1. Updating Ambiguity Averse Preferences Eran Hanany Peter Klibanoff Tel Aviv UniversityNorthwestern University Faculty of EngineeringKellogg School of Management Managerial Economics and Decision Sciences (MEDS)

2. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Introduction Concerned with how one should update preferences upon receiving new information. Approach will be to investigate update rules delivering the following properties: Dynamic Consistency –Ex-ante optimal information contingent choices are respected by updated preferences Closure –If limit consideration ex-ante to a certain model, then updated preferences should remain within that model Update beliefs, not tastes (when separated)

3. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Introduction In Hanany and Klibanoff (2007), we pursued these issues in the context of a specific model of preferences: Maxmin expected utility (Gilboa and Schmeidler 1989) In the current paper, we provide a characterization of dynamically consistent update rules that applies to a significantly broader set of preferences. We also apply this general result to some specific models of recent interest in the ambiguity literature, while delivering the first rules for updating these models in a consistent way.

4. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Closely Related Literature Non-expected utility updating and consistency (Machina 1989, McClennen 1990, Machina and Schmeidler 1992, Epstein and Le Breton 1993, Eichberger and Grant 1997, Segal 1997, Wakker 1997, Grant, Kajii, Polak 2000) DC and updating MEU preferences (Hanany and Klibanoff 2007) Recursive Smooth Ambiguity preferences (Klibanoff, Marinacci, Mukerji 2006) Dynamic Variational preferences (Maccheroni, Marinacci, Rustichini 2006) (includes Recursive MEU (Epstein and Schneider 2003 Consistent Planning under ambiguity (Siniscalchi 2006)

5. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Preferences Anscombe-Aumann framework –Objects of choice are acts, Consider non-degenerate, continuous, monotonic preference relations that, when restricted to constant acts, obey the vNM expected utility axioms. is quasi-concave if and only if preferences satisfy Schmeidler's Uncertainty (Ambiguity) Aversion axiom:

6. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Preferences (cont.) Early models satisfying ambiguity aversion include the now-classic: –Maxmin Expected Utility (Gilboa and Schmeidler 1989) –concave Choquet Expected Utility (Schmeidler 1989); concave Cumulative Prospect Theory over gains (Tversky and Kahneman, 1992) Recent advances have led to several more flexible models of ambiguity aversion, including: –concave Smooth Ambiguity preferences (Klibanoff, Marinacci, Mukerji 2005; Nau 2006; Seo 2006) –Variational preferences (Maccheroni, Marinacci, Rustichini 2006)

7. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Update Rules Quadruple: –A preference representation, –A convex, compact feasible set of acts, B –An (interior) act, g, optimal in B –A non-null event, E  S An update rule is a function defined on a set of such quadruples that maps each quadruple to a new preference representation that makes E c a null event. Restrict attention to rules for which updated preferences are ambiguity averse quasiconcave) and risk preferences are unchanged.

8. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences “An ex-ante optimal act cannot be feasibly improved conditional on any planned-for event.” Formally (Hanany & Klibanoff 2007): DC: For each quadruple in the domain of the update rule, Preferences for acts other than g need not be preserved Comparison with feasible acts identical on E-complement Helps separate dynamic consistency from consequentialism Dynamic Consistency of an Update Rule

9. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences A General Characterization Definition: For a quasiconcave V, the measures supporting the conditional indifference curve at h on E are: T E,h (V) ≡ {q ∈ Δ(E) ∫(u ∘ f)dq ≥ ∫(u ∘ h)dq for all f such that V(u ∘ f)>V(u ∘ h)}. Definition: The measures supporting the conditional optimality of h are: Q E,h,B ≡ {q ∈ Δ ∣ q(E)>0 and ∫(u ∘ h)dq ≥ ∫(u ∘ f)dq for all f ∈ B with f=h on E c }. Denote by Q E,h,B (E) the set of Bayesian conditionals on E of measures in Q E,h,B. Theorem: An update rule satisfies DC if and only if T E,g (V E,g,B ) ∩ Q E,g,B (E) ≠  for each quadruple in the domain of the update rule.

10. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences A General Characterization This gives us a test for dynamic consistency of an update rule for any quasiconcave model: 1.Calculate T E,g (V E,g,B ) 2.Calculate Q E,g,B (E) 3.See if they intersect. If V E,g,B is concave, T E,g (V E,g,B ) can be replaced by a set of measures derived from the superdifferential of V E,g,B at. If differentiable then look at gradient. Similarly, smoothness of the relevant slice of B at simplifies calculation of Q E,g,B (E).

11. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Application: Updating Smooth Ambiguity preferences Definition: A smooth ambiguity preference (KMM 2005) over acts has the following representation: where  is an increasing real-valued function and captures ambiguity attitude (  concave = ambiguity aversion) and  is a subjective probability over probability measures  on the state space. Ambiguity is captured through disagreement in the support of  about the probability of an event..

12. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences DC updating of SA preferences Assume  differentiable. Consider update rules that: 1.Leave  unchanged 2.Do not depend on the feasible set B given ( ,u, ,E,g). Theorem: Such rules are dynamically consistent iff

13. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Application: Expected Utility Let’s take a step back – to expected utility (EU). Expected utility corresponds to the special case where  is affine, so  ’ is constant. The only beliefs that matter in this case are the reduced measures p = and p E,g = Corollary U is dynamically consistent iff p E,g is the Bayesian update of p.

14. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Ambiguity Consider again ambiguity averse Smooth Ambiguity (SA) preferences. Will Bayesian updating work here? No! Easy to show in a simple dynamic Ellsberg example. Also follows from our Theorem.

15. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Ambiguity Since Bayesian updating is dynamically inconsistent, to find a rule that satisfies consistency for SA preferences like Bayes’ rule does for EU preferences we need to look elsewhere. We will look at reweightings of Bayes’ rule.

16. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Bayes’ Rule Reweighted s

17. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Ambiguity Proposition A reweighted Bayes’ rule is dynamically consistent iff  E,g is the measure generated by setting

18. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Properties of the Smooth Rule This rule generalizes Bayesian updating of beliefs. Exactly coincides with Bayesian updating when: –preferences are ambiguity neutral, or –the ex-ante optimum, g, is constant in utilities. In both cases the distortion factor vanishes. Departs from Bayesian updating in a manner that depends on: –The ex-ante optimum, g. –the ambiguity attitude of the decision-maker

19. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Properties of the Smooth Rule Compared to Bayes’ Rule, our rule overweights measures that result in higher conditional valuations of g relative to unconditional valuations of g.

20. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Properties of the Smooth Rule The Smooth rule is commutative. Commutativity means that the order of information received does not affect updating. Specifically, for any non-null events E and F having non-null intersection:

21. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Conclusion General result characterizing dynamic consistency for convex (ambiguity/uncertainty averse) preferences. Provide characterization of dynamically consistent updating for ambiguity averse Smooth Ambiguity preferences and identify a natural generalization of Bayes’ rule. Provide characterization of dynamically consistent updating for Variational preferences (also have further results on specific rules). Show Bayesian updating is consistent for Multiplier preferences. Substantially generalizes previous results of ours on updating MEU.

22. The End E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences

23. Properties of the Smooth Rule The Smooth rule also satisfies a strict version of dynamic consistency when there is strict ambiguity aversion ( is strictly concave): Strict DC: Add to DC that if g is ex-ante strictly preferred (indifferent) to some, then g remains strictly preferred (indifferent) to f after learning E.

24. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Ambiguity Model (cont.) Only model of decision-making under ambiguity currently available that simultaneously allows: 1.separation of ambiguity attitude from perception of ambiguity 2.flexibility in ambiguity attitude 3.subjective and flexible perception of which events are ambiguous 4.the tractability of smooth preferences 5.expected utility as a special case for any given ambiguity attitude.

25. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences 3-Color Ellsberg Paradox Bet on Black > Bet on Red Bet against Black > Bet against Red BlackRedYellow Bet on Black100 Bet on Red010 Bet against Black011 Bet against Red101 30 + ? + ? = 90

26. Dynamic Consistency vs. Recursion Assume (1,0,0) ≻ (0,1,0) and (0,1,1) ≻ (1,0,1) as in Ellsberg r (1,0, · ) b EEcEc (·,·,0) (0,1, · ) EEcEc (·,·,0) B R B R r' b' (0,1,0)(1,0,0)

27. Dynamic Consistency vs. Recursion Assume (1,0,0) ≻ (0,1,0) and (0,1,1) ≻ (1,0,1) as in Ellsberg r (1,0, · ) b EEcEc (·,·,1)(·,·,1) (0,1, · ) EEcEc (·,·,1)(·,·,1) B R B R r' b' (0,1,1)(1,0,1)

28. d r+  EEcEc BR (1- ,- , · ) (·,·,0) (0,1,·) (1,0,·) b-  EEcEc (·,·,-  )( ,1+ , · ) EEcEc (·,·,)(·,·,) B or R B R Dynamic consistency vs. inconsistency Assume (1,0,0) ≻ (1-ε,-ε,-ε) ≻ (ε,1+ε,ε) ≻ (0,1,0) in Ellsberg

29. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Variational preferences Definition: A variational preference (MMR 2006) over acts has the following representation: where u is a vN-M utility function and c :Δ→[0,∞] is grounded, convex and lower semicontinuous.

30. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Variational preferences Special cases include: Maxmin EU (Gilboa and Schmeidler 1989) Robust Control Preferences (Hansen and Sargent 2001)

31. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences DC updating of Variational preferences Consider update rules that: 1.Leave u unchanged, and 2.Make E c a null event. Proposition: Such rules are dynamically consistent iff

32. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Expected Utility Q: Why use Bayes’ rule to update beliefs under EU? A: It is the unique such update rule under EU that is dynamically consistent.

33. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Bayesian Update Rule

34. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Ambiguity Unfortunately, Bayesian updating is not dynamically consistent for SA preferences.

35. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Under concavity & differentiability of and convexity of the feasible set, optimality of g is equivalent to a unique hyperplane separating the upper contour set from the feasible set at g. DC requires conditional optimality of g among feasible acts agreeing with g on E c. Sufficient to show the gradients of the unconditional and conditional representations are proportional on E at g. Proof Sketch Part 1: Smooth Rule satisfies DC

36. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Smooth Rule satisfies DC (cont.) For s  E,

37. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Consider a feasible set with smooth boundary at g. Now necessary for DC that the gradients of the unconditional and conditional representations are proportional on E at g. Considering a  with two-point support and some moderately messy algebra shows the only reweighted Bayes’ rule satisfying this proportionality is the Smooth rule. Proof Sketch Part 2: DC implies Smooth Rule

38. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences DC updating of MEU preferences Consider update rules that: 1.Leave u unchanged, and 2.Make E c a null event. Proposition (Prop. 1, Hanany and Klibanoff (2007)): Such rules are dynamically consistent iff

39. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Dynamic Ellsberg example with our rule

40. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Dynamic Ellsberg example with our rule (cont.)

41. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Dynamic Ellsberg example with our rule (cont.)

42. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Introduction (cont.) We begin by proposing an update rule for ambiguity averse Smooth Ambiguity preferences and show it is the unique dynamically consistent rule among a large class of rules generalizing Bayes’ rule. We then provide a general result characterizing dynamically consistent updating for any ambiguity averse preferences and apply it to Smooth Ambiguity preferences and to Variational preferences

43. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Urn with 90 balls, exactly 30 are Black, no information about the proportions of Red and Yellow. One ball drawn from Urn at random. Consider Preferences over bets on the color of the drawn ball. Modal preferences: Bet on Black > Bet on Red ~ Bet on Yellow Bet against Black > Bet against Red ~ Bet against Yellow Reflects aversion to ambiguity about probabilities in this context. 3-Color Ellsberg Paradox

44. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences 3-Color Ellsberg Paradox Bet on Black > Bet on Red Bet against Black > Bet against Red Conditional on ball drawn being {not Yellow}, a Bet on Black is identical to a Bet against Red. Similarly, a Bet on Red is identical to a Bet against Black. BlackRedYellow Bet on Black100 Bet on Red010 Bet against Black011 Bet against Red101 30 + ? + ? = 90

45. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Comparison with Recursive preferences As our rule is non-consequentialist, it is not recursive. As our example showed, no recursive (or even consequentialist) rule is capable of dynamic Ellsberg behavior.

46. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences DC updating of MEU preferences Corollary (HK (2007)): Such rules are dynamically consistent iff Proposition (HK (2007)): The unique ambiguity maximizing dynamically consistent update rule is:

47. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences Toward a General Result Thus far, we have identified a relatively simple rule for updating Smooth Ambiguity preferences that results in dynamic consistency and other nice properties. Q: Is there a way to describe the set of "all" dynamically consistent update rules for these preferences? For other ambiguity averse preference models (even ones with kinks)? A: Yes!!

48. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences A General Result Definition: If V is a concave function of utility-acts then its superdifferential at x is: ∂V(x)≡{r ∈ Δ ∣ V(z)-V(x)≤ ∫(z-x)dr for all z} Definition: For V, the measures supporting the conditional indifference curve at h on E are: G E,h (V) ≡ {q ∈ Δ(E) ∣∃ λ>0 such that q ∈ λ∂V(a) evaluated at a=u ∘ h}. Definition: The measures supporting the conditional optimality of h are: Q E,h,B ≡ {q ∈ Δ ∣ q(E)>0 and ∫(u ∘ h)dq ≥ ∫(u ∘ f)dq for all f ∈ B with f=h on E c }. Denote by Q E,h,B (E) the set of Bayesian conditionals on E of measures in Q E,h,B.

49. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences A General Result Proposition: Fix a preference relation, ≿ E,g,B, on Anscombe-Aumann acts, a vN-M utility u representing ≿ E,g,B on constant acts, and an event E ∈ Σ. If there exists a concave V E,g,B such that 1.f ≿ E,g,B h if and only if V E,g,B (u ∘ f) ≥ V E,g,B (u ∘ h) 2. a(s) = b(s) for all s ∈ E implies V E,g,B (a) = V E,g,B (b), and 3. a(s) > b(s) for all s ∈ E implies V E,g,B (a) > V E,g,B (b), Then, [h ≿ E,g,B f for all f ∈ B with f = h on E c ]  G E,h (V E,g,B ) ∩ Q E,h,B (E) ≠ .

50. E. Hanany and P. Klibanoff ES-NASM 08 Updating Ambiguity Averse Preferences A General Result This gives us a test for dynamic consistency of an update rule for any concave model: 1.Calculate G E,g (V E,g,B ) 2.Calculate Q E,g,B (E) 3.See if they intersect. What happens when we apply this to concave Smooth Ambiguity preferences?