es deep er

The observed conjunction of preferences R B and B Y R Y is

paradoxical in that it violates the Sure Thing Principle Savage The

preference of R to B in the rst problem should not change if one gets Euro

not only for the color b etted up on but also in case yellow turns up regardless

of whether one b etted on Red or on Black Note that this Euro is not an

additional amountthatyou might get in addition to the prize of Euro for

b etting on the right color You get the extra prize only if you would have lost

b oth of the two b ets R and B whatever your actual b et was consult Table

It is a kind of consolation prize that you get if neither a red nor a black ball

is drawn and this consolation prize should not interfere with your decision

whether to b et on Red or on Black Thus R B should imply R Y B Y

The fact that most p eople in their b ehavior seem to contradict this almost

logical implication is seen to b e paradoxical at least by those that see the

Sure Thing Principle as a selfevident principle of decision making Note that

this argument do es not make use of any probability assertions it do es not

even use the concept of probability

Rep etitive draws

The situation describ ed in Section changes dramatically if instead of having

just one draw from the urn several indep endent draws are considered After

a b et has b een made a ball is drawn from the urn n times with replacements

Each time the color b etted up on shows up the amountofEuroispayed to

the decision maker Let X b e the average gain ie the amount gained after

n rep eated draws divided by n Then nX Binn P C where Bin stands

P C is the probabilityof drawing a ball for the binomial distribution and

with color C in a single draw When C is BlackorYellow this probability

dep ends in an obvious way on the prop ortion p of black balls within the set

of black and yellow balls ie the numb er of black balls divided by

We assume that the b etting acts are decided up on by a rational p er

son This means that the decisions of this p erson are governed by a sub jective

probability distribution But now the sample space consists not of the three

colors but rather of the various outcomes of n rep eated draws Aconvenient

way to mo del probabilities for these outcomes is to assign a sub jective prob

ability to the prop ortion p of black balls and then to compute in the usual

way conditional ob jective probabilities for the outcomes of n draws condi

tional on p As we are only interested in the average gain X wemay use the

conditional probability distribution of X given p f xjp This pro cedure is

based on the obvious assumption that the sub jective probabilityofanevent

coincides with its ob jective probability if the latter exists and is known to the

decision maker So the only truly sub jective probability distribution is the

one for p Let its distribution function b e denoted by F p Any rational

decision maker has a distribution F p which is indep endent of the b etting

act chosen The unconditional distribution of X is then given by

f x f xjpdF p

In particular the distribution of X for the four b etting acts is given by the

following expressions for P X x nx n

nx nx

B p p dF p

nx nx

p p dF p R Y

nx nx

B Y

A rational p erson following SEU theory bases his decisions on a Neumann

Morgenstern utility function uX which in our case should b e increasing

That act is preferred for which the exp ected utility E fuX g is largest For

a risk averse p erson the utility function u is concave We assume for

simplicity that u is quadratic The results should however hold true also

for more general concave utility functions For a quadratic utility function

E fuX g is a function of E X and V X If u is chosen so

that u and u and is taken to be increasing for x then

uxa x ax a and for any random variable X with range

E fuX g is increasing in and decreasing in The parameters

and can be easily computed for the four b etting acts see Table We

Table Mean and variance of average gain X

b etting act lim

B E p M p V p V p

R Y E p M p V p V p

B Y

E p M pE p

indicate their derivation for the b etting act B Wehave

p E p E fE X jpg E

E fV X jpg V fE X jpg

V p p p E

E p E p V p

The preferences b etween b ets are determined bythevalue of

E fuX g a a

where a and

For n is the probability of the b et to come true and E fuX g

This means that of two b ets that one is preferred which has the higher

probability of coming true Therefore R B implies R Y B Y as shown

in Section This is the b etting b ehavior of a rational p erson if only one

draw is considered

Let us return to rep eated draws If n b ecomes large then will be

negligible for the unambiguous b ets R and B Y but will b e large and will not

go to zero for the ambiguous b ets B and R Y see the last column of Table

Thus one may exp ect a tendency in eachof the two comparisons to prefer

the unambiguous b et to the ambiguous one ie R B and B Y R Y

Of course whether this will in fact be true dep ends on the probability

distribution of p in particular on E p and V p

To illustrate let us supp ose that the decision maker has a symmetric dis

For n the exp ected utilities of the tribution F p so that E p

four b ets b ecome as in Table

Table Exp ected utilities

a a aV p R B

B Y R Y a a aV p

From Table it is obvious that for anyvalues of a and V p R B and at

the same time B Y R Y These are exactly the preferences observed

for the ma jority of participants in Ellsb ergs exp eriment They follow from

Bayesian arguments and are therefore the preferences of a rational per

son It is however a p erson that determines his preferences from viewing the

drawing of balls as b eing rep eated an indenite numb er of times

One can generalize this result by allowing the number of red balls to

vary Let r be the ratio of red balls in the urn which is supp osed to be

known to the decision maker In Ellsb ergs original exp eriment r A

simple computation analogous to the previous one leads to the exp ected

utilities for n ofTable where we assumed a symmetric distribution

for p as b efore Noting that V p it is easy to see by letting r

Table Exp ected utilities with general r

a a

R r ar r B r r

a r V p

a a

B Y r ar R Y r r

a r V p

go to zero that B R and B Y R Y for suciently small r where

the p oint of switching from R B to B R dep ends on the sub jectively

p erceived amount of risk V p and on the measure of aversion of risk a

Such a switching strategy has actually b een p ostulated by Ellsb erg himself

Note that under a minimax criterion no switching of the kind describ ed is

allowed to take place

Similar arguments lead to analogous conclusions for Ellsb ergs rst exp er

iment describ ed in the Intro duction Let p b e the prop ortion of black balls

in urn I and let as b efore X be the average gain of n rep eated indep en

dent draws from the urn chosen and for the color b etted up on Then mean

and variance of X are determined as in Table corresp onding to Table

For n E fuX g which is just the probability of winning the bet

It follows for a rational p erson that RII RI implies B I BII

which however typically is not observed On the other hand for n large

n and assuming E p the exp ected utilities for RII and BII

a a

and for RI and BI they are aV p It follows that are b oth

a rational p erson will have preferences RII RI and at the same time

BII BI which corresp onds to the b ehavior of the ma jority of p eople in

Ellsb ergs exp eriment

Table Mean and variance of X in Ellsb ergs rst exp eriment

b etting act lim

RII

RI E p E fp pg V p V p

BI E p E fp pg V p V p

BII

Conclusion

We can explain Ellsb ergs paradoxby assuming that p eople sub consciously

and erroneouslyevaluate the p ossible outcomes of the exp erimentasif the

draws from the urn were rep eated an indierent number of times They

are assumed to have a concave p ossibly quadratic utility function and a

sub jective probability distribution over the unknown states of the world

which in this case are the unknown values of the prop ortion of balls in the

urn They then compute the sub jective exp ected utility of the average gain

from a b et on a particular color and cho ose the b et with the biggest exp ected

utility In doing so they are in accordance with Bayesian SEU as well as

with the actual b ehavior observed in Ellsb ergs exp eriment It turns out that

ambiguityaversion is nothing but risk aversion in a SEU framework

I m ust admit though that this explanation replaces one contradiction

of rational decision theory with another one In the original interpretation

where p eople are supp osed to fully understand the exp erimental set up which

is that indeed just one ball is to b e drawn their b ehavior is in conict with

the Sure Thing Principle and so they are seen to b ehave irrationally In

the interpretation prop osed in this pap er their b ehavior is irrational in that

they misconceive the imp ortant alb eit a bit articial trait of the exp eriment

that only one draw is to b e executed In the rst interpretation p eople show

ambiguityaversion in the second they showriskaversion

Is there a rational explanation why p eople in a situation like Ellsb ergs

exp eriment intuitively think of rep eated draws even though they are told

that only one draw will b e executed I should like to argue that in practical

decision situations under uncertainty the rep etitive element is prevalent

Consider an entrepreneur who wants to invest in an enterprise with un

certain prots Supp ose the entrepreneur can buy shares of a rm that for

a long time in the past showed varying yearly prots with constant mean

and variance and that will supp osedly continue with that p erformance in the

future To be more sp ecic yearly prots of that rm are iid N

with known and Now supp ose there is another p ossibilityforinvest

ment The entrepreneur can invest in a newly founded rm whichwillhave

a rather constant hardly varying stream of yearly prots the amountof

whichhowever is unknown Let us for the sake of simplicity assume that is

constant but unknown Dep ending on the future development of the mar

ket the rm has business in can turn out to b e high or low but whatever

value it will have in the future this value will stay constant

If the entrepreneur is a rational decision maker in the Bayesian sense

he will assign a probability distribution to Supp ose N where

it so happ ens that the mean of this sub jective distribution is the same as

the mean of the ob jective distribution in the rst case but where

If variances were also equal the twoinvestments would only dier

in their ambiguity the rst one b eing unambiguous the second one b eing

very ambiguous Now if the entrepreneur invested his money just to receive

a prot after one year and then went out of business he would prefer the

uncertain second alternative to the rst one b ecause the second alternative

has the lower variance and b oth have equal means for the prot of this one

year However such a situation is rather unrealistic An entrepreneur cannot

so easily quit his engagement esp ecially not in the second case If prots

turn out to b e low he will of course come to knowthisvery so on in fact

after one year but so will everyb o dy else So he will be able to disengage

from his investment only with a loss of money Therefore the entrepreneur

has to consider not just the prot of one year but rather the whole stream of

prots over the years He might wish to evaluate an average prot or rather

an average of discounted prots a case where we could argue in a similar

but more complicated way and that we shall not follow up here In doing

so he will most probably prefer the rst investment even though it has the

higher variance the means b eing equal The reason is that for the average

prot the variance in the rst investment go es to zero if the number of years

increases but remains p ositive and constant for the second investment no

matter how many years the investor takes into account What lo oks as a

case of ambiguityaversion actually is risk aversion

As this kind of situation seems to arise quite often in practice we should

not b e surprised to discover that p eople b ehave in Ellsb ergs exp erimentas

if they considered not just one draw but several draws and thus come to a

conclusion which seems to contradict the axioms of rationality but is in fact

in accordance with these axioms if seen from the p ersp ective of a long term

investment

Acknowledgement

I thank Thomas Augustin for some helpful comments on an earlier draft

of the pap er

After the pap er was nished I learned ab out an unpublished pap er by Halevy

and Feltkamp that in part deals with the same problem using a sim

ilar approach as in the present pap er

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