Decisions under risk: Dispersion and skewness

Abstract

When people take decisions under risk, it is not only the expected utility that is important, but also the shape of the distribution of utility: clearly the dispersion is important, but also the skewness. For given mean and dispersion, decision-makers treat positively and negatively skewed prospects differently. This paper presents a new behaviourally-inspired model for decision making under risk, incorporating both dispersion and skewness. We run a horse race of this new model against six other models of decision making under risk and show that it outperforms many in terms of goodness of fit and shows a reasonable performance in predictive ability. It can incorporate the prominent anomalies of standard theory such as the Allais paradox, the valuation gap, and preference reversals, and also the behavioural patterns observed in experiments that cannot be explained by Rank Dependent Utility Theory.

Appendices

Appendix A

Estimated Preference Functionals

1) EU: Expected Utility—subjects choose on the basis of expected utility:

$$ V(p)={p}_2{u}_2+{p}_3{u}_3+{p}_4{u}_4 $$

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Parameters estimated: u₂, u₃ and u₄.

2) DA: Disappointment Aversion—subjects choose on the basis of expected (modified) utility—where utility is modified ex post to take account of any disappointment or delight experienced:

$$ V(p)=\min \left({W}_1,{W}_2,{W}_3\right) $$

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$$ {W}_1=\left(1+\beta \right){p}_2{u}_2+\left(1+\beta \right){p}_3{u}_3+{p}_4{u}_4+\beta {p}_1+\beta {p}_2+\beta {p}_3 $$

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$$ {W}_2=\left(1+\beta \right){p}_2{u}_2+{p}_3{u}_3+{p}_4{u}_4+\beta {p}_1+\beta {p}_2 $$

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$$ {W}_3={p}_2{u}_2+{p}_3{u}_3+{p}_4{u}_4+\beta {p}_1 $$

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Parameters estimated u₂, u₃, u₄ and β.

3) DS: See Section 2.

Parameters estimated u₂, u₃, u₄ and k.

4) PR: Prospective Reference—subjects choose on the basis of a weighted average of the expected utility calculated using the correct probabilities and the expected utility calculated using equal probabilities for all the non-null outcomes:

$$ V(p)=\lambda \left({p}_2{u}_2+{p}_3{u}_3+{p}_4{u}_4\right)+\left(1-\lambda \right)\left({a}_2{u}_2+{a}_3{u}_3+{a}_4{u}_4\right) $$

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a_i = 1/n(p) andn(p) is the number of non-zero elements in p.

Parameters estimated: u₂, u₃, u₄ and λ.

5) RL: Rank dependent with Prelec weighting function—subjects choose on the basis of expected utility where the (cumulative) probabilities are distorted by a weighting function which takes the power function form:

$$ V(p)=w\left({p}_2+{p}_3+{p}_4\right){u}_2+w\left({p}_3+{p}_4\right)\left({u}_3-{u}_2\right)+w\left({p}_4\right)\left({u}_4-{u}_3\right) $$

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wherew(.) is the Prelec functionw(p) = exp(−β ⋅ (− ln p)^α).

Parameters estimated u₂, u₃, u₄, αandβ..

6) ST: Salience Theory—When choosing between (x,p) and (y,q) (each of which has at most four states) subjects consider the 16 possible states: with probabilityp_iq_j(i = 1..4, j = 1..4) getx_i if (x,p) chosen and gety_jif (y,q) chosen. Then decide on the basis of whether V is positive (choose (x,p)) or negative (choose (y,q)) where V is given by:

$$ V=\sum \limits_{i=1}^4\sum \limits_{j=1}^4\mu \left({x}_i,{y}_j\right)\left[u\left({x}_i\right)-u\left({y}_i\right)\right]{p}_i{q}_j $$

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where u(x₁) = 0, u(x₂) = u₂, u(x₃) = u₃,andu(x₄) = u₄,andμ(x, y) = |x − y|^β..

This is the Salience Function.

Parameters estimated: u₂, u₃, u₄ andβ.

7) WU: Weighted Utility—subjects choose on the basis of expected weighted utility:

$$ V(p)=\frac{w_2{p}_2{u}_2+{w}_3{p}_3{u}_3+{p}_4{u}_4}{p_1+{w}_2{p}_2+{w}_3{p}_3+{p}_4} $$

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Parameters estimated: u₂, u₃, u₄, w₂andw₃.

Appendix B

Figure B1:

Histogram of k values (Version C: No editing, Luce Error)

Appendix C

A Possible Editing Phase

Ideally, we would prefer to include an editing phase conducted by the DM before the evaluation phase of the theory: in this editing phase, the DM looks to see if one lottery first-order dominates the other. In such a case, the DM chooses the dominant option without proceeding with the evaluation phase of the theory. In the other problems the DM proceeds as in the theory.

While we think that this editing phase is an important feature of our theory, we want to be fair to the other theories in our ‘horse race’. It seems that the only other theory which also has an explicit editing phase is Rank Dependent Expected Utility, at least in its earliest incarnation—Prospect Theory. However, recent descriptions of both it and the other theories seem to omit descriptions of such an editing phase—perhaps regarding it as implicit.

So, to be fair to the other theories, we omit any editing phase in any of the models in all the econometric analyses reported in the main body of the paper. For those readers interested in the effects of the incorporation of an editing phase in DS we include online Appendices E and F which report the results. There are two Versions reported there: Version A in which the DM trembles in the dominating problems with an exogenous probability; and Version B in which the DM trembles in the dominating problems with an endogenous probability. Not surprisingly DS does much better with this editing phase included. These are in addition to the model without editing, which is referred to as Version C.