基于差分的等级依赖效用模型行为决策权重扩展

李伟兵　王金山　谢英超

摘要 基于等级依赖期望效用模型（RDEU），提供了一个简单但有效的行为决策权重扩展.因结果序列具有直线增长趋势，介绍一种基于一阶差分的新权重，并证明其对拆分效应的有效性.差分权重和RDEU权重的凸组合构成最终决策权重命名为D'RDEU权重，它不仅可继承RDEU的优点，也可克服RDEU的两个不足.特别是，它通过拆分获得随机优势，可从理论上解释拆分效应.也提供了连续形式的D'RDEU模型，连续模型的存在证明D'RDEU比DRDEU更实用.

关键词 权重；等级依赖效用模型；差分；拆分效应；随机优势

中图分类号 F019 文献标识码 A

AbstractBased on the underlying rankdependent expected utility （RDEU）， we provide a simple but effective extension to the behavioral decision weight. As outcome series satisfies right a linear growth trend， we introduce a new type of weight based on firstorder difference， and have proved its effectiveness on event-splitting effect. New weight and RDEU weight constitute final combined decision weight named D'RDEU weight as final decision weight， which can not only inherit the advantages of RDEU but also can successfully overcome RDEU's two stubborn issues. Especially， it can theoretically explain event-splitting effect by making event-splitting obtaining stochastic dominance. At last， we provide the continuous form of D'RDEU model. The existence of continuous D'RDEU proves D'RDEU is more functional than DRDEU.

Key wordsdecision weight； RDEU； firstorder difference； eventsplitting effect； stochastic dominance1Introduction

As a normative theory in the behavioral sciences， the expected utility （EU） model 1 is widely used for predicting and describing choice under risk and uncertainty. However， because of lots of no accidental but systematical violations to empirical behavioral research， e.g.， Allais paradox 2， common consequence effect 3 and common ratio effect 4， 5， EUs usefulness is limited as a descriptive theory. We know EU has two main features. It is （a） linear in probabilities without probability weighting function and （b） utility is defined on final wealth levels without consideration of gain or loss or compare. The most common extension to EU is the relaxation of linearity in probabilities. Among a series of descriptive models is Quiggins （1982） 6 rankdependent expected utility （RDEU） model which replaces probabilities with decision weight by probability weighting function. With this modification to EU， RDEU model can perfectly explain Allais paradox etc.， withal， Machina （1994） 7 describes RDEU as the most natural and effective alternative to EU.

Although RDEU model is widely used in behavioral decision choices and prediction， and has solved a mass of practical problem up to now， it still has two stubborn issues. It is （a） unable to explain eventsplitting effect （ESE） 8-11 including branchsplitting independence and violation of monotonicity 12 and （b） has an incontrovertible flaw on the setting of decision weight. The flaw implies a property called comonotonic independence by Wakker， Erev and Weber （1994） 13 which is regard to be incongruent. Therefore， RDEU also needs a modification like EU. Wang and Li （2014） 14 proposes DRDEU model， it can both solve abovementioned two issues， but it has no continuous form. Making a better extension is the main work of our paper.

The paper is organized as follow. Section II reviews these two incontrovertible flaws in RDEU model. In Section III， we present the feasibility and availability of firstorder difference influencing behavioral decision， and present a new model named D'RDEU. We will test our new model aimed to eventsplitting effect in Section IV. Section V offers the continuous form of D'RDEU. Last is conclusion.

2Two Stubborn Issues of RDEU

1）Unable to Explain Eventsplitting Effect

The eventsplitting effect occurs when an event， which yields a positive outcome in one lottery but zero under another， is separated into two subevents and this increases the relative attractiveness of the former lottery. RDEU is proved unable to explain it.

And we also realize that the primary cause is the setting of decision weight is not dependent on the magnitude of the outcome xi but on the rank of the series.

3）Weight on FirstOrder Difference and D'rdeu

We know if a time series has a linear growth trend， if we apply single exponential smoothing model to predict， there will be obvious lagging deviation. The best solution is applying firstorder difference （FD） exponential smoothing model. So difference plays an important role in time series and similar series. We are curious to if difference can produce a type of appropriate decision weight in behavioral decision to solve two stubborn issues in Section II of RDEU.

The RDEUs series of outcomes satisfies right the linear growth trend as these outcomes are incremental along with the subscript i=1， 2，…， n. So the idea to produce a type of appropriate decision weight through firstorder difference is feasible. We have validated that the change trend of FD is consistent with the empirical behavioral evidence， see Eq. （7）. If the normalized FD can hold abovementioned consistence， then our idea is available.

We regard the role of di as that of pi， and Eq. （21） shows that event after splitting obtains stochastic dominance 16， 17. In terms of probabilities of obtaining an outcome， stochastic dominance implies that an elementary shift of probability from a lower ranked outcome xi to an adjacent better outcome xi+1 improve a lottery， so eventsplitting can increase the relative attractiveness of the former lottery.

So we get the third property of FD weight di （adjacent to the former two properties in Section III）

Unlike unaltered pi which is useless to explain eventsplitting effect， smart di is theoretically consistent to eventsplitting effect.

Thus far， we have confirmed that D'RDEU model of Eq. （13） and （14） can explain eventsplitting effect， i.e. the first stubborn issue （a） of RDEU is solved， too.

D'rdeus Continuous Form

Section III has given the discrete form of D'RDEU， while the continuous form is also necessary to practical problem. While Wang and Lis DRDEU model has no continuous form. Next we will give the continuous form of D'RDEU. We begin at continuous form of RDEU， and then get our result by contrast.

According to RDEU， we consider the continuous outcome x from a to b （Let b-a=c）， whose probability density function is f. Let probability distribution function （PDF） is F. We introduce function G=1-F named probability tail distribution function （PTDF）. For an agent， if his utility function is u， then his RDEU is （Quiggin 6， 1982）

RDEU（X）=∫bau（x）φ（1）G（x）f（x）dx

=-∫bau（x）dφG（x）. （22）

Where φ is still probability weighting function as in Eq. （1） but which affects on the probability tail distribution function.

We divide x into n-1 parts from x1=a to xn=b， and keep consistent with the differences of Eq. （7） except Δxn+1=0. We have

3Conclusion

Our main objective in this paper has been to provide an extension to behavioral decision weight which not only inherits the advantages of original RDEU but also can overcome its disadvantages. We have presented the disadvantages of RDEU， and both of them can be overcome in the new model， D′RDEU. Our conclusion indicates that eventsplitting brings stochastic dominance and improves a lottery.

So D′RDEU can theoretically explain eventsplitting effect.

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