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/ Projektpartner 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