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Stronger utility

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Abstract

Empirical research often requires a method how to convert a deterministic economic theory into an econometric model. A popular method is to add a random error term on the utility scale. This method, however, ignores stochastic dominance. A modification of this method is proposed to account for stochastic dominance. The modified model compares favorably to other existing models in terms of goodness of fit to experimental data. The modified model can rationalize the preference reversal phenomenon. An intuitive axiomatic characterization of the modified model is provided. Important microeconomic concept of risk aversion is well defined in the modified model.

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Notes

  1. See Camerer (1989), Starmer and Sugden (1989), Hey and Orme (1994), Wu (1994), Ballinger and Wilcox (1997), Loomes and Sugden (1998), Hey (2001), Schmidt and Hey (2004), Schmidt and Neugebauer (2007).

  2. If a decision maker has a strict preference over all outcomes (for any \(x, y \in X\) either \(x\succ y\) or \(y \succ x\)), then the second case is possible only when \(L = L^{\prime }\).

  3. Alternatively, choice probability may be left undefined in this case.

  4. Setting \(F(1) = 1\) is appealing on normative grounds. From a descriptive perspective, it may be desirable to set \(F (1)< 1\). This allows model (10) to account for rare violations of stochastic dominance that are observed in the data. See Loomes et al. (2002) for a more detailed discussion.

  5. Note that Fishburn (1978) model with a power function coincides with Blavatskyy (2011a) model with a function \(\varphi (v)= v ^{\mu }\) for all \(v \ge 0\).

  6. See also graphical representation in MacCrimmon and Smith (1986) and Butler and Loomes (2007).

  7. This special case may be considered as nearly conventional economic theory based on a revealed preference relation: \(L^{\prime }\succsim L\) when \(P (L, L^{\prime })=0\), \(L\succsim L^{\prime }\) when \(P (L,L^{\prime })=1\), and \(L \sim L^{\prime }\) when \(P (L, L^{\prime }) = 0.5\). This is “nearly conventional” because in standard economic theory a choice probability \(P (L, L^{\prime }\)) is not defined when \(L \sim L^{\prime }\).

  8. A popular class of QRE is logit QRE, which is based on model (6) with error term \(\xi \) drawn from the logistic distribution. This specification of strong utility is sometimes called Luce’s choice model (cf. Luce (1959).

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  1. Innsbruck, Austria

    Pavlo R. Blavatskyy

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Correspondence to Pavlo R. Blavatskyy.

Appendix

Appendix

1.1 Proof of Proposition 3

Consider an arbitrary outcome \(x \in X\) and a lottery \(L \in \mathcal{{L}}\). We shall prove that if conditions of proposition 3 hold.

If outcome \(x\) is less preferred than the worst possible outcome in lottery \(L\), then model (10) implies that and . Since , it must be the case that . Similarly, if outcome \(x\) is more preferred than the best possible outcome in lottery \(L\), then we have . What remains is to consider the case when outcome \(x\) lies between the worst and the best possible outcome in lottery \(L\).

If inequality (22) holds for all \(x, y, z \in X\) such that \(z \succsim x \succsim y\) then

(24)

If inequality (24) holds then the following inequality holds as well:

(25)

Since for all \(v \in [-1,1]\) and it is a non-decreasing function, we can rewrite inequality (25) as

(26)

According to formula (10), the left-hand side of (26) is equal to and the right-hand side of (26) is equal to . Thus, . \(\square \)

1.2 Matlab R2009a programing code used in Sect. 3 to estimate model (10) with functional form (11)

% this program searches for the maximum likelihood estimates of model (10) with functional form (11) and rank-dependent utility (expected utility is a special case when we set parameter gamma \(=\) 1)

figure a

% the program calls a user-defined function RDU_RC presented below. The function uses an input \(100\times 8\) array L1 of lottery probabilities. Probability of the lowest, two middle and the highest outcome in the left lottery is given in vectors L1(:,1), L1(:,2), L1(:,3) and L1(:,4) correspondingly. Probability of the lowest, two middle and the highest outcome in the right lottery is given in vectors L1(:,5), L1(:,6), L1(:,7) and L1(:,8) correspondingly. The function also uses an input \(100\times 53\) array A of probabilities with which each subject chooses the right lottery in each decision problem.

figure b

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Blavatskyy, P.R. Stronger utility. Theory Decis 76, 265–286 (2014). https://doi.org/10.1007/s11238-013-9366-3

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