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Do bets reveal beliefs?

A unified perspective on state-dependent utility issues

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This paper examines the preference-based approach to the identification of beliefs. It focuses on the main problem to which this approach is exposed, namely that of state-dependent utility. First, the problem is illustrated in full detail. Four types of state-dependent utility issues are distinguished. Second, a comprehensive strategy for identifying beliefs under state-dependent utility is presented and discussed. For the problem to be solved following this strategy, however, preferences need to extend beyond choices. We claim that this a necessary feature of any complete solution to the problem of state-dependent utility. We also argue that this is the main conceptual lesson to draw from it. We show that this lesson is of interest to both economists and philosophers.

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  1. 1.

    See e.g. Genest and Zidek (1986) and Clemen and Winkler (2007) for reviews of the literature on probabilistic opinion aggregation. See footnote 56 regarding the necessity of identifying beliefs when the goal is to aggregate preferences (rather than simply opinions) on uncertain outcomes. Regarding interactive epistemology, see in particular the agreement theorem literature, originating in Aumann (1976) and reviewed e.g. in Bonanno and Nehring (1997).

  2. 2.

    This is the school of Koopman (1940), Good (1950) or Kraft et al. (1959). Although they all belong to the probabilistic tradition more specifically, we use the term likelihood to stay neutral regarding the structure of the beliefs to be identified.

  3. 3.

    This is the school of Ramsey (1931), Finetti (1937), Savage (1954/1972), and followers.

  4. 4.

    See e.g. Schlag et al. (2014) for a review of those techniques.

  5. 5.

    Admittedly, there is at least one other issue which can be considered preliminary to the ones above. When preferences are interpreted as choices, the betting approach to the identification of beliefs is confronted with the problem of moral hazard (defined in Arrow 1965, p. 142 and anticipated in Drèze 1961, p. 78). Section 3.1 introduces this problem and discusses how it relates to that of state-dependent utility, the topic of this paper.

  6. 6.

    The introduction of state-dependent utility in theoretical insurance economics is due to Arrow (1953, 1973). See e.g. Finkelstein et al. (2013) for recent empirical work on insurance demand under state-dependent utility.

  7. 7.

    Relatedly, state-dependent utility is a source of complication for pragmatic arguments for probabilism (the claim according to which rational beliefs should take the form of a unique probability measure over a given set space), the expected utility rule, and the like. Some such arguments are known as the Dutch book arguments (see Hájek 2008 for a review). It is beyond the scope of this paper to assess how these arguments fare in light of state-dependent utility issues (a topic Nau has investigated, see in particular 1995). Section 3.4, however, relates the conceptual discussion of state-dependent utility issues to the better-known philosophical discussions of pragmatic arguments for probabilism.

  8. 8.

    It was sketched by Aumann in private correspondence with Savage. Thanks to Drèze, the letters have been made public in Savage and Aumann (1987).

  9. 9.

    Under those conditions, it is also the case that S is infinite, that \(\pi ^{\succcurlyeq }\) is non-atomic and finitely additive, and that u is bounded (see e.g. Fishburn 1970, Chap. 14, and further technical clarifications in Wakker 1993). These restrictions on u and on \(\pi ^{\succcurlyeq }\) indicate that Savage’s conditions do not characterize all preference relations representable according to the subjective expected utility model.

  10. 10.

    Specifically, we suppose that \(\pi ^*\) is a non-atomic finitely additive probability function on \(2^S\), the \(\sigma \)-algebra of all subsets of \(S=(0,1]\), such that the probability of any interval in (0, 1] simply is its length. It follows from a classical result in Horn and Tarski (1948) that such a function exists.

  11. 11.

    For the sake of concreteness, the case of Mr. Smith is presented with a dynamic twist, but this is inessential to the argument. For instance, you could be with him in the hospital waiting room and willing to identify his beliefs regarding the outcome of the operation on his wife at a time at which this operation has occurred. The analysis would be unchanged.

  12. 12.

    See Sect. 3.1. Our message at this point is essentially the following: before debating of re-description, let us first have a comprehensive look at all there would be to re-describe.

  13. 13.

    Philosophers might think that this kind of example requires the assumption that, as he evaluates bets, Mr. Smith agrees with his future (more or less cautious) self. First, as we argued in footnote 11, the dynamic twist of the story is conceptually inessential. Second, notice that even if it mattered, no more agreement with one’s future self would be required in state-dependent utility cases like the one above, than in any state-independent utility counterpart (as when Mr. Smith anticipates to stay just the same, were his wife to die). The same observation applies to the other variations on the case of Mr. Smith.

  14. 14.

    Consider (2), as fully specified with \(u_{\overline{E^*}}(x)=\sqrt{x}\) and \(E^*=(0,\frac{1}{2}]\). First, check that \(x > y \Leftrightarrow x \succ y\). Then, taking e.g. \(x=100,\,y=0=y',\,x'=9,\,E=(0,\frac{1}{10}],\,E'=(\frac{1}{2},1]\), check that \(x \succ y,\,x'\succ y',\,xEy \succ xE'y\) but \(x'Ey' \prec x'E'y'\).

  15. 15.

    In the Bayesian context we are focusing on here, “risk attitude” refers to the concavity properties of the utility function. From a mathematical point of view, when all the other postulates of Savage’s theorem are respected, Savage’s fourth postulate amounts to requiring that all (non-constant) conditional utility functions are related by a positive affine transformation. This appears more explicitly in a related theorem, due to Anscombe and Aumann (1963). This paper draws several comparisons between Savage’s theorem and the Anscombe–Aumann theorem. The framework of the latter result is adopted in Sect. 3.

  16. 16.

    This is but an implication of the postulate. The content of the postulate is far more general, in particular because like all of Savage’s postulates, it applies to arbitrary sets of consequences, non-numerical ones included (in which case, risk attitudes cannot be defined).

  17. 17.

    See Finetti 1937 (for a modern exposition, see e.g. Gilboa 2009, Chap. 9).

  18. 18.

    See the Appendix for a proof. Taken together, the cautious husband case and the ascetic husband case establish the logical independence of Savage’s third and fourth postulates. The framework of Anscombe and Aumann would prove less flexible than Savage’s on this topic. Endowed with the usual axioms, it would not let the difference between the cautious husband case and the ascetic husband case be fully expressed.

  19. 19.

    Consider (2), as fully specified with \(u_{\overline{E^*}}(x)=-x\) and \(E^*=(0,\frac{2}{3}]\). Taking e.g. \(x=1,\,y=0\), check that \(x \succ y\) and that \(x\varnothing y \succ x \overline{E^*} y\). This means that \(\varnothing \) is revealed to be held strictly more likely to happen than \(\overline{E^*}\). This disqualifies the likelihood order induced by Mr. Smith’s preferences as a qualitative probability relation (as defined in Fishburn 1970, p. 195). It is the most consensual property of qualitative probability relations, sometimes called “non-negativity”, which fails here to be satisfied.

  20. 20.

    Take e.g. \(x=1,\,y=0,\,E=\overline{E^*}\). First, check that \(x \succ y\), and that E is non-null. Then, taking \(h=x\) for example, check that \(xEh \prec yEh\), which violates the postulate.

  21. 21.

    See for instance Savage’s own (meteorological) case in Savage (1954/1972, p. 25).

  22. 22.

    For this terminology, see e.g. Wakker and Zank (1999, p. 10) or Hill (2010, p. 2045). It proves more relevant in the framework of Anscombe and Aumann, in which it originates.

  23. 23.

    Consider (2), as fully specified with \(u_{\overline{E^*}}(x)=c\) and \(E^*=(0,\frac{1}{2}]\). Taking e.g. \(x=1\) and \(y=0\), check that \(x E^* y \succ x \overline{E^*} y\) and that \( x \overline{E^*} y \sim x\varnothing y\). Noteworthily, in Savage’s own presentation of his theorem, the claim that an event is null if and only if it is associated with a null subjective probability value is stated on a par with the general representation in (1).

  24. 24.

    It is readily checked that the two concepts are logically independent.

  25. 25.

    This postulate requires that there exists consequences xy such that \(x \succ y\).

  26. 26.

    Besides, it would be impossible in Savage’s model because his postulates imply that all states, viz. degenerate events, are null (this is why it is required that there are infinitely many of them, so that the non-triviality postulate above can nonetheless be satisfied).

  27. 27.

    The kind of uniqueness clause to follow is not standard in discussions of Savage’s theorem or related results. In this context, the only explicit statement we are aware of is to be found in Wakker (1987, p. 293). Albeit in a different analytical framework, it has also been considered in recent models of awareness (see Karni and Vierø 2013, p. 2802). Essentially, this clause describes a renormalization of the probability values of non-null events. It amounts to making these values unique as on a ratio scale, rather than absolutely unique as they normally are.

  28. 28.

    We stress that this uniqueness clause is targeted at the problem of null events specifically. In effect, the “only if” direction of the proposition below will be questioned by the next variation on the case of Mr. Smith.

  29. 29.

    The new probability function \(\pi '^{\succcurlyeq }\) would contribute to representing the same underlying preferences, as stated in (1). But it would not represent the qualitative probability relation those preferences induce, following Savage’s construction. This is precisely this route to the identification of beliefs that is being questioned: the existence of an induced qualitative probability relation is one thing, its relevance is another.

  30. 30.

    Consider (2), as fully specified with \(u_{\overline{E^*}}(x)=\frac{1}{2}x\) and \(E^*=(0,\frac{1}{2}]\). Taking any consequences xy such that \(x \succ y\), check that \(x E^* y \succ x \overline{E^*} y\).

  31. 31.

    Ever since this paper, Karni has tirelessly tried to make decision theorists aware of the issue and to solve it one way or another. This paper is the closest to ours in the literature. Unlike our paper, however, it focuses on this particular issue only, instead of discussing together all state-dependent utility issues.

  32. 32.

    The proof of the Anscombe–Aumann theorem illustrates the fact that a collection of state-dependent utility functions is admissible here if and only if all (non-constant) state-dependent utility functions are related by a positive affine transformation. It is then always possible to interpret the transformation coefficients as probability weights. Among others, Karni often makes the following algebraic observation (see e.g. Karni and Mongin 2000, p. 238) which is suggestive of how arbitrary such a decomposition is. Consider a preference relation \(\succcurlyeq \) that is represented according to the subjective expected utility model, with \(\pi \) the probability function on a (for the sake of convenience, finite) state space S, and u the utility function. Pick an arbitrary function \(a : S \rightarrow {\mathbb {R}}^{+*}\). Define a new utility w based on \(w_s(x)=u_s(x)\) / a(s), for all x and all s, and a new probability \(\sigma \) based on \(\sigma (s)=\pi (s)a(s)\) / \(\sum _{t\in S}\pi (t)a(t)\), for all s. It is readily checked that the product of \(\sigma \) and w represents the same preferences. Focusing on the probability function in all these representations, this illustrates the extent to which preferences underdetermine beliefs.

  33. 33.

    See e.g. Karni (1996, p. 259). In light of this issue, there is no reason to think that the Ramseyan approaches to the identification of beliefs (see Ramsey 1931 and e.g. Bradley 2004 for a modern exposition) are not exposed to the problem of state-dependent utility. More precisely, either these approaches are not able to express this problem (this will be the case if their framework is such that no consequence is available in two different states of nature, see Bradley 2004, p. 494 for a discussion), or they are also exposed to it. Either way, they do not solve this problem.

  34. 34.

    In decision theory under risk, the respect of von Neumann and Morgenstern independence is sometimes defended against the Allais-type behavior on such grounds of simplicity.

  35. 35.

    For a model accommodating Allais’ paradox and relying on Savage’s third and fourth postulates, see e.g. Machina and Schmeidler (1992). For a model accommodating Ellsberg’s paradox and relying on these postulates as well, see e.g. Gilboa (1987) (Savage’s third postulate is weakened in this case, but the key implication highlighted in Sect. 2.3 is still in place).

  36. 36.

    Thus, the problem of state-dependent utility is yet another topic of discussion for the methodological literature on scoring rules. Deviations from risk-neutrality already prove challenging for standard scoring rules (for a presentation of the problem, see e.g. Schlag et al. 2014, Sect. 2.4, and see Karni 2009 for a solution). State-dependent utility, which includes variations in risk attitudes as illustrated in Sect. 2.2, is therefore even more challenging for these rules (see Karni 1999 for a partial investigation).

  37. 37.

    Extreme risks, such as exceptional natural cataclysms, epidemics, or terrorist attacks, raise distinctive problems for the economic theory of insurance (see e.g. Gollier 1997).

  38. 38.

    See in particular Wakker and Zank (1999) and Hill (2010). As these references discuss, it is particularly challenging to obtain a general additively separable representation in Savage’s framework (recall that, in this framework, the state space must be infinite and notice that, without a probability function, integration is undefined). As the rest of our paper illustrates, this is a much simpler task in the framework of Anscombe and Aumann.

  39. 39.

    See Nau (2001) for a detailed defense of this claim.

  40. 40.

    We would dismiss by the same argument the counter-objections to our last two variations on the case of Mr. Smith according to which, when all of Savage’s postulates are satisfied, the state-independent decomposition in (1) is as good as any alternative state-dependent decomposition because it is sufficient to predict individual behavior accurately. This last point is correct (although it is important that it can be incorrect in some dynamic strategic settings, as detailed in Karni 2008). Yet, if the goal is to identify beliefs, this observation is a non-starter and the problem of state-dependent utility remains open.

  41. 41.

    See Savage (1954/1972, p. 25) and Savage and Aumann (1987, pp. 78–80).

  42. 42.

    This is essentially Drèze’s seminal example, see e.g. Drèze (1987, p. 25).

  43. 43.

    See Drèze (1961) for the pioneering version of this approach, Karni (2011a, b) for a more recent version. A key difference between these models is that, unlike the former, the latter explicitly articulates the actions by which the agent affects the likelihood of events. This leads to significantly different representations and interpretations of subjective probability.

  44. 44.

    It is stated in print as Theorem 1.4 in Karni (1985). This result has been significantly generalized in different directions (see Karni 2003 and Grant and Karni 2004). These generalizations, however, do not matter for the philosophical discussion to come.

  45. 45.

    From a mathematical point of view, it is also possible to think of Savage’s framework as a variant of Anscombe and Aumann’s framework in which the consequences are all degenerate lotteries. In the rest of this paper, for the sake of convenience, S is assumed to be finite (see Fishburn 1970, p. 179, for a generalization of the Anscombe–Aumann theorem when S is infinite). As mentioned before, this assumption is not compatible with Savage’s postulates, but it is compatible with the new set of postulates which we need from now on.

  46. 46.

    They are the conditions of the von Neumann–Morgenstern theorem. Nevertheless, in the proof of the representation theorem, another supposition is needed. It can be interpreted as a form of indifference regarding the order in which uncertainty is resolved. Under this interpretation, it has been criticized by Drèze in relation with the problem of moral hazard (see e.g. 1987, p. 27).

  47. 47.

    This is what is required by the key linkage postulate of the Karni–Schmeidler theorem, which consists in coordinating the conditional preferences of \(\succcurlyeq \) and \(\widehat{\succcurlyeq }\). Some notation is needed to introduce it formally. Denote by \(\widehat{F}\) the set \(\Delta (S\times Z)\). For a generic element \(\widehat{f}\) of \(\widehat{F}\), denote by \(\widehat{f}(s,z)\) the probability value associated by \(\widehat{f}\) to the hypothetical outcome (sz). Denote by \(\widehat{\succcurlyeq _s}\) the hypothetical preference of the decision-maker conditional on state s obtaining, defining it as follows: \(\widehat{f}\,\widehat{\succcurlyeq _s}\,\widehat{g}\) if \(\widehat{f}\,\widehat{\succcurlyeq }\,\widehat{g}\), with \(\widehat{f}\) and \(\widehat{g}\) such that for all \(t \ne s \in S\), and all \(z \in Z,\,\widehat{f} (t,z)=\widehat{g} (t,z)\). (Given that \(\widehat{\succcurlyeq }\) respects the so-called von Neumann–Morgenstern independence, such conditional preferences are well-defined.) Denote by \(\widehat{F^*}\) the set of all elements of \(\widehat{F}\), the marginal probability of which have full support on S. On the other hand, consider F and the relation \(\succcurlyeq \) which is defined over it. For a generic element f of F, denote by \(f_s(z)\) the probability value associated by f to outcome z in state s. Denote by \(\succcurlyeq _s\) the preference of the decision-maker on \(\Delta (Z)\), conditional on state s obtaining, defining it as usual. A bijection H can be defined between \(\widehat{F^*}\) and F, as follows: for all \(\widehat{f^*}\), let \(H(\widehat{f^*})\) denote the element \(f \in F\) such that for all \(z \in Z\), and all \(s \in S\), \(f_s(z)=\widehat{f^*}(s,z)\) / \(\sum _{y \in Z}\widehat{f^*}(s,y)\). The key linkage postulate of the Karni–Schmeidler theorem (labeled “strong consistency” by the authors) requires that for all \(\widehat{f^*},\,\widehat{g^*} \in \widehat{F^*}\), and for any non-null \(s \in S\), we have \(\widehat{f^*}\,\widehat{\succcurlyeq }_s\,\widehat{g^*}\) if and only if \(H( \widehat{f^*}) \succcurlyeq _s H(\widehat{g^*})\).

  48. 48.

    It is adapted from Karni et al. (1983, p. 1025). These authors, however, have a terminology less informative than ours on this topic. They oppose “evidently null” events to the “indeterminate case[s]”. They would be more specific in opposing “cognitively” null events to “conatively” null ones instead.

  49. 49.

    However, those two kinds of exogenous probabilities are unequally essential to the hypothetical preference approach. As illustrated in Karni (2003) following previous work by Wakker, under some conditions, the preliminary additively separable representation can be obtained without exogenous probabilities of the first kind. Exogenous probabilities of the second kind, by contrast, are instrumental in neutralizing the decision-maker’s beliefs and identifying state-(in)dependent utility. It is more difficult to imagine how they could be dispensed with (but see the alternative approach mentioned in footnote 57).

  50. 50.

    See Karni and Mongin (2000, Sect. 4.3), for another methodological discussion of hypothetical preferences. By contrast with ours, Karni and Mongin’s discussion is more concerned with assessing whether ordinary preferences really are immune to the criticisms usually levied against hypothetical preferences. We focus here on the preliminary step of clarifying those criticisms more completely than elsewhere in the current literature.

  51. 51.

    Notice that in some cases, \(\pi ^{\circ }\) may correspond to a Bayesian updating of \(\pi ^*\) after the reception of some information by the decision-maker. The restriction of hypothetical preferences to a generic cell \(\Delta ^{\circ }\) has been investigated in a theorem related to the Karni–Schmeidler theorem (see Karni et al. 1983). It has been proved that it would not be sufficient for beliefs to be identified uniquely (see Karni and Mongin 2000, Sect. 3.4). To that end, it is indispensable to let hypothetical preferences be defined across different cells of the partition, i.e., necessary to introduce an even stronger form of hypotheticality than the one of counterfactual choices.

  52. 52.

    Thus, hypothetical preferences are reminiscent of extended preferences which have been considered in social choice theory (see in particular Mongin 2001, Sects. 4 and 6). Such preferences typically read as follows: “from the point of view of individual i, it is better to be individual j in social state x, than to be individual k in social state y”. Like hypothetical preferences, in general, extended preferences cannot be related to ordinary or counterfactual choices.

  53. 53.

    Notice the following contrast with Savage’s own proposal regarding state-dependent utility issues. Savage’s approach introduces impossible options and, to that extent, impossible choices. Karni and Schmeidler’s approach introduces impossible choices, but no impossible option (recall the difference between \(\Delta (S\times Z)\) and \((S\times Z)^S\), which we highlighted above).

  54. 54.

    Consider for example the result in Hill (2009), which is set in Savage’s framework. It is the only such result we are aware of which proposes an identification of subjective probability applicable when neither Savage’s third postulate, nor his fourth is satisfied. It relies on generalizations of these two postulates, which would be violated in the case above. Admittedly, in this case, the state space might be partitioned in such a way that, conditional on any event within each cell of the partition (by contrast with events across such cells), Savage’s third postulate holds, and likewise for his fourth. This last case is investigated in Karni and Schmeidler (1993). However, the partition relevant for Savage’s third postulate might differ from the one relevant to his fourth, e.g., the third postulate could be respected on either \(E_1\cup E_3\) or \(E_2 \cup E_4\), while the fourth postulate would be respected on either \(E_1\cup E_2\) or \(E_3 \cup E_4\). Karni and Schmeidler’s 1993 result does not cover this case nor indicates how to do so. We conjecture that it is intractable in Savage’s setup.

  55. 55.

    See especially Samuelson (1950) for the pioneering original competitive consumer version of those results, and e.g. Sen (1971) for the later more abstract set-theoretical version. Sen (1973) illustrates the now familiar philosophical discussions of the revealed preference semantics according to which, in the results above, preference is defined in terms of choices (see Hausman 2000 for a more recent example of such discussions). Our discussion of the revealed preference framework is distinct. In particular, the methodological issues considered below are relevant even if “choice” and “preference” are treated as distinct concepts.

  56. 56.

    For example, some results in portfolio theory rely on assumptions about the investor’s beliefs regarding the return of assets (see Karni and Schmeidler 1993, p. 272 for a discussion of a classical theorem by Arrow in light of the problem of state-dependent utility with state-dependent preferences). On the other hand, many results rely on a shared prior assumption, especially in the interactive epistemology literature which we mentioned in the introduction. Of special interest in our context is the use of such assumption to circumscribe the correct use of Pareto conditions in the literature about the aggregation of preferences under uncertainty (see Gilboa et al. 2004, and a discussion in Karni 2007, Sect. 3.1 in light of the problem of state-dependent utility without state-dependent preferences). If beliefs are unidentified, none of these results is applicable.

  57. 57.

    For instance, another important stream in the literature is the conditional expected utility approach (see Luce and Krantz 1971; Fishburn 1973, and more recently Karni 2007). In this approach, preferences are defined over acts conditioned on different events, as when Mr. Smith prefers a certain act, knowing that the operation succeeds, to another act, knowing that it fails. This amounts to acting across conflicting beliefs, and large parts of the discussion of Sect. 3.3 would apply there as well. This is why we claimed that focusing on Karni and Schmeidler’s result entailed no loss of generality for our topic.

  58. 58.

    For example, it is sometimes argued that, in order to carry out welfare evaluations, but also to have a more unified theory of decision-making, economics needs a cardinal utility applicable to decisions under certainty (see e.g. the discussion in Wakker 1994, Sect. 2). In general, however, preferences over certain options cannot deliver a cardinal utility, unless one introduces a notion of preference differences in this context. Nevertheless, such preference differences are generally held to be incompatible with the revealed preference framework (see Fishburn 1970, Sect. 6.1 for a representative statement of this view).

  59. 59.

    See e.g. the discussion about outcome individuation in Joyce (1999, Sect. 2.2).

  60. 60.

    See e.g. Joyce (1998) for a critique of the Dutch Book argument along the line mentioned. Regarding the wider understanding of the methodological topic under discussion, see e.g. Joyce (1999, p. 89), where “pragmatism” is defined as the claim according to which “we can learn everything we need to know about epistemology by doing decision theory”.


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The author thanks the Editor-in-Chief Wiebe van der Hoek, two referees, Mikaël Cozic, Éric Danan, Vincent Éli, Itzhak Gilboa, Raphaël Giraud, Brian Hill, Marcus Pivato, Fanyin Zheng and especially Philippe Mongin for most helpful comments and discussions. The author is sole responsible for all remaining errors or omissions. Research for this paper was funded by the Université de Cergy-Pontoise, and before that by the École Normale Supérieure - Ulm (grants ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*).

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In this appendix, it is proved that preferences representable as in (2) with \(u_{\overline{E^*}}(x)=-x\) and \(E^*=(0,\frac{2}{3}]\) respect Savage’s fourth postulate, namely:

$$\begin{aligned} \forall E,E' {\in } 2^S, \forall x,y,x',y' {\in } X \text { s.t. } x \succ y, x' \succ y' : xEy \succ xE'y \Leftrightarrow x'Ey' \succ x'E'y'. \end{aligned}$$

First, it is readily checked that for those preferences, \(x \succ y \Leftrightarrow x > y\). Next, suppose that \(x \succ y\) and \(xEy \succ xE'y\). Using (2), this is true if and only if:

$$\begin{aligned}&\pi ^*(E) \cdot u_E(x)+\pi ^*(\overline{E}) \cdot u_{\overline{E}}(y) > \pi ^*(E') \cdot u_{E'}(x)+\pi ^*(\overline{E'}) \cdot u_{\overline{E'}}(y) \\&\Leftrightarrow \pi ^*(E\cap E^*)x - \pi ^*(E\cap \overline{E^*})x + \pi ^*(\overline{E}\cap E^*)y - \pi ^*(\overline{E}\cap \overline{E^*})y \\&>\pi ^*(E'\cap E^*)x - \pi ^*(E'\cap \overline{E^*})x + \pi ^*(\overline{E'}\cap E^*)y - \pi ^*(\overline{E'}\cap \overline{E^*})y \\&\Leftrightarrow x\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&>y\left[ \pi ^*(\overline{E'}\cap E^*) - \pi ^*(\overline{E}\cap E^*) + \pi ^*(\overline{E}\cap \overline{E^*}) - \pi ^*(\overline{E'}\cap \overline{E^*})\right] . \end{aligned}$$

Noticing that \(\pi ^*(E'\cap \overline{E^*})+\pi ^*(\overline{E'}\cap \overline{E^*})=\pi ^*(\overline{E^*})=\pi ^*(E\cap \overline{E^*})+\pi ^*(\overline{E}\cap \overline{E^*})\) and that \(\pi ^*(E'\cap E^*)+\pi ^*(\overline{E'}\cap E^*)\, =\pi ^*(E^*)=\pi ^*(E\cap E^*)+\pi ^*(\overline{E}\cap E^*)\), using the fact that \(x \succ y \Leftrightarrow x > y\) and \(x' \succ y' \Leftrightarrow x' > y'\), the supposition is true if and only if:

$$\begin{aligned}&x\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&>y\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&\Leftrightarrow \left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] > 0 \\&\Leftrightarrow x'\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&>y'\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&\Leftrightarrow x'\left[ \pi ^*(E\cap E^*) -\pi ^*(E'\cap E^*) + \pi ^*(E'\cap \overline{E^*}) - \pi ^*(E\cap \overline{E^*})\right] \\&>y'\left[ \pi ^*(\overline{E'}\cap E^*) - \pi ^*(\overline{E}\cap E^*) + \pi ^*(\overline{E}\cap \overline{E^*}) - \pi ^*(\overline{E'}\cap \overline{E^*})\right] \\&\Leftrightarrow x'Ey' \succ x'E'y'. \end{aligned}$$

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Baccelli, J. Do bets reveal beliefs?. Synthese 194, 3393–3419 (2017).

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  • Beliefs
  • Subjective probability
  • State-dependent utility
  • Hypothetical preferences
  • Revealed preference