A new tool is presented for measuring beliefs/likelihoods under uncertainty. It will simplify: preference axiomatizations; quantitative measurements; testing and characterizing qualitative properties.

A Purely Uncertainty-Oriented Approach to Subjective Expected

Utility and its ExtensionsPeter P. Wakker (& Mohammed Abdellaoui)

July 1, 2004; FUR 11, Cachan, France This lecture will be on my homepage on July 8; paper is

there already now.

1. Introduction and History

Savage (1954): First full-blown decision model for uncertainty (restricted to SEU).Savage‘54 is uncertainty-oriented: - measurement of uncertainty/beliefs was central; - richness (continuity) imposed on state space;- measurement of utility is by-product.

Following Debreu (1959) and Arrow (1963), most modern analyses of uncertainty are outcome-oriented:- measurement of utility is first (cf. micro-economics);- richness (continuity) imposed on outcome space;- measurement of uncertainty is indirect.

No surprise because: Savage was statistician.Most formal-decision theorists

today are economist.

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3Uncertainty-oriented references:1. Expected utility:

Savage ‘54, von Neumann-Morgenstern ‘44 (+ Herstein & Milnor ‘53 + Jensen ‘67).

2. Nonexpected utility: - Uncertainty: Gilboa ‘87, Jaffray ‘89, Machina & Schmeidler ’92, Karni ’93, Grant ’95, Epstein & Zhang ’01, Ergin & Gul ’04, Kopylov ‘04

- Risk: Abdellaoui ‘02, Nakamura ‘95.

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For uncertainty (risk and "ambiguity"):uncertainty-oriented is most natural.

We do it systematically.Most general, and simplest, axioms you ever saw!

Extra, mathematical, reason for simplicity + generality:It naturally exploits set-theoretic structure on the state space.

The idea of our “revealed-likelihood” method can be recognized in Gilboa’s (’87) axiom P2*; our work builds on it, and aims to give intuition to it.

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2. Notation

A Ac

event A

Act, yielding under A, under Ac.

complementaryevent Ac

outcome outcome

Convention: is a good outcome, a bad one, .

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A Ac

Act, yielding under A, under Ac.' ' Act, yielding ' under A, ' under Ac.

Using one matrix to denote several acts:

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3. The Basic Idea, Uncertainty-Oriented Measurements

8Savage/de Finetti revealed likelihood orderings between events, through gambles on these events.Point of departure is outcome level .

A Ac

B Bc

=?

(remember: )

Def. If stict preference for left gamble then: A b B;Def. If indifference ~ then: A ~b B;

~

Def. If stict preference for right gamble then: A b B.

Gambleon A Gamble

on B

Quantifiers: We formally defineA b B if there exist such that ... etc.

Choose between two improvements.

A is revealed more likelythan B (in a basic sense)

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Under subjective expected utility (SEU):

SEU-gain is P(A)(U() – U())

SEU-gain is P(B)(U() – U())

SEU: A b B P(A) > P(B)A ~b B P(A) = P(B)A b B P(A) < P(B)

A Ac

B Bc

=

Conclusion: Basic revealed likelihood relations elicit probability orderings.

10You don’t want contradictions in your measurements: Savage's P4 excludes them for b. We rename (and weaken) P4 as basic likelihood consistency:

NOT [A ~b B and A b B]

In preferences, for all and ' ':

A Ac B Bc

= ~

' ' ' ' =' ' ' ' and

NOT

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Obviously:Lemma. SEU basic likelihood consistency.

Many other models imply basic likelihood consistency also.

Not identity, but equivalence, as point of departure.

?

12An extension now.

A Ac

B Bc

=

Def. If indifference ~ then: A ~ B;

~

Def. If stict preference for right gamble then: A B.

Under Bc not , but any outcomes of any act g result.

f ~ g

We require ~ (instead of =) as point of departure.

Under Ac not , but any outcomes of any act f result.

f g

Quantifiers: there exist , f, g, such that ... etc.

Def. If stict preference for left gamble then: A B;

A is revealed more likelythan B

I hope these defs have some first-sight plausibility …

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Will definitions always reveal a sensible likelihood ordering?Or will they reveal contradictions (and then signal problems)? Let us try, work on them, and see where they take us.

SEU-gain is P(A)(U()–U())

SEU-gain is P(A)(U()–U())

SEU-gain is P(B)(U()–U())

Preceding (basic) elicitation:

A Ac

B Bc

=?

14Comparison of new with preceding revelation method.

Stict preference for left gamble then: A B; etc.

SEU-gain is P(B)(U()–U())

?

A Ac

B Bc

=f ~ gf g

New elicitation:

A b B P(A) > P(B)A ~b B P(A) = P(B)A b B P(A) < P(B)

SEU: Conclusion: Revealed likelihood orderings elicit probability orderings, as do the basic orderings b.Also if we drop the subscripts b.

Stict preference for left gamble then: A b B; etc.

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Under SEU, revealed likelihood orderings give desirable results. Let us now investigate a general criterion, the analog of Savage’s P4, that revealed likelihood orderings should not run into contradictions.

A Ac B Bc

f g, ~ f g, ~

' f ' ' g', ~' f ' ' g' and

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Likelihood consistency: NOT [A ~ B and A B]

Same as basic likelihood consistency ( P4), but with subscript b dropped.

Directly in terms of preferences:For all , ' ', f, f ', g, g‘,

NOT

17Lemma. SEU likelihood consistency.

General question: When, besides SEU, is revealed likelihood free of contradictions; i.e., when is our measurement instrument OK?

Questions to the audience:We assume “usual things,” - weak ordering, - monotonicity on outcomes/events as much as you want.- continuity on outcomes/events as much as you want.Then: how strong is likelihood consistency?Does it imply more than Savage’s P4?If so, what?

18Hypothesis 1. Likelihood consistency P4.(No addition; implies no more than dominance.)

Hypothesis 2. Likelihood consistency probabilistic sophistication.(Implies compatibility with disjoint unions.)

Hypothesis 3. Likelihood consistency Choquet expected utility.(Implies compatibility within cumulative events.)

Hypothesis 4. Likelihood consistency Subjective expected utility. go to next slide

A Ac B Bc

f g, ~ f g, ~

' f ' ' g', ~' f ' ' g' and

Likelihood consistency: NOT [A ~ B and A B]

Same as basic likelihood consistency ( P4), but with subscript b dropped.

Directly in terms of preferences:For all , ' ', f, f ', g, g‘,

NOT

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20Answer:

Hypothesis 4 is correct!

Likelihood consistency subjective expected utility.

(Bayesians: Good news! A new foundation of Bayesianism!Non-Bayesians: Disappointing? Instrument doesn’t bring anything interesting? Please wait. NonEU will

result from natural modifications. Comes later.)

For now, we get very simple axiomatization of SEU!

To state it, we turn to technical axioms. Here uncertainty-orientedness also gives improvements.

Richness assumed on state space through solvability (as in Gilboa ’87):

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Df

A

Lf

g

Df

A

Lf

If

~Df

A

Lf

A

direction of increa-sing preference

then ~ holds for some partition A,A of A:

both preferences hold,

“If improving on the whole event is too much to get ~, then you can improve on a subevent.”

Archimedeanity: If , and A (A “nonnull”),and

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… … A R E1 … Ei1EiEi+1 … L

… … ~for all i, then the sequence Ei must be finite.

Archimedean axiom:The can be no infinite sequences of equally likely disjoint nonnull events.

~A R E1 … Ei1EiEi+1 … L A R E1 … Ei1EiEi+1 … L

… … … …

~ … … … … we see that Ei ~ Ei+1 for all i.

Clarification: From

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Monotonicity:fg if f(s) g(s) for each state s.

Only weak form needed!

That’s all. Outcomes can be general; only richness on state space.

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Theorem [generalizing Savage ’54]. Assume “nondegeneracy“ and solvability, with acts measurable w.r.t. a general algebra of events. Then

Subjective expected utility holds

if and only if:

(i) Weak ordering;(ii) Monotonicity;(iii) Archimedeanity;(iv) Likelihood consistency.

Further, P is unique; U is unique up to unit and origin.

25More general than Savage in a structural sense:

Every structure satisfying Savage’s axioms also satisfies our axioms. Not vice versa.

We can handle:general algebras (Kopylov earlier & more general here).Equally-likely finite state spaces (with atoms); and no richness in outcomes contrary to Chew & Karni’94, Grodal’76/’03, Gul’92, Wakker’84..If atomless, then still no convex-rangedness of P.

We are not more general in a “logical” sense. Savage’s theorem is not a corollary of ours. Axioms per se are logically independent.

26Strange thing: Where did cardinality get in???Outcome-oriented approaches always have to do something extra (linearity, mixtures, midpoints, tradeoffs, ...) to get cardinality in. We didn’t.Likelihood consistency seems to concern only ordinal comparisons of likelihood.How come??

Explanation:Likelihood consistency is the dual version of tradeoff consistency for outcomes. The latter concerned differences of utility.Our condition concerns differences of probabilities (or, later, capacities). Those can be restricted to nested events, after which the differences reduce to decision weights, i.e. singular empirical concepts.

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How use likelihood revelations for nonEU?

For those of you who like nonEU:Every violation of SEU can be signaled through a contradictory likelihood revelation. We can classify the contradictions, exclude the ones we don’t like (say the comonotonic violations) and allow for others and, thus, get characterizations and measurements of all those models.

So, for generalizing SEU: we have to restrict the permitted likelihood revelations.

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4. Violations of Likelihood Consistency and Rank-Dependence

Yet another axiomatization of rank-dependence!?Well-known; so novelty of our general measurement instrument for uncertainty will be clearer.You can judge it on its didactical merits, i.e. how easy it is.If you don’t know rank-dependence yet: A very simple explanation is coming up.We replace comonotonicity restrictions by “ranking position” conditions.

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Example. Ellsberg paradox. Urn:

$ $ 0 $ $ 0

B Y R$ 0 0 $ 0 0

R Y B

30 R

$1+ $ 0 $ $ 0~

B Y R$1+ 0 0 $ 0 0

R Y B

We can get >0 s.t.:

Reveals Y Y. Signals that there is a problem! This and Y ~ Y: likelihood consistency is violated.Signals that SEU is violated.

60 B/Y (unknown proportion)

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We investigate more closely what is going on,from perspective of rank-dependence.Familiar to many of you. Therefore, suited to demonstrate the applicability of revealed likelihood, and uncertainty-orientedness.

3130 R 60 B/Y

$ $ 0 $ $ 0

B Y R$ 0 0 $ 0 0

R Y B

Left acts: Y ranked worse than B.Right acts: Y ranked worse than R.

Decision weight of Y depends on ranking position. Y “adds” more to B (giving known probability) than to R.This may explain why Y is weighted more for left acts than for right acts.Basic idea of rank-dependence: the weight of an event can depend on its “ranking position,” i.e. the event yielding better outcomes.Notation: YB YR.

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In general:Because weight of A depends on its ranking position, we should specify ranking position and write AD instead of A. AD is ranked event.Let us reconsider revealed likelihoods from this perspective.

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?

~

Def. If indifference ~ then: AD ~ BD’;

~

Def. If stict preference for right gamble then: AD BD’.

Def. If stict preference for left gamble then: AD BD’;

AD is revealed more likelythan BD’.

Again, the concept of study should be ranked events AD, and not just events.Our relation directly reveals decision-weight orderings of ranked events!

D A Lf f

f f

D’ B L’g g

g g

increasingly preferred outcomes

increasingly preferred outcomes

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Ellsberg is fine now. Contradictory Y Y has been replaced bynoncontradictory YB YR; (YB) > (YR).

Lemma. Under Choquet expected utility:AD BD’ (AD) > (BD'

), i.e. || || W(AD) > W(BD') – – W(D) W(D')

AD ~ BD’ (AD) = (BD') AD BD’ (AD) < (BD')

Let us reconsider likelihood consistency, now for ranked events.

g’ ’ g’f’ ’ f’ increasingly preferred outcomes

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likelihood consistency: [A ~ B and A B ]cannot be. Necessary for expected utility.

Directly in terms of preferences:For all , ' ', f, f ', g, g‘,

Rank-dependentD’D DD’

increasingly preferred outcomes

f f ~ g gg g

D A Lf f

D' B L'~

g’ ’ g’f’ ’ f’ ~

Choquet

NOT

Further, P is unique; U is unique up to unit and origin.

Theorem [generalizing Savage ’54]. Assume “nondegeneracy“ and solvability, with acts measurable w.r.t. a general algebra of events. Then

Subjective expected utility holds

if and only if:

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(i) Weak ordering;(ii) Monotonicity;(iii) Archimedeanity;(iv) likelihood consistency.

Gilboa ‘87

Choquet

Rank-dependent

W

More general than Savage in structural senseand also in logical sense:His assumptions and axioms directly imply ours.

37Gilboa ‘87

Not vice versa.

We can handle:general algebras (Kopylov earlier & more general here).Equally-spaced finite state spaces (with atoms).If no atoms, then still no convex-rangedness of P.

We are not more general in a “logical” sense. Savage’s theorem is not a corollary of ours. Axioms per se are logically independent.

W

Likelihood consistency aims to popularize Gilboa's P2*.

Applications for Rank-Dependent Models

Rank-dependent revealed likelihood directly observes orderings of decision weights, and is, therefore, a useful tool for analyzing properties of Choquet expected utility.Better-suited than earlier tools because uncertainty-oriented.

Quantitative measurements of capacity W:Take "equally likely" partition A1,...,An such that Ai

A1…Ai-1 ~ A1b for all i.

All have decision weight 1/n.Such equally likely ("uniform") partitions could not be defined easily heretofore.

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Testing qualitative properties of capacity W:

Convexity of W falsified ifAD ADE.

Characterizing qualitative properties of W:

W is convex iffnever AD ADE.

W2 more convex/pessimistic/amb.av. than W1 if (AF)D ~1 ADE (AF)D 2 ADE;

etc.

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Applications to Other Studies of Ambiguity

Machina & Schmeidler '92, probabilistic sophistication.Let .

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A B R B A R

h h

A B R B A R

' ' h'' ' h' h h ~ ' ' h'' ' h' ~

Not:

A ms B A ms B

Epstein & Zhang 2001:T is linearly unambiguous if, for all ,

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g f

T R T R g f

NOT:

T T I.e., NOT:

Same for Tc

T A B R T A B R

h h h h

NOT:

Epstein & Zhang 2001:T is unambiguous if, for ,

T T if measured under … specify the restrictions such and such …

I.e., not:

Same for Tc

Conclusion

For studying uncertainty, uncertainty orientedness seems to be optimal.Revealed likelihoods: general tool for measuring uncertainty. Everything becomes nicer: preference axiomatizations; quantitative measurements; testing and characterizing qualitative properties; etc.We showed some applications: Generalizing and simplifying Savage’s ’54 SEU. Generalizing and simplifying Gilboa’s ’87 CEU. Quantitative measurements of capacities. Transparent characterization of convex capacities and more-convex-than. New interpretations of probabilistic sophistication of M&S’92, unambiguity of E&Z’01.

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