Risk Neutrality Incompleteness and Degenerate Indifference

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Then, x ∼ y ⇐⇒ x = y (2) The proof appears in the Appendix. In the case where n = 2 this result can be easily illustrated with an example: Suppose x ∼ y for x, y ∈ R 2 . It follows from (1) and the nonempty-interior assumption that there exist distinct probability measures (p, 1 − p) and (q, 1 − q) in P such that px 1 + (1 − p)x 2 = py 1 + (1 − p)y 2 qx 1 + (1 − q)x 2 = qy 1 + (1 − q)y 2 Moreover, since P has a nonempty interior, (p, 1 − p) and (q, 1 − q) can be chosen to be strictly positive. In this case, dividing the first equation by p and the second by q, and then subtracting the former from the latter yields x 2 = y 2 and x 1 = y 1 . The proof of the general case generalizes this argument mainly by showing that there exists an n-tuple of strictly positive probability measures that span X ≡ R n . 3 Strict Bewley Preferences As in Bewley (2002), we first explore the implications of Proposition 1 in the case where the primitive is a strict preference relation ≻ on X. The following four axioms are standard in this context and were also employed in the statement of Bewley’s Theorem 1, which is of direct relevance here. Strict Partial Order ≻ is asymmetric and transitive. Independence For all x, y, z ∈ X and all α ∈ (0, 1), x ≻ y implies αx + (1 − α)z ≻ αy + (1 − α)z. Strict Monotonicity For all x, y ∈ X, x ≥ y and x ≠ y implies x ≻ y. Upper Hemicontinuity For all x ∈ X, the set U ≻ (x) := {y ∈ X : y ≻ x} is open. We restate Bewley’s Theorem 1 below in a way that reflects the indifference-degeneracy implications of Proposition 1 and also allows for one additional clarification to be made. Corollary 2. A binary relation ≻ on X is a Strict Partial Order that satisfies Independence, Strict Monotonicity and Upper Hemicontinuity if and only if there exists a compact, convex set P ⊆ rint(∆(S)) such that, for all x, y ∈ X, x ≻ y ⇐⇒ px > py for all p ∈ P (3) x = y ⇐⇒ px = py for all p ∈ P (4) Although Bewley (2002) did not assume an indifference relation as part of the model’s primitives, he noted that once a set of priors P that represents ≻ in the sense of (3) is known to exist, an 2
indifference relation ∼ on X could be defined by x ∼ y ⇔ px = py for all p ∈ P . Driven by Proposition 1, part (4) of this result clarifies that, under the conditions assumed in Bewley’s (2002) Theorem 1, this indifference relation is actually degenerate. Moreover, the statement of Corollary 2 also clarifies that the relevant conditions in Bewley’s (2002) Theorem 1 are not merely sufficient but also necessary for ≻ to be represented as in (3). We note, finally, that when a strict relation ≻ is assumed to be the primitive in a model of incomplete preferences, Galaabaatar and Karni (2013) suggested to define an indifference relation ∼ from ≻ by letting x ∼ y if and only if U ≻ (x) = U ≻ (y). 2 Since this ∼ is an equivalence relation, it follows from Proposition 1 that this too is degenerate in the present context whenever the set P that represents ≻ has a nonempty interior. 4 Weak Bewley Preferences A conceptual difficulty with the strict-preference version of Bewley’s unanimity model is that it does not specify a relevant “mild preference” label for the cases where all priors weakly favour an act x over another act y and some but not all strictly favour x. An approach that addresses this issue was followed by Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), Gilboa, Maccheroni, Marinacci, and Schmeidler (2010) and Ok, Ortoleva, and Riella (2012), who took as the primitive a (possibly incomplete) weak preference relation . Unlike Bewley’s (2002) framework, under this approach the relations ≻ and ∼ that define strict preference and indifference are derived from as its asymmetric and symmetric parts in the usual way. While this is an advantage of the weakover the strict-preference approach, a well-known difficulty that arises here is that even though the relation is continuous, its strict part ≻ cannot be unless is complete (see below). To study the analogue of Corollary 2 in a weak-preference framework we state a few axioms that are particular to such a relation. Preorder is reflexive and transitive. Incompleteness There exist x, y ∈ X such that x ̸ y and y ̸ x. Restricted Hemicontinuity The set U (0) := {y ∈ X : y 0} is closed. The former two axioms ensure that is a transitive and incomplete weak preference relation. It will be clear in the statement of the result below why requiring incompleteness explicitly is important for our purposes. Although the third axiom is weaker, in principle, than the standard axiom of closed-graph continuity, it turns out that in the present context the two are equivalent. 3 As far as the Independence axiom for is concerned, it coincides with the one that was stated in the previous section provided that ≻ is replaced by . 2 Bewley (2002) called two acts x and y equivalent when, in addition, to the above condition of Galaabaatar and Karni (2013) that requires equality between the strict upper contour sets of x and y, their strict lower contour sets are also equal. 3 The reader is referred to Theorem 1 in Gerasimou (2015) and p. 156 in Ghirardato, Maccheroni, and Marinacci (2004) for more details on this. 3
Page 1 and 2: Risk Neutrality, Incompleteness and
Page 3: 1 Introduction In Bewley’s (2002)
Page 7 and 8: Observe that the set {q 1 , . . . ,

Risk Neutrality Incompleteness and Degenerate Indifference

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