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Social Choice and Welfare

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01-09-2015

State dependent choice

Authors: Paola Manzini, Marco Mariotti

Published in: Social Choice and Welfare | Issue 2/2015

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Abstract

We propose a theory of choices that are influenced by the psychological state of the agent. The central hypothesis is that the psychological state controls the urgency of the attributes sought by the decision maker in the available alternatives. While state dependent choice is less restricted than rational choice, our model does have empirical content, expressed by simple ‘revealed preference’ type of constraints on observable choice data. We demonstrate the applicability of simple versions of the framework to economic contexts. We show in particular that it can explain widely researched anomalies in the labour supply of taxi drivers.

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See Mandler et al. (2012) for a discussion of the psychological foundations of this procedure.

A sample of publications is: Capra (2004), Erber et al. (2004), Ifcher and Zarghamee (2011), Isen (2000), Kahn and Isen (1993), Kirchsteiger et al. (2006), Mayer et al. (1992), Mittal and Ross (1998), Nygren (1998), Nygren et al. (1996), Oswald et al. (2009), Thayer (2000), Thayer (2001) and Williams and Voon (1999).

Some of the relevant literature is cited in Sect. 5.

While in this paper we confine ourselves to finite sets, the definitions are written so as to immediately extend to the infinite cases.

A weak order is a a complete transitive relation.

This result is proved in MMM in greater generality than for the domain considered here. The connections with the classical theory of lexicographic preferences are explained in that paper.

The easy proof of this assertion is left as an exercise for the reader.

As it often happens, choice data alone cannot be the ultimate arbiter in the selection of the model. For example, ‘rationalisation’ in the sense of Cherepanov et al. (2013) could be the ‘true’ explanation of the behaviour just described. Context and non-choice data will help model selection.

The proof is available from the authors.

Formally, \(c\) must satisfy the following weakening of WARP, where for any alternative \(x\in X\) we define \(\Sigma _{x}=\left\{ S\subseteq X:x\in S\right\} \) and \(\Sigma _{\lnot x}=\Sigma \backslash \Sigma _{x}\):

t-WARP (trigger WARP): There exists \(t\in X\) such that for \(i=x,\lnot x\): For all \(A,B\in \Sigma _{i}\): \(\left[ x\in A\backslash c\left( A\right) , y\in c\left( A\right) , y\in B\right] \Rightarrow x\notin c\left( B\right) \).

Mandler (2009) shows how indifference can be distinguished from incompleteness by observing the trades an agent is willing to carry out.

As in the previous section, our setup is static. Laibson (2001) studies a dynamic model of what we would call an ‘environment driven state’ triggered by binary cues.

While we are working for simplicity in the full domain, so that IIA and WARP are equivalent, the result is independent of this domain restriction.

Below (Sect. 6) we explore the relationship with Salant and Rubinstein’s (2008) in more detail. Bernheim and Rangel (2009) also allow choice to depend on information beyond the feasible set, although their focus is not on the properties of \(c\) but rather on the welfare inferences that could be made by observing \(c\).

In Appendix 8.4 we also establish, as an homage to aficionados of choice theory, a connection between our result and an old result from the Russian school.

To be precise, KRS deal with choice functions. We are referring to the obvious extension of their ideas to choice correspondences, in which a weak order is maximised in each menu.

The claim is trivial for single-valued state dependent choices.

See e.g. Gul and Pesendorfer (2005) for a surprising application of the revealed preference method to a psychological model.

A classical technical treatment of the relationship between deterministic choice correspondences and stochastic choice functions is Fishburn (1978).

See for example, Dalton and Ghosal (2008), who resolve the interaction through an elegant equilibrium analysis, and Dillenberger and Rozen (2013), where disappointment and elation are moods that affect risk aversion, and are determined endogenously based on how risk unfolds over time (generating a history of disappointment and elation states).

To see this, let sd-WARP hold, and suppose that \(C\left( A\right) \subset B\) but that in contradiction there exists some \(x\) such that \(x\in \left( C\left( B\right) \cap A\right) \backslash C\left( A\right) \). Since \(C\left( A\right) \subset B\) and \(x\in A\backslash C\left( A\right) \), sd-WARP requires \(x\notin C\left( B\right) \), contradiction. For the other direction, let sd-WARP (restated) hold, and suppose that \(x\in A\backslash C\left( A\right) , C\left( A\right) \subset B\) but that in contradiction \( x\in C\left( B\right) \). Then \(x\in \left( C\left( B\right) \cap A\right) \backslash C\left( A\right) \), an immediate contradiction of sd-WARP (restated).

A small choice theoretic observation: this formulation makes it clear that sd-WARP is a stronger version of the classic axiom by Aizerman (see Aizerman and Malishevski 1981), which adds to the premise in sd-WARP the requirement that the sets \(A\) and \(B\) are nested, i.e. \(B\subset A\).

This example is loosely based upon Wansink (1994) who studies the determinants of ‘eating bouts’. An eating bout is a splurge on food that is some multiple (three times the usual amount in this study). Both external cues (i.e. menu composition) and internal states (i.e. moods) are invoked as alternative triggers for such eating bouts.

Contrast again the restriction imposed by All or Nothing with that imposed by WARP, which can be written as \(A\cap c\left( B\right) \ne \varnothing \Rightarrow c\left( A\right) \cap c\left( B\right) =c\left( A\right) \cap B\) (the same conclusion of All or Nothing from a weaker premise).

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Title: State dependent choice
Authors: Paola Manzini
Marco Mariotti
Publication date: 01-09-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0894-3

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