DEF 23, 121 – 132 (2000)
Decisions in Economics and Finance © Springer-Verlag 2000
A uniqueness theorem for convex-ranged probabilities Massimo Marinacci Dipartimento di Statistica e Matematica Applicata, Università di Torino e-mail:
[email protected] Received: 18 December 1999 / Accepted: 17 July 2000
Abstract. A finitely additive probability measure P defined on a class of subsets of a space is convex-ranged if, for all P (A) > 0 and all 0 < α < 1, there exists a set, B ⊆ A, such that P (B) = αP (A). Our main result shows that, for any two probabilities P and Q, with P convex-ranged and Q countably additive, P = Q whenever there exists a set A ∈ , with 0 < P (A) < 1, such that (P (A) = P (B) ⇒ Q(A) = Q(B)) for all B ∈ . Mathematics Subject Classification (2000): 28A10, 91B06 Journal of Economic Literature Classification: C60, D81 1. Introduction and results A finitely additive probability measure P defined on a class of subsets of a space , is convex-ranged if, for all P (A) > 0 and all 0 < α < 1, there exists a set, B ⊆ A, such that P (B) = αP (A).1 As is well-known, if P is countably additive and a σ -algebra, P is convex-ranged if and only if it is non-atomic, that is, if for all P (A) > 0 there exists a set, B ⊆ A, such that 0 < P (B) < P (A). In this paper we prove a uniqueness theorem for convex-ranged measures. To better appreciate the strength of our theorem, we first present a simple uniqueness result. I wish to thank the editor, two anonymous referees and Paolo Ghirardato for useful comments. The financial support of MURST is gratefully acknowledged. 1 Convex-ranged probabilities are called strongly non-atomic by Bhaskara Rao and Bhaskara Rao (1982).
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Proposition 1. Let P and Q be two finitely additive probabilities on a class of subsets of a space , and suppose that P is convex-ranged. If, for all A ∈ , we have P (A) = P (B) ⇒ Q(A) = Q(B)
(1)
whenever B ∈ , then P = Q. This result is, for example, implicitly proved on p. 39 of Aumann and Shapley (1974). Its proof is simple: since condition (1) holds for all A ∈ , it follows that Q is a function of P , i.e., Q = g (P ) for some function g : [0, 1] → [0, 1] with g(0) = 0 and g(1) = 1. Since both P and Q are additive, we then have g (x1 + x2 ) = g (x1 ) + g (x2 ) for all x1 , x2 ∈ [0, 1] such that 0 ≤ x1 + x2 ≤ 1. Hence, by a standard result in the theory of functional equations, it holds that g(x) = x for all x ∈ [0, 1], and so P = Q. This is a simple and nice result. However, the requirement that condition (1) holds for all A ∈ is very demanding. The next theorem, which is our main result, considerably weakens such an assumption, at the cost of two extra hypotheses: (i) Q has to be countably additive; (ii) the class must be at least a λ-system, that is, (i) ∈ , c (ii) A ∈ if A ∈ , (iii) n≥1 An ∈ for any sequence {An }n≥1 of pairwise disjoint events contained in .2 We can now state our main result. Theorem 1. Let P and Q be two finitely additive probabilities on a λ-system . Suppose that P is convex-ranged and that Q is countably additive. If there exists a set A ∈ , with 0 < P (A) < 1, such that P (A) = P (B) ⇒ Q(A) = Q(B) whenever B ∈ , then P = Q. The claimed weakening is, therefore, substantial: we now require the existence of just a single set A ∈ , with 0 < P (A) < 1, for which condition (1) is satisfied. Such a considerable weakening seems to more than offset the cost of the two extra conditions, and it shows a remarkable property of range convexity: a minimal agreement between two probabilities, at least one of them being convex-ranged, forces them to be equal. 2 Recently, λ-systems have been used in Bayesian decision theory to model unambiguous events (see Zhang (1996), Epstein (1999) and Ghirardato and Marinacci (1997)).
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1.1. Some motivation The motivation for this uniqueness result comes mostly from Bayesian decision theory. In this literature, the basic notion is a binary relation defined on an event σ -algebra . The relation represents the decision makers’ subjective beliefs about the events’ relative likelihoods, and, for any two events A, B ∈ , A B reads as follows: “the decision maker believes that event A is at least as likely as event B”. The binary relation is called a likelihood preference relation. We write A ∼ B if the two events A and B are equally likely, i.e., if we have both A B and B A. Given an event A ∈ , let [A] be the collection of all events judged to be equally likely to A, i.e., [A] = {B ∈ : A ∼ B}. These collections [A] are called likelihood indifference classes. In this setting, a probability P always represents an underlying likelihood preference relation , i.e., P (A) ≥ P (B) if and only if A B. Starting from the seminal contributions of de Finetti (1931), (1937) and Savage (1954), it is often assumed that these representing probabilities are convexranged. Theorem 1 shows that this assumption is quite demanding in the context of Bayesian decision theory. Indeed, consider two different decision makers whose beliefs are described by the likelihood relations 1 and 2 , in turn represented by P1 and P2 , respectively.Assume that P1 is convex-ranged and that P2 is countably additive, a mild assumption. We can read Theorem 1 as saying that, if there exists a non-trivial A ∈ such that [A]1 ⊆ [A]2 , then P1 = P2 , and so 1 = 2 . That is, if the two decision makers have just two nested likelihood indifference classes, they must share the same beliefs. This unpleasantly strong implication shows that convex-rangedness is a demanding assumption on subjective probabilities that unduly restricts the room for diversity in beliefs. This observation is further elaborated in Ghirardato and Marinacci (2000). Finally, aside from its conceptual interest, Theorem 1 turned out to be useful as a technical tool in Epstein and Zhang (1999) and Marinacci (1999).
1.2. Two variations We close with two variations on Theorem 1. It is sometimes of interest to look at classes of sets that are not closed under countable operations. The next result – based on standard Stone space techniques – provides a uniqueness theorem for algebras of sets, which are closed under complementation and finite unions, but not necessarily under countable unions. Notice that, in general, algebras and λ-systems are different notions. Actually, a class of sets is both an algebra and a λ-system if and only if it is a σ -algebra.
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Before stating the result we recall that a finitely additive probability P : → [0, 1] is strongly continuous if, for every ε > 0, there exists a finite partition {Ai }ni=1 ⊆ such that P (Ai ) ≤ ε for all i, 1 ≤ i ≤ n. On algebras, this is a weaker condition than convex-rangedness, while on σ -algebras they are equivalent, a result essentially due to Savage (1954) (see, e.g., Bhaskara Rao and Bhaskara Rao (1982), p. 259).3 Proposition 2. Let P and Q be two finitely additive probabilities on an algebra . Suppose that P is strongly continuous on . If there exists a monotone sequence {An }n≥1 ⊆ , with 0 < limn P (An ) < 1, such that lim P (An ) = lim P (Bn ) ⇐⇒ lim Q (An ) = lim Q (Bn ) n
n
n
n
whenever {Bn }n≥1 is a monotone sequence contained in , then P = Q. Next we present a second variation on Theorem 1 in which Q is no longer required to be countably additive, but condition (1) is significantly strenghtened. Its proof is omitted as it is basically already contained in that of Theorem 1. A similar variation holds for Proposition 2. Proposition 3. Let P and Q be two finitely additive probabilities on a λsystem , and suppose that P is convex-ranged. If there exists a set A ∈ , with 0 < P (A) < 1, such that P (A) = P (B) ⇐⇒ Q(A) = Q(B) whenever B ∈ , then P = Q. 2. Proof of Theorem 1 Set P (A) = α and Q (A) = β. By hypothesis, α ∈ (0, 1) and P (E) = α ⇒ Q(E) = β
(2)
for all E ∈ . Lemma 1. We have β ∈ (0, 1). Proof. Suppose, per contra, that β ∈ {0, 1}. Then there exists α ∗ ∈ {α, 1 − α} such that, for all E ∈ , P (E) = α ∗ ⇒ Q (E) = 0.
(3)
3 Though in his classic book the result is proved for power sets (the case of decision-
theoretic interest for him), it is immediate to see that Savage’s proof holds for general σ -algebras.
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∗ Hence, Q (E) > 0 implies P (E) > α ∗ . For, if P (E) ≤ α , there is, ∗ E ⊇ E, such that P E = α . Hence, by (3), Q E = 0, and so Q (E) = 0, a contradiction. We now tacitly use several times the range convexity of P . Let E1 ∈ be such that P (E1 ) = α ∗ . By (3), Q (E1 ) = 0, and so Q ( − E1 ) = 1. In turn, this implies that 1 − α ∗ = P ( − E1 ) > α ∗ , and so α ∗ < 1/2. If α ∗ ≥ 1/2 we have a contradiction and so we are done. Otherwise, let, E2 ⊆ − E1 , be such that P (E2 ) = α ∗ . Again by (3), Q (E2 ) = 0, and so Q (( − E1 ) − E2 ) = 1. This implies that 1 − 2α ∗ = P (( − E1 ) − E2 ) > α ∗ , and so α ∗ < 1/3. If α ∗ ≥ 1/3 we are done. Otherwise, setting 1 ∗ ∗ n ≡ max n ≥ 1 : α < , (4) 1+n ∗
we can go on constructing a sequence {Ei }n1=1 ⊆ of pairwise disjoint events such that, for each 1 < k ≤ n∗ k, we have Ek ⊆ (. . . (( − E1 ) − E2 ) · · · − Ek−1 ) , P (Ek ) = α ∗ , Q (. . . (( − E1 ) − E2 ) · · · − Ek ) = 1 and P (. . . (( − E1 ) − E2 ) · · · − Ek ) > α ∗ .
(5)
By (5), there is, En∗ +1 ⊆ (. . . (( − E1 ) − E2 ) · · · − En∗ ), such that P (En∗ +1 ) = α ∗ . By (3), Q (En∗ +1 ) = 0. Then Q (. . . (( − E1 ) − E2 ) · · · − En∗ +1 ) = 1, and so 1 − n∗ + 1 α ∗ = P (. . . (( − E1 ) − E2 ) · · · − En∗ +1 ) > α ∗ . Hence α∗
β ⇒ P (E) > α, Q (E) > 1 − β ⇒ P (E) > 1 − α, Q (E) < β ⇒ P (E) < α, Q (E) < 1 − β ⇒ P (E) < 1 − α.
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Proof. Suppose, to the contrary, that Q (E) > β and P (E) ≤ α. If P (E) = α, then Q (E) = β, a contradiction. Suppose P (E) < α. Since P is convexranged, there exists, B ⊇ E, such that P (B) = α. Hence, Q (B) = β, so that Q (B) = β < Q(E), which contradicts B ⊇ E. We conclude that Q (E) > β ⇒ P (E) > α. By additivity, P (E) = 1−α ⇒ Q (E) = 1−β for all E ∈ , and so, by proceeding in the same way, Q (E) > 1 − β ⇒ P (E) > 1 − α. Next, suppose that Q(E) < β. Then Q (E c ) > 1 − β, so that P (E c ) > 1 − α. In turn this implies that P (E) < α. Finally, suppose that Q (E) < 1 − β. Then Q (E c ) > β, which implies that P (E c ) > α, and so P (E) < 1 − α. Lemma 3. For all E ∈ , Q (E) > 0 implies that P (E) > 0. Proof. If Q(E) > β, then, by Lemma 2, P (E) > α. Hence, assume that 0 < Q(E) ≤ β. Suppose, to the contrary, that P (E) = 0. Since P is convexranged, there exists B ⊇ E such that P (B) = α. Hence, Q (B) = β, so that Q (B − E) < β. Then, by Lemma 2, P (B − E) < α, so that α = P (B) = P (B) − P (E) = P (B − E) < α, a contradiction. Lemma 4. If Q(E) > 0, there exists B ⊆ E such that 0 < Q(B) < Q(E). Proof. By Lemma 3, P (E) > 0. Since P is exists convex-ranged, 1 there 1 1 1 1 , B P (E). If ⊆ of E such that P E = P B = a partition E 2 1 0 < Q E 1 < Q(E) or 0 < Q B < Q(E), we are done. Suppose, in contrast, that either E 1 = Q(E) or Q B 1 = Q(E). Without loss Q of generality, let Q E 1 = Q(E). Let E 2 , B 2 ∈ be a partition of 2 2 2 1 1 1 E 1 such that 2P E = 1 P B = 2 P (E ). If 0 < Q E < Q(E ) or 0 0. n≥1 E ∈ , we have P n≥1 E n≥1 E By Lemma 3, this is impossible. Hence, there exists set, B ⊆ E, such that 0 < Q(B) < Q(E). Lemma 5. For each Q(E) > 0 and every ε > 0, there exists a finite partition {Ei }ni=1 ⊆ of E such that Q (Ei ) < ε for all i, 1 ≤ i ≤ n. Proof. Without loss of generality, let ε > 0 be such that both ε < β and ε < 1 − β. Let Q(E) = β, so that Q (E c ) = 1 − β. By Lemma 4, there exists E1 ⊆ E c such that 0 < Q (E1 ) < Q(E c ). Without loss of generality,
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assume that Q (E1 ) ≤ 21 Q (E c ). Proceeding in this way, we can construct a chain {En }n≥1 ⊆ such that Q (En ) ≤ 21n Q (E c ). Therefore, there exists, B ⊆ E c , such that 0 < Q (B) < ε. Then, 0 < (1 − β) − ε < Q E c − B < Q(E c ) = 1 − β. By Lemmas 2 and 3, 0 < P (E c − B) < 1 − α. Since P is convex-ranged, we can construct a partition {Bi }ni=1 ⊆ of E such that for each i we have P ((E c − B) ∪ Bi ) ≤ 1 − α. By Lemma 2, Q ((E c − B) ∪ Bi ) ≤ 1 − β, so that Q (Bi ) ≤ (1 − β) − Q (E c − B) < (1 − β) − (1 − β − ε) = ε. Hence, Q (Bi ) < ε for all i, 1 ≤ i ≤ n. Next suppose that 0 < Q (E) < β. By Lemma 2, P (E) < α. Since P is convex-ranged, there exists, B ⊇ E, such that P (B) = α, so that Q (B) = β. By what has just been proved, there exists a partition {Bi }ni=1 ⊆ of B such that for each i we have P (Bi ) < ε. Hence, {E ∩ Bi }ni=1 is the desired partition of E. Finally, that Q(E) > β. We want suppose m to show that there exists a partition Ei i=1 ⊆ of E such that Q Ei ≤ β for all i, 1 ≤ i ≤ m. By Lemma 2, P (E) > α. Since P is convex-ranged, there exists, B0 ⊆ E, such that P (B0 ) = α, so that Q(B0 ) = β. If Q (E − B0 ) ≤ β we are done since Q (E − B0 ) > 0. Suppose that, in contrast, Q (E − B0 ) > β. Then P (E − B0 ) > α, and so there exists, B1 ⊆ E − B0 , such that P (B1 ) = α . Hence, Q (B1 ) = β. If Q ((E − B0 ) − B1 ) ≤ β we are done. If Q ((E − B0 ) − B1 ) > β, we can go on constructing a sequence {Bi }ni=1 ⊆ of pairwise disjoint events such that, for all i, 0 ≤ i ≤ n − 1, we have Q (B i+1 ) = β and Bi+1 ⊆ (((E − B0 ) − B1 ) − . . . ) − Bi . Then n−1
∗ Q i=0 Bi = nβ, so that there exists n ≥ 1 such that 0 < Q ((((E − B0 ) − B1 ) − . . . ) − Bn∗ ) ≤ β. By setting Ei = Bi for each i, 0 ≤ i < n∗ , and En ∗ = (((E − B0 ) − B1 ) − . . . ) − Bn∗ , m we obtain the desired partition Ei i=1 of E, where m ≡ n∗ . Now, since Q Ei ≤ β for all i, 1 ≤ i ≤ m, by what has been proved before, each Ei ni
has a partition Bji ⊆ such that Q Bji < ε for all j , 1 ≤ j ≤ ni . j =1 i ni Hence, m is a partition of E whose all elements take their values B j i=1 j =1
in the interval (0, ε).
Lemma 6. Q is convex-ranged.
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Proof. Let 0 < c < Q(E). We want to show that there exists a set, E ⊆ E, with Q E = c. Let n ≥ 1 be such that n1 < c. By Lemma 5, k1 there exists a partition Bi1 i=1 ⊆ of E such that 0 < Q Bi1 < n1 for all 1 1 BK1 1 ∈ . i, 1 ≤ i ≤ k1 . For 1 ≤ K1 ≤ k1 , set BK1 1 = K i=1 Bi . We have Set K1+ = max K1 : Q BK1 1 ≤ c and K1− = min K1 : Q BK1 1 > c .
Clearly, BK1 + ⊆ BK1 − and c − Q BK1 + ≤ n1 . If c = Q BK1 + we are 1 1 1
1 1 1 done. Suppose that c > Q BK + and set c1 = c − Q BK + . There exists 1
1
m ≥ n + 1 such that m1 < c1 . By the definitions of K1+ and K1− , there k1 exists B ∈ Bi1 i=1 such that BK1 + ∪ B = BK1 − and BK1 + ∩ B = ∅. We 1 1 1 k2 ⊆ next consider B. Again by Lemma 5, there exists a partition Bi2 i=1 2 1 of B such that 0 < Q Bi < m for all i, 1 ≤ i ≤ k2 . For 1 ≤ 2 2 2 + ≤ c and K2 ≤ k2 , set BK2 2 = K 1 K i=1 Bi . Set K2 = max K2 : Q B
2 K2− = min K2 : Q BK2 2 > c1 . We have c1 − Q BK2 + ≤ m1 and, as 2
BK1 + ∪ BK2 + ∈ , 1
2
0 ≤ c − Q BK1 + ∪ BK2 + = c1 + Q BK1 + − Q BK1 + − Q BK2 + 1 2 1 1 2
1 1 ≤ . = c1 − Q BK2 + ≤ 2 m n+1
If c − Q BK1 + ∪ BK2 + = 0 we are done. If c > Q BK1 + ∪ BK2 + , we can 1 2 1 2 k2 such that BK2 + ∪B = BK2 − and BK2 + ∩B = ∅. consider the set B ∈ Bi2 i=1 2 2 2 By in this way we can construct induction an increasing chain proceeding by Cj j ≥1 ⊆ such that 0 ≤ c − Q Cj ≤ n1j ≤ j1 for each j ≥ 1. Hence, ∞ 0 ≤ c − Q Cj = c − lim Q Cj j =1
j →∞
1 = lim c − Q Cj ≤ lim = 0. j →∞ j →∞ j
∞ Since ∞ j =1 Cj ∈ , we have Q j =1 Cn = c, and this completes the proof. The final lemma proves that at least part of the classic Lyapunov Theorem holds also on λ-systems. The proof is a simple adaptation for λ-systems of the proof of the classic result provided by Bhaskara Rao and Bhaskara Rao (1982), and it is therefore omitted.
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Lemma 7. If P1 , ..., Pn are convex-ranged finitely additive probabilities on a λ-system , then t (P1 (E) , ..., Pn (E)) ∈ R (P1 , ..., Pn ) , for all 0 ≤ t ≤ 1 and all E ∈ , where R (P1 , ..., Pn ) is the range of (P1 , ..., Pn ). Proof of Theorem 1. By Lemma 6, Q is convex-ranged. Hence, by Lemma 7, there exists E such that P (E) = Q (E) = α. Therefore, α = β, that is P (E) = α ⇒ Q(E) = α
(6)
for all E ∈ . In turn, this implies that P = Q. In fact, let E ∈ be such α that P (E) = Q (E). Assume first that 1 > P (E) ≥ α. Then 0 < P (E) ≤ 1, so that, by Lemma 7, α αQ(E) ∈ R (P , Q) . (P (E), Q(E)) = α, P (E) P (E) Thus, αQ(E) = α, and so P (E) = Q(E). Suppose now that 0 < P (E) < α. P (E) Then P (E c ) > 1−α and, by proceeding as before, we get P (E c ) = Q(E c ) since (6) implies that P (E) = 1 − α ⇒ Q(E) = 1 − α for all E ∈ . Hence, P (E) = Q(E). We conclude that 0 < P (E) < 1 implies that P (E) = Q(E) for all E ∈ . By Lemma 3, P (E) = 0 implies that Q(E) = 0 for all E ∈ . On the other hand, by considering complements, we have that P (E) = 1 implies Q (E) = 1. This completes the proof that P = Q.
3. Proof of Proposition 2 We first state two simple standard lemmas, whose proofs are omitted (for similar results, see, e.g., Armstrong and Prikry (1981)). Lemma 8. Let P be a strongly continuous and finitely additive probability defined on an algebra . For each A ∈ with P (A) > 0, each t ∈ (0, 1) and each ε > 0, there exists a set, B ⊆ A, with |tP (A) − P (B)| < ε. Lemma 9. Let P be a strongly continuous and finitely additive probability defined on an algebra . For every α ∈ [0, P (A)] and every A ∈ , there exists an increasing sequence {An }∞ n=1 ⊆ A such that lim P (An ) = α.
n→∞
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Proof of Proposition 2. Let Ult() be the Stone space of , Clop(Ult()) the dual algebra of in Ult(), s : → Ult() the Stone map (i.e., the isomorphism between and Clop(Ult())) and σ (Clop (Ult ())) the σ -algebra generated by Clop(Ult()). Let Ps be defined on Clop(Ult()) by Ps (s(B)) = P (B) for B∈. It is known that Ps is a strongly continuous countably additive probability (see Halmos (1963)). By the Caratheodory Extension Theorem, Ps has a unique σ -additive extension Ps on σ ( Clop (Ult())), which is strongly continuous and, thus, convex-ranged. In a similar way, Q induces a countably additive probability Qs . For the sequence {An }n≥1 in the statement of the proposition, set α = limn P (An ) and β = limn Q (An ). Then, for all monotone sequences {Bn }n≥1 in , we have that lim P (Bn ) = α ⇒ lim Q (Bn ) = β,
n→∞
n→∞
(7)
lim P (Bn ) = 1 − α ⇒ lim Q (Bn ) = 1 − β.
n→∞
n→∞
Let F ∈ σ ( Clop(Ult())) be closed. Then ∞there exists a decreasing ⊆ Clop(Ult()) such that sequence {Fn }∞ n=1 n=1 Fn = F (see Halmos (1963), p. 99). Suppose that Ps (F ) ≤ α, so that lim Ps (Fn ) = lim P s −1 (Fn ) ≤ α. n→∞
n→∞
We want to show that limn→∞ Q s −1 (Fn ) ≤ β. If limn→∞ P s −1 (Fn ) = −1 α, this is true by hypothesis. (Fn ) < α n→∞ P s Now, suppose that lim −1 ∞ while limn→∞ Q s (Fn ) > β. Let Fn0 ∈ {Fn }n=1 be such that P s −1 Fn0 < min α, 1 − α + lim P s −1 (Fn ) . n→∞
It is easy to check that such an Fn0 exists. Hence, c 0 < α − lim P s −1 (Fn ) < P s −1 Fn0 . n→∞
⊆ s −1 (Fn )c such By Lemma 9, there exists a decreasing sequence {An }∞ 0 n=1 −1 that limn→∞ P (An ) = α − limn→∞ P s (Fn ) . Hence, lim P s −1 (Fn ) ∪ An = α, n→∞
so that limn→∞ Q s −1 (Fn ) ≤ limn→∞ Q s −1 (Fn ) ∪ An = β. We conclude that
Qs (F ) = lim Qs (Fn ) ≤ β. n→∞
(8)
Using (7), a similar argument shows that Ps (F ) ≤ 1 − α implies that Qs (F ) ≤ 1 − β.
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Let G ∈ σ ( Clop(Ult())) be an open set such that Ps (G) ≥ α. Then (Gc ) ≤ 1 − α, so that Qs (Gc ) ≤ 1 − β. Hence, Qs (G) ≥ β.
(9)
Since σ ( Clop(Ult())) is the Baire σ -algebra on the compact Hausdorff space with basis Clop(Ult()) (see Halmos (1963), p. 97), the probability measures Ps and Qs are regular. Let A ∈ σ ( Clop(Ult())) be such that Ps (A) = α. Then α = Ps (A) = sup Ps (F ) : σ ( Clop(Ult())) F ⊆ A and F closed = inf Ps (G) : σ ( Clop(Ult())) G ⊇ A and G open , so that Ps (F ) ≤ α for all these closed sets F and Ps (G) ≥ α for all these open sets G. By (8) and (9), we then have Qs (F ) ≤ β and Qs (G) ≥ β, so that Qs (A) = sup Ps (F ) : σ ( Clop(Ult())) F ⊆ A and F closed ≤ β, Qs (A) = inf Qs (G) : σ ( Clop(Ult())) G ⊇ A and G open ≥ β. Therefore, Qs (A) = β. By Theorem 1, Ps = Qs , so that P = Q.
References Armstrong, T., Prikry, K. (1981): Liapounoff’s theorem for non atomic finitely-additive, bounded finite-dimensional vector-valued measures. Transactions of the American Mathematical Society 266, 499–514 Aumann, R., Shapley, L. (1974): Values of non-atomic games. Princeton University Press, Princeton Bhaskara Rao, K.P.S., Bhaskara Rao, M. (1983): Theory of charges: a study of finitely additive measures. Academic Press, London de Finetti, B. (1931): Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17, 298–329. [English translation in: de Finetti, B. (1993): Probabilità ed induzione. Bologna: Cooperativa Libraria Universitaria Editrice Bologna] de Finetti, B. (1937): La prevision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 7, 1–68. [English translation in: Kyburg, H.E., Smokler, H.E. (eds.) (1980): Studies in Subjective Probability, 2nd edition. Huntington, NY: Krieger, pp. 53–118] Epstein, L.G. (1999): A definition of uncertainty aversion. The Review of Economic Studies 66, 579–608 Epstein, L.G., Zhang, J. (2000): Subjective probabilities on subjectively unambiguous events. Econometrica, to appear Ghirardato, P., Marinacci, M. (1997): Ambiguity made precise: a comparative foundation and some implications. Social Science Working Paper 1026. Pasadena: California Institute of Technology
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Ghirardato, P., Marinacci, M. (2000): The impossibility of compromise: some uniqueness properties of expected utility preferences. Economic Theory 16, 245–258 Halmos, P. (1963): Lectures on Boolean algebras. Van Nostrand, Princeton Marinacci, M. (1999): Upper probabilities and additivity. Sankhy¯a (Series A) 61, 358–361 Savage, L. (1954): The foundations of statistics. Wiley, New York Zhang, J. (1996): Subjective ambiguity, probability, and capacity. University of Toronto, Toronto
