Rationality and Uncertainty

Rationality and Uncertainty Based on papers by Itzhak Gilboa, Massimo Marinacci, Andy Postlewaite, and David Schmeidler Warwick Aug 23, 2013

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager)

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager) 1921 Knight, Keynes

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager) 1921 Knight, Keynes 1931 Ramsey, de Finetti

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager) 1921 Knight, Keynes 1931 Ramsey, de Finetti 1954 Savage "The crowning glory"

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager) 1921 Knight, Keynes 1931 Ramsey, de Finetti 1954 Savage "The crowning glory" Ambiguity = Knightian Uncertainty = (sometimes) Uncertainty

Risk and Uncertainty Dual use of probability: empirical frequencies in games of chance and a subjective tool to quantify beliefs Dates back to Pascal and Leibniz (cf. Pascal s Wager) 1921 Knight, Keynes 1931 Ramsey, de Finetti 1954 Savage "The crowning glory" Ambiguity = Knightian Uncertainty = (sometimes) Uncertainty Objectivity interpersonal concept, convincing others

Rationality 1 Economic decision is rational if it optimizes the agent s preferences,

Rationality 1 Economic decision is rational if it optimizes the agent s preferences, As long as the preferences are consistent

Rationality 1 Economic decision is rational if it optimizes the agent s preferences, As long as the preferences are consistent De gustibus non est disputandum

Rationality 1 Economic decision is rational if it optimizes the agent s preferences, As long as the preferences are consistent De gustibus non est disputandum In case of risk or uncertainty the agent should maximize expected utility with respect to the known or subjective probability

Rationality 1 Economic decision is rational if it optimizes the agent s preferences, As long as the preferences are consistent De gustibus non est disputandum In case of risk or uncertainty the agent should maximize expected utility with respect to the known or subjective probability This is the accepted view of economic theory

Rationality 1 Economic decision is rational if it optimizes the agent s preferences, As long as the preferences are consistent De gustibus non est disputandum In case of risk or uncertainty the agent should maximize expected utility with respect to the known or subjective probability This is the accepted view of economic theory or majority of economic theorists and game theorists.

Rationality 2 The definition we use:

Rationality 2 The definition we use: A mode of behavior is irrational for a given decision maker, if, when the decision maker behaves in this mode, and is then exposed to the analysis of her behavior, she regrets it (feels embarrassed).

Rationality 2 The definition we use: A mode of behavior is irrational for a given decision maker, if, when the decision maker behaves in this mode, and is then exposed to the analysis of her behavior, she regrets it (feels embarrassed). In other words, an act is rational (or objectively rational) if the decision maker can convince others that she optimized her goals.

Rationality 2 The definition we use: A mode of behavior is irrational for a given decision maker, if, when the decision maker behaves in this mode, and is then exposed to the analysis of her behavior, she regrets it (feels embarrassed). In other words, an act is rational (or objectively rational) if the decision maker can convince others that she optimized her goals. Like Objectivity this is an interpersonal concept convincing others

Rationality 2 The definition we use: A mode of behavior is irrational for a given decision maker, if, when the decision maker behaves in this mode, and is then exposed to the analysis of her behavior, she regrets it (feels embarrassed). In other words, an act is rational (or objectively rational) if the decision maker can convince others that she optimized her goals. Like Objectivity this is an interpersonal concept convincing others An act is subjectively rational if the decision maker can not be convinced by others that she failed to optimize her goals.

The Bayesian approach Four tenets of Bayesianism in economic theory

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty Prior Probability: (i) Whenever a fact is not known, one should have probabilistic beliefs about its possible values.

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty Prior Probability: (i) Whenever a fact is not known, one should have probabilistic beliefs about its possible values. (ii) These beliefs should be given by a single probability measure defined over the state space

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty Prior Probability: (i) Whenever a fact is not known, one should have probabilistic beliefs about its possible values. (ii) These beliefs should be given by a single probability measure defined over the state space Updating of the prior according to Bayes rule

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty Prior Probability: (i) Whenever a fact is not known, one should have probabilistic beliefs about its possible values. (ii) These beliefs should be given by a single probability measure defined over the state space Updating of the prior according to Bayes rule When facing a decision problem, one should maximize expected utility

The Bayesian approach Four tenets of Bayesianism in economic theory Formulation of a state space, where each state resolves all uncertainty Prior Probability: (i) Whenever a fact is not known, one should have probabilistic beliefs about its possible values. (ii) These beliefs should be given by a single probability measure defined over the state space Updating of the prior according to Bayes rule When facing a decision problem, one should maximize expected utility (ii)* Sometimes the prior is posited on the consequences.

Background Undoubtedly, the Bayesian approach is immensely powerful and successful

Background Undoubtedly, the Bayesian approach is immensely powerful and successful It is very good at representing knowledge, belief, and intuition Indeed, it is a first rate tool to reason about uncertainty (cf. paradoxes )

Background Undoubtedly, the Bayesian approach is immensely powerful and successful It is very good at representing knowledge, belief, and intuition Indeed, it is a first rate tool to reason about uncertainty (cf. paradoxes ) Used in statistics, machine learning and computer science, philosophy (mostly of science), and econometrics...

Background Undoubtedly, the Bayesian approach is immensely powerful and successful It is very good at representing knowledge, belief, and intuition Indeed, it is a first rate tool to reason about uncertainty (cf. paradoxes ) Used in statistics, machine learning and computer science, philosophy (mostly of science), and econometrics... However, in most of these, only when the prior is known.

Background Undoubtedly, the Bayesian approach is immensely powerful and successful It is very good at representing knowledge, belief, and intuition Indeed, it is a first rate tool to reason about uncertainty (cf. paradoxes ) Used in statistics, machine learning and computer science, philosophy (mostly of science), and econometrics... However, in most of these, only when the prior is known. Typically, for a restricted state space where the set of parameters does not grow with the database

Background Undoubtedly, the Bayesian approach is immensely powerful and successful It is very good at representing knowledge, belief, and intuition Indeed, it is a first rate tool to reason about uncertainty (cf. paradoxes ) Used in statistics, machine learning and computer science, philosophy (mostly of science), and econometrics... However, in most of these, only when the prior is known. Typically, for a restricted state space where the set of parameters does not grow with the database By contrast, in economics, it has been applied to very large spaces; almost to the Grand State Space

Non-Bayesian decisions A B = A C a 7 0 b 0 7 c 3 3

Ellsberg s Paradox One urn contains 50 black and 50 red balls Another contains 100 balls, each black or red Do you prefer a bet on the known or the unknown urn? Many prefer the known probabilities. People often prefer known to unknown probabilities This is inconsistent with the Bayesian approach

Ellsberg s Paradox One urn contains 50 black and 50 red balls Another contains 100 balls, each black or red Do you prefer a bet on the known or the unknown urn? Many prefer the known probabilities. People often prefer known to unknown probabilities This is inconsistent with the Bayesian approach Still, many insist on this choice even when the inconsistency and Savage s axioms are explained to them

Symmetry and Reality Ellsberg s paradox may be misleading If one wishes to be Bayesian, it is easy to adopt a prior in this example (due to symmetry)

Symmetry and Reality Ellsberg s paradox may be misleading If one wishes to be Bayesian, it is easy to adopt a prior in this example (due to symmetry) But this is not the case in real life examples of wars, stock market crashes, etc.

Symmetry and Reality Ellsberg s paradox may be misleading If one wishes to be Bayesian, it is easy to adopt a prior in this example (due to symmetry) But this is not the case in real life examples of wars, stock market crashes, etc. Indeed, my critique was based on the cognitive implausibility of the Bayesian approach, and not on the results of an experiment

Tabloid Summary To Bayesians:

Tabloid Summary To Bayesians: If you do not know the prior, do not pretend that you know it.

Tabloid Summary To Bayesians: If you do not know the prior, do not pretend that you know it. To economists:

Tabloid Summary To Bayesians: If you do not know the prior, do not pretend that you know it. To economists: If you do not know, do not pretend that you know.

Anscombe-Aumann s Framework X outcomes, a convex set

Anscombe-Aumann s Framework X outcomes, a convex set S states of the world

Anscombe-Aumann s Framework X outcomes, a convex set S states of the world A = { a : S X... } acts

Anscombe-Aumann s Framework X outcomes, a convex set S states of the world A = { a : S X... } acts on A ( and on X )

Anscombe-Aumann s Axioms on : Three axioms of von Neumann and Morgenstern utility theory

Anscombe-Aumann s Axioms on : Three axioms of von Neumann and Morgenstern utility theory Monotonicity (pointwise)

Anscombe-Aumann s Axioms on : Three axioms of von Neumann and Morgenstern utility theory Monotonicity (pointwise) Non-degeneracy (there are a, a such that a a )

Anscombe-Aumann s Theorem Theorem Let be a preference defined on A, The following conditions are equivalent: (i) satisfies the five axioms; (ii) there exists an affi ne function u : X R and a probability measure P : Σ [0, 1] such that, for all a, b A, a b iff S (u(a (s)) dp S (u(b (s)) dp.moreover, P is unique and u is cardinally unique.

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom vn&m (and AA) Independence: a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom vn&m (and AA) Independence: a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c Consider two states S = {1, 2} a = (1, 0) b = (0, 1)

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom vn&m (and AA) Independence: a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c Consider two states S = {1, 2} a = (1, 0) b = (0, 1) Out of ignorance, a b

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom vn&m (and AA) Independence: a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c Consider two states S = {1, 2} a = (1, 0) b = (0, 1) Out of ignorance, a b But 1 2 a + 1 b = (0.5, 0.5) 2 provides hedging and is better than both a and b

Ambiguity in Anscombe-Aumann s Framework The culprit: the Independence axiom vn&m (and AA) Independence: a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c Consider two states S = {1, 2} a = (1, 0) b = (0, 1) Out of ignorance, a b But 1 2 a + 1 b = (0.5, 0.5) 2 provides hedging and is better than both a and b Or the other way for ambiguity lovers

Comonotonic Independence Schmeidler (1989): restricts the Independence axiom

Comonotonic Independence Schmeidler (1989): restricts the Independence axiom a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c whenever a and c, and b and c are pairwise comonotonic

Comonotonic Independence Schmeidler (1989): restricts the Independence axiom a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c whenever a and c, and b and c are pairwise comonotonic i.e., it never happens that one increases between two states while the other decreases

Comonotonic Independence Schmeidler (1989): restricts the Independence axiom a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c whenever a and c, and b and c are pairwise comonotonic i.e., it never happens that one increases between two states while the other decreases there are no s, t S s.t., a(s) c(s) and a(t) c(t)

Comonotonic Independence Schmeidler (1989): restricts the Independence axiom a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c whenever a and c, and b and c are pairwise comonotonic i.e., it never happens that one increases between two states while the other decreases there are no s, t S s.t., a(s) c(s) and a(t) c(t)... and thus hedging is ruled out

Choquet EU 1 (Schmeidler, 1989) v : Σ [0, 1] is a capacity if ν( ) = 0 and ν(s) = 1 E E implies ν(e ) ν(e )

Choquet EU 1 (Schmeidler, 1989) v : Σ [0, 1] is a capacity if ν( ) = 0 and ν(s) = 1 E E implies ν(e ) ν(e ) Choquet integral of φ : S R + S φ dν = 0 ν ({s S : φ (s) t}) dt

Choquet EU 2 Schmeidler (1986, 1989): Theorem Let be a preference defined on A. The following conditions are equivalent: (i) satisfies the AA axioms except Independence. The latter is replaced with Comonotonic Independence; (ii) there exists a function u : X R and a capacity ν : Σ R such that, for all a, b A, a b if and only if S (u(a(s)) dν S (u(b (s)) dν 1. Moreover, ν is unique and u is cardinally unique.

Another Restriction on Independence Certainty Independence, Gilboa-Schmeidler (1989)

Another Restriction on Independence Certainty Independence, Gilboa-Schmeidler (1989) a, b, A, x X, α (0, 1) a b αa + (1 α) x αb + (1 α) x where x is the constant act with x(s) = x for all s.

Another Restriction on Independence Certainty Independence, Gilboa-Schmeidler (1989) a, b, A, x X, α (0, 1) a b αa + (1 α) x αb + (1 α) x where x is the constant act with x(s) = x for all s. Add the Uncertainty Aversion (Schmeidler, 1986) axiom: a b 1 2 a + 1 2 b a, b

Maxmin EU (MEU or MMEU) Theorem (GS_1989) Let be a preference relation defined on A. The following conditions are equivalent: (i) satisfies the AA axioms except Independence. The latter is replaced with Certainty Independence and Uncertainty Aversion; (ii) there exists a function u : X R and a convex and compact set K (Σ) of probability measures such that, for all a, b A a b V (a) V (b), where V (a) = min (u(a (s)) dp, P K S Moreover, K is unique and u is cardinally unique.

Weak Certainty Independence A further weakening of the Independence axiom by Maccheroni, Marinacci, and Rustichini (2006)

Weak Certainty Independence A further weakening of the Independence axiom by Maccheroni, Marinacci, and Rustichini (2006) a, b, c, d A, x, y X, α (0, 1) αa + (1 α) x αb + (1 α) x αa + (1 α)ȳ αb + (1 α)ȳ

Variational Preferences Also MRR (2006) V (a) = [{ }] min (u(a (s)) dp + Cost(P) P (S ) S where Cost : (S) R { } is convex (with min Cost = 0).

Variational Preferences Also MRR (2006) V (a) = [{ }] min (u(a (s)) dp + Cost(P) P (S ) S where Cost : (S) R { } is convex (with min Cost = 0). It extends GS MEU where Cost(P) = 0 for P K. Otherwise Cost(P) =

Variational Preferences Also MRR (2006) V (a) = [{ }] min (u(a (s)) dp + Cost(P) P (S ) S where Cost : (S) R { } is convex (with min Cost = 0). It extends GS MEU where Cost(P) = 0 for P K. Otherwise Cost(P) = This generalizes the multiplier preferences used by Hansen and Sargent (2001,...,2008) Cost(P) = θr (P Q),

The MMR (2006) representation theorem Theorem Let be a preference relation defined on A. The following conditions are equivalent: (i) satisfies the GS axioms except Certainty Independence. The latter is replaced with Weak Certainty Independence (ii) there exists a function u : X R (cardinally unique) such that, for all a, b A, a b V (a) V (b).

Unambiguity 1 In MEU model if an event E is unambiguous one expect that P(E ) = P (E ) for all P, P K.

Unambiguity 1 In MEU model if an event E is unambiguous one expect that P(E ) = P (E ) for all P, P K. In such a case also E c = S E is unambiguous.

Unambiguity 1 In MEU model if an event E is unambiguous one expect that P(E ) = P (E ) for all P, P K. In such a case also E c = S E is unambiguous. Suppose that a set of unambiguous events is given. It is closed under complements and contains S.

Unambiguity 1 In MEU model if an event E is unambiguous one expect that P(E ) = P (E ) for all P, P K. In such a case also E c = S E is unambiguous. Suppose that a set of unambiguous events is given. It is closed under complements and contains S. Xiangyu Qu (2011): satisfies Unambiguity Independence if a, b, c A, α (0, 1) a b αa + (1 α)c αb + (1 α)c whenever c is measurable w.r. to the unambiguous events.

Unambiguity 2 Theorem (Qu 2011) Let be a preference relation defined on A. The following conditions are equivalent: (i) satisfies the GS axioms except Certainty Independence. The latter is replaced with Unambiguity Independence. (ii) There exists representation like in MEU model such that for any unambiguous event E : P(E ) = P (E ) for all P, P K.