Abstract
We present a new theory of decision under uncertainty: third-generation prospect theory (PT3). This retains the predictive power of previous versions of prospect theory, but extends that theory by allowing reference points to be uncertain while decision weights are specified in a rank-dependent way. We show that PT3 preferences respect a state-conditional form of stochastic dominance. The theory predicts the observed tendency for willingness-to-accept valuations of lotteries to be greater than willingness-to-pay valuations. When PT3 is made operational by using simple functional forms with parameter values derived from existing experimental evidence, it predicts observed patterns of the preference reversal phenomenon.
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Notes
Some descriptive limitations of prospect theory are discussed by Birnbaum and Bahara (2007).
In Bar-Hillel and Neter’s ‘Experiment 2’, subjects are unwilling to exchange lottery tickets even if they know that the number of the winning ticket will not be announced. This suggests that subjects use a state-contingent conceptualisation of gains and losses even if they will never know which state is realised.
The restriction v(f[s i ], h[s i ]) = u(f[s i ] − h[s i ]) prevents cumulative prospect theory from taking account of income effects. Kahneman and Tversky (1979, pp. 277–278) comment that ‘strictly speaking’, value should be defined as a function in two arguments—changes in wealth relative to a current asset position, and that position itself. The simpler function they use is presented as ‘a satisfactory approximation’.
Notice that the derivation of this result uses two symmetries between the treatments of gains and losses that are specific to our parameterisation: the weighting function is the same for gains and losses, and the exponent of u(.) is the same for gains and losses.
Our analysis of PR does not depend on the cumulative transformation of probabilities. The acts that we analyse have no more than one strictly positive consequence and no more than one strictly negative one. For such acts, the cumulative transformation is observationally equivalent to Handa’s simple transformation.
In addition, f P ~ h f $ ⇔ w(p) / w(q) = (pr / q)a. From now on, to avoid cluttering the exposition, we will not state conditions for indifference explicitly. In all cases, the condition for indifference can be constructed from the condition for weak preference by substituting an equality for a weak inequality.
The reader may wonder why, in all the cases represented in Fig. 2, the valuation boundary intersects the choice boundary at λ = 1. It can be shown that this property is induced by a special feature of the benchmark case, namely that the P and $ bets are symmetrical in the sense that q = 1 – p. If the bets are symmetrical and if β = 1, the valuation boundary passes through the point (ln[p / q] / ln[rp / q], 1). Whether the bets are symmetrical or not, this point lies on the choice boundary. (Since this result has little substantive significance, we leave the proof to sufficiently curious readers.) If the assumption of symmetry is relaxed, the two boundaries may intersect at positive or negative values of λ. If r > 1 and if the intersection is at a positive value of λ, there is a (typically small) region of the empirically plausible quadrant at which non-standard PR occurs.
The reader may have noticed that, in all three cases, the choice and valuation boundaries intersect at (β, 1). On the assumption that r = 1, it can be shown that, for all admissible values of p and q, the valuation boundary passes through (β, 1). The choice boundary passes through the same point if and only if q = 1 – p (compare note 9). Figures 4 and 5, discussed in the next paragraph, illustrate some cases in which this symmetry condition does not hold.
In each case, the valuation boundary passes through (β, 1), illustrating the general result stated in note 10. In general, if β < 1 and r = 1, the choice boundary lies to the right of (respectively passes through, lies to the left of) the point (β, 1) if p + q is greater than (equal to, less than) 1. The proof of this result is omitted for brevity.
If λ = 1 and a = 1, Eq. 15 reduces to 1 − w[q] > w(1 − q). This inequality holds if and only if β < 1.
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Acknowledgements
The authors thank Serge Blondel, Louis Lévy-Garboua, William Neilson, Peter Wakker, Horst Zank and anonymous referees for helpful comments on earlier drafts. We have also benefitted from discussions with participants at various conferences and seminars where we have presented this work. Sugden’s work was supported by the Economic and Social Research Council (award no. RES 051 27 0146).
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Schmidt, U., Starmer, C. & Sugden, R. Third-generation prospect theory. J Risk Uncertainty 36, 203–223 (2008). https://doi.org/10.1007/s11166-008-9040-2
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DOI: https://doi.org/10.1007/s11166-008-9040-2