基于差分的等級依賴效用模型行為決策權重擴展

李偉兵　王金山　謝英超

摘要 基于等級依賴期望效用模型（RDEU），提供了一個簡單但有效的行為決策權重擴展.因結果序列具有直線增長趨勢，介紹一種基于一階差分的新權重，并證明其對拆分效應的有效性.差分權重和RDEU權重的凸組合構成最終決策權重命名為D'RDEU權重，它不僅可繼承RDEU的優點，也可克服RDEU的兩個不足.特別是，它通過拆分獲得隨機優勢，可從理論上解釋拆分效應.也提供了連續形式的D'RDEU模型，連續模型的存在證明D'RDEU比DRDEU更實用.

關鍵詞 權重；等級依賴效用模型；差分；拆分效應；隨機優勢

中圖分類號 F019 文獻標識碼 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.

References

1J V NEUMANN， O MORGENSTERN. Theory of games and economic behavior M. New Jersey： Princeton University Press， 1944.

2M ALLAIS. Le comportement de thomme rationnel devant le risqué： critique des postulats et axioms de IEclo Americaine J. Econometrica， 1953， 21（4）： 503-546.

3M ALLAIS. The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American School. In M. Allais & O. Hagen （Eds.）， Expected utility hypothesis and the Allais paradox M. Dordrecht， The Netherlands： Reidel， 1979.

4D KAHNEMAN， A TVERSKY. Prospect theory， an analysis of decisionmakingunder risk J. Econometrica ， 1979（47）： 263-292.

5S M ZHANG. Reexploring the common ratio effect in the unit triangle for anticipated utility theory J. Journal of Quantitative Economics， 2009， 31（1）： 1-7.

6J QUIGGIN. A theory of anticipated utility J. Journal of Economic Behavior and Organization 1982， 3（4）： 323–343.

7M J MACHINA. Review of “generalized expected utility theory： the rankdependent model” J. Journal of Economic Literature. 1994， 32 （3）： 1237-1238.

8M H BIRNBAUM. Tests of rankdependent utility and cumulative prospect theory in gambles represented by natural frequencies： effects of format， event framing， and branch splitting J. Organizational Behavior and Human Decision Processes，2004（ 95）： 40-65.

9S J HUMPHREY. Are eventsplitting effects actually boundary effects？ J. Journal of Risk and Uncertainty， 2001，22（1）： 79-93.

10M H BIRNBAUM. New paradoxes of risky decision making J. Psychological Review. 2008， 115（2）： 463-501.

11M H BIRNBAUM. Tests of branch splitting and branchsplitting independence in Allais paradoxes with positive and mixed consequences J. Organizational Behavior and Human Decision Processes. 2007（102）： 154-173.

12A TVERSKY， D KAHNEMAN. Rational choice and the framing of decisions J. Journal of Business. 1986， 59（4）： 251-278.

13P WAKKER， I EREV， E WEBER. Comonotonic independence： the critical test between classical and rankdependent utility theories J. Journal of Risk and Uncertainty. 1994， 9（3）： 195-230.

14J S Wang， W B Li. Differencebased RankDependent Expected Utility Model J. Journal of Mathematics in Practice and Theory. 2014， 44（23）： 297-321.

15D PRELEC. The probability weighting function J. Econometrica. 1998（66）： 497-527.

16C S WEBB， H ZANK. Accounting for optimism and pessimism in expected utility J. Journal of Mathematical Economics. 2011（47）： 706-717.

17A TVERSKY， D KAHNEMAN. Advances in prospect theory： cumulative representation of uncertainty J. Journal of Risk and Uncertainty. 1992， 5（4）： 297-323.