Forthcoming in Erkenntnis COUNTEREXAMPLES TO SOME CHARACTERIZATIONS OF DILATION MICHAEL NIELSEN AND RUSH T. STEWART 1. Introduction Pedersen and Wheeler (2014) and Pedersen and Wheeler (2015) offer a wide-ranging and in-depth exploration of the phenomenon of dilation. We find that these studies raise many interesting and important points. However, purportedly general characterizations of dilation are reported in them that, unfortunately, admit counterexamples. The purpose of this note is to show in some detail that these characterization results are false. We start by rehearsing the relevant definitions (Section 2) and some known results about dilation (Section 3). Section 4 addresses the results in Pedersen and Wheeler (2014), and Section 5 addresses the results in Pedersen and Wheeler (2015). We conclude, in Section 6, by pointing out where Pedersen and Wheeler's proofs go wrong. 2. Definitions We use (Ω,F) to denote a measurable space. A probability measure P on (Ω,F) is a non-negative, finitely additive set function with domain F and P (Ω) = 1. P denotes a set of probability measures on (Ω,F), and P∗ denotes the weak*-closure of the convex hull of P. The lower probability of and event A in P is P (A) = inf{P (A) : P ∈ P}, and the upper probability of and event A in P is P (A) = sup{P (A) : P ∈ P}. We call (Ω,F ,P, P ) a lower probability space. If P (E) > 0, then the conditional probability of A given E is defined, as usual, by P (A | E) = P (A ∩ E)/P (E). (1) Upper and lower conditional probabilities are defined, respectively, by P (A | E) = inf{P (A | E) : P ∈ P} and P (A | E) = sup{P (A | E) : P ∈ P}. Let E = {Ei : i ∈ I} be a positive, measurable partition of Ω. Measurability of the partition means that E ∈ F for all E ∈ E , while by "positive" we mean that P (E) > 0 for all E ∈ E . Say that E dilates an event A ∈ F just in case, for each E ∈ E P (A | E) < P (A) ≤ P (A) < P (A | E). (2) The next definitions are used in the alleged characterizing conditions for dilation. If A,E ∈ F , let SP (A,E) = P (A ∩ E) P (E)P (A) (3) provided P (A)P (E) > 0. If P (A)P (E) = 0, then we can let SP (A,E) = 1, as in Seidenfeld and Wasserman (1993), but we won't encounter that situation in this note. Now let S−P (A,E) = {P ∈ P : SP (A,E) < 1} and S + P (A,E) = {P ∈ P : SP (A,E) > 1}. (4) Date: June 28, 2019. 1 2 NIELSEN AND STEWART If ε > 0, then let P(A | E, ε) = {P ∈ P : |P (A | E)− P (A | E)| < ε} (5) and P(A | E, ε) = {P ∈ P : |P (A | E)− P (A | E)| < ε}. (6) Finally, let R+ be the set of positive real numbers, and, if I is a set, let RI+ be the set of functions from I into R+. We also call members of RI+ I-sequences. 3. Some Known Results The following proposition summarizes some known necessary and sufficient conditions for dilation. See Pedersen and Wheeler (2014) and Seidenfeld and Wasserman (1993). Proposition 1. Let (Ω,F ,P,P) be a lower probability space, let E = {Ei : i ∈ I} be a positive, measurable partition of Ω, and let A ∈ F . The following condition is sufficient for E to dilate A: (a) P(A) ∩ S−P (A,Ei) 6= ∅ and P(A) ∩ S + P (A,Ei) 6= ∅ for all i ∈ I. And the following conditions are necessary for E to dilate A: (b) P(A | Ei) ⊆ S−P (A,Ei) and P(A | Ei) ⊆ S + P (A,Ei) for all i ∈ I. (c) There is (εi)i∈I ∈ RI+ such that for every i ∈ I: P(A | Ei, εi) ⊆ S−P (A,Ei) and P(A | Ei, εi) ⊆ S + P (A,Ei). (7) Remark 1. The proof that dilation implies (c) in Pedersen and Wheeler (2014) uses several concepts from functional analysis. But an elementary proof is possible. Indeed, if dilation occurs, then P (A | Ei) < P (A) for all i ∈ I. Let 0 < εi < P (A) − P (A | Ei). If P ∈ P(A | Ei, εi), then P (A | Ei)−P (A | Ei) < εi. Hence, P (A | Ei) < P (A) ≤ P (A), which means that P ∈ S−P (A,Ei). An analogous proof, from the fact that dilation implies P (A) < P (A | Ei) for all i ∈ I, yields, for each i, an εi > 0 such that P(A | Ei, εi) ⊆ S+P (A,Ei). We then let εi = min(εi, εi), and (c) is proved. Proposition 1 lists two necessary conditions and one sufficient condition for dilation, but no jointly necessary and sufficient conditions. In Section 6, we will refer to a result due to Wasserman and Seidenfeld that provides a condition that is both necessary and sufficient for dilation in simple models (1994, Result 1). Our leading example, Example 1, is an instance of such a model. 4. Counterexamples to the 2014 Results We will now record the main results from Pedersen and Wheeler (2014) and show that they are false. Claim 1 (Pedersen and Wheeler, 2014, Proposition 5.1). Let (Ω,F ,P,P) be a lower probability space such that P is weak*-closed and convex, let E = {Ei : i ∈ I} be a positive, measurable partition of Ω, and let A ∈ F . Then the following are equivalent: (1i) E dilates A. (1ii) There is (εi)i∈I ∈ RI+ such that for every i ∈ I: P(A | Ei, εi) ⊆ S−P (A,Ei) and P(A | Ei, εi) ⊆ S + P (A,Ei). (8) (1iii) There are (εi)i∈I ∈ RI+ and (εi)i∈I ∈ RI+ such that for every i ∈ I: P(A | Ei, εi) ⊆ S−P (A,Ei) and P(A | Ei, εi) ⊆ S + P (A,Ei). (9) COUNTEREXAMPLES TO SOME CHARACTERIZATIONS OF DILATION 3 Furthermore, each radius εi may be chosen to be the unique positive minimum of |P (A | Ei) − P (A | Ei)| attained on C+i := {P ∈ P : SP (A,Ei) ≥ 1}, and similarly each radius εi may be chosen to be the unique positive minimum of |P (A | Ei) − P (A | Ei)| attained on C−i := {P ∈ P : SP (A,Ei) ≤ 1}. On the most natural reading of Claim 1, it is a conjunction: it asserts that the statements (i)-(iii) are equivalent and conjoins to that assertion an additional assertion about how εi and εi may be chosen. To falsify this claim, then, it is enough to show that the assertion about the equivalence of (i)-(iii) is false. In personal communication, Pedersen and Wheeler have indicated that "the statement of our theorem in the Erkenntnis paper [Pedersen and Wheeler (2014)] is not as explicit as our generalization of this result, Theorem 1, in our ISIPTA 2015 paper [Pedersen and Wheeler (2015)]" and that "'Furthermore, each radius εi may be chosen' ought to read 'Furthermore, each radius εi should be chosen' [emphasis added]." But the replacement of "may" with "should" changes the content of Claim 1 significantly. Since the result is stated with "may," we think that it is worth showing that Claim 1 is false as stated. Later, we will show that Pedersen and Wheeler's result remains false even when "may" is replaced by "should", as is done more explicitly in their 2015 paper. Before giving the counterexample, note that (1ii) and (1iii) are clearly equivalent. That (1ii) implies (1iii) is obvious, and the implication from (1iii) to (1ii) is proved by setting εi = min(εi, εi) for all i ∈ I. Moreover, by Proposition 1, (1i) implies (1ii) whether or not P is weak*-closed and convex. To show that (1i)-(1iii) are not equivalent, then, we must show that (1iii) does not imply (1i): Proposition 2. In Claim 1, (1iii) is not sufficient for E to dilate A. In the proof of this proposition and throughout the rest of the note, we will use the following simple example. Example 1. Let P0 and P1 be probabilities on the discrete measurable space Ω = {ω1, ω2, ω3, ω4} defined by ω1 ω2 ω3 ω4 P0 1/8 1/2 1/3 1/24 P1 5/24 1/24 1/24 17/24 Let P = {P0, P1}. Let A = {ω1, ω2}, E1 = {ω1, ω3}, E2 = {ω2, ω4} and E = {E1, E2}. 4 Proof of Proposition 2. We show that (Ω,P(Ω),P∗, P ), E , and A, as defined in Example 1 satisfy (1iii) but not (1i). We begin by noting that P∗ is identical to the convex hull of {P0, P1}. For each α ∈ [0, 1] let Pα = (1− α)P0 + αP1. (10) Due to the one-to-one correspondence between [0, 1] and P∗, it is useful to think of P∗ as a set of probabilities indexed by the unit interval. Some straightforward calculations show that Pα(A) = 5− 3α 8 , Pα(A | E1) = 3 + 2α 11− 5α , Pα(A | E2) = 12− 11α 13 + 5α . (11) Since, P (A) = 1/4 < 3/11 = P (A | E1), it follows that E does not dilate A. Thus, (1i) does not hold. To finish, we must show that (1iii) holds. An easy way to see that (1iii) does indeed hold is to graph the curves in (11). 4 NIELSEN AND STEWART 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 α Pα(A) Pα(A | E1) Pα(A | E2) Observe that for each E ∈ {E1, E2} all Pα ∈ P∗ that are sufficiently close to minimizing (resp. maximizing) {P ′(A | E) : P ′ ∈ P∗} are members of S−P∗(A,E) (resp. S + P∗(A,E)). In other words, observe that for some small ε1, ε1 > 0, the blue curve lies below the red line on [0, ε1) and above the red line on (1− ε1, 1]. Similarly, there are small ε2, ε2 > 0 such that the green curve lies below the red line on (1− ε2, 1] and above the red line on [0, ε2). In case the graph is not transparent, we now check all of the relevant calculations in detail. We begin by showing that there exists ε1 > 0 such that P∗(A | E1, ε1) ⊆ S−P∗(A,E1). Let ε1 = 1/11. Then, Pα ∈ P∗(A | E1, ε1) iff 3 + 2α 11− 5α − 3 11 < 1 11 , which holds iff 0 ≤ α < 11 42 . Now, Pα ∈ S−P∗(A,E1) iff 3 + 2α 11− 5α < 5− 3α 8 , which holds iff 0 < 31− 74α+ 15α2. The function f(α) = 31 − 74α + 15α2 is decreasing in the interval [0, 1] because f ′(α) = 30α − 74 < 0 if α ∈ [0, 1]. Since f(11/42) ≈ 12.6 > 0, it follows that for all α ∈ [0, 11/42), f(α) > 0. Hence, if Pα ∈ P∗(A | E1, ε1), then Pα ∈ S−P∗(A,E1). Next, we show that there exists ε1 > 0 such that P∗(A | E1, ε1) ⊆ S+P∗(A,E1). Let ε1 = 1/6. Note that P (A | E1) = 5/6. Then, Pα ∈ P∗(A | E1, δ2) iff 5 6 − 3 + 2α 11− 5α < 1 6 , which holds iff 13 16 < α ≤ 1. Now, Pα ∈ S+P∗(A,E1) iff 3 + 2α 11− 5α > 5− 3α 8 , COUNTEREXAMPLES TO SOME CHARACTERIZATIONS OF DILATION 5 which holds iff 0 > 31− 74α+ 15α2. As we noted in the first case, the function f(α) = 31−74α+15α2 is decreasing in the interval [0, 1]. Since f(13/16) ≈ −19.2 < 0, it follows that for all α ∈ (13/16, 1], f(α) < 0. Hence, if Pα ∈ P∗(A | E1, ε1), then Pα ∈ S+P∗(A,E1). Next, we show that there exists ε2 > 0 such that P∗(A | E2, ε2) ⊆ S−P∗(A,E2). Let ε2 = 1/18. Note that P (A | E2) = 1/18. Then, Pα ∈ P∗(A | E2, ε2) iff 12− 11α 13 + 5α − 1 18 < 1 18 , which holds iff 95 104 < α ≤ 1. Now, Pα ∈ S−P∗(A,E2) iff 12− 11α 13 + 5α < 5− 3α 8 , which holds iff 0 > 31− 74α+ 15α2. As usual, the function f(α) = 31 − 74α + 15α2 is decreasing in the interval [0, 1]. Since f(95/104) ≈ −24.1 < 0, it follows that for all α ∈ (95/104, 1], f(α) < 0. Hence, if Pα ∈ P∗(A | E2, ε2), then Pα ∈ S−P∗(A,E2). Finally, we show that there exists ε2 > 0 such that P∗(A | E2, ε2) ⊆ S+P∗(A,E2). Let ε2 = 1/13. Note that P (A | E2) = 12/13. Then, Pα ∈ P∗(A | E2, ε2) iff 12 13 − 12− 11α 13 + 5α < 1 13 , which holds iff 0 ≤ α < 13 198 . Now, Pα ∈ S+P∗(A,E2) iff 12− 11α 13 + 5α > 5− 3α 8 , which holds iff 0 < 31− 74α+ 15α2. By now the reader will anticipate our assertion that the function f(α) = 31 − 74α + 15α2 is decreasing in the interval [0, 1]. Since f(13/198) ≈ 26.2 > 0, it follows that for all α ∈ [0, 13/198), f(α) > 0. Hence, if Pα ∈ P∗(A | E2, ε2), then Pα ∈ S+P∗(A,E2). We have now shown that (εi)i∈{1,2} and (εi)i∈{1,2} satisfy (9) for each i ∈ {1, 2}, thus (1iii) holds.  Remark 2. Let ε = min(ε1, ε1, ε2, ε2) > 0 in the proof of Proposition 2. Then, by what we have just shown, the following condition-which is, in general, stronger than (1iii), but, in the case |I| <∞, equivalent to (1iii)-is not sufficient for dilation: (1iii') There exists ε > 0 such that for all i ∈ I P(A | Ei, ε) ⊆ S−P (A,Ei) and P(A | Ei, ε) ⊆ S + P (A,Ei). 6 NIELSEN AND STEWART The corollaries to Pedersen and Wheeler's Proposition 5.1 are also false. To show this, we need an additional definition. Let S−∗ (A,E) = {P ∈ P∗ : SP (A,E) < 1} and S+∗ (A,E) = {P ∈ P∗ : SP (A,E) > 1}. (12) (We note that in the proof of Proposition 2, we used the notation S−P∗(A,E) (resp. S + P∗(A,E)) instead of S−∗ (A,E) (resp. S + ∗ (A,E)). The only reason we introduce the latter, redundant notation now is in order to follow Pedersen and Wheeler as closely as possible.) Claim 2 (Pedersen and Wheeler, 2014, Corollary 5.2). Let (Ω,F ,P,P) be a lower probability space, let E = {Ei : i ∈ I} be a positive, measurable partition of Ω, and let A ∈ F . Then the following are equivalent: (2i) E dilates A. (2ii) There is (εi)i∈I ∈ RI+ such that for every i ∈ I: P∗(A | Ei, εi) ⊆ S−∗ (A,Ei) and P∗(A | Ei, εi) ⊆ S+∗ (A,Ei). (13) (2iii) There are (εi)i∈I ∈ RI+ and (εi)i∈I ∈ RI+ such that for every i ∈ I: P∗(A | Ei, εi) ⊆ S−∗ (A,Ei) and P∗(A | Ei, εi) ⊆ S+∗ (A,Ei). (14) Furthermore, each radius εi may be chosen to be the unique positive minimum of |P (A | Ei)− P (A | Ei)| attained on C+i := {P ∈ P∗ : SP (A,Ei) ≥ 1}, and similarly each radius εi. Claim 3 (Pedersen and Wheeler, 2014, Corollary 5.3). Let (Ω,F ,P,P) be a lower probability space, let E = {Ei : 1 ≤ i ≤ n} be a finite positive, measurable partition of Ω, and let A ∈ F . Then the following are equivalent: (3i) E dilates A. (3ii) There is ε > 0 such that for every i = 1, ..., n: P∗(A | Ei, ε) ⊆ S−∗ (A,Ei) and P∗(A | Ei, ε) ⊆ S+∗ (A,Ei). (15) Proposition 3. In Claim 2, (2iii) does not imply (2i). In Claim 3, (3ii) does not imply (3i). Proof. Let (Ω,P(Ω),P,P), E = {E1, E2}, and A be defined as in Example 1. In the proof of Proposition 2, we have already shown that (2iii) holds. Now note that P (A) = P1(A) = 1/4 < 3/11 = P0(A | E1) = P (A | Ei), so (2i) doesn't hold. Thus, (2iii) does not imply (2i). As E is finite, the conditions of Claim 3 are met, and we have just shown that (2i) does not hold, so (3i) does not hold. In Remark 2, we have already shown that (3ii) holds. Thus, (3ii) does not imply (3i).  We now turn to the results in Pedersen and Wheeler (2015). 5. Counterexamples to the 2015 Results As we noted above, the "Furthermore..." clauses of Claims 1 and 2 of the previous section are incorporated more explicitly into the characterizing conditions of dilation in Pedersen and Wheeler's later work. In this section, we show that the resulting claims are still false. Claim 4 (Pedersen and Wheeler, 2015, Theorem 1). Let (Ω,F ,P,P) be a lower probability space, let E = {Ei : i ∈ I} be a positive, measurable partition of Ω, and let A ∈ F . Then the following are equivalent: COUNTEREXAMPLES TO SOME CHARACTERIZATIONS OF DILATION 7 (4i) E dilates A. (4ii) There is (εi)i∈I ∈ RI+ such that for every i ∈ I: P∗(A | Ei, εi) ⊆ S−∗ (A,Ei) and P∗(A | Ei, εi) ⊆ S+∗ (A,Ei). (16) (4iii) There is (εi)i∈I ∈ RI+ such that for every i ∈ I: P(A | Ei, εi) ⊆ S−P (A,Ei) and P(A | Ei, εi) ⊆ S + P (A,Ei), (17) where for each i ∈ I, εi ≤ min(εi, εi) and εi is the unique minimum of |P (A | Ei) − P (A | Ei)| attained on C+i := {P ∈ P∗ : SP (A,Ei) ≥ 1} and εi is the unique minimum of |P (A | Ei)− P (A | Ei)| attained on C−i := {P ∈ P∗ : SP (A,Ei) ≤ 1}. We note that (4iii) is stronger than the mere existence claim in (2ii), since it requires that εi be no greater than the minimizers εi and εi. More analysis of Example 1 shows, however, that (4iii) is still not strong enough to imply dilation. Proposition 4. In Claim 4, (4iii) does not imply (4i). Proof. Let (Ω,P(Ω),P∗, P ), E , and A, be defined as in Example 1. In the proof of Proposition 2, we have already shown that (2i) does not hold, so (4i) does not hold. At this point, there is a somewhat unfortunate overlap of notations. εi and εi have different meanings in the proof of Proposition 2 and (4iii). To resolve this difficulty, let us rewrite the quantities from the proof of Proposition 2 as δ1 = ε1, δ2 = ε1, δ3 = ε2, δ4 = ε2. To be clear, then, in the remainder of the current proof εi and εi refer to the minimizers defined in (4iii). Now, to complete the proof, we must show that (4iii) holds. To that end, it suffices to show, for i = 1, 2, that min(εi, εi) > 0, because we can then let εi = min(εi, εi, δ2i−1, δ2i) > 0, i = 1, 2. By what we have already shown in the proof of Proposition 2, the resulting (εi)i∈{1,2}, will satisfy condition (4iii). First, consider ε1. Observe that the functions f1(α) = |Pα(A | E1) − P (A | E1)| and g1(α) = SPα(A,E1) are both strictly increasing and continuous in the interval [0, 1]. (We note that this, and all the inferences that follow, can be seen quite easily by graphing the relevant functions.) Let α1 be the unique point in [0, 1] such that g1(α1) = 1. Then, C + 1 = [α1, 1]. Thus, the unique minimum, ε1, of f1 on C + 1 is attained at α1. Solving the relevant quadratic equation, we find that α1 ≈ 0.462. So, ε1 = f(α1) > 0 because f1 is strictly increasing on [0, 1] and f 1 (0) = 0. Second, consider ε1. Observe that the function f1(α) = |Pα(A | E1)−P (A | E1)| is strictly decreasing and continuous in the interval [0, 1]. With α1 defined as in the previous paragraph, we have C−1 = [0, α1]. Thus, the unique minimum, ε1, of f1 on C − 1 is attained at α1. As above, we have α1 ≈ 0.462, so ε1 = f1(α1) > 0 because f1(1) = 0. We now draw the intermediate conclusion that min(ε1, ε1) > 0 and proceed to the cases involving E2. Consider ε2. Observe that the functions f2(α) = |Pα(A | E2) − P (A | E2)| and g2(α) = SPα(A,E2) are both strictly decreasing and continuous in the interval [0, 1]. Let α2 be the unique point in [0, 1] such that g2(α2) = 1. Then, C + 2 = [0, α2]. Thus, the unique minimum, ε2, of f2 on C + 2 is attained at α2. Solving, we again find that α2 ≈ 0.462, so ε2 = f2(α2) > 0 because f 2 (1) = 0. 8 NIELSEN AND STEWART Finally, consider ε2. Observe that the function f2(α) = |Pα(A | E2)−P (A | E2)| is strictly increasing and continuous in the interval [0, 1]. With α2 defined as in the previous paragraph, we have C−2 = [α2, 1]. Thus, the unique minimum, ε2, of f2 on C − 2 is attained at α2. As before, we find that ε2 = f2(α2) > 0 because f2(0) = 0. We conclude that min(ε2, ε2) > 0, which completes the proof.  Finally, we show that the following claim is false. Claim 5 (Pedersen and Wheeler, 2015, Corollary 1). Let (Ω,F ,P,P) be a lower probability space, let E = {Ei : 1 ≤ i ≤ n} be a finite, positive, measurable partition of Ω, and let A ∈ F . Then the following are equivalent: (5i) E dilates A. (5ii) There is ε > 0 such that for each i = 1, ..., n: P∗(A | Ei, ε) ⊆ S−∗ (A,Ei) and P∗(A | Ei, ε) ⊆ S+∗ (A,Ei). (18) (5iii) There is ε > 0 such that for each i = 1, ..., n: P(A | Ei, ε) ⊆ S−P (A,Ei) and P(A | Ei, ε) ⊆ S + P (A,Ei), (19) where ε ≤ min(εi, εi) and εi is the unique minimum of |P (A | Ei) − P (A | Ei)| attained on C+i := {P ∈ P∗ : SP (A,Ei) ≥ 1} and εi is the unique minimum of |P (A | Ei)− P (A | Ei)| attained on C−i := {P ∈ P∗ : SP (A,Ei) ≤ 1}. Proposition 5. In Claim 5, (5iii) does not imply (5i). Proof. Let (Ω,P(Ω),P∗, P ), E , and A, be defined as in Example 1. Continuing from the proof of Proposition 4, we note that we have shown that (4i) does not hold, so (5i) does not hold. Moreover, we showed that (εi)i∈{1,2} satisfies (4iii). It is easily seen, then, that ε = min(ε1, ε2) > 0 satisfies (5iii).  6. Remarks on Pedersen and Wheeler's Proof We conclude our note with a few remarks about the mistakes in Pedersen and Wheeler's alleged proof of the Claims above. We focus on their proof that (1iii) implies (1i) (Pedersen and Wheeler, 2014, p. 1340).1 Here is the relevant portion of their proof with a few harmless changes to the notation: Suppose that E does not dilate A. Then there is i ∈ I such that P (A | Ei) ≥ P (A) or P (A) ≥ P (A | Ei) . We may assume without loss of generality that P (A | Ei) ≥ P (A) for some i ∈ I. First, if P (A) ≤ P (A) ≤ P (A | Ei), then choosing a minimizer p ∈ P of P (A | Ei), we see that SP (A,Ei) ≥ 1. Second, if P (A) < P (A | Ei) < P (A), then for every ε > 0 we can find a convex combination P ∈ P of P ′, P ′′ ∈ P assigning a probability to A within ε-distance below P (A | Ei), where P (A) ≤ P ′(A) < P (A | Ei) < P ′′(A) ≤ P (A), so SP (A,Ei) > 1. Third, if P (A) = P (A | Ei) < P (A), then choosing a minimizer P ∈ P of P (A), we see that SP (A,Ei) ≥ 1. Evidently, the conditions of the main claim cannot be jointly satisfied. 1Note that there is a typo in their paper where they describe the direction of implication; "(iii) ⇐ (i)" should be replaced by "(iii) ⇒ (i)". COUNTEREXAMPLES TO SOME CHARACTERIZATIONS OF DILATION 9 To prove the contrapositive of the desired implication, Pedersen and Wheeler must show that the negation of (1iii) follows from the assumption that E doesn't dilate A. In the passage above, it seems that they are attempting to argue, from the assumption that there is no dilation, that there is some i ∈ I such that for all ε > 0 there is some P ∈ P(A | Ei, ε) such that P /∈ S−P (A,Ei). The argument errs in the second and third cases that it considers. What is needed in both cases is a P whose conditional value for A given Ei is ε-close to P (A | Ei). That is the meaning of P ∈ P(A | Ei, ε). What Pedersen and Wheeler provide, in both cases, is a P whose unconditional value for A is ε-close to P (A | Ei). As it is not true in general that a P with unconditional value for A that is ε-close to P (A | Ei) will also have a conditional value for A given Ei that is ε-close to P (A | Ei), the proof is fallacious. In particular, in the third case, it is not true in general that a P that attains P (A) also attains P (A | Ei). The reader can verify that our Example 1 illustrates this fact. We are grateful to an anonymous referee for showing us that the mistake in Pedersen and Wheeler's proof can also be explained using Result 1 in Wasserman and Seidenfeld (1994). That result provides a condition that is both necessary and sufficient for dilation in a binary partition. Since our central example uses only a binary partition and does not exhibit dilation, it follows that it violates Seidenfeld and Wasserman's characterizing condition. We will not go into further details but simply note that the mistake in Pedersen and Wheeler's proof that we identified just above amounts to erroneously restricting attention to a special case of Wasserman and Seidenfeld's condition, namely, the case in which P attains both P (A) and P (A | Ei). 10 NIELSEN AND STEWART References Pedersen, A. P. and G. Wheeler (2014). Demystifying dilation. Erkenntnis 79 (6), 1305–1342. Pedersen, A. P. and G. Wheeler (2015). Dilation, disintegrations, and delayed decisions. In Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications, pp. 227– 236. Seidenfeld, T. and L. Wasserman (1993). Dilation for sets of probabilities. The Annals of Statistics 21 (3), 1139–1154. Wasserman, L. and T. Seidenfeld (1994). The dilation phenomenon in robust Bayesian inference. Journal of Statistical Planning and Inference 40 (2), 345–356.