Size Doesn t Really Matter

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1 B.D. Pulford Experimetal & A.M. Psychology 2008 Colma: Hogrefe 2008; Ambiguity & Vol. Huber 55(1):31 37 Publishers Aversio Size Does t Really Matter Ambiguity Aversio i Ellsberg Urs with Few Balls Brioy D. Pulford ad Adrew M. Colma Uiversity of Leicester, UK Abstract. Whe attemptig to draw a ball of a specified color either from a ur cotaiig 50 red balls ad 50 black balls or from a ur cotaiig a ukow ratio of 100 red ad black balls, a majority of decisio makers prefer the kow-risk ur, ad this ambiguity aversio effect violates expected utility theory. I a experimetal ivestigatio of the effect of ur size o ambiguity aversio, 149 participats showed similar levels of aversio whe choosig from urs cotaiig 2, 10, or 100 balls. The occurrece of a substatial ad sigificat ambiguity aversio effect eve i the smallest ur suggests that ifluetial theoretical iterpretatios of ambiguity aversio may eed to be recosidered. Keywords: ambiguity effect, Ellsberg paradox, payoff variace, ur size effect Itroductio Ambiguous prospects, with outcome probabilities that caot be calculated from first priciples or estimated from empirical evidece, preset a major challege to decisio theory. The issue was first highlighted by Kight (1921), who itroduced a distictio betwee decisios made uder risk ad ucertaity, ad simultaeously by Keyes (1921), who drew a parallel distictio betwee probability ad weight of evidece. I Kight s more familiar termiology, a decisio is risky whe the decisio maker does ot kow what outcome will occur but kows the outcome probabilities, or ca judge them with some cofidece, ad ucertai whe the decisio maker is igorat eve of the probabilities. Kight illustrated this distictio with a example of two people makig blid drawigs from a ur cotaiig balls of two colors: Oe ma kows that there are red ad black balls, but is igorat of the umbers of each; aother kows that the umbers are three of the former to oe of the latter (pp ). The first faces a decisio uder (umeasurable) ucertaity, owadays more commoly called ambiguity i psychological literature; the secod faces a decisio uder (measurable) risk. This distictio was igored or rejected by most subsequet decisio theorists, partly because ambiguity is relatively itractable, ad partly because decisio theorists (e.g., Raiffa, 1961) were quick to poit out that we ca alwaysapplythepriciple of isufficiet reaso ad assig equal probabilities to the outcomes of a ambiguous choice. Kight (1921) evidetly believed that people hadle ucertaity i this way: It must be admitted that practically, if ay decisio as to coduct is ivolved, such as a wager, the first ma [choosig from the ambiguous ur] would have to act o the suppositio that the chaces are equal (p. 219). Savage (1954), i his ifluetial axiomatic subjective expected utility (SEU) theory, brushed ambiguity aside o the grouds that, i order to icorporate it ito decisio theory, secod-order probabilities would be required, ad the itroductio of a edless hierarchy seems iescapable. Such a hierarchy seems very difficult to iterpret, ad it seems to make the theory less realistic, ot more (p. 58). Secod-order probabilities ca be calculated whe outcome probabilities are ot kow directly but probabilities of probabilities ca be iferred, as whe a decisio maker does ot kow the ratio of red ad black balls i a ur but kows that every possible ratio is equally probable. Ambiguity Aversio I spite of its theoretical itractability, ambiguity is commo i everyday decisios. Furthermore, it is psychologically distiguishable from risk, ad empirical evidece cotradicts Kight s (1921) assumptio that huma decisio makers merely assig equal probabilities to the outcomes of ambiguous prospects. A substatial body of evidece has show that most decisio makers prefer risky prospects with equal outcome probabilities to ambiguous oes. This is the ambiguity aversio effect, ad the classic demostratio of it is the Ellsberg paradox (Ellsberg, 1961; Feller, 1961). I Ellsberg s simplest illustratio, two urs are filled with red ad black balls, Ur A cotaiig 50 red ad 50 black balls, radomly mixed, ad Ur B cotaiig a ukow ratio of 100 red ad black balls, radomly mixed. A decisio maker chooses either color (red or black) ad either ur (A or B) for a blid drawig ad wis a prize if a ball of the chose color is draw. Most decisio makers 2008 Hogrefe & Huber Publishers Experimetal Psychology 2008; Vol. 55(1):31 37 DOI /

2 32 B.D. Pulford & A.M. Colma: Ambiguity Aversio strictly prefer the kow-risk Ur A to the ambiguous Ur B, irrespective of the preferred color. Ambiguity aversio violates the axioms of subjective expected utility (SEU) theory, for the followig reaso. Suppose a decisio maker tries to draw a red ball ad strictly prefers Ur A to Ur B. Because the decisio maker kows that the probability of drawig a red ball from Ur A is 1/2, it ca be iferred from SEU theory that the subjective probability of drawig a red ball from Ur B must be less tha 1/2. It follows that this decisio maker s subjective probability of drawig a black ball from Ur B must be greater tha 1/2, because the two probabilities must sum to uity i the ur, give that the ball must be either red or black. This suggests that the decisio maker prefers drawig a black ball from Ur B to drawig a red ball from Ur A, ad the decisio to try for a red ball from Ur A is therefore icosistet with the decisio maker s ow prefereces. It fails to maximize SEU ad therefore violates SEU theory. Nevertheless, most decisio makers prefer the kow-risk Ur A for both red ad black balls, thereby maifestig ambiguity aversio. Sice Ellsberg (1961) discovered this ituitively compellig violatio of SEU theory, empirical evidece has cofirmed that ambiguity aversio is a powerful ad robust pheomeo (Camerer, 1995, pp ; Camerer & Weber, 1992; Curley & Yates, 1989; Frisch & Baro, 1988; Kere & Gerritse, 1999; Rode, Cosmides, Hell, & Tooby, 1999). It has bee foud eve whe decisio makers, without beig told the actual ratio of red to black balls i Ur B, were told that every possible ratio is equally likely, although this iformatio about secod-order probabilities makes the objective chaces equal i both urs. Theoretical Iterpretatios Ambiguity aversio is easy to demostrate but surprisigly hard to explai. It has bee suggested (e.g., Krähmer & Stoe, 2006; Tetlock, 1991) that it arises from a desire to avoid the aticipated regret that would follow from drawig a losig ball from a ambiguous ur. These ad other curret theories have o obvious implicatios regardig the umber of balls i the kow-risk ad ambiguous urs, ad the effects of ur size, icludig very small urs, do ot appear to have bee systematically ivestigated. However, there are at least two promiet theories that have strog implicatios for ur size. The first is Eihor ad Hogarth s (1985) descriptive model based o the achorig ad adjustmet heuristic (Slovic & Lichtestei, 1971). Accordig to this model, a decisio maker faced with a ambiguous prospect begis with a provisioal probability estimate ad the adjusts it up or dow o the basis of a metal simulatio i which all probability distributios that might apply are imagied, ad those that are judged to be iapplicable are excluded. The adjustmet is affected by two factors, represeted by parameters i the model: the amout of perceived ambiguity Experimetal Psychology 2008; Vol. 55(1):31 37 ( ), causig a liear icrease i the size of the adjustmet, ad the decisio maker s attitude toward ambiguity i the give circumstaces (β), causig a further oliear adjustmet that may vary for differet probability values. Eihor ad Hogarth reported four experimets, based o Ellsberg ur choices, i which the model parameters show most decisio makers adjustig their subjective probabilities of success i ambiguous urs dow from 1/2, causig them to prefer kow-risk urs. Accordig to Eihor ad Hogarth (1985), there are special circumstaces i which decisio makers are likely to prefer ambiguous to kow-risk optios (pp ), but the amout of ambiguity is a icreasig fuctio of the umber of distributios that are ot ruled out (or made implausible) by oe s kowledge of the situatio (p. 435). The model does ot iclude ay effect of ur size o the iitial estimate, although such a effect is quite coceivable. However, the amout of perceived ambiguity reflects the cogitive simulatio process (p. 450), or more specifically the degree to which oe simulates values of p that might be (p. 438), ad whe a decisio maker has relatively sparse relevat iformatio, oe would expect ambiguity to be high because few distributios are ruled out (p. 442). The model seems to imply greater ambiguity aversio i larger urs, because there are costs of ivestig i imagiatio, icreased metal effort ad the discomfort that results from greater ucertaity (p. 459), ad above all because the size of the adjustmet parameter icreases mootoically with the umber of distributios that eed to be imagied ad ot ruled out (p. 435) there are more of these distributios i larger urs. This suggests a clear predictio about ur size. I a metal simulatio of a ambiguous ur cotaiig 100 red ad black balls i a ukow ratio, there are 101 possible distributios to be imagied, from o red balls to 100 red balls, ad oe of these is ruled out or excluded. But i a metal simulatio of a ambiguous ur cotaiig just two red ad black balls i a ukow ratio, there are oly three distributios to be imagied ad ot excluded, amely o red balls, oe red ball, ad two red balls. I terms of the model, the larger ur is therefore perceived to be vastly more ambiguous tha the smaller oe. Because oly three distributios are imagied ad oe excluded i the two-ball ur, the model assigs a miute value to the parameter that quatifies the amout of perceived ambiguity ad determies the size of the ambiguity aversio effect. A secod theoretical approach with strog implicatios for ur size was put forward by Rode et al. (1999). They suggested that ambiguity aversio arises from decisio makers associatig ambiguous outcomes with high payoff variability although our calculatios (see Appedix) suggest that variability is ot ecessarily greater i ambiguous optios. Accordig to this approach, ambiguity aversio is a by-product of the applicatio of a risk-sesitive cogitive architecture, adapted by evolutio for optimal foragig, that takes accout of both the mea ad the variace of expected payoffs to miimize the probability that the out Hogrefe & Huber Publishers

3 B.D. Pulford & A.M. Colma: Ambiguity Aversio 33 come will fail to satisfy the orgaism s eed (Stephes & Krebs, 1986). If a orgaism eeds X calories of food to survive, ad if two resource patches have the same mea calorie payoff but differet payoff variaces, ad the mea payoff of the low-variace patch is above X, the the orgaism should forage i that patch; but if the mea payoff i the low-variace patch is below X, the it should forage i the high-variace patch. I the extreme case i which the low-variace patch has zero variace (the payoff is certai), the orgaism is certai to satisfy its eed by foragig i that patch if the mea payoff is above the threshold X ad certai to fail if the mea is below X; but if the mea is below X, the foragig i the high-variace patch yields a small but positive probability of satisfyig its eed. Hece, accordig to Rode et al., decisio makers are ot avoidig ambiguity per se: istead, they are avoidig the high variace of outcomes of ambiguous optios (p. 296). Accordig to Rode et al. (1999), ambiguity aversio arises from a overgeeralizatio of this policy to decisios i which oe of the optios has a uspecified distributio. They provided evidece to support the cojecture that huma decisio makers ted to avoid ambiguous optios oly whe kow-risk optios meet their eeds. First, they showed that the size of the ambiguity-aversio effect teds to icrease with the probability of success i the kowrisk optio. Secod, i a experimet i which decisio makers had to choose betwee a kow-risk optio with obviously high payoff variability ad a ambiguous optio with obviously low payoff variability, most chose the lowvariability optio. These results reversed the stadard ambiguity aversio effect ad provided further corroboratio for this iterpretatio. I ambiguous urs i which every possible distributio is equally likely, as they were i our experimet ad those of Rode et al. (1999), it is possible to calculate expected payoff variaces exactly (see Appedix). The expected variace i a ambiguous ur cotaiig balls, calculated as the mea of the equiprobable variaces that might apply to the ur, turs out to be 1/6 1/6. Hece, the expected payoff variace icreases rapidly for small values of ad ever exceeds 1/6. The expected payoff variace is very small for a 2-ball ambiguous ur (0.08) ad much larger i 10-ball ad 100-ball ambiguous urs (0.15 ad 0.16, respectively). These expected payoff variaces are smaller, ot larger, tha the variaces for kow-risk urs. I a kow-risk ur of ay size with 50% balls of each color, the payoff variace is This result seem to be at odds with the theory of Rode et al. (1999), which iterprets ambiguity aversio as a cosequece of variace avoidace. Ur size clearly has relevace to the iterpretatio of ambiguity aversio. If the effect turs out to be uaffected by ur size, ad particularly if ambiguity aversio is foud eve i very small urs, the a explaatio of it will have to iclude somethig i additio to the cogitive mechaisms suggested by Eihor ad Hogarth (1985) ad Rode et al. (1999). Evidece from other areas of research suggests that decisio makers are sometimes sesitive to ur sizes. For example, Dees-Raj ad Epstei (1994) showed that may people preferred drawig from a large ur tha from a smaller oe with fewer wiig balls but a larger proportio of wiig balls, eve whe they uderstood that the probability of wiig was greater i the small ur. Typically, they preferred to draw from a ur cotaiig seve wiig balls amog 100, rather tha from a ur cotaiig oe wiig ball amog 10. Itrospective reports suggested that they preferred the larger ur because it offered more ways of wiig. To clarify the possible effects of ur size o ambiguity aversio, we therefore ivestigated choices i a stadard Ellsberg urs task, usig urs of widely differet sizes, from the covetioal 100 balls dow to just two balls. Method Participats The sample cosisted of 151 udergraduate studets ad members of the geeral public (100 wome ad 51 me) with a mea age of years (SD = 10.24, rage 16 to 76). Prizes of 30 sterlig were awarded to three lottery wiers, with etry to the lottery beig depedet o drawig a blue ball from a ur cotaiig red ad blue balls. The resposes of two participats were illegible ad were discarded, reducig the usable sample size to 149. Materials Three pairs of urs were used, each ur cotaiig red ad blue balls. The pairs differed accordig to the umber of balls i each ur: 2, 10, or 100 balls. For each ur size, oe of the urs cotaied 50% red ad 50% blue balls (the kow-risk ur), ad the other cotaied a radomly selected ratio of red ad blue balls (the ambiguous ur). Desig ad Procedure This experimet was desiged to examie the effects of ur size (2, 10, or 100 balls) o ur choice (kow-risk or ambiguous ur) usig a idepedet-groups experimetal desig. The ratio of red to blue balls was kow to the decisio makers i the kow-risk ur ad was ukow i the ambiguous ur. Participats were told that if they picked a blue ball, they would be etered ito a lottery with the chace of wiig oe of three 30 prizes. They were free to choose from either the kow-risk or the ambiguous ur. Participats were radomly assiged to these three treatmet coditios, ad they bega by fillig i coset forms ad providig demographic ad cotact details. Those assiged to the 100-ball coditio were the preseted with the followig writte istructios: 2008 Hogrefe & Huber Publishers Experimetal Psychology 2008; Vol. 55(1):31 37

4 34 B.D. Pulford & A.M. Colma: Ambiguity Aversio Cosider the followig problem carefully, the write dow your decisio. O the table are two urs, labeled A ad B, cotaiig red ad blue marbles, ad you have to draw a marble from oe of the urs without lookig. If you get a blue marble, you will be etered ito a 30 lottery draw. Ur A cotais 50 red marbles ad 50 blue marbles. Ur B cotais 100 marbles i a ukow color ratio, from 100 red marbles ad 0 blue marbles to 0 red marbles ad 100 blue marbles. The mixture of red ad blue marbles i Ur B has bee decided by writig the umbers 0, 1, 2,..., 100 o separate slips of paper, shufflig the slips thoroughly, ad the drawig oe of them at radom. The umber chose was used to determie the umber of blue marbles to be put ito Ur B, but you do ot kow the umber. Every possible mixture of red ad blue marbles i Ur B is equally likely. You have to decide whether you prefer to draw a marble at radom from Ur A or Ur B. What you hope is to draw a blue marble ad be etered for the 30 lottery draw. Cosider very carefully from which ur you prefer to draw the marble, the write dow your decisio below. You will draw a marble from your chose ur straight afterwards. I prefer to draw a marble from Ur A/Ur B... Mior alteratios were made for the treatmet coditios with smaller urs, replacig the umber 100 with either 2 or 10. So, for example, the two-ball coditio read the mixture of red ad blue marbles i Ur B has bee decided by writig the umbers 0, 1, 2 o separate slips of paper, shufflig the slips thoroughly, ad the drawig oe of them at radom ad the 10-ball coditio read the mixture of red ad blue marbles i Ur B has bee decided by writig the umbers 0, 1, 2,..., 10, o separate slips of paper, shufflig the slips thoroughly, ad the drawig oe of them at radom. Each participat drew a ball from the chose ur, ad those who drew blue balls were etered ito the prize lottery. The ratios of red to blue balls i the ambiguous urs were decided radomly, as described i the writte istructios. I the two-ball coditio, the kow-risk ur cotaied oe red ad oe blue ball, ad i the ambiguous ur the radomizatio procedure resulted i two red balls. I the 10-ball coditio, the kow-risk ur cotaied five red ad five blue balls, ad the ambiguous ur eight red ad two blue balls. I the 100-ball coditio, the kow-risk ur cotaied 50 red ad 50 blue balls, ad the ambiguous ur 53 red ad 47 blue balls. Results Of the 149 decisio makers who participated i the experimet, 106 (71%) chose the kow-risk ur, ad 43 (29%) chose the ambiguous ur. This fidig replicates the basic ambiguity aversio effect across ur sizes, χ² (1, N = 149) = 26.64, p <.001, effect size w =.42 (medium). I this experimet, the same umber of me ad wome chose the ambiguous urs. Results for differet ur sizes are show i Table 1. Experimetal Psychology 2008; Vol. 55(1):31 37 Table 1. Choices of kow-risk ad ambiguous urs of three differet sizes Ur chose Ur size Kow risk Ambiguous 2 29 (64.44%) 16 (35.56%) (78.26%) 10 (21.74%) (70.69%) 17 (29.31%) Total 106 (71.14%) 43 (28.86%) Strog ambiguity aversio effects occurred i all ur sizes, ad ur choice was ot sigificatly iflueced by ur size: χ² (2, N = 149) = 2.12, p =.35, s. To provide a more severe test of the effect of ur size, data from ur sizes of 10 ad 100 were collapsed to determie whether ur choice differed sigificatly betwee smallest (2) ad larger sizes (10 or 100), but the associatio remaied osigificat: χ² (1, N = 149) = 1.41, p =.24, s. Furthermore, if ur sizes 2 ad 10 are collapsed ad compared with ur size 100, the associatio is still osigificat: χ² (1, N = 149) = 0.92, p =.53, s. Take together, these results provide clear-cut cofirmatio of the fidig that the umber of balls, ad hece the umber of possible permutatios of colors i the ambiguous ur, had o sigificat effect o ur choice ad hece ambiguity aversio. Discussio Oly 29% of the decisio makers chose ambiguous urs, replicatig the fudametal ambiguity aversio effect across ur sizes. The participats kew that every possible distributio of balls i the ambiguous urs was equally probable, with the obvious implicatio that the objective chaces were equal i the kow-risk ad ambiguous urs, but a medium-sized ambiguity aversio effect occurred oetheless. This is hardly a ew fidig, but the occurrece of a substatial ad sigificat ambiguity aversio effect eve i the smallest ur fails to cofirm predictios implied by two leadig theoretical iterpretatios of ambiguity aversio (Eihor & Hogarth, 1985; Rode et al., 1999). Accordig to Eihor ad Hogarth s (1985) model, smaller urs should ecessarily be perceived as less ambiguous tha larger oes, because far fewer distributios eed to be imagied ad excluded i the metal simulatio that is hypothesized to occur durig the process of judgmet ad decisio makig, ad accordig to the model s equatios, this should reduce the size of the ambiguity aversio effect. We foud o ur size effect, ad a sigificat ambiguity aversio effect occurred eve i the smallest ur, cotaiig just two balls. With oly three distributios to simulate, amely o red balls, oe red ball, ad two red balls, compared to 101 distributios i the largest 100-ball ur, ad oe to exclude i either case, the ambiguity aversio 2008 Hogrefe & Huber Publishers

5 B.D. Pulford & A.M. Colma: Ambiguity Aversio 35 effect should have bee elimiated or at least greatly atteuated, yet it remaied sigificat eve i this very small ur. The risk-sesitive foragig theory of Rode et al. (1999) also predicts very little ambiguity aversio i the smallest ur, because expected payoff variace is close to zero whe there are oly two balls i the ur, ad our fidigs are therefore icosistet with that theory also. If ambiguity aversio is related to variace avoidace, the our fidigs suggest that decisio makers have a tolerace for payoff variace up to some threshold above σ 2 = 0.25 ad that, for all ur sizes, they ted to prefer kow-risk alteratives for some other reaso. Whatever accouts for ambiguity aversio, our fidig that the effect remaied sigificat i the smallest ur seems difficult to recocile with the purely cogitive theories that we have cosidered i this article. I spite of the vastly smaller umber of distributios i the smallest ur, careful data aalysis failed to reveal evidece of ay dimiutio of ambiguity aversio. Oe possibility is that ambiguity aversio is drive by the rage of probabilities of success the rage was from zero to uity i urs of all three sizes i our experimet rather tha the umber of distributios that eed to be metally simulated (as suggested by Eihor & Hogarth, 1985) or the expected payoff variace (as suggested by Rode et al., 1999). What is most revealig is the positive fidig of a sigificat effect i the smallest ur, ad this eeds to be take ito accout i ay iterpretatio of ambiguity aversio. We have provided prelimiary rather tha coclusive evidece that we hope will ispire further research ito ur size effects. Our experimetal desig was restricted to betwee-subjects ur size comparisos, to avoid cofoudig ur size with subject-expectacy effects, although a withi-subjects desig might possibly have made ur size more saliet ad caused decisio makers to have bee more sesitive to these differeces (cf. Dees-Raj & Epstei, 1994). If decisio makers were preseted i a future study with choices betwee ambiguous urs of differet sizes, the a sigificat preferece for smaller urs would provide evidece i favor of theories of ambiguity aversio, such as those of Eihor ad Hogarth (1985) ad Rode et al. (1999), that imply differet degrees of aversio i urs of differet sizes. O the other had, a absece of ay sigificat ur size prefereces would be cosistet with theories that have o obvious implicatios for ur size, icludig the iterpretatio that we suggest below. However, irrespective of ay betwee-subjects or withisubjects ur size comparisos, our fidig of sigificat ambiguity aversio i the smallest ur is icosistet with purely cogitive theories that imply that ambiguity aversio arises from the effort ivolved i metally simulatig the possible distributios or avoidig high-variace optios. Our research was also restricted to comparig prefereces for kow-risk urs with ambiguous urs cotaiig ukow umbers of wiig balls betwee 0 ad 100 per cet, although we ackowledge that prefereces for restricted-rage ambiguous urs with (for example) betwee 40 ad 60 per cet wiig balls also deserve ivestigatio. With these caveats i mid, we believe that existig theories of ambiguity aversio may eed to be recosidered i the light of our fidigs. Our results show that ambiguity aversio occurs whe decisio makers are uable to quatify the risks ivolved i ambiguous optios, eve whe the outcome sets are easily cogitively simulated ad the expected payoff variace is very small. We agree with Rode et al. s (1999) fidig that ambiguity aversio is caused by aversio to the ukow probability parameter ad is ot due to a comparative process. We suggest that ambiguity aversio may arise from a more geeral itolerace of ucertaity, ad i particular from the aversive ad disturbig effects of ucertaity, irrespective of ur size. Most people prefer to avoid exposig themselves to evets ad circumstaces that they do ot uderstad (Becker & Browso, 1964; Freesto, Rhéaume, Letarte, Dugas, & Ladouceur, 1994; Furham, 1994; Ghosh & Ray, 1997), ad ambiguity aversio may be a particular maifestatio of this. Ucertaity iduces a disturbig ad aversive psychological state. However, there are large idividual differeces i itolerace of ucertaity. Habitual worriers tolerate ucertaity less well tha others ad appear to be especially proe to defie ambiguous prospects as threateig (Butler & Mathews, 1983, 1987), ad this may explai why some people display more ambiguity aversio tha others. Purely cogitive iterpretatios that igore such affective processes are ulikely to provide a complete explaatio of ambiguity aversio. Ackowledgmets The research reported i this article was fuded by research grat RES from the Ecoomic ad Social Research Coucil of the UK. The authors are grateful to Clare Davies for assistace with the recruitmet of participats ad data collectio ad to Ali al-nowaihi, Sajit Dhami, ad Kostatios Katsikopoulos for suggestios for improvemet of earlier drafts. We dedicate this article to the memory of Ia Poutey, who died while it was beig revised, o 12 July Refereces Becker, S.W., & Browso, F.O. (1964). What price ambiguity? Or the role of ambiguity i decisio-makig. Joural of Political Ecoomy, 72, Butler, G., & Mathews, A. (1983). Cogitive-processes i axiety. Advaces i Behaviour Research ad Therapy, 5, Butler, G., & Mathews, A. (1987). Aticipatory axiety ad risk perceptio. Cogitive Therapy ad Research, 11, Camerer, C.F. (1995). Idividual decisio makig. I J.H. Kagel 2008 Hogrefe & Huber Publishers Experimetal Psychology 2008; Vol. 55(1):31 37

6 36 B.D. Pulford & A.M. Colma: Ambiguity Aversio & A.E. Roth (Eds.), Hadbook of experimetal ecoomics (pp ). Priceto, NJ: Priceto Uiversity Press. Camerer, C.F., & Weber, M. (1992). Recet developmets i modelig prefereces: Ucertaity ad ambiguity. Joural of Risk ad Ucertaity, 5, Curley, S.P., & Yates, J.F. (1989). A empirical evaluatio of descriptive models of ambiguity reactios i choice situatios. Joural of Mathematical Psychology, 33, Dees-Raj, V., & Epstei, S. (1994). Coflict betwee ituitive ad ratioal processig: Whe people behave agaist their better judgmet. Joural of Persoality ad Social Psychology, 66, Eihor, H.J., & Hogarth, R.M. (1985). Ambiguity ad ucertaity i probabilistic iferece. Psychological Review, 92, Ellsberg, D. (1961). Risk, ambiguity, ad the savage axioms. Quarterly Joural of Ecoomics, 75, Feller, W. (1961). Distortio of subjective probabilities as a reactio to ucertaity. Quarterly Joural of Ecoomics, 75, Frisch, D., & Baro, J. (1988). Ambiguity ad ratioality. Joural of Behavioral Decisio Makig, 1, Freesto, M.H., Rhéaume, J., Letarte, H., Dugas, M.J., & Ladouceur, R. (1994). Why do people worry? Persoality ad Idividual Differeces, 17, Furham, A. (1994). A cotet, correlatioal ad factor aalytic study of four tolerace of ambiguity questioaires. Persoality ad Idividual Differeces, 16, Ghosh, D., & Ray, M.R. (1997). Risk, ambiguity, ad decisio choice: Some additioal evidece. Decisio Scieces, 28, Kere, G., & Gerritse, L.E.M. (1999). O the robustess ad possible accouts of ambiguity aversio. Acta Psychologica, 103, Keyes, J.M. (1921). A treatise o probability. Lodo: Macmilla. Kight, F.H. (1921). Risk, ucertaity ad profit. Bosto: Houghto, Schaffer & Marx. Krähmer, D., & Stoe, R. (2006). Regret ad ambiguity aversio. Retrieved Jue 15, 2006, from the Freie Uiversität Berli web site: Raiffa, H. (1961). Risk, ambiguity, ad the savage axioms: Commet. Quarterly Joural of Ecoomics, 75, Rode, C., Cosmides, L., Hell, W., & Tooby, J. (1999). Whe ad why do people avoid ukow probabilities i decisios uder ucertaity? Testig some predictios from optimal foragig theory. Cogitio, 72, Savage, L.J. (1954). The foudatios of statistics. New York: Wiley. Slovic, P., & Lichtestei, S. (1971). Compariso of Bayesia ad regressio approaches to the study of iformatio processig i judgmet. Orgaizatioal Behavior ad Huma Performace, 6, Stephes, D.W., & Krebs, J.R. (1986). Foragig theory. Priceto, NJ: Priceto Uiversity Press. Tetlock, P.E. (1991). A alterative metaphor i the study of judgmet ad choice: People as politicias. Theory ad Psychology, 4, Received May 2, 2006 Revisio received October 24, 2006 Accepted October 27, 2006 Brioy D. Pulford School of Psychology Uiversity of Leicester Leicester LE1 7RH UK bdp5@le.ac.uk Appedix Expected Payoff Variace i Ambiguous Urs A ambiguous ur cotais balls, of which k (k =0,1,..., ) are red ad the rest black, with every value of k equally likely. A decisio maker draws a ball ad receives a payoff of x = 1 if it is red. The probability of a red ball is 1/2, by symmetry. Formally, 1 +1 k 1 = k (+1) = 1 (+1) 1 2 (+1) = 1 2. I a ur cotaiig exactly k red balls, the expected payoff E(x)=μ = k/. By defiitio, the variace σ 2 = E(x μ) 2 = Experimetal Psychology 2008; Vol. 55(1):31 37 E(x 2 2μx + μ 2 ). Because μ is a costat, σ 2 = E(x 2 ) 2μE(x) +μ 2, ad because μ = E(x), σ 2 = E(x 2 ) 2[E(x)] 2 +[E(x)] 2 = E(x 2 ) [E(x)] 2 = k 12 k2 = k 2 k2. 2 We first prove by iductio that k 2 = [(+1)(2+1)]/6. For = 0, the formula reduces to 0 = 0, which is true. We the prove that if it holds for, the it must also hold for k 2 = k 2 +(+1) Hogrefe & Huber Publishers

7 B.D. Pulford & A.M. Colma: Ambiguity Aversio 37 Usig the expressio for k 2 assumed above, this is equal to [( + 1)(2 + 1)]/6 + ( +1) 2 =[( + 1)/6][(2 + 1) + 6( + 1)], which simplifies to [( + 1)/6][ ] = [( + 1)/6][( + 2)(2 + 3)] = [( + 1)/6]( + 2)[(2 + 1) + 1], ad this is equal to {[( + 1)][( + 1) + 1][2( + 1) + 1]}/6, as required. Therefore, for all, k 2 = [(+1)(2+1)]/6. The sum of variaces for all values of k is k k2 k 2 = k 2 2. This is equal to 1 k 1 k 2 = 1 (+1) (+1)(2+1), 2 6 which simplifies to ( + 1)/2 [1/][( + 1)(2 + 1)]/6. The expected variace is thus E(σ 2 )= 1 ( (+1)(2+1) 6. Therefore, E(σ 2 ) = 1/2 (2 + 1)/6 = 1/2 1/3 1/6 = 1/6 1/6. This expressio measures the expected payoff variace i a ambiguous ur. The expected variace teds to 1/6 as. For =2, E(σ 2 ) = 1/6 1/12 = 1/ ; for = 10, E(σ 2 ) = 9/60 = 0.150; ad for = 100, E(σ 2 ) = 99/600 = Hogrefe & Huber Publishers Experimetal Psychology 2008; Vol. 55(1):31 37