Cautious and Globally Ambiguity Averse

Transcription

1 Ozgur Evren New Economic School August, 2016

2 Introduction Subject: Segal s (1987) theory of recursive preferences. Ambiguity aversion: Tendency to prefer risk to ambiguity. (Ellsberg paradox) risk objective, ambiguity subjective

3 Introduction Subject: Segal s (1987) theory of recursive preferences. Ambiguity aversion: Tendency to prefer risk to ambiguity. (Ellsberg paradox) risk objective, ambiguity subjective A lottery is objectively defined.

4 Introduction Subject: Segal s (1987) theory of recursive preferences. Ambiguity aversion: Tendency to prefer risk to ambiguity. (Ellsberg paradox) risk objective, ambiguity subjective A lottery is objectively defined. Attitudes towards ambiguity and compound risk are connected.

5 Introduction Subject: Segal s (1987) theory of recursive preferences. Ambiguity aversion: Tendency to prefer risk to ambiguity. (Ellsberg paradox) risk objective, ambiguity subjective A lottery is objectively defined. Attitudes towards ambiguity and compound risk are connected. Attitudes towards ambiguity and (simple) risk are also connected.

6 Introduction Subject: Segal s (1987) theory of recursive preferences. Ambiguity aversion: Tendency to prefer risk to ambiguity. (Ellsberg paradox) risk objective, ambiguity subjective A lottery is objectively defined. Attitudes towards ambiguity and compound risk are connected. (Halevy, 2007; Dean and Ortoleva, 2015) Attitudes towards ambiguity and (simple) risk are also connected. (Dean and Ortoleva, 2015)

7 Segal (1987) focuses on Rank Dependent Utility (RDU) model for risk preferences (Quiggin, 1982).

8 Segal (1987) focuses on Rank Dependent Utility (RDU) model for risk preferences (Quiggin, 1982). Dillenberger (2010), on compound vs simple risk: Preference for one-shot resolution of uncertainty (PORU) is equivalent to the negative certainty independence axiom (NCI). (NCI) is compatible with common ratio and consequence effects of Allais.

9 Results Theorem 1: Risk preferences that satisfy (NCI) is the only class that robustly predicts ambiguity aversion. (Global ambiguity aversion.) Any other risk preference (including RDU) will produce non-ambiguity averse behavior for some specification of second-order beliefs.

10 Results Theorem 1: Risk preferences that satisfy (NCI) is the only class that robustly predicts ambiguity aversion. (Global ambiguity aversion.) Any other risk preference (including RDU) will produce non-ambiguity averse behavior for some specification of second-order beliefs. Theorem 2: An analogous but finer characterization of expected utility preferences with only three states and binary acts.

11 A dual question: How can we increase relative ambiguity aversion by manipulating a second-order belief, irrespective of the details of risk preferences? (Assuming NCI.) Theorem 3: Characterization of a mean-preserving spread notion for second-order beliefs.

12 Literature Artstein-Avidan and Dillenberger (2011): (NCI) implies global ambiguity aversion.

13 Literature Artstein-Avidan and Dillenberger (2011): (NCI) implies global ambiguity aversion. Dillenberger and Segal (2015a,b): Recursive preferences and ambiguity.

14 Setup X := [x, x ] R, x < x. (X ) the space of (finitely supported) monetary lotteries: p, q. Simple or one-shot lotteries.

15 Setup X := [x, x ] R, x < x. (X ) the space of (finitely supported) monetary lotteries: p, q. Simple or one-shot lotteries. S is a finite state space. (S) is the set of probability distributions over S. (First-order distributions.) π denotes a generic first-order distribution.

16 Setup X := [x, x ] R, x < x. (X ) the space of (finitely supported) monetary lotteries: p, q. Simple or one-shot lotteries. S is a finite state space. (S) is the set of probability distributions over S. (First-order distributions.) π denotes a generic first-order distribution. µ 2 (S), the set of (finitely supported) distributions over (S). µ(π) represents the probability that the correct distribution of states is given by π.

17 The paper works with the original Anscombe-Aumann setup. Here, let s focus on purely subjective acts and (one-shot) lotteries.

18 The paper works with the original Anscombe-Aumann setup. Here, let s focus on purely subjective acts and (one-shot) lotteries. A (purely subjective) act, denoted as f, maps S into X. H X is the set of all acts. is a binary relation over H X (X ).

19 The paper works with the original Anscombe-Aumann setup. Here, let s focus on purely subjective acts and (one-shot) lotteries. A (purely subjective) act, denoted as f, maps S into X. H X is the set of all acts. is a binary relation over H X (X ). Notation for degenerate probability measures: δ x δ s D π Lives in: (X ) (S) 2 (S)

20 Representation Definition. A certainty equivalence function c is a continuous map, in the topology of weak convergence, from (X ) onto X such that: (i) p fosd (> fosd )q c(p) (>)c(q). (ii) c(δ x ) = x for every x X. Think of c as a utility function over (X ) s.t. c(δ c(p) ) = c(p). Existence of such a utility function is guaranteed for every continuous weak-order on (X ) that is monotonic w.r.t. fosd.

21 Given f H X and π (S), let π f := π(s)δ f (s), s S

22 Given f H X and π (S), let and π f := π(s)δ f (s), s S µ f := µ(π)δ c(πf ). π (S) Interpretation: π f is the lottery induced by f when the states are distributed according to π. µ(π) is the probability of this event, and c(π f ) is the perceived return. Thus, f µ f.

23 Definition. A recursive representation for is a utility representation over H X (X ) that consists of a µ 2 (S) and a certainty equivalence function c such that: c (µ f ) is the utility of any f H X, c (p) is the utility of any p (X ).

24 Definition. A recursive representation for is a utility representation over H X (X ) that consists of a µ 2 (S) and a certainty equivalence function c such that: c (µ f ) is the utility of any f H X, c (p) is the utility of any p (X ). In this case, say that is a recursive preference.

25 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X.

26 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π

27 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π is more ambiguity averse than if (i) f δ x f δ x f H X and x X. (ii) p q p q p, q (X ).

28 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π is more ambiguity averse than if (i) f δ x f δ x f H X and x X. (ii) p q p q p, q (X ). c = c

29 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π is more ambiguity averse than if (i) f δ x f δ x f H X and x X. µ f µ f (ii) p q p q p, q (X ). c = c

30 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π is more ambiguity averse than if (i) f δ x f δ x f H X and x X. µ f µ f (ii) p q p q p, q (X ). c = c is ambiguity averse if it is more ambiguity averse than an ambiguity neutral preference: π (S) s.t. π f µ f f H X.

31 Ambiguity Neutrality and Aversion is ambiguity neutral if π (S) s.t. f π f f H X. µ = D π is more ambiguity averse than if (i) f δ x f δ x f H X and x X. µ f µ f (ii) p q p q p, q (X ). c = c is ambiguity averse if it is more ambiguity averse than an ambiguity neutral preference: π (S) s.t. π f µ f f H X. Ambiguity loving defined symmetrically.

32 Global Ambiguity Aversion Definition. A preference relation c on (X ) represented by a certainty equivalence function c has the global ambiguity aversion property if the recursive preference represented by (µ, c) is ambiguity averse for any (finite) state space S and any µ 2 (S).

33 Global Ambiguity Aversion Definition. A preference relation c on (X ) represented by a certainty equivalence function c has the global ambiguity aversion property if the recursive preference represented by (µ, c) is ambiguity averse for any (finite) state space S and any µ 2 (S). A robustness criterion. Example. An Ellsberg-type experiment with two control variables: the number of different colors and what the subject knows about the composition of the ambiguous urn.

34 NCI Negative Certainty Independence (NCI). α [0, 1] p c δ x αp + (1 α)q c αδ x + (1 α)q.

35 NCI Negative Certainty Independence (NCI). α [0, 1] p c δ x αp + (1 α)q c αδ x + (1 α)q. Interpretation: Degenerate lotteries have a certainty appeal.

36 Main Result Theorem 1. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following statements are equivalent. (i) c satisfies (NCI). (ii) c has the global ambiguity aversion property.

37 Non-robustness of Recursive RDU Example 2: With four states, there exist second-order beliefs that produce non-ambiguity averse behavior for any c Strictly Convex RDU.

38 CEU Representation Cautious Expected Utility (CEU) Representation (Cerreia-Vioglio, Dillenberger, Ortoleva, 2015): c(p) := inf v W v 1 (E p (v)), where W C (X ), the space of strictly increasing, continuous functions on X.

39 Some Remarks µ(s) := µ(π)π(s) s S. π (S) µ (S) is the mean (or reduced-form) of µ.

40 Some Remarks µ(s) := µ(π)π(s) s S. π (S) µ (S) is the mean (or reduced-form) of µ. It is straightforward to show that c satisfies (NCI) iff µ f c µ f S, µ 2 (S), f H X. (1)

41 Some Remarks µ(s) := µ(π)π(s) s S. π (S) µ (S) is the mean (or reduced-form) of µ. It is straightforward to show that c satisfies (NCI) iff µ f c µ f S, µ 2 (S), f H X. (1) However, in Theorem 1, the benchmark π is allowed to be distinct from µ.

42 What do we gain by letting π be free?

43 What do we gain by letting π be free? According to current definition, in a two-urn (two-color) experiment, Ellsberg paradox implies ambiguity aversion irrespective of the composition of the risky urn.

44 What do we gain by letting π be free? According to current definition, in a two-urn (two-color) experiment, Ellsberg paradox implies ambiguity aversion irrespective of the composition of the risky urn. Demanding π = µ Demanding the risky urn to be distributed according to µ.

45 What do we gain by letting π be free? According to current definition, in a two-urn (two-color) experiment, Ellsberg paradox implies ambiguity aversion irrespective of the composition of the risky urn. Demanding π = µ Demanding the risky urn to be distributed according to µ. Example 1: µ = distribution because of some asymmetric info about the ambiguous urn. Ellsberg Paradox observed with distributed risky urn. But not if the risky urn were distributed according to µ.

46 Global Ambiguity Neutrality Corollary. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following statements are equivalent. (i) c satisfies the independence axiom. (ii) c has the global ambiguity neutrality property.

47 Global Ambiguity Neutrality Corollary. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following statements are equivalent. (i) c satisfies the independence axiom. (ii) c has the global ambiguity neutrality property. Global ambiguity neutrality ambiguity neutrality irrespective of S and µ. Global ambiguity loving ambiguity loving irrespective of S and µ.

48 Global Ambiguity Neutrality Corollary. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following statements are equivalent. (i) c satisfies the independence axiom. (ii) c has the global ambiguity neutrality property. Global ambiguity neutrality ambiguity neutrality irrespective of S and µ. Global ambiguity loving ambiguity loving irrespective of S and µ. Any recursive non-expected utility preference will exhibit a paradoxical form of ambiguity aversion or the opposite.

49 An Experimental Characterization Two urns, three colors (b, o, w), only binary bets (on a given color). Prizes involved: x, x

50 An Experimental Characterization Two urns, three colors (b, o, w), only binary bets (on a given color). Prizes involved: x, x Ellsberg Paradox: For all three colors, the subject strictly prefers betting on the risky urn. Anti-Ellsberg Paradox: The opposite.

51 An Experimental Characterization Two urns, three colors (b, o, w), only binary bets (on a given color). Prizes involved: x, x Ellsberg Paradox: For all three colors, the subject strictly prefers betting on the risky urn. Anti-Ellsberg Paradox: The opposite. A recursive subject of type c evaluates the bets according to a recursive representation (µ, c) for some µ 2 ({b, o, w}).

52 Theorem 2. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following two statements are equivalent. (i) c satisfies the independence axiom. (ii) A recursive subject of type c exhibits neither Ellsberg nor anti-ellsberg paradox for any µ 2 (S) and any distribution of the three colors in the risky urn.

53 Theorem 2. Let c be a preference relation on (X ) represented by a certainty equivalence function c. The following two statements are equivalent. (i) c satisfies the independence axiom. (ii) A recursive subject of type c exhibits neither Ellsberg nor anti-ellsberg paradox for any µ 2 (S) and any distribution of the three colors in the risky urn. Alternatively: Any recursive non-expected utility model will produce either the Ellsberg or the anti-ellsberg paradox in an experiment of the form above.

54 Increasing Second-Order Uncertainty How can we increase relative ambiguity aversion by manipulating µ, irrespective of the details of the risk preferences? (Assuming NCI.)

55 Increasing Second-Order Uncertainty How can we increase relative ambiguity aversion by manipulating µ, irrespective of the details of the risk preferences? (Assuming NCI.) Definition. Given a state space S, µ is a mean-preserving spread of µ if there exists an α [0, 1] s.t. µ = αµ + (1 α)d µ.

56 Increasing Second-Order Uncertainty How can we increase relative ambiguity aversion by manipulating µ, irrespective of the details of the risk preferences? (Assuming NCI.) Definition. Given a state space S, µ is a mean-preserving spread of µ if there exists an α [0, 1] s.t. µ = αµ + (1 α)d µ. µ embodies a larger amount of second-order uncertainty.

57 Suppose µ (δ s ) = 0 s S. δ s3 (S) δ s1 δ s2

58 Theorem 3. Suppose µ (δ s ) = 0 s S. Then, the following two statements are equivalent. (i) µ is a mean-preserving spread of µ. (ii) The preference represented by (µ, c) is more ambiguity averse than that represented by (µ, c) for every c that satisfies (NCI). Moreover, the same conclusion obtains if we focus only on risk-averse c (or only on risk-loving c) that satisfy (NCI).

59 Why do I assume µ (δ s ) = 0? Time-neutrality implies D µ s S µ(s)d δs.

60 Why do I assume µ (δ s ) = 0? Time-neutrality implies D µ s S µ(s)d δs. Extending the definition of a mean-preserving spread accordingly would lead to a less elegant formulation. What would we gain: A by-product of time-neutrality?

61 Why do I assume µ (δ s ) = 0? Time-neutrality implies D µ s S µ(s)d δs. Extending the definition of a mean-preserving spread accordingly would lead to a less elegant formulation. What would we gain: A by-product of time-neutrality? The present definition can also be useful in alternative models without time-neutrality (Klibanoff, Marinacci, Mukerji, 2005; Seo, 2009).