Uncertainty RUD, June 28, 2006, Peter P. Wakker Paper: Chapter for New Palgrave. Its didactical beginning is not optimal for this specialized audience. Paper does contain opinions about current and future directions in the field. Opinions and discussion is the purpose of this lecture. Abstract:

1. Probabilistic sophistication does not signal ambiguity neutrality, but ambiguity uniformity. An endogenous definition of unambiguous seems to be impossible. 2. Unambiguity is best defined through given objective probabilities. 3. Convexity (say; or inverse-S) of capacities for unknown probabilities is primarily implied by the Allais paradox and not by the Ellsberg paradox. 4. The Ellsberg paradox generates deviations from expected utility in a relative sense, not suggesting that capacities for unknown probabilities are convex, but that capacities for unknown probabilities are more convex than they are for known probabilities. 5. In Schmeidler (1989), where capacities are linear for known probabilities, "more-convex-for-unknown-probabilities" happens to coincide with "convex," which has led to many misunderstandings. 6. The main novelty of the Ellsberg paradox is not the introduction of ambiguity aversion, but the introduction of within-person comparisons (as opposed to between-person comparisons as by Pratt, Arrow, Yaari) of behavior across different sources of uncertainty. 2

Survey Part I. Endogenous versus Exogenous Ambiguity Part II. Group Discussion on Terminology Part III. New Views on Ambiguity, Ellsberg, and Allais 3

1. Ambiguity; Definitions Urn 1... Urn i... Each urn contains 1000 balls ("=" continuum). Each ball has # between 1 and Contents can be irregular, with some #'s double, others missing (partially unknown compositions). One ball drawn at random, its # inspected; # i : the number drawn from urn i. For each urn i, info symmetric w.r.t. 1,…,1000 ("exchangeable", "strongly uniform partitions"). We define a uniform "mathematical" probability measure P on each urn i (P(j) = for each j). 4

Outcomes: $ Acts: as usual. Throughout, I assume Choquet expected utility (CEU). Always U is linear. For each urn i, W i = w i o P. Probabilistic sophistication within each urn! w i may be nonlinear. As soon as some w i 's are different, there cannot be a "joint" probabilistic sophistication. 5 Contrary to misunderstandings in the literature (strongly uniform partitions) E&Z'01, footnote 18)

CEU: w 1 (0.5)  100 = 0.5  100 = 50 6 (# 1  500: $100) (# 2  500: $100) EXAMPLE. w 1 (p) = p; w 2 (p) = p 2. w 2 lower, more convex, more pessimistic, than w 1. (# 1  500: $100) (# 2  500: $100)!  > ? Similarly: (# 1 > 500: $100) (# 2 > 500: $100)  Ellsberg-type dislike for urn 2. Tversky: source preference for urn 1. CEU: w 2 (0.5)  100 =  100 = 25

Question: How determine whether urns are (un)ambiguous? 2. Only Endogenous Info 2.1. Observe choices for one urn only My answer: (Un)ambiguity cannot be determined. No way to separate ambiguity from ambiguity attitude unless strong extra assumptions. E&Z: Epstein (& Zhang 2001) G&M: Ghirardato & Marinacci (2002) 7 Example a minute ago was two urns. Forget about that. Now only one urn.

E&Z:  i urn i is unambiguous. Remember: only 1 urn is observed! G&M: urn i is unambiguous iff w i is linear (EU) Observe Two (or More) Urns with Different w's (see previous example) Then no "joint" probabilistic sophistication. I: still can't disentangle ambiguity from ambiguity attitude. G&M: It is the one with w linear (if any); EU! Problem: Then all nonEU is ambiguity!? NonEU may also occur in absence of ambiguity (Allais etc.). 8 G&M is Internally coherent: If -only normatively interested -Assume that EU is normative for given probs. -Assume that EU is not normative for unknonw probs. Then terminology OK, does not harm. However, whoever disagrees with normative claims cannot follow this terminology. Descriptive work cannot follow this terminology either. G&M is Internally coherent: If -only normatively interested -Assume that EU is normative for given probs. -Assume that EU is not normative for unknonw probs. Then terminology OK, does not harm. However, whoever disagrees with normative claims cannot follow this terminology. Descriptive work cannot follow this terminology either. The fact that you can't observe whether it is ambiguous or unambiguous does not justify calling it unambiguous. Should say "I don't know." Ask Larry?

E&Z: You can't tell as long as no  's of different- urn events are considered. To wit: 9 is not violated for any event T on domain of no  's of different-urn events. So how can E&Z determine on (un)ambiguity? As follows. TcTc A x* B xB x E 1 ….E n h 1 ….h n Not Let x*>x. TcTc A xA x B x* E 1 ….E n h 1 ….h n  c´c´ x* x h 1 ….h n c´c´ x x* h 1 ….h n  T is unambiguous if (also for not-T):

10 1. Inspect acts for  's of different-urn events. E.g. for A,B, E j in: A: set of all events passing test (Dynkin system). 2. If A satisfies some extra axioms (mainly 4: richness; 6: strong-partition-neutrality), implying "global" probabilistic sophistication on A, then all of A is unambiguous. TcTc A x* B xB x E 1 ….E n h 1 ….h n Not Let x*>x. TcTc A xA x B x* E 1 ….E n h 1 ….h n  c´c´ x* x h 1 ….h n c´c´ x x* h 1 ….h n  T is unambiguous if (also for not-T): Really get complete algebra of events.

11 My opinion: There are difficulties for E&Z. - In many plausible situations, the extra axioms are not satisfied. Then E&Z does not help. - Extension to  's of different-urn events is crucial (contrary to Savage 1954), but often not natural. - In many situations, E&Z's definition of unambiguous does not capture that concept. - E&Z's theorem is an axiomatization of probabilistic sophistication, not more entitled to capture (un)ambiguity than any other axiomatization thereof on Dynkin systems. Later: Prob. soph. = uniformity of ambiguity,  absence of ambiguity.

3. Exogenous Info Available Again, assume two urns. EXAMPLE. Urn 1 : all #'s are present in urn; known composition; objective probabilities (I: "risk"). w 1 (p) =  p (optimism); w 2 (p) = (  p) 2 = p. I: Urn 1 is unambiguous. E&Z: ? G&M: ? 12

5. Probabilistic Sophistication as Uniform Ambiguity Each urn is a source (of uncertainty). (Tversky; Ergin & Gul: Issue; Chew & Sagi, 2004: small world). Strong source preference of Urn 1 over Urn 2 : For all partitions (A 1,…,A n ) for Urn 1 and (B 1,…,B n ) for Urn 2, A i  B i (in sense of betting on) for all i & A i  B i for some i may happen; converse cannot happen. 13

Uniform ambiguity of a source: the source exhibits strong source preference with respect to itself; so, "strong source indifference:" For all partitions (A 1,…,A n ) and (B 1,…,B n )  i  n  1: B j ~ A i  B n ~ A n. IMPLICATION: A 1 ~ B 1 & A 2 ~ B 2  A 1  A 2 ~ B 1  B 2 (disjunctions). PROOF: (A 1  A 2 ) c ~ (B 1  B 2 ) c 14 as reflexivity would have it

Implication: Probabilistic sophistication! We also have reversed implication. It is an equivalence. Conclusion: Probabilistic sophistication = uniform ambiguity. Can serve well to define sources ("small worlds"). 15

Btw: Above reasoning comprises not only source preference but also source sensitivity. 16