Preference Cloud Theory: Imprecise Preferences and Preference Reversals Oben Bayrak and John Hey

Size: px

Start display at page:

Download "Preference Cloud Theory: Imprecise Preferences and Preference Reversals Oben Bayrak and John Hey"

Marvin Moore
5 years ago
Views:

1 Preference Cloud Theory: Imprecise Preferences nd Preference Reversls Oben Byrk nd John Hey This pper presents new theory, clled Preference Cloud Theory, of decision-mking under uncertinty. This new theory provides n explntion for empiriclly-observed Preference reversls. Centrl to the theory is the incorportion of preference imprecision which rises becuse of individuls vgue understnding of numericl probbilities. We combine this concept with the use of the Alph model (which builds on Hurwicz s criterion) nd construct simple model which helps us to understnd vrious nomlies discovered in the experimentl economics literture tht stndrd models cnnot explin. 1. Introduction Suppose you re sked to stte your subjective vlue for lottery ticket which gives 10 dollrs with probbility of nd zero otherwise: Cn you pin your vlue down to single precise number or do you end up with rnge of vlues? Our preference imprecision rgument sttes tht individuls most likely end up hving n intervl; preferences re not s precise s stndrd theory presumes (for evidence, see: Butler nd Loomes, 1988, 2007, 2011; Dubourg et l, 1994, 1996; nd Morrison, 1998). Such imprecision might explin observed nomlies of stndrd theory (Butler nd Loomes 2011). Most importntly, imprecision might be the explntion of the nomlies, despite reserchers efforts, in the lst four decdes, focusing on precise but non-stndrd preferences such s loss version. Suppose now you re ssigned to be buyer. You re most likely to stte vlue close to the lower bound of this rnge; the converse is true when you re ssigned to be seller. The model presented in this pper nswers two key questions: wht is the psychologicl mechnism behind the formtion 1

2 of this rnge?; how do individuls select one vlue from this rnge? It lso shows how the incorportion of imprecision cn provide n explntion for preference reversls. For the first question our theory proposes tht imprecision rises due to the decision-mker s vgue understnding of the probbilities involved. The empiricl support for this ssertion comes from the psychophysics literture; see Budescu et l (1988). In n experiment reported by them subjects were sked to stte bids for lotteries; in the lotteries the probbilities were represented numericlly, grphiclly or verblly. The results suggested tht bids nd ttrctiveness rtings re lmost identicl under the different representtions (See Budescu nd Wllsten (1990) nd Bisntz et l (2005) for further evidence). Wllsten nd Budescu (1995) explin tht the similrity of behvior under different representtion modes is due to similrities in the vgue understnding of probbilities. We therefore rgue tht numericl, objective, probbility corresponds to rnge of probbilities nd subjects use this rnge in their clcultions 1. There is lso implicit evidence from Plott nd Zeiler (2005) nd Isoni et l (2011) who find tht the endowment effect is observed only for the lottery tickets, but not for ordinry mrket goods such s mugs nd cndies. Zimmer (1984) introduced useful insight from n evolutionry perspective: he noted tht the probbility concept in numericl sense is reltively new concept, ppering s recently s the 17 th century. However, people were communicting uncertinty vi verbl expressions long before probbility ws codified in mthemticl terms. Zimmer further suggested tht people process uncertinty in verbl mnner nd mke their decisions bsed on this processed informtion, not on the numericl informtion. We therefore ssume tht decision mkers mp ny given objective probbility into n intervl. This implies tht people end up with rnge of expected utilities (EUs) nd they do not hve prior knowledge bout their true EU from this rnge. For the second question, pinning down this rnge to single vlue cn be modelled s decision problem under mbiguity. We use the Alph Model (embodying Hurwicz s criterion) to provide vlution of the 1 Verbl expressions include rre, very likely etc. Ech expression cn be interpreted s rnge of probbilities tht my vry from individul to individul. 2

3 prospect, given s the weighted verge of the worst nd the best possible EU. The next section formlises these ides. 2. Theory In order to give the intuition of the theory we focus on two-outcome lotteries. Let S be finite stte spce, with elements s1 nd s 2 ; let x 1 nd x 2 be the corresponding consequences. We ssume tht ux, whereu. u x 1 2 is the individul s Neumnn-Morgenstern utility function. For ech stte there is n objective probbility; cll them p nd1 p. The crucil point of our theory is tht individuls perceive these probbilities vguely, nd hence p is mpped to rnge:p, p. As fr s the individul is concerned, the worst sitution is when probbility p is llocted to stte s 1 (nd probbility1p to stte s 2, nd the best sitution is when probbility p is llocted to stte s 1 (nd probbility1p to stte s 2 ). So the worst nd best expected utilities re p ux 1 p ux nd p ux 1 p ux 1 2 respectively. 1 2 At this point PCT uses the Alph Mx-Min criterion to evlute the lottery. This gives EU p u x p u x p p u x p u x (1) One cn think of such decision-mker s mixture of pessimist nd n optimist: of the time he or she ssumes the worst, nd1 of the time the best. The prmeter we cll the imprecision level; our theory postultes tht it is function of p (the objective probbility) nd (the individul-specific sophistiction level). A reltively unsophisticted individul would disply reltively high imprecision. For exmple, stock brokers nd gmblers who re expected to be more fmilir with the concept of probbility exhibit lower imprecision thn the ordinry mn. As fr s the shpe of,p is concerned we ssume tht individuls exhibit no imprecision if the probbility is 0 or 1 since the events occurring with these probbilities re not probbilistic events in 3

4 dily lnguge, tht is, the event either never hppens or lwys hppens. Secondly, imprecision reches mximum t becuse it implies the event is neither likely nor unlikely, this incommensurbility mkes it difficult to derive mening from this probbility. Finlly, for simplicity we ssume tht,p is symmetric round p. 3. Explining Preference Reversls We show how preference reversls cn rise with our model. We illustrte using the CRRA utility function for u. u() z z (2) For 0, the function is concve, implying risk version. For simplicity, we focus on P nd $ bets tht give either positive pyoff or zero: P-bet xp, ;0 nd $-bet yq, ;0 with yx0 nd 1 p q 0 where p nd q re the winning probbilities, nd x nd y re the winning prizes of the P-bet nd $-bet, respectively. If the individul prefers the P-bet over the $-bet in the choice tsk, we cn write: $ EU P bet EU bet (2) 1 1 p x p x q y p y (3) Thus the criticl vlue for in determining whether the P-bet preferred to the $-bet is: y q x p * 2 y x (4) If the ctul is greter thn * the individul chooses the P-bet; if lower the $-bet. When we come to vlutions we need to tell different story. Let us consider willingness-to-ccept. Tke the P-bet. Imgine tht the individul owns the P-bet nd is sked the minimum mount for which he or she would sell it the WTA. If it is sold the individul hs WTA (presumbly n mount less thn x ), nd the worst thing tht cn hppen to the individul is tht the P-bet would hve pid out x, nd the best thing tht cn hppen to the individul is tht the P-bet would hve pid out 0. 4

5 So the pessimist ttches weight p to the possibility of getting x, while the optimist ttches weight p. So we get Similrly, The criticl vlue for is: 1 1 u WTA P bet p x p x (5) $ bet 1 1 u WTA q y q y (6) x p y q ** 2 y x (7) If is greter ** the individul vlues the $-bet higher thn the P-bet; if it is lower the P-bet is vlued higher. We explore the prmeters of PCT in three cses: risk neutrl, risk verse nd risk loving. Consider Figure 2, where r set to 1; P-bet 1.25,;0, $-bet 5,0.2;0 Figure 1 2 P-bet chosen nd vlued more Stndrd PR non-stndrd PR β Stndrd PR non-stndrd PR Risk Averse, =0.7 Risk Neutrl, =1 Risk Lover, =1,3 β $-bet chosen nd vlued more Vlution (bove: $-Bet vlued more) Choice (bove: P-Bet chosen) Stndrd PR non-stndrd PR β The dshed line shows the ** boundry nd the solid line is the * boundry. Above the dshed line the $-bet is vlued higher nd bove the solid gry line the P-bet is chosen; the region between 2 For the imprecision level, we use p p p, 1, there is no prticulr reson behind choosing this, except it is simple nd stisfies the ssumptions of the theory. We normlize the expected vlue (EV) of the P bet by setting its pyoff equl to 1 / p; r is the EV of the $ bet s rtio of the EV of the P-bet. Therefore the winning prize of the $ bet equls r / p. 5

6 the two lines is clled the consistency rnge where the chosen bet is vlued more. For risk-neutrl individul, in the cse of imprecision( 0), stndrd Preference reversl occurs if, when it is lower thn, the model predicts non-stndrd Preference reversl. For risk-verse individul, in the consistency rnge individul chooses the P-Bet nd vlues it higher; wheres for the risk loving cse the $-Bet is chosen nd vlued more. This mkes sense: the P-bet would be more ttrctive for risk-verse individul. Overll, in the cse of imprecision ( 0), sufficiently high level of pessimism results in stndrd Preference Reversl while optimism implies non-stndrd Preference Reversl. Next we consider the cse in which the winning probbilities remin the sme, but the winning prize of the $-Bet vries. Figure 3 shows the criticl bounds for three cses: Figure 2 Stndrd PR Vlution (r=) Vlution (r=1) Consistency Rnge non-stndrd PR β 1 0 Vlution (r=1.2) Choice (r=) Choice (r=1) 0 Choice (r=1.2) 1 Risk Averse, =0.7 The dshed lines show the vlution boundry, nd the solid lines show the choice boundry, for three levels of r (,1,1.2). For risk-verse decision-mker, the consistency rnge shrinks s r increses up to certin level. The prmeter vlues to induce stndrd nd non-stndrd preference reversls converge to the risk neutrlity bseline cse. However, bove this criticl level of r, the consistency rnge fvors the $-bet nd it expnds s r increses. Even if we increse the reltive ttrctiveness of the $-bet to extreme vlues, the model predicts tht both stndrd nd non-stndrd preference reversls cn be observed. 6

7 Conclusion We demonstrted the intuition of the Preference Cloud Theory nd explined the possible preference reversls. In the sme mnner other nomlies of EUT cn be explined such s Endowment Effect nd Allis Prdox. To best of our knowledge, this is the only theory tht cn explin nomlies in Expected Utility theory without incorporting the loss version notion of the Prospect Theory. 7

8 REFERENCES Arrow, Kenneth J. nd Leonid Hurwicz, An Optimlity Criterion for Decision-Mking under Ignornce. In Crter, C. F. nd J. L. Ford, eds., Uncertinty nd Expecttions in Economics. Oxford: Bsil Blckwell, Bisntz, Ann M., Stephnie Schinzing Mrsiglio, nd Jessic Munch Displying Uncertinty: Investigting the Effects of Disply Formt nd Specificity. Humn Fctors: The Journl of the Humn Fctors nd Ergonomics Society 47 (4): Budescu, Dvid V., Shlv Weinberg, nd Thoms S. Wllsten Decisions Bsed on Numericlly nd Verblly Expressed Uncertinties. Journl of Experimentl Psychology: Humn Perception nd Performnce 14 (2): 281. Budescu, Dvid V., nd Thoms S. Wllsten Dydic Decisions with Numericl nd Verbl Probbilities. Orgniztionl Behvior nd Humn Decision Processes 46 (2): Butler, Dvid, nd Grhm Loomes Decision Difficulty nd Imprecise Preferences. Act Psychologic 68 (1 3): Butler, Dvid J., nd Grhm C. Loomes Imprecision s n Account of the Preference Reversl Phenomenon. Americn Economic Review 97 (1): Butler, Dvid, nd Grhm Loomes Imprecision s n Account of Violtions of Independence nd Betweenness. Journl of Economic Behvior & Orgniztion 80 (3): Dubourg, W. R., M. W. Jones-Lee, nd Grhm Loomes Imprecise Preferences nd the WTP-WTA Disprity. Journl of Risk nd Uncertinty 9 (2): Dubourg, Jones-Lee, nd Grhm Loomes Imprecise Preferences nd Survey Design in Contingent Vlution. Economic 64 (256):

9 Isoni, Andre, Grhm Loomes, nd Robert Sugden. The Willingness to Py Willingness to Accept Gp, the Endowment Effect, Subject Misconceptions, nd Experimentl Procedures for Eliciting Vlutions: Comment. The Americn Economic Review 101, no. 2 (2011): Morrison, Gwendolyn C Understnding the Disprity between WTP nd WTA: Endowment Effect, Substitutbility, or Imprecise Preferences? Economics Letters 59 (2): Plott, Chrles R., nd Kthryn Zeiler. The Willingness to Py-Willingness to Accept Gp, the Endowment Effect, Subject Misconceptions, nd Experimentl Procedures for Eliciting Vlutions. The Americn Economic Review 95, no. 3 (June 1, 2005): Wllsten, Thoms S, Smuel Fillenbum, nd Jmes A Cox Bse Rte Effects on the Interprettions of Probbility nd Frequency Expressions. Journl of Memory nd Lnguge 25 (5): Wllsten, Thoms S., nd Dvid V. Budescu A Review of Humn Linguistic Probbility Processing: Generl Principles nd Empiricl Evidence. The Knowledge Engineering Review 10 (01): Zimmer, Alf C A Model for the Interprettion of Verbl Predictions. Interntionl Journl of Mn- Mchine Studies 20 (1):

10 Appendix A Figure 4 Stndrd PR Stndrd PR Stndrd PR Consistency Rnge Consistency Rnge Consistency Rnge non-stndrd PR non-stndrd PR non-stndrd PR β β Risk Averse, =0.7 Risk Neutrl, =1 Risk Lover, =1,3 β Vlution (r=) Vlution (r=1) Vlution (r=1.2) Choice (r=) Choice (r=1) Choice (r=1.2) 10

A ppendix to. I soquants. Producing at Least Cost. Chapter

A ppendix to. I soquants. Producing at Least Cost. Chapter A ppendix to Chpter 0 Producing t est Cost This ppendix descries set of useful tools for studying firm s long-run production nd costs. The tools re isoqunts nd isocost lines. I soqunts FIGURE A0. SHOWS