larry blume - sites.santafe.edu

Expressive Rationality

Larry Blume

Cornell University & The Santa Fe Institute & IHS, Vienna

What Caused the Financial Crisis?

Blume Expressive Rationality 1


MATH!Blume Expressive Rationality 1


When it comes to the all-too-human problem of recessions anddepressions, economists need to abandon the neat but wrongsolution of assuming that everyone is rational and markets workperfectly.

Paul Krugman, NYT



When it comes to the all-too-human problem of recessions anddepressions, economists need to abandon the neat but wrongsolution of assuming that everyone is rational and markets workperfectly.

Paul Krugman, NYT

Copula image source:http://www.wired.com/techbiz/it/magazine/17-03/wp quant.

What is Probability?

Coin Flip



Horse Race



Horse Race

This is the Objective Subjective Distinction


Electron Cloud



Galton Board



f (x) =

{µx if 0 ≤ x ≤ 1/2,

µ(1− x) if 1/2 < x ≤ 1.

Tent Map



FinancialMarket



FinancialMarket

The ontic/aleatory — epistemic distinction.

Quantum mechanics gives a probalistic description of the location of anelectron around the nucleus of an atom. It is inherently probalistic.

The Dalton board would be deterministic if we knew the initial conditions, butwe don’t. Same with Coin flips.

For√

2 ≤ µ ≤ 2 the Tent map has a unique invariant measure νµ which isabsolutely continuous with respect to Lebesgue measure, and that measure isergodic. If you do not know the initial condition, and have any prior belief p0

on x0, then your prior belief about x1000 should be p1000 ≈ νµ. So even thoughbeliefs about one run of the tent map should be thought of as epistemic, thesystem forces your beliefs about x1000. There are physical systems with thisproperty — the location of a gas molecule, solutions to the n-body problem,etc.

Financial Markets?

Classification of Probability Types

Objective Subjective

OnticElectron Tent

Cloud Map

EpistemicCoin Horse

Flip Race


Measurement and Meaning

Definition: A measurement structure M =< S ,R1, . . . ,Rn > is aset of objects S together with n mi -ary relations Ri on S .

Definition: A real-valued representation of M is a set Σ of realnumbers and n mi -ary relations Ri on Σ, together with a functionφ : S → Σ such that (s1, . . . , smi ) ∈ Ri iff

(φ(s1), . . . , φ(smi )

)∈ Ri .

The problem of measurement is to find a representation.

The problem of meaning is to determine which properties of< Σ,R1, . . . ,Rn > have meaning for M.


Measurement and Meaning

Definition: A measurement structure M =< S ,R1, . . . ,Rn > is aset of objects S together with n mi -ary relations Ri on S .

Definition: A real-valued representation of M is a set Σ of realnumbers and n mi -ary relations Ri on Σ, together with a functionφ : S → Σ such that (s1, . . . , smi ) ∈ Ri iff

(φ(s1), . . . , φ(smi )

)∈ Ri .

The problem of measurement is to find a representation.

The problem of meaning is to determine which properties of< Σ,R1, . . . ,Rn > have meaning for M.

What does it mean to measure something? What are the experiments.

• The relation being measured: warmer than, better than, more likely than;together with operations on objects, such as concatenation, piling up onone side of a balance beam, . . .

• These relations collectively define experiments.

Frequentism

1. Probability is assigned only to collectives. These are repetitive events ormass phenomena.

2. Collectives are modeled as infinite sequences.

3. Relative frequencies converge not just for the infinite sequence, but forany infinite subsequence.

So what is the experiment that we can perform on, say, financial markets?

Probability as a Theory of Measurement

An algebra A of sets of elements of a ground set S (of states).

A complete relation � B on A: A � B means “A is at leastas likely as B.”

S � A � ∅ for all A ∈ A, and S � ∅.

A representation of � is a function p : A → [0, 1] such that

A � B iff p(A) ≥ p(B).


Probability as a Theory of Measurement: Finite States

When is a representation a probability?

A ∩ C = ∅ ⇒ p(A ∪ C ) = p(A) + p(C ).

Disjoint Union Property:

Suppose C is disjoint from A, B. A � B iff A ∪ C � B ∪ C .


Probability as a Theory of Measurement: Finite States


A ∩ C = ∅ ⇒ p(A ∪ C ) = p(A) + p(C ).

Cancellation:

If A1, . . . ,AN and B1, . . . ,BN are sets such that for all s,

#{n : s ∈ An} = #{n : s ∈ Bn}

and for all n ≤ N − 1, An � Bn, then BN � AN .


Probability as a Theory of Measurement: Many States


1. � is complete and transitive on the power set S of S .

2. For A, B disjoint from C , A � B iff A ∪ C � B ∪ C .

3. If A � B, there is a finite partition {C1, . . . ,CM} of S suchthat for all m, A � B ∪ C .

Theorem (Savage): If so there is a unique probability measure pon S such that A � B iff p(A) ≥ p(B). Furthermore, for all A � ∅and 0 ≤ ρ ≤ 1 there is a B ⊂ A such that p(B) = ρp(A).


Probability as a Theory of Measurement: Many States


1. � is complete and transitive on the power set S of S .

2. For A, B disjoint from C , A � B iff A ∪ C � B ∪ C .

3. If A � B, there is a finite partition {C1, . . . ,CM} of S suchthat for all m, A � B ∪ C .

Theorem (Savage): If so there is a unique probability measure pon S such that A � B iff p(A) ≥ p(B). Furthermore, for all A � ∅and 0 ≤ ρ ≤ 1 there is a B ⊂ A such that p(B) = ρp(A).

• If you accept the continuum hypothesis, then p must be only finitelyadditive.

• S can be a σ-algebra, but not just an algebra.

• Savage’s axioms does not imply that the state space be uncountable.

Alternative Measures of Probability

Sets of Probabilities

Non-additive Probabilities

Belief Functions

Inner and Outer Measure

Lexicographic Probabilities

Possibility Measures

Plausibility Measures

Ranking Functions


Why Axiomatic Decision Theory?

Are representations compelling?

Axioms characterize preferences in terms of choice behavior.

Important Models

SEU

LSEU

Probabilistic Sophistication

Wald citerion

Minimax regret

CEU

MMEU

Prospect Theory

Hurwicz α rule


Why Axiomatic Decision Theory?

Are representations compelling?

Axioms characterize preferences in terms of choice behavior.

Important Models

SEU

LSEU


Wald citerion

Minimax regret

CEU

MMEU

Prospect Theory

Hurwicz α rule

Representations per se are not compelling.

• Characterize representations in terms of choice behavior.

• A decision model is normatively appropriate iff its characterizing axiomshave normative appeal

• A decision model is descriptively appropriate iff its characterizing axiomshave descriptive appeal

• Axioms give us a handle on verifying or falsifying, justifying or criticizinggiven models.

• vN-M - preferences over probability distributions

• SEU

A Common Framework

States: A finite state space S .

Outcomes: A finite set O, with best and worst outcomesx∗ and x∗.

Roulette Wheels: The set R of probability distributions on O.

Horse Lotteries: The set H of functions h : S → O.

Preferences: A preference relation � on H.

Anscombe and Aumann (1963).


Subjective Expected Utility

SEU 1. � is complete and transitive.

SEU 2. Independence: If h � k then for all g and 0 ≤ α ≤ 1,αg + (1− α)h � αg + (1− α)k .

SEU 3. Archimedean axiom.

SEU 4. State independence.

There is a payoff function u : O → R and a unique probabilitydistribution p on S which define a functional V on H,

V (h) =∑

s

p(s)∑o

u(o)h(s)(o)

such that h � k iff V (h) ≥ V (k).



Savage framework: S , O, acts f : s 7→ o. P0 set of finite-supportprobabilities on O

Definition: An individuali s said to be probabilisticallysophisticated if there exists a probability measure p on S and apreference functional V (x1, p1, . . . , xm, pm) on P0 satisfyingmixture continuity and monotonicity with respect to stochasticdominance, such that preferences on acts are represented by thefunctional f 7→ V

(x1, p(f −1(x1)), . . . , xn, p(f −1(xn)

)where

{x1, . . . , xn} is the range of f .



Savage framework: S , O, acts f : s 7→ o. P0 set of finite-supportprobabilities on O

Definition: An individuali s said to be probabilisticallysophisticated if there exists a probability measure p on S and apreference functional V (x1, p1, . . . , xm, pm) on P0 satisfyingmixture continuity and monotonicity with respect to stochasticdominance, such that preferences on acts are represented by thefunctional f 7→ V

(x1, p(f −1(x1)), . . . , xn, p(f −1(xn)

)where

{x1, . . . , xn} is the range of f .

• The separation of tastes and beliefs.

• Mixture-closed,

• Stochastic dominance: p stochastically dominates q iff for everyincreasing function f ,

Rf dp ≥

Rf dq.

Why Non-Additive Probabilities?

Ellsberg’s Urns

Schmeidler’s Coins


Why Non-Additive Probabilities?

Ellsberg’s Urns

Schmeidler’s Coins

Image: http://scholar.lib.vt.edu/ejournals/SPT/v8n2/grinbaum.html.

Choquet Expected Utility

Definition: Acts f and g are comonotonic if f (s) � f (t) impliesg(s) � g(t) for all s, t ∈ S .

CEU 1. � is complete and transitive.

CEU 2. Comonotonic independence.

CEU 3. Archimedean axiom.

CEU 4. State independence.

There is a payoff function u : O → R and a unique capacity φ on Swhich define a functional V on H,

V (h) =

∫S

∑o

u(o)h(s)(o)dφ(s)



MMEU

MMEU 1. � is complete and transitive.

MMEU 2. Certainty independence.

MMEU 3. Archimedean axiom.

MMEU 4. State independence.

MMEU 5. f ∼ g implies that for all 0 < α < 1,αf + (1− α)g � f .

There is a payoff function u : O → R and a set P of probabilitydistributions on S which define a functional V on H,

V (h) = infp∈P

∑s

p(s)∑o

u(o)h(s)(o)



MMEU

MMEU 1. � is complete and transitive.

MMEU 2. Certainty independence.

MMEU 3. Archimedean axiom.

MMEU 4. State independence.

MMEU 5. f ∼ g implies that for all 0 < α < 1,αf + (1− α)g � f .

There is a payoff function u : O → R and a set P of probabilitydistributions on S which define a functional V on H,

V (h) = infp∈P

∑s

p(s)∑o

u(o)h(s)(o)


• Connection between MMEU, CEU. The core of a capacity is the set of allprobability measures that dominate it. If φ is convex,φ(A ∪ B) + φ(A ∩ B) ≤ φ(A) + φ(B), then the Choquet integral of anyreal-valued function f with respect to φ is the minimum of the integralswith respect to probability distributions in the core.

• Notice that with equality, this is inclusion/exclusion, and is alwayssatisfied by any probability measure.

The Problem with Conditioning

Savage Conditioning: f �A g iff for some h, f |Ah � g |Ah.

In SEU, any h gives the same answer, and �A is represented by uand p( · |A).




S = {x , y , z}. φ(x) = 1/4, φ(y) = φ(z) = 1/8, φ(x , z) = 1/2,φ(y , z) = 3/4, φ(x , y) = 1/2. Let A = {x , y}

f (x) = 1 on x , else 0, g(x) = 1 on y , else 0.





f (x) = 1 on x , else 0, g(x) = 1 on y , else 0.∫f dφ = 0 + 1φ(x) = 1/4

∫g dφ = 0 + 1φ(y) = 1/8






∫g dφ = 0 + 1φ(y) = 1/8

h = 0∫f |Ah dφ = 0 + 1φ(x) = 1/4

∫g |Ah dφ = 0 + 1φ(y) = 1/8






∫g dφ = 0 + 1φ(y) = 1/8

h = 0∫f |Ah dφ = 0 + 1φ(x) = 1/4

∫g |Ah dφ = 0 + 1φ(y) = 1/8

h = 1∫f |Ah dφ = 0+1φ(x , z) = 1/2

∫g |Ah dφ = 0+1φ(y , z) = 3/4




MMEU version

µ1 = (3/8, 1/8, 1/2) µ2 = (1/4, 1/2, 1/4) µ3 = (1/4, 5/8, 1/8)

f |A0 7→ 1/4 g |A0 7→ 1/8

f |A1 7→ 3/8 g |A1 7→ 5/8




MMEU version

µ1 = (3/8, 1/8, 1/2) µ2 = (1/4, 1/2, 1/4) µ3 = (1/4, 5/8, 1/8)

f |A0 7→ 1/4 g |A0 7→ 1/8

f |A1 7→ 3/8 g |A1 7→ 5/8

• Updating is the problem of defining conditional preference.

• Define fAh.

• Same issue arises with MMEU. This is what I will talk about most.

The Conditioning Tradeoff

Dynamic Consistency: If f �A g in all ex post situations A, thenf � g .

Consequentialism: Conditional preferences given A �A onlydepend on what happens in A.

Fact of Life: If the CEU updating rule satisfies dynamicconsistency and conequentialism, then the capacity is a probability(and updating is Bayes).


f -Bayesian Updating

Definition: An updating rule is a map (�,A)→�A that assigns toan unconditional preference relation and an event A the preferencerelation given A.

Definition: For act f , the f -Bayesian updating rule is g �fA h iff

g |Af � hAf .





g |Af � hAf .

The optimistic rule: f ≡ o∗. A is good news.

φA(B) = φ(B ∩ A)/φ(A).

The Bayesian rule.

Select and update all priors assigning maximal probability to A.





g |Af � hAf .

The pessimistic rule: f ≡ o∗. A is bad news.

φA(B) =[φ((B ∩ A) ∪ Ac

)− φ(Ac)

]/(1− φ(Ac)

).

The Dempster-Shafer rule for belief functions.





g |Af � hAf .

The pessimistic rule: f ≡ o∗. A is bad news.

φA(B) =[φ((B ∩ A) ∪ Ac

)− φ(Ac)

]/(1− φ(Ac)

).

The Dempster-Shafer rule for belief functions.

• Affine u.

• |S | ≥ 4.

• Fagin-Halpern is full updating.

MMEU: Full Bayesian Updating

F1. The updating rules (�,A) 7→�A takes MMEU preferences inoMMEU preferences.

F2. For all non-null events A and outcomes o, if f ∼ o thenf |Ao ∼ o.

Theorem: If u and P are an MMEU representation for � andp(A) > 0 for all p ∈ P, then u and {q = p( · |A), p ∈ P} give anMMEU representation for �A.

Fagin and Halpern (1989), Pires (2002).


MMEU: Full Bayesian Updating

F1. The updating rules (�,A) 7→�A takes MMEU preferences inoMMEU preferences.

F2. For all non-null events A and outcomes o, if f ∼ o thenf |Ao ∼ o.

Theorem: If u and P are an MMEU representation for � andp(A) > 0 for all p ∈ P, then u and {q = p( · |A), p ∈ P} give anMMEU representation for �A.

Fagin and Halpern (1989), Pires (2002).

• MMEU is inherently pessimist, selecting priors which put the most weighton the worst outcomes. F2 guarantees that no matter the weights on Evs. E c , the relative weights on states within E have to cohere with theunconditional weights.

Easy Implications: Portfolios0 net positions in portfolios over a range of prices.

An asset pays off 1 in state H and 3 instate T . Trader beliefs areφ(H) = 0.3 and φ(T ) = 0.4. The expected payoff of a unit longposition at price p is

vb = (1− p) + 0.4(2) = 1.8− p.

The value of a unit short position is

vs = p − 3 + 0.3(2) = p − 2.4.

For 1.8 < p < 2.4, the 0 position is better than both.

Home Bias ParadoxEquity Premium Puzzle


Easy Implications: Portfolios0 net positions in portfolios over a range of prices.

An asset pays off 1 in state H and 3 instate T . Trader beliefs areφ(H) = 0.3 and φ(T ) = 0.4. The expected payoff of a unit longposition at price p is

vb = (1− p) + 0.4(2) = 1.8− p.

The value of a unit short position is

vs = p − 3 + 0.3(2) = p − 2.4.

For 1.8 < p < 2.4, the 0 position is better than both.

Home Bias ParadoxEquity Premium Puzzle

• WIth SEU preferences, the 0 zone is a point, and there is indifferencebetween buying and selling.

FINIS!


Definition: Non-Additive Probability

Definition: A non-additive probability ν on S is a functionmapping subsets of S to [0, 1] such that

N.1. ν(∅) = 0,

N.2. ν(S) = 1,

N.3. If A ⊂ B, then ν(A) ≤ ν(B).

For example, S = {s1, s2}.

να(∅) = 0

να(s1) = να(s2) = α

να(S) = 1


Definition: Non-Additive Probability

Definition: A non-additive probability ν on S is a functionmapping subsets of S to [0, 1] such that

N.1. ν(∅) = 0,

N.2. ν(S) = 1,

N.3. If A ⊂ B, then ν(A) ≤ ν(B).

Integration

Suppose the values of f are x1 < · · · < xn. Then

Eν f = x1 + (x2 − x1)ν(f > x1) + · · · (xn − xn−1)ν(f > xn−1).


Dempster-Shafer Belief Functions

Definition: A belief function β on S is a function mapping subsetsof S to [0, 1] such that

B.1. β(∅) = 0,

B.2. β(S) = 1,

B.3. β(∪ni=1Ai ) ≥

∑ni=1

∑I⊆{1,...,n}:|I |=i (−1)i+1β(∩j∈IAj).




B.1. β(∅) = 0,

B.2. β(S) = 1,

B.3. β(∪ni=1Ai ) ≥

∑ni=1

∑I⊆{1,...,n}:|I |=i (−1)i+1β(∩j∈IAj).

B.3 is like inclusion-exclusion:

Pr(A ∪ B) = Pr(A) + Pr(B)− Pr(A ∩ B),

Pr(A ∪ B ∪ C ) = Pr(A) + Pr(B) + Pr(C )− Pr(A ∩ B)−Pr(A ∩ C )− Pr(B ∩ C ) + Pr(A ∩ B ∩ C ).




B.1. β(∅) = 0,

B.2. β(S) = 1,

B.3. β(∪ni=1Ai ) ≥

∑ni=1

∑I⊆{1,...,n}:|I |=i (−1)i+1β(∩j∈IAj).

β(A) = inf{p(A) : p ≥ β}




B.1. β(∅) = 0,

B.2. β(S) = 1,

B.3. β(∪ni=1Ai ) ≥

∑ni=1

∑I⊆{1,...,n}:|I |=i (−1)i+1β(∩j∈IAj).

β(A) = inf{p(A) : p ≥ β}

• Belief functions are tight capacities.

• m(A) is the weight of evidence for A not assigned to any of its subsets.

• Theorem due to Shafer.

Mass Functions & Belief Functions

Definition: A mass function m on S is a function mapping subsetsof S to [0, 1] such that

M.1. m(∅) = 0,

M.2.∑

A∈S(A) = 1.

Theorem: If S is finite and S = 2S , then beta is a belief functionif and only if there is a (unique) mass function m such that for allA, β(A) =

∑B⊂A m(B).


Mass Functions & Belief Functions

Definition: A mass function m on S is a function mapping subsetsof S to [0, 1] such that

M.1. m(∅) = 0,

M.2.∑

A∈S(A) = 1.

Theorem: If S is finite and S = 2S , then beta is a belief functionif and only if there is a (unique) mass function m such that for allA, β(A) =

∑B⊂A m(B).

• Belief functions are tight capacities.

• m(A) is the weight of evidence for A not assigned to any of its subsets.

• Theorem due to Shafer.

Definition: Lexicographic Probability

Definition: A lexicographic probability on S is a vector ofprobabilities µ = (µ1, . . . , µn) on S such that A � B iff the vectorµ(A) lexicographically dominates µ(B).

LSEU

U(f ) =

(Eµi

{∑o

u(o)f (s)(o)})n

i=1

.

f � g if and only if U(f ) ≥ U(g).

Blume Brandenburger and Dekel (1993a,b)


References

Blume, L., A. Brandenburger and E. Dekel (1993). Lexicographic probabilities and choice under uncertainty.Econometrica.

Cerreia-Vioglio,S., et. al., (2008). Uncertainty averse preferences. Unpublished.

Fagin, R. and Halpern, J. (1991). A new approach to updating belief, in Uncertainty in Artificial Intelligence 6.

Gelman, A. (2006). The boxer, the wrestler, and the coin flip: A paradox of robust Bayesian inference and belieffunctions. The American Statistician.

Gilboa, I. and D. Schmeidler (1989). Maxmin expected utility with a non-unique prior. Journal of MathematicalEconomics.

Gilboa, I. and D. Schmeidler (1993). Updating ambiguous beliefs. Journal of Economic Theory 59.

Kast, R., A. Lapied and P. Toquebeuf (2008). Updating Choquet integrals, consequentialism and dynamicconsistency. Unpublished.

Machina, M. and D. Schmeidler (1992). A more robust definition of subjective probability Econometrica.

Marinacci, M. and L. Montrucchio (2004). Introduction to the mathematics of ambiguity. In Uncertainty inEconomic Theory: A collection of essays in honor of David Schmeidlers 65th birthday.

Pires, C (2002). A rule for updating ambiguous beliefs. Theory and Decision

Schmeidler, D. (1989). Subjective probability and expected utility without additivity, Econometrica.


http://www.carloalberto.org/files/no.77.pdf

http://www.cs.cornell.edu/home/halpern/papers/DS_update1.pdf

http://www.stat.columbia.edu/~gelman/research/published/augie4.pdf

http://www.stat.columbia.edu/~gelman/research/published/augie4.pdf

http://www.icer.it/docs/wp2008/ICERwp04-08.pdf

http://www.icer.it/docs/wp2008/ICERwp04-08.pdf

http://www.carloalberto.org/people/intro.pdf

larry blume - sites.santafe.edu

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