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A genuine foundation for prospect theory

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Abstract

In most models of (cumulative) prospect theory, reference dependence of preferences is imposed beforehand and the location of the reference point is determined exogenously. This paper presents principles that provide critical tests and foundations for prospect theory preferences without assuming reference-dependent preferences a priori. Instead, reference dependence is derived from behavior and the reference point arises endogenously.

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Notes

  1. These include Edwards (1953, 1954), Hogarth and Einhorn (1990), Tversky and Kahneman (1992), Abdellaoui (2000), Bleichrodt et al. (2001), Etchart-Vincent (2004), Payne (2005), and Abdellaoui et al. (2005, 2010).

  2. Kahneman and Tversky (1979), Tversky and Kahneman (1991, 1992), Hershey and Schoemaker (1985), Budescu and Weiss (1987), Camerer (1989), Currim and Sarin (1989), Fennema and van Assen (1999), Luce (2000), Abdellaoui (2000) and Abdellaoui et al. (2005, 2007 ,2008); studies in neuroeconomics include Dickhaut et al. (2003) and de Martino et al. (2006).

  3. Battalio et al. (1990), Tversky and Kahneman (1992), Camerer (1998), Kahneman and Tversky (2000), Schmidt and Traub (2002), Camerer et al. (2004), Brooks and Zank (2005) and Abdellaoui et al. (2007, 2008).

  4. This applies, among others, to the axiomatizations of Luce (1991, 2000), Luce and Fishburn (1991), Tversky and Kahneman (1992), Wakker and Tversky (1993), Chateauneuf and Wakker (1999), Zank (2001), Wakker and Zank (2002), Köbberling and Wakker (2003), Schmidt (2003), Schmidt and Zank (2009), Wakker (2010) and Kothiyal et al. (2011).

  5. Our results can be extended to infinite state spaces by using tools presented in Wakker (1993). Identical results for the case of decision under risk, that is, under probabilistic sophistication or when (objective) probabilities are given, can be derived by applying the procedure of Köbberling and Wakker (2003, Section 5.3).

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Acknowledgements

We thank Peter Wakker for extensive comments. The current version has also profited from useful suggestions obtained from Mohammed Abdellaoui and a referee.

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Authors and Affiliations

  1. Department of Economics, Christian-Albrechts-Universität zu Kiel, 24098, Kiel, Germany

    Ulrich Schmidt

  2. Kiel Institute for the World Economy, Hindenburgufer 66, 24105, Kiel, Germany

    Ulrich Schmidt

  3. Economics, School of Social Sciences, The University of Manchester, Oxford Road, M13 9PL, Manchester, UK

    Horst Zank

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  1. Ulrich Schmidt
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Correspondence to Horst Zank.

Appendix: Proofs

Appendix: Proofs

Proof of Theorem 1

To prove Theorem 1 we remark that deriving statement (ii) from statement (i) is standard in conjunction with the comments preceding Theorem 1 regarding CDS. Next we assume statement (ii) and derive statement (i). We distinguish three cases:

Case 1

For all outcomes x we have condition (I) of CDS satisfied. In this case the comonotonic tradeoff consistency of Köbberling and Wakker (2003) holds and it follows from their Theorem 8 that CEU holds (with uniqueness results for utility and capacity as noted in Observation 9 (c) of Köbberling and Wakker). Further, locally, we can always find indifferences x j f~y j g and z j f~w j g for acts f,g a state j and outcomes w,z,y > x. Substitution of CEU and subtraction of the first resulting equality from the second implies

$$ U(y)-U(x)=U(w)-U(z). $$

CDS for outcomes demands w − z > y − x in this case. Because this implication holds for all outcomes x (and corresponding outcomes w,z,y > x), it follows, first locally and then globally, that the utility function is concave.

Case 2

For all x we have condition (II) of CDS satisfied. Similar to the previous case, the results of Köbberling and Wakker (2003) hold and we obtain CEU. Further, CDS implies, first locally and then globally, that the utility function is convex. Uniqueness results for utility and capacity apply as noted in Observation 9 (c) of Köbberling and Wakker (2003).

Case 3

There exists an outcome x  +  for which condition (I) of constant diminishing sensitivity holds and an outcome x  −  for which condition (II) of CDS holds. It then follows that there exists a unique outcome r for which both (I) and (II) hold, which is the reference point for the preference \(\succcurlyeq \). In this case CDS implies the sign-comonotonic tradeoff consistency of Köbberling and Wakker (2003), and from their Theorem 12 we obtain that PT holds. By Proposition 8.2 in Wakker and Tversky (1993) the gain-loss consistency requirement can be dropped from statement (ii) in Theorem 12 in Köbberling and Wakker’s (2003) if the number of states of nature exceeds 2, which is the case here. Similar to cases 1 and 2 above we derive strict concavity of utility for outcomes above r and strict convexity for utility for outcomes below r, first locally and then globally. Uniqueness results for capacities and utility follow from Observation 13 in Köbberling and Wakker (2003).

Together Cases 1–3 cover all possibilities and thus statement (i) follows in conjunction with the uniqueness results. This completes the proof of Theorem 1.□

Proof of Theorem 2

The proof of this theorem follows from the analysis preceeding the theorem in the main text.□

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Schmidt, U., Zank, H. A genuine foundation for prospect theory. J Risk Uncertain 45, 97–113 (2012). https://doi.org/10.1007/s11166-012-9150-8

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