Abstract
Consider a subjective expected utility preference relation. It is usually held that the representations which this relation admits differ only in one respect, namely, the possible scales for the measurement of utility. In this paper, I discuss the fact that there are, metaphorically speaking, two additional dimensions along which infinitely many more admissible representations can be found. The first additional dimension is that of state-dependence. The second—and, in this context, much lesser-known—additional dimension is that of act-dependence. The simplest implication of their usually neglected existence is that the standard axiomatizations of subjective expected utility fail to provide the measurement of subjective probability with satisfactory behavioral foundations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Besides, although the Savage theorem can be invoked only when S is infinite, reasonably close variants of this theorem can be invoked when S is finite (see, e.g., Köbberling and Wakker 2003 and the references therein). In these variants, unlike in the Savage theorem, instead of being imposed on S, richness requirements are typically imposed on X.
- 2.
In general, seeing u as a function from X to \(\mathbb {R}^{|S|}\), a state-dependent increasing affine transformation is of the form αu + β, with \(\alpha \in \mathbb {R}^{|S|}_{>0}\) and \(\beta \in \mathbb {R}^{|S|}\). For simplicity, we here take β = 0 and a suitably restricted domain for α; but the problem applies more generally, with an order-preserving renormalization. Notice the parallel with the case of cardinal non-comparable utilities in social choice theory (e.g., d’Aspremont and Gevers 2002, p. 485).
- 3.
I will eventually return (on p. 23) to the ellipsis in the above quotation.
- 4.
First, apply (9.2), thus inducing (9.3) with \(v_{i^*}\big (x\big )=0\) for all x ∈ X. All expected utility levels being preserved, so is the representation. Next, add the constant \(q(s_{i^*})c_{i^*}\) to both sides of the inequality in (9.3). The inequality being preserved, so is the representation.
- 5.
In the first case, if q(s i) = 0, then for any state-dependent utility function v i it holds that q(s i)v i(x) = 0 for all x ∈ X, while to preserve the representation of the relation given in (9.1) with p(s i) > 0, it must be that q(s i)v i(x) ≠ q(s i)v i(y) for some x, y ∈ X. In the second case, if q(s i) = 1, then for any state-dependent utility function v i it holds that q(s i)v i(x) ≠ q(s i)v i(y) for some x, y ∈ X, while to preserve the representation of the relation given in (9.1) with p(s i) = 0, it must be that q(s i)v i(x) = q(s i)v i(y) for all x, y ∈ X. Clearly, the converse of the two implications thus established do not hold; to see this, consider the case of a constant v i and that of p and q having the same support, respectively.
- 6.
This is contrary to what is suggested by some passages in Schervish et al. 1990, 2014, such as the one quoted at the beginning of this development. The authors are nonetheless clearly aware of the necessary qualifications (see, e.g., Seidenfeld et al. 1995, p. 2174: “only when p(s i) ≠ 0 is it worth restricting U j in a decomposition of a linear utility V ”).
- 7.
On which see Nau (2001). Nau draws many other interesting implications from the key underlying fact here, viz. that up to a standard affine transformation of u, the compound summands or state-values in (9.1) are uniquely identified, irrespective of how each of them is to be further decomposed, state by state, in probability and utility values, respectively.
- 8.
- 9.
See Baccelli (2021) for a partial answer and a discussion.
- 10.
The above analysis could be sharpened based on the following fact. Most non-expected utility models can be construed as departing from expected utility exactly in that they operate with act-dependent, rather than act-independent, probability values (see, e.g., Chambers and Echenique 2016, p. 126; and more fundamentally Cerreia-Vioglio et al. 2011, Cor. 3). For example, maxmin expected utility is the particular case of (9.4) where
for some state-independent u over X, and Π a closed, convex (act-independent) set of priors over 2S. - 11.
I am aware of the fact that, considering decision-making from a first-person and a prescriptive point of view, rather than from the third-person and the descriptive point of view adopted here, act-dependence raises a different set of questions (see, e.g., Joyce 1999). There is no reason to consider these questions as exclusive from the one focused on here.
- 12.
To the best of my knowledge, the idea that (9.4), (9.5), or the like can be compatible with (9.1) has appeared only once before—and rather indirectly—in the literature, namely, in Drèze and Rustichini 1999, Section 5. Drèze and Rustichini’s motivations, assumptions, and conclusions, which I cannot present here, are sufficiently different from mine to justify that I offer the discussion to follow.
- 13.
David Dillenberger and co-authors do not discuss moral hazard, a particular form of which is behaviorally indistinguishable from what they call “optimism”. Besides, because their focus is different, they provide an analysis of act-dependence that is less detailed than the one offered in the present paper. Finally, as the end of the present paper will make clear, their suggested analysis of the links between act-dependence and state-dependence (Dillenberger et al. 2017, fn. 14, p. 1171) is, in several important respects, incomplete.
- 14.
This is typically the case in the axiomatizations of subjective expected utility over a finite state space (see fn. 1), so as to make the uniqueness of the representation tractable.
- 15.
Notice that, by the defining properties of certainty equivalent functions, if f is constant, then, whatever the function u f, there is no constraint on p f in (9.8).
- 16.
David Dillenberger and co-authors sketch the geometric analysis of Fig. 9.1, but they do not provide the complementary algebraic analysis presented for the case |S| = 3 in the Appendix. (The general case |S| = n can be proved following a constrained optimization analysis.)
- 17.
Although its usage is typically more specialized, I freely borrow the phrase “multi-prior expected multi-utility representation” from the literature on incomplete preferences (see, e.g., Galaabaatar and Karni 2013 and the references therein, including Seidenfeld et al. 1995). Observationally and literally speaking, this is justified inasmuch as (9.5) features multiple probability measures and multiple utility functions.
- 18.
For an elaboration on this non-sufficiency, see Baccelli 2021. Admittedly, both Drèze and Karni are well aware of the fact that, with actions that are unequally costly to the agent, moral hazard may not suffice to overcome the challenges posed by state-dependent utility to the behavioral identification of beliefs. However, neither points out let alone elaborates on the fact that this is even compatible with the respect of the Savage axioms.
for some state-independent u over X, and Π a closed, convex (act-independent) set of priors over 2S.References
Arrow, K. 1963. Uncertainty and the welfare economics of medical care. American Economic Review 53(3):941–973.
Arrow, K. 1974. Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal 1974(1):1–42.
Baccelli, J. 2017. Do bets reveal beliefs? Synthese 194(9):3393–3419.
Baccelli, J. 2021. The problem of state-dependent utility: a reappraisal. British Journal for the Philosophy of Science 72(2):617–634.
Baccelli, J. 2021. Moral Hazard, the savage framework, and state-dependent utility. Erkenntnis 86, 367–387.
Chambers, C., and F. Echenique. 2016. Revealed preference theory. New York: Cambridge University Press.
Cerreia-Vioglio, S., P. Ghirardato, F. Maccheroni, M. Marinacci, and M. Siniscalchi. 2011. Rational preferences under ambiguity. Economic Theory 48(2–3):341–375.
d’Aspremont, C., and L. Gevers. 2002. Social welfare functionals and interpersonal comparability. In Handbook of social choice and welfare, eds. Arrow, K., Sen, A., and Suzumura, K. vol. 1, 459–541. Amsterdam: North-Holland.
Dillenberger, D., A. Postlewaite, and K. Rozen. 2017. Optimism and pessimism with expected utility. Journal of the European Economic Association 15(5):1158–1175.
Drèze, J. 1961. Les fondements logiques de la probabilité subjectives et de l’utilité. In La décision. colloques internationaux du centre national de la recherche scientifique, 73–87. Paris: Centre National de la Recherche Scientifique édition.
Drèze, J. 1987. Decision theory with moral hazard and state-dependent preferences. In Drèze, J. essays on economic decisions under uncertainty, 23–89. Cambridge: Cambridge University Press.
Drèze, J., and A. Rustichini. 1999. Moral hazard and conditional preferences. Journal of Mathematical Economics 31(2):159–181.
Eeckhoudt, L., C. Gollier, and H. Schlesinger. 2005. Economic and financial decisions under risk. Princeton: Princeton University Press.
Galaabaatar, T., and E. 2013. Karni subjective expected utility with incomplete preferences. Econometrica 81(1):255–284.
Gilboa, I., and D. Schmeidler. 1989. Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18(2):141–153.
Hart, O., and B. Holmström. 1987. The theory of contracts. In Advances in economic theory, ed. Bewley, T., 71–155. Cambridge: Cambridge University Press.
Joyce, J. 1999. The foundations of causal decision theory. New York: Cambridge University Press.
Karni, E. 1996. Probabilities and beliefs. Journal of Risk and Uncertainty 13(3):249–262.
Karni, E. 2011. Subjective probabilities on a state space. American Economic Journal: Microeconomics 3(4):172–185.
Köbberling, V., and P. Wakker. 2003. Preference foundations for nonexpected utility: a generalized and simplified technique. Mathematics of Operations Research 28(3):395–423.
Laffont, J.-J., and D. Martimort, The theory of incentives: the principal-agent model. Princeton: Princeton University Press.
Levi, I. 1974. On indeterminate probabilities. The Journal of Philosophy 71(13):391–418.
Levi, I. 1980. The enterprise of knowledge. Cambridge: MIT Press.
Machina, M. 1982. ‘Expected utility’ analysis without the independence axiom. Econometrica 50(2):277–323.
Marschak, J. 1950. Rational behavior, uncertain prospects, and measurable utility. Econometrica 18(2):111–141.
Nau, R. 2001. de Finetti was right: probability does not exist. Theory and Decision 51(2–4):89–124.
Pratt, J. 1964. Risk aversion in the small and in the large. Econometrica 32(1):122–136.
Savage, L. 1954. The foundations of statistics. New York: Wiley.
Savage, L. 1972. The foundations of statistics. New York: Dover.
Savage, L., and R. Aumann. 1987. Letters between Leonard Savage and Robert Aumann [January 1971]. In Drèze, J. essays on economic decisions under uncertainty, 76–81. Cambridge: Cambridge University Press.
Schervish, M., T. Seidenfeld, and J. Kadane. 1990. State-dependent utilities. Journal of the American Statistical Association 85(411):840–847.
Schervish, M., T. Seidenfeld, and J. Kadane. 2014. The effect of exchange rates on statistical decisions. Philosophy of Science 80(4):504–532.
Seidenfeld, T., M. Schervish, and J. Kadane. 1990. When fair betting odds are not degrees of belief. Philosophy of Science 169(1):517–524.
Seidenfeld, T., M. Schervish, and J. Kadane. 1990. Decisions without ordering. In Acting and reflecting, ed. Sieg, W., 143–170. Boston: Kluwer Academic Press.
Seidenfeld, T., M. Schervish, and J. Kadane. 1995. A representation of partially ordered preferences. The Annals of Statistics 23(6):2168–2217.
Sugden, R. 2004. Alternatives to expected utility: foundations. In Handbook of Utility Theory, ed. Barbera, S., Hammond, P., and Seidl, C., vol. 2, 685–755. Boston: Kluwer Academic Press.
Acknowledgements
I am grateful to Thomas Augustin, Fabio Gagliardi Cozman, and Gregory Wheeler for inviting this contribution and offering me the opportunity to celebrate Teddy Seidenfeld’s work. I have learnt immensely from Teddy’s writings, and I have also come to owe him a lot for his time and advice. I am especially thankful for the many inspiring conversations he had the kindness to have with me during the year 2019–2020, while I was visiting Pittsburgh.
For helpful comments on earlier drafts of the present paper, I am indebted to two anonymous reviewers, Richard Bradley, Florian Brandl, Franz Dietrich, Raphaël Giraud, Brian Hill, Jay Lu, Marcus Pivato, Rush Stewart, Jiji Zhang, Fanyin Zheng, as well as the late and regretted Arthur Merin and Philippe Mongin. All errors and omissions are mine.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Appendix
Appendix
Proposition 1
Assume that |S| = 2 and f(s 1) ≠ f(s 2). For any concave transform u f of u, there is one and only one p f that satisfies constraint (9.8), and it is such that \(\widehat {p}^f\) first-order stochastically dominates \(\widehat {p}\).
Proof
Given the definition u f ≡ h f ∘ u and the properties of function composition, (9.8) can be uniquely solved as detailed next. For brevity, in this proof, (u 1, u 2) stands for (u(f(s 1)), u(f(s 2))), \((u^f_1, u^f_2)\) for (u f(f(s 1)), u f(f(s 2))), and (h f(u 1), h f(u 2)) for (h f ∘ u(f(s 1)), h f ∘ u(f(s 2))).
It is readily checked that p f(s 1) ∈ [0, 1], so that the solution defines a probability measure. Next, assume without loss of generality that f(s 1) > f(s 2). Jensen’s inequality and the solution given above for p f(s 1) imply that \(\widehat {p}^f\) first-order stochastically dominates \(\widehat {p}\), for h f is (strictly) concave if and only if:
□
Proposition 2
Assume that |S| = 3 and f(s 1) ≠ f(s 2) ≠ f(s 3). For any concave transform u f of u, if p f minimizes (9.9) under constraint (9.8), then, p f is such that \(\widehat {p}^f\) first-order stochastically dominates \(\widehat {p}\).
Proof
Henceforth, for brevity, let (u 1, u 2, u 3) stand for (u(f(s 1)), u(f(s 2)), u(f(s 3))), (h f(u 1), h f(u 2), h f(u 3)) for (h f ∘ u(f(s 1)), h f ∘ u(f(s 2)), h f ∘ u(f(s 3))), \((\widehat {p}_1, \widehat {p}_2, \widehat {p}_3)\equiv \widehat {p}\) for \(\Big (\widehat {p}\big (f(s_1)\big ), \widehat {p}\big (f(s_2)\big ), \widehat {p}\big (f(s_3)\big )\Big )\), and \((\widehat {p}^f_1, \widehat {p}^f_2, \widehat {p}^f_3)\equiv \widehat {p}^f\) for \(\Big (\widehat {p}^f\big (f(s_1)\big ), \widehat {p}^f\big (f(s_2)\big ), \widehat {p}^f\big (f(s_3)\big )\Big )\).
I start with the following preliminary observation. Given a (strictly) concave h f, \(\widehat {p}^f\) cannot satisfy (9.8) and be first-order stochastically dominated by \(\widehat {p}\). Assume, by way of contradiction, that such is the case. Then, by the properties of expected utility, we have \(\widehat {p}_1h^f\big (u_1\big )+\widehat {p}_2h^f\big (u_2\big )+\widehat {p}_3h^f\big (u_3\big ) > \widehat {p}^f_1h^f\big (u_1\big )+\widehat {p}^f_2h^f\big (u_2\big )+\widehat {p}^f_3h^f\big (u_3\big )\). By the concavity of h f, we also have \(h^f(\widehat {p}_1u_1+\widehat {p}_2u_2+\widehat {p}_3u_3)>\widehat {p}_1h^f\big (u_1\big )+\widehat {p}_2h^f\big (u_2\big )+\widehat {p}_3h^f\big (u_3\big )\). Therefore, we have \(h^f(\widehat {p}_1u_1+\widehat {p}_2u_2+\widehat {p}_3u_3)>\widehat {p}^f_1h^f\big (u_1\big )+\widehat {p}^f_2h^f\big (u_2\big )+\widehat {p}^f_3h^f\big (u_3\big )\), thus contradicting (9.8). Similarly, in Fig. 9.1, the line on which \(\widehat {p}^f\) is to be found cannot pass by \(\widehat {p}\) itself. This is because (9.8) would then require that \(\widehat {p}_1h^f\big (u_1\big )+\widehat {p}_2h^f\big (u_2\big )+\widehat {p}_3h^f\big (u_3\big )=h^f(\widehat {p}_1u_1+\widehat {p}_2u_2+\widehat {p}_3u_3)\), while concavity requires that \(h^f(\widehat {p}_1u_1+\widehat {p}_2u_2+\widehat {p}_3u_3)>\widehat {p}_1h^f\big (u_1\big )+\widehat {p}_2h^f\big (u_2\big )\widehat {p}_3h^f\big (u_3\big )\)—a contradiction.
Next, without loss of generality, assume that u 1 > u 2 > u 3, as in Fig. 9.1. Then, \(\widehat {p}^f\) first-order stochastically dominates \(\widehat {p}\) if and only if \(\widehat {p}^f_1\ge \widehat {p}_1\) and \(\widehat {p}^f_1+\widehat {p}^f_2\ge \widehat {p}_1+\widehat {p}_2\), with one of these inequalities being strict. Because, as explained in the preliminary observation, it is excluded that \(\widehat {p}\) first-order stochastically dominates \(\widehat {p}^f\), it remains to be shown that if \(\widehat {p}^f_1\ge \widehat {p}_1\) (respectively, \(\widehat {p}^f_1+\widehat {p}^f_2\ge \widehat {p}_1+\widehat {p}_2\)) and the minimal Euclidean distance condition is satisfied, then, \(\widehat {p}^f_1+\widehat {p}^f_2\ge \widehat {p}_1+\widehat {p}_2\) (respectively, \(\widehat {p}^f_1\ge \widehat {p}_1\)). I now show, by contraposition, that if \(\widehat {p}^f_1\ge \widehat {p}_1\) but \(\widehat {p}^f_1+\widehat {p}^f_2< \widehat {p}_1+\widehat {p}_2\), then, the minimal Euclidean distance condition is not satisfied. (The other case is similar.) With reference to Fig. 9.1, this amounts to showing that, if one picks a point r, corresponding to \(\widehat {p}^f\), that is on the line and northeast of \(\widehat {p}\), then, one can always find another point r ′ that is also on the line, but closer—as measured by the Euclidean distance (adapted from measures over the algebra of events to lotteries over the set of consequences in the obvious way)—to \(\widehat {p}\).
I now show how to construct r
′ from r. As is clear from Fig. 9.1 and as I now detail algebraically, in general, this can be done by transferring to f(s
2) appropriately small probability weights 𝜖
1, 𝜖
3 from f(s
1) and f(s
3), respectively. First, notice that since (i) \(\widehat {p}^f_1\ge \widehat {p}_1\) and \(\widehat {p}^f_1+\widehat {p}^f_2< \widehat {p}_1+\widehat {p}_2\) and (ii) as explained in the preliminary observation, \(\widehat {p}\) cannot first-order stochastically dominate \(\widehat {p}^f\), it must be not only that \(\widehat {p}^f_1\ge \widehat {p}_1\), but more specifically that \(\widehat {p}^f_1> \widehat {p}_1\); hence, that \(\widehat {p}^f_1> 0\). Second, notice that if it also holds that \(\widehat {p}^f_3>0\), points r and r
′ will both satisfy (9.8) if and only if 
, which is true if and only if \(\epsilon _3=\left (\Big (h^f\big (u_1\big )-h^f\big (u_2\big )\Big ) / \Big (h^f\big (u_2\big )-h^f\big (u_3\big )\Big )\right )\epsilon _1\). Next, assuming \(\widehat {p}^f_3>0\) still, consider the Euclidean distance between \(\widehat {p}\) and r and r
′, respectively, defining it like in (9.9). For commodity, examine more specifically the squared Euclidean distances, denoting them by d
r and \(d_{r^\prime }\), respectively. Notice that, by (9.9), we have that:
Therefore, to have \(d_{r^\prime } - d_r < 0\), i.e., to find a point r ′ on the line but at a lesser Euclidean distance to \(\widehat {p}\) than point r, it suffices to pick any 𝜖 1 such that the following two conditions are satisfied:
It is the case that α, β > 0 since: (i) \(\widehat {p}^f_1\ge \widehat {p}_1\) and \(\widehat {p}^f_1+\widehat {p}^f_2< \widehat {p}_1+\widehat {p}_2\) hold by assumption; (ii) for the reasons previously detailed, not only \(\widehat {p}^f_1\ge \widehat {p}_1\), but more specifically \(\widehat {p}^f_1> \widehat {p}_1\) must hold; (iii) h f(u 1) > h f(u 2) > h f(u 3) holds by assumption. Thus, if \(\widehat {p}^f_3>0\), any \(\epsilon _1 \in (0,\min \{\alpha ;\beta \})\) will define a point r ′ that, like r, satisfies (9.8), while generating, by (9.9), a lesser distance than r. If \(\widehat {p}^f_3=0\), there is no need to preliminarily express 𝜖 3 in terms of 𝜖 1, and one may directly compare the relevant squared Euclidean distances simply by setting 𝜖 3 = 0 in the preceding equalities; then, with α as defined above, any 𝜖 1 ∈ (0, α) has the desired properties. This establishes in all cases that under constraint (9.8), if \(\widehat {p}^f_1\ge \widehat {p}_1\) but \(\widehat {p}^f_1+\widehat {p}^f_2< \widehat {p}_1+\widehat {p}_2\), then, the minimal Euclidean distance condition is not satisfied. By contraposition, under constraint (9.8), if \(\widehat {p}^f_1\ge \widehat {p}_1\) and the minimal Euclidean distance condition is satisfied, then, \(\widehat {p}^f_1+\widehat {p}^f_2\ge \widehat {p}_1+\widehat {p}_2\), i.e., \(\widehat {p}^f\) first-order stochastically dominates \(\widehat {p}\). □
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Baccelli, J. (2022). Expected Utility in 3D. In: Augustin, T., Cozman, F.G., Wheeler, G. (eds) Reflections on the Foundations of Probability and Statistics. Theory and Decision Library A:, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-031-15436-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-15436-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-15435-5
Online ISBN: 978-3-031-15436-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)