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The Shape of Incomplete Preferences

Robert Nau

Duke University

Presented at ISIPTA ‘03

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Background of research

• I have long studied the role of no-arbitrage (“coherence”) arguments and separating hyperplane theorems in rational choice theory (decisions, games, markets…)

• Like many, I feel that the completeness-of-preferences axiom is the most questionable postulate of normative decision theory

• When completeness is not assumed, preferences are naturally represented by imprecise probabilities and/or utilities

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Other related papers

• “Indeterminate (confidence-weighted) Probabilities on Finite Sets”, Ann. Statist. 1992

• “Coherent Behavior in Noncooperative Games”: JET 1990

• “Uncertainty Aversion with Second-Order Probabilities and Utilities”: ISIPTA ’01

• “De Finetti Was Right: Probability Does Not Exist”: Theory & Decision 2002

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Motivation of this paper

• The theory of Subjective Probability (decision theory with linear, state-independent utility) can be easily axiomatized via no-arbitrage arguments and the separating hyperplane theorem (a la de Finetti)

• The theory of Expected Utility (decision theory with objective probabilities) be axiomatized in exactly the same way—simply reverse the roles of money & probabilities

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Motivation, continued

• When completeness is dropped from the SP or EU axioms, the result is that preferences are naturally represented by convex sets of probability distributions or convex sets of utility functions (e.g., Koopman 1940, Aumann 1963)

• What is the corresponding result for Subjective Expected Utility theory (decision theory with subjective probabilities and nonlinear or state-dependent utilities)???

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Subjective expected utility

• SEU theory uses a richer set of objects of choice: “Savage acts” or Anscombe-Aumann “horse lotteries”, which map states of nature to (probability distributions over) consequences

• An additional axiom of state-independence is used to separate probabilities from utilities

• Actually, “true” subjective probability can never be uniquely separated from utility by this approach, but that’s another story…

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What happens when completeness is dropped from SEU theory??

• Are coherence arguments and separating-hyperplane theorems still applicable, and what is the “shape” of the representing set of probabilities/utilities?

• In particular, does the representation consist of a set of “probability/utility pairs” (i.e., a multi-Bayesian Pareto compromise), and what is the geometry of that set?

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Antecedent literature

• Seidenfeld, Schervish & Kadane (Ann. Statist. 1995) derive such a representation for the case of strict preferences, and argue that “the set of probability/utility pairs for the problem of two distinct Bayesians is not connected and therefore not convex. Hence, a common method of proof—separating hyperplanes—is not available…”

• Instead they use transfinite induction and “indirect reasoning” in their proof.

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This paper…

• Axiomatizes non-strict preferences among horse lotteries, minus the completeness axiom

• Shows that the separating hyperplane theorem is still applicable in this setting

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Main results

• When completeness is merely dropped from Anscombe-Aumann’s SEU axioms, the result is that preferences are represented by a convex set of state-dependent expected utilities (probabilities paired with state-dependent utilities), via the usual separating hyperplane theorem

• State-independence requires (only) that the set must contain at least one probability/utility pair

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Main results, continuedThe representation collapses to the convex hull

of a set of probability/utility pairs if one of two additional assumptions is made:

(i) A constructive axiom of “stochastic substitution” that strengthens the state-independence assumption in the presence of background risk

(ii) SSK’s “indirect reasoning” postulate: wherever a weak preference is precluded, the opposite strict preference must be affirmed (“no shades of gray”)

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Summary & conclusions

For the case of non-strict primitive preferences, Anscombe-Aumann’s axioms of SEU theory minus completeness and plus one additional condition lead easily to a representation of preferences by (the convex hull of) a set of probability/utility pairs, namely those p/u pairs that are consistent with the “basis” of the preference relation (the directly asserted preferences)

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Why is this important?

• Completeness is a normatively controversial and behaviorally unrealistic preference axiom, so it is important to explore relaxations of it.

• It’s nice if the resulting preference representation has a multi-Bayesian interpretation, as if the decision maker is “of several minds” (each of them Bayesian).

• It’s also nice if coherence arguments and separating hyperplane theorems can still be invoked, providing connections to results in other branches of choice theory.

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Technical details

• S = {1, 2, …, M) a finite set of states

• C = {0, 1, …, K) = a finite set of consequences

• X:SC→ = a horse lottery, where X(s,c) is the objective probability of receiving consequence c when state s occurs

• Hc = the horse lottery that yields consequence c with probability 1 in every state (c= 0, 1, …)

• Hp = the horse lottery that yields consequence 1 with probability p & consequence 0 otherwise (u(0,1))

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Primal representation of incomplete preferences

is a preference relation on {X}A1 (quasi order): is transitive and reflexiveA2 (indep.): X Y X+(1-)Z X+(1-)Y

A3 (continuity): Xn Yn lim Xn lim Yn

A4 (best & worst): H1 Hc H0 for all c

A5 (coherence or non-triviality): H1 > H0

Theorem 1: satisfies A1-A5 iff there exists a closed convex cone B* receding from the origin such at that X Y X-YB*

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State-dependent expected utilities and probability/utility pairs

• v:SC→ is a state-dependent expected utility (sdeu) function where v(s,c) denotes the expected utility assigned by v to consequence c in state s.

• Uv(X) = s,c X(s,c)v(s,c) is the utility assigned to X by v

• v is a probability/utility pair if it can be expressed as a product of a probability distribution and a state-dependent utility function: v(s,c) = p(s)u(c)

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Dual representation of incomplete preferences (via sep.-hyp. theorem)

• v agrees with if X Y Uv(X) Uv(Y)

• A set V of sdeu functions represents if X Y Uv(X) Uv(Y) for all v V

Theorem 2: satisfies A1-A5 iff it is represented by a non-empty closed convex set V* of sdeu functions assigning (w.l.o.g.) a utility of 0 to H0 and a utility of 1 to H1

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The state-independence axiom

• HE denotes the horse lottery yielding consequence 1 if E, consequence 0 otherwise.

• E is not potentially null if HE Hp for some p> 0.

• EX EY means “X is preferred to Y conditional on E”, i.e., EX+(1-E)Z EY+(1-E)Z for all Z.

A6 (state-independence): If X & Y are constant and not potentially null, EX EY EX E Y for any other event E

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Basic representation of state-independent preferences

Theorem 2: satisfies A1-A6 iff it is represented by a non-empty closed convex set V** of sdeu functions of which at least one element is a probability/utility pair

Note: this representation admits “shades of gray” in preferences. X Y may be precluded, yet Y > X cannot be inferred from A1-A6.

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How to eliminate shades of gray?

Indirect approach (SSK): assume that wherever a weak preference is precluded, the opposite strict preference must be affirmed.

Direct approach: strengthen the state-independence axiom so that it holds in the presence of “background risk”

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A strengthening of state-independence

A7 (stochastic substitution): If

X+(1-)(EX+(1-E)Z) Y+(1-)(EY+(1-E)Z)

for some (0,1) where X, Y and Z are constant lotteries and E is not potentially null, then

X+(1-)(pX+(1-p)Z) Y+(1-)(pY+(1-p)Z)

…for some p (0,1]

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In other words…

…the subjective mixtures of the constant lotteries X and Y with Z can be replaced by non-trivial objective mixtures against the background of a comparison between the nonconstant lotteries X and Y.

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Main theorem

Theorem 4: satisfies A1-A7 iff it is represented by a non-empty closed convex set V*** of sdeu functions that is the convex hull of a set of a probability/utility pairs

Note: the same result is obtained with A1-A6 plus indirect reasoning

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Recipe for constructing the geometrical representation

• Every preference assertion X Y determines a linear constraint Uv(X) Uv(Y) in the space of sdeu functions.

• The intersection of those constraints, over all assertions in the basis of the preference relation, is a convex polytope.

• Take the intersection of that polytope with the (nonconvex) set of probability/utility pairs.

• Take the convex hull of that intersection: this is the set V***.