Expected utility within a generalized concept of probability - a comprehensive framework for decision making under ambiguity

Transcription

1 Statistical Papers 43, 5-22 (2002) Statistical Papers 9 Springer-Verlag 2002 Expected utility within a generalized concept of probability - a comprehensive framework for decision making under ambiguity Thomas Augustin Department of Statistics, University of Munich, AkademiestraBe 1, MUnchen, Germany (e-maih thomas@stat.uni-muenchen.de) Received: March 2000; revised version: July 2001 Abstract It had often been complained that the standard framework of decision theory is insufficient. In most applications, neither the maximin paradigm (relying on complete ignorance on the states of nature) nor the classical Bayesian paradigm (assuming perfect probabilistic information on the states of nature) reflect the situation under consideration adequately. Typically one possesses some, but incomplete, knowledge on the stochastic behaviour of the states of nature. In this paper first steps towards a comprehensive framework for decision making under such complex uncertainty will be provided. Common expected utility theory will be extended to interval probability, a generalized probabilistic setting which has the power to express incomplete stochastic knowledge and to take the extent of ambiguity (non-stochastic uncertainty) into account. Since two-monotone and totally monotone capacities are special cases of general interval probability, where Choquet integral and interval-valued expectation correspond to one another, the results also show, as a welcome by-product, how to deal efficiently with Choquet Expected Utility and how to perform a neat decision analysis in the case of belief functions. Key words Ambiguity - Interval probability - Decision Making - Expected Utility - Choquet Expected Utility - Imprecise Probabilities - Capacities - Lower and Upper Probabilities - Non-additive Measures - Belief Functions - Choquet Integral - Multiple Priors - Linear Programming - Ambiguity Aversion - Ambiguity Seeking.

2 6 Thomas Augustin 1 Introduction Consider the basic problem of decision analysis: One has to choose an action (treatment, investment decision,... ) from a non-empty, finite set IA = {al,..., as,...,an} (1) of possible actions. The consequences of every action depend on which one of a given non-empty set O = {~1,..., ~j,...,~m} (2) of states of nature (true disease, development of the market... ) comes true. Their utility is described by the utility function u: (Ih x O) -, 1R (a, u(a, (3) and by the associated random variable u(a) on (O, 7)o(0)) taking the values u(a, If the states of nature are produced by a perfectly known random mechanism (e.g. a lottery) described by a probability measure ~r(.) on (O, Po(O)), then it is widely accepted to choose that action a* which maximizes the expected utility ]Enu(a) = ~-'~jm 1 u(a, ~j). ~r((~j)) among all a 9 IA. As in the examples mentioned above, in most practical applications, however, the true state of nature can not be understood as arising from an ideal random mechanism. And even if so, the corresponding probability distribution will be not known exactly. There are two main directions to proceed in this situation. Since for a classical subjectivist, or Bayesian, every situation under uncertainty can be described by a single, precise probability measure lr(.), the lack of such a known random mechanism does not make any important difference to the decision maker. (S)he acts according to subjective expected utility. Then r(.) is called prior probability, and the optimal actions are called Bayes actions with respect to r(.). In contrast, from the viewpoint of an 'objectivist', now it does not make any sense at all to think of a probability on (O, Po(O)). Hence, the objectivist concludes, that the decision maker is completely ignorant about which state of nature will occur; (s)he has to act according to a criterion based on complete ignorance (like the maximin rule). It has often been complained that both ways to proceed, relying on subjective expected utility as well as acting according to a criterion based on complete ignorance, are inappropriate, because they both have to destroy the partial nature of the knowledge on the decision maker's hand. The objectivist's criteria treat partial knowledge like complete ignorance, often leading to unsatisfactory solutions. Subjective utility theory identifies partial knowledge with complete probabilistic knowledge. This conflicts, for

3 Generalized Expected Utility 7 instance, with Ellsberg's (1961) experiments, which made it perfectly clear that ambiguity (i.e. the deviation from ideal stochasticity) plays a constitutive role in decision making -- neglecting it may lead to deceptive conclusions. In the meantime substantial generalizations of the concept of probability and its mathematical formalization emerged which have the power to take ambiguity into account appropriately. Summaries of the state of the art are provided by decooman, Fine, Moral & Seidenfeld (2001), Weichselberger (2001, Ch. 1), and the Imprecise Probability Web Page (decooman & Walley (2001)). As a generalization of expected utility theory mainly the socalled Choquet expected utility received attention. (See Chateauneuf, Cohen & Meilijson (1997) and the corresponding chapters in Grabisch, Nguyen & Walker (1995) for surveys.) In this paper expected utility theory will be extended to the case where the decision maker's knowledge is described by interval probability, which provides a natural and comprehensive framework to handle ambiguity. Section 2 briefly introduces some basics of the theory of interval probability. In Section 3 a precise definition of generalized expected utility will be given. In the main part of the paper general algorithms will be developed to derive optimal actions. Firstly, in Section 4, optimal actions under strict ambiguity aversion will be considered. Then some attention is paid to the special case of two-monotone capacities and Choquet expected utility. In Section 5 the setting will be extended to the whole range of ambiguity attitudes (all degrees of ambiguity aversion as well as of ambiguity seeking). For the whole argumentation linear programming plays an essential role. On the one hand, it is powerful for efficient calculation of optimal solutions even by standard software, on the other hand, it provides valuable theoretical insights. 2 Interval Probability A very natural idea to represent ambiguity is to use interval probabilities. As the name suggests, the probability P(A) of every event A is described by an interval [L(A), U(A)] C_ [0, 1] instead of a real number p(a). 1 The difference between upper and lower interval limits provides a means to describe the amount of ambiguity -- the higher the ambiguity, the larger this difference. In the extreme case of complete ignorance (full ambiguity) the interval [0, 1] is appropriate, while under ideal probabilistic information without any ambiguity this interval shrinks to a single point. Hence, the usual formalization of probability as a real number is contained as a border case, which is only understood as appropriate for describing the ideal stochastic situation with an perfectly working random mechanism and complete knowledge on its stochastic behaviour. i For the sake of clarity, the following convention is made: throughout the paper, capital P is used for interval-valued assignments, while small p stands for classical probability.

4 8 Thomas Augustin 2.1 Basic Definitions With respect to the intended application the whole consideration is restricted here to the case of finitely generated algebra `4 based on a sample space ~2. Then, without loss of generality, ~ is finite, and `4 is the power set of ~2 = {oj1,...,ojk}. To distinguish in terminology, every probability measure in the usual sense, i.e. every set function p(.) satisfying Kolmogorov's axioms is called a classical probability. The set of all classical probabilities on the measurable space (~2, `4) will be denoted by C (I2, `4). Axioms for interval-valued probabilities P(-) = [L(.), U(-)] can be obtained by looking at the relation between the non-additive set-function L(.) and U(-) and the set of classical probabilities being in accordance with them. On a finite sample space, as considered throughout this paper, several concepts of interval probability coincide. They all are concerned with set-functions e(.):.4 -, {[L, U] l0 _< L <_ U _< 1} A ~ P(A) = [L(A), U(A)] with and A4 := {p(-) 9 C(12,.4) IL(A) _< p(a) <_ U(A), VA 9 A} # 0. (4) inf p(a) = L(A) p(.)e~ sup p(a) = U(A) VA 9 `4. (5) Such P(-), and the corresponding set functions L(-) and U(-), are called lower and upper probability (Huber & Strassen (1973)), envelopes (Walley & Fine (1982), Denneberg (1994)), coherent probability (Walley (1991)) and F-probability (Weichselberger (1995, 2000, 2001)). In game theory AJ is the core. Here Weichselberger's terminology is used, calling A/[ structure of the F-probability P (. ). Note that in case of F-probability there is a one-to-one correspondence between structure and interval limits. Therefore, in some situations, it is very helpful to think of an F-probability as a set of classical probabilities. For every F-probability, L(.) and U(.) are conjugate, i.e. L(A) = 1 - U(AC), VA 9,4. Therefore, every F-probability is uniquely determined either by L(-) or by U(-) alone. Then one obtains for its structure 2r : I f14 = {p(.) 9 C(/2,,4) I L(A) < p(a), VA 9.4} = {p(.) 9 C(/2,.4) I p(a) < U(A), VA 9.4}. (6) Here L(-) is used throughout, and ~- = (~2,.4, L(.)) is called an F-probability field. Specifying an F-probability field (/2,.4, L(-)), it is implicitly required that the conjugate set function U(.) = 1 - L(.e) describes the upper bound of the interval.

5 Generalized Expected Utility Special Cases: Two-monotone Capacities and Belief Functions In the literature two special cases of F-probability have extensively been studied. Definition 1 Two-monotone capacities and belief functions a) A set-function L(-) :.4 --* [0, 1] w/th L(12) = 1 and LiA ) < LIB), A C B, is called capacity. The corresponding set-function #(.) defined such that L( A) = ~-~SC_A#( B) is called its Moebius inverse. 2 b) A capacity L(. ) is called two-monotone 3 if L i A U B) + LiA n B) > L(A) + LIB), VA, B E.4, and totally monotone (or belief function with basic probability assignment #(.)), if L (. q) z Ai > ( - 1 ) IIt+1L r (n/ A~. \~ei / Accordingly, also the interval-valued assignment P(') = [Li'),U(')] with U(A) = 1 - L(AC), A E `4, will be called two-monotone probability and totally monotone probability, respectively. Totally monotone capacities are the basic entity of the Dempster-Shafer theory of belief functions (e.g., Shafer (1976), Yager, Fedrizzi and Kacprzyk (1994)), which are characterized by the non-negativity of their Moebins inverse /z(.). T h e n / z ( A ) is understood as the weight of the evidence which is contained solely in the event A. - - Clearly, every totally monotone assignment is two-monotone. Two-monotone probabilities are special cases of F-probabilities (Huber & Strassen (1973, Lemma 2.5, p. 254)). The class of two-monotone probabilities plays an important role in robust statistics as a superstructure comprising the commonly used neighborhood models (cf. the review in Augustin (2001A, Section 4.2)). Two-monotone probabilities possess some very characteristic properties (cf. also Proposition 2), but in many situations t h e y are understood to be not general enough to provide a neat basis for describing situations under ambiguity Interval-valued Expectation, Choquet Integral Many concepts of classical probability theory can appropriately be generalized to interval probability. For decision making the notion of expectation is the most important one. Looking at the classical expectation 2 It can be shown that p(.) is uniquely defined, and that there is an explicit formula to calculate p(.), see Sharer (1976). 3 For the property of two-monotonicity m a n y different names are common. In particular, it is also called 'supermodularity' (Denneberg (1994)) or 'convexity' (Jaffray (1989)).

6 10 Thomas Augustin ]EpX = f Xdp = f p({w I X(w) > t})dt of a non-negative random variable X on (~2,,4) with respect to the classical probability p(.), two ways to proceed had been suggested. The first one replaces the additive set-function p(-) by the capacity L(.) leading to the Choquet integral (or fuzzy integral) (e.g, Grabisch, Nguyen & Walker (1995)) of a non-negative random variable X on (12,.A) defined by IELX := L({w I X(~) > t})dt. (7) ~0 ~ There are two reasonable ways to extend this definition to random variables taking arbitrary real values (cf. the asymmetric integral or Choquet integral (in the narrower sense) and the symmetric integral in Denneberg (1994, Ch. 5 and Ch. 7)). These way of defining expectation are mainly applied to two-monotone capacities; their meaning for more general non-additive set-functions is not unanimously accepted. Another natural way to define expectation for arbitrary F-probabilities is to rely directly on the structure and to define expectation via the infimum and the supremum 4 over all classical expectations with respect to elements of the structure. Definition 2 (Expectation with respect to an F-probability field) For every F-probability field ~ = ($2,,4, L(.)) with structure j~4 and every random variable X on (~2,.A) ]E~X :--- [LIE~X,t~X] := [ inf ]EpX, sup EpX] (8) p(.)~.m p(.)e~ is the (interval-valued) expectation of X (with respect to ~). Remark 1 The results derived later on will be based on expectation in the sense of Definition 2. Note, however, that for two-monotone probabilities and belief functions, both ways of generalizing expectation are equivalent (cf., e.g., Denneberg (1994, Prop. 10.3, p.126)). Therefore, in this case, all results are Mso valid for the Choquet integral. 2.5 Some Properties of the Structure In the case of finite spaces ~2 = {wl,..., wk} considered throughout this paper, the structure can be given a simple geometric interpretation: Every classical probability p(.) is uniquely determined by the vector p({wl}),..., p({wk}) of probability components of the singletons wl,..., wk. Therefore, every classical probability may be identified with a point in [0, 1] k, and every structure with a certain subset of [0, 1] k. 4 It can be shown that indeed one has minimum and maximum. Note further that, because Y~ is finite, integrability is trivially satisfied.

7 Generalized Expected Utility II Proposition 1 (Extreme points) The structure.h4 of an F-probability field is (isomorphic to) a convex polyhedron. In particular, the set E(Jt4) of the extreme points (vertices) is non-empty, finite, and it uniquely determines A~. Proof: Shapley (1971, p. 16), Weichselberger (2001, Ch. 2). [] This relation to convex polyhedra makes linear programming a power~ ful tool for working with interval probability. Weichselberger (1996) gives characterizations of F-probabilities in terms of linear programs. By linear programming also efficient algorithms are available to calculate the structure f14 and E(f14) for a given F-probability field. For two-monotone probabilities an alternative, closed form can be derived which additionally gives an elegant characterization of two-monotone probabilities. Proposition 2 Let J: = (12,.4, L(. )) be an F-probability field with structure M4, and denote the set of all permutations of {1,...,k} by T. The lower interval limit L(.) is a two-monotone capacity iff E(A4) = {pc(.) I r E T} withpr = L(U' j=l {we(j) }) - L (U j=l '-1 {w~(i)}), for aui= 1,..., k. Proof: It is easy to check that defining L(.) := minp(.)ez(~) p(.) leads to a two-monotone capacity. For the other direction compare, e.g., Dempster (1967, p ) and Shapley (1971, p.19). [] Given a random variable X, calculating 1E~X means to maximize or minimize the expression ~-~,~ea X(w). p({w}) over all p(.) E M/I, i.e. to calculate extremes of a linear function on a convex polyhedron. This can easily be done by linear programming. From linear programming one also knows that it is sufficient for the calculation to concentrate on the extreme points C(A4) of the structure f14. 5 Lemma 1 Let ~ = (12; ~4; L(.)) be an F-probability field with structure and extreme points E(f14). Then f ] IE~X min F, px; max IEpX (9) [P()ez(~) p(.)ez(~) J In the decision theoretic setting the following immediate conclusion will become very important. Corollary 1 (Reduction lemma) In the situation of Lemma 1, for every real g, LIE~ X >- g ", ~." ]EvX >- g, Vp(.) E ~(J$4). (10) 5 This result can be generalized to arbitrary measurable spaces (f~, ~4), with I"2 infinite, by introducing an additional continuity condition which guarantees compactness in an appropriate weak* topology, cf. Augustin (1998, Satz 2.12, p. 77).

8 12 Thomas Augustin 3 The Generalized Decision Theoretic Framework 3.1 Some Preliminaries In this paper the decision problem as described in the Relations (1) - (3) will be analyzed. For brevity of reference, the relevant components, the set IA of actions, the set O of states of nature and the precise utility function 6 u(.), are collected in the triple (IA, O, u(.)) which is called basic decision problem. For many applications it will prove helpful to extend the problem by allowing for randomized actions. Formally, every randomized action can be identified with a classical probability 7 )~(-) on (IA, :Po(IA)) where 1({a}), a E IA, is interpreted as the classical probability to perform action a. The set of all randomized actions will be denoted by A(IA). Pure (or unrandomized) actions, i.e. elements a of IA itself, are identified with the Dirac measure in the point {a}, and therefore are also understood as elements of A(IA). The utility function is extended in a canonical way to randomized actions, i.e. to the domain A(IA) O, by u(.h, vqj) := ~-]~'~=z )~(as). u(as, v~j). Analogously to above, u(1) is understood as that random variable which gives the utility of )~ in dependence on the true state ~. 3.P Generalized Expected Utility Now all the ingredients are prepared needed to cast the extended decision theoretic framework where the ambiguous prior knowledge is described by interval probability. With this generalization at hand, it becomes possible to formulate the decision maker's knowledge in a substantially more flexible way. The partial character of the prior knowledge can be expressed appropriately and need no longer be denied; the constitutive role ambiguity plays is reflected in the model. With the consideration in Section 2.3 on interval-valued expectation one immediately obtains the basic elements of a generalized decision theory. Definition 3 (Generalized expected utility) Given the basic decision problem (IA, O, u(.) ) and an F-probability II (.) on (IA, 7Po(IA) ) with structure rid. For every pure action a E IA and for every randomized action )~ in A(IA), IE~u(a) and ]E~u(1) are the generalized expected utility (with respect to the prior II(.)). 6 Throughout the paper it is assumed that a (precise) utility function is given. On the construction of utility functions in the presence of ambiguity, generalizing the Neumann Morgenstern approach, see Jaffray (1989) and Schmeidler (1989). 7 This paper argues in favor of generalized probabilities. Hence, it may appear reasonable, at a first glance, to consider also interval-valued randomization probabilities. Though, formally, this would be possible without the need of substantial modifications, it is not quite coherent with the theory, because the concept of randomization is based on ideal random experiments, for which classical probabilities are still understood as appropriate.

9 Generalized Expected Utility 13 Note that ]E~u(a) and le~u(a) are interval-valued quantities. In most cases, comparing the generalized expected utilities of actions directly will lead only to partial orders on IA and A(IA). But typically, in decision analysis, a linear (complete) order of actions is desired. This can be achieved by choosing an appropriate representation, i.e. by a mapping from IR IR to lit which evaluates intervals by real numbers to make use of the natural order on ]R for distinguishing optimal actions. The representation is strongly connected with the attitude towards ambiguity; the more ambiguity averse the decision maker is, the higher is the influence of the lower interval limit of generalized expected utility. In Section 4 and 5 calculation of optimal actions and their properties will be discussed for two important representations. These results are based on the characteristic properties of the structure of F-probabilities and of two-monotone probabilities collected in Section Optimal Actions under Strict Ambiguity Aversion 3.1 The Criterion Under strict ambiguity aversion every action is evaluated by its minimal expected utility over the structure; therefore, the interval-valued expectations [LIE~u()Q, VIEjvtu(A)] are represented by the lower interval limit alone. An action ),* is optimal iff LIEA4u(A*) >_ L]EA4u()~), VA 9 A(IA). (11) This criterion corresponds to the Gamma-Minimax criterion (e.g. Berger (1984, Section 4.7.6), Vidakovic (2000)), to the Maxmin expected utility model (Gilboa & Schmeidler (1989)), and to the MaxEMin criterion considered by Kofler & Menges (1976; cf. also Kofler (1989) and the references therein.) In the case of two-monotone capacities it is equivalent to maximizing Choquet expected utility. Remark P It should be noted that the criterion considered here contains the two classical decision criteria as border cases. If there is perfect probabilistic information and therefore no ambiguity, then A4 consists only of one single classical prior probability ~r(.), and (11) coincides with Bayes optimality with respect to ~r(.). On the other hand, in the case of completely lacking information, II(B) = [0, 1], for every B 9 Po(Y2) \ {0, O}, ('non-selective or vacuous prior'). Then one can show that LIEA4u(,k ) = minje{1... m} u(a, zgj), and (11) leads to the maximin criterion. 4.2 Optimal Actions - Calculation and Basic Properties Searching for an optimal action means to maximize m LIEA4u(A) = rain ~-~ (u(a;zgj). 7r({tgj})) (12)

10 14 Thomas Augustin among all A(-) E A(IA), i.e. to solve min (u(as;tgj)a(aa)). lr({zgj}) --* max ~r(-)ea4 ~ s (13) under the constraints -~A(as) = 1, A(as) ~ O, s = 1,...,n. (14) s----1 This originally non-linear problem can be made linear by introducing an auxiliary variable g for LIE~.tu(A): The maximization in (13) is equivalent to g --* max under the additional constraints (u(as, >g, w(-) (15) Now both, the objective function as well as the constraints, are linear in a finite number of variables, A(al),..., A(as),..., A(an) and g. Nevertheless this is no linear programming problem in the usual sense. Except in the case of a classical prior probability, the structure A4 consists of non-countable many elements. Hence, in the way as it is written, (15) contains an infinite number of constraints. At this point the reduction lemma (Corollary 1) becomes very valuable: it states that Relation (15) is equivalent to requiring the constraints )-~jrn----1()-~:----1(u(as,~j). A(as)). ~l'({tgj})) ~ g only for ah lr(.) in the set s of extreme points of f14. Since, according to Proposition 1, the cardinality of the set s is finite, the original decision problem was successfully transformed into a linear programming problem. This result is of double interest. Firstly, it means that the calculation of optimal actions can be done by standard numerical routines even implemented in most software packages. Secondly, it allows to apply general results from the theory of linear programming to the decision problem under consideration. Summing up one obtains Theorem 1 Consider a basic decision problem (IA, O, u(.)), an F-probability H (. ) on ((9, 7~ o( 0 ) ) with structure All, and the linear programming problem under the constraints (14) and g ---}max (16)

11 Generalized Expected Utility 15 a) Let (A*({al}),...,A*({an}),g*) be an optimal solution, then the randomized action s ~'(.) = (A'({al}),...,)~*({an}) is maximizing generalized expected utility with respect to the prior 1I(.) and the criterion (11). Furthermore, LIE~u()~*) = g*. b) Every action,k*(.) maximizing generalized expected utility uniquely corresponds to an optimal solution of this optimization problem. c) There always exists an action maximizing generalized expected utility. d) The set A* of actions maximizing generalized expected utility is convex and compact. Proof: Part a) and b) directly follow from the considerations preceding the theorem. Part c) and d) are applications of general results from linear optimization: With respect to Part c) it is important that for the existence of an optimal solution of a bounded linear programming problem the existence of an arbitrary admissible solution not contradicting the constraints is sufficient. (A0({al}),...,A0({an},g0) with A0({al}) = 1, A0({as}) = 0, s ,...,n, and 90 = minje{1... m} u(al, Oj) is such an admissible solution. Part d) uses the fact that the set of optimal solutions (and therefore also the projection on the first n-components) is convex and closed. Since A(IA) is bounded, A* is bounded and therefore, by Heine and Borel's Theorem, compact. Remark 3 If one explicitly wants to concentrate on pure actions a E IA, Theorem 1 still is useful for efficient calculation of optimal solutions: it then can be interpreted as a (partially) Boolean optimization problem where the variables A({al},. :.,A({an}) are only allowed to take the values zero or one. 4.3 Special Cases: Two-monotone Capacities and Choquet Integral Of particular interest is the special case where the decision maker's prior knowledge can be expressed by two-monotone or totally monotone capacities (belief functions). Then, following Remark 1, the lower interval limit of generalized expected utility in the sense of Definition 2 and the Choquet integral (cf. Relation (7)) coincide, and actions maximizing generalized expected utility are also maximizing Choquet expected utility. Note further that, according to Proposition 2, it then becomes possible to describe the elements of s in closed form. Therefore, Theorem 1 immediately offers a very efficient procedure to calculate optimal actions. Corollary 2 Consider a basic decision problem (Ilk, (9, u(.) ), and an F-probability Hi. ) = [L(-), U(.)] on (O,:Po(~)) with structure )~4 where L(.) is a two-monotone capacity. Let T be the set of all permutations of {1,..., rn}. s Here, and later on, every element,k(.) of A(IA) is identified with the vector (~({al)),..., ~({a,}))'.

12 16 Thomas Augustin a) The following three statements are equivalent: i) ~*(.) is maximizing generalized expected utility with respect to the prior II(.) and the criterion (11). ii) )r is maximizing Choquet expected utility with respect to L(.), i.e., lelu()~*) _> IELU(A), for all ~ 9 A(IA). iii) A*(.) corresponds to an optimal solution (A*({al}),...,A*({an},g*) of the linear programming problem g"* max under the constraints (14) and (0,) -(U,)) V~ E :F. j:l s=l 1: b) If p(.) is the Moebius inverse of L(.), then the relation above can be written as m n ~-'~-~u(a,,#i).a(a,). E #(muzgr V~eT. (17) j:l s=l j-1 AC_UI= l t%(o Proof." Part a) of Corollary 2 immediately follows from Theorem 1 by using the special form of the extreme points (cf. Proposition 2). Additionally taking into account that ]z(.) is the Moebius inverse of L(-) yields Part b).[:] Part b) is in particular attractive for belief functions which are often directly assigned via the function ]z(.). It may be noted that (17) is also valid for two-monotone capacities but not for arbitrary F-probabilities. The Moebius inversion still may be applied to the lower interval limit of the latter, but the extreme points of the structure are no longer of the form described in Proposition Some Further Properties Actions maximizing expected utility with respect to a classical prior probability lr(-) have two characteristic properties which deserve a brief look in the generalized setting considered here. a) Essentially completeness of pure actions: For every optimal A* (.) E A(IA) there exists an a* E IA with 1E~u(a*) --- IE~u(~*). b) Compatibility with the dominance principle: Optimal actions A*(-) are not uniformly dominated by other actions A(-), i.e, there is no action A0(-) with u(a0, ~j) > u(a*, tgj) for all j = 1,..., k. The first statement does not apply to maximin actions. Since, according to Remark 2, maximin solutions are a special case, namely actions maximizing generalized expected utility with respect to the non-selective or vacuous

13 Generalized Expected Utility 17 prior, this property can not generally be satisfied under interval-valued priors: under ambiguity, introducing ideal randomness by randomization may improve the utility. The second property may be understood as a minimal requirement for optimality criteria. Admissibility in the sense of being not uniformly dominated is related to Pareto optimality: one can not improve in any state without getting worse somewhere else. Indeed, it is straightforwardly shown by contradiction that actions maximizing generalized expected utility are admissible. 5 Optimal Actions under General Ambiguity Attitudes 5.1 The Criterion Being strictly ambiguity averse is a very pessimistic attitude towards ambiguity, which inherits - even though in a weakened way - some of the counterintuitive properties of maximin solutions. Already Ellsberg had emphasized the importance of considering other ambiguity attitudes. In many decision situations it is quite reasonable to take also into account the maximal expected utility compatible with the prior. Indeed, a plenty of empirical evidence strongly suggests that many decision makers are not strictly ambiguity averse, some of them indeed are even ambiguity seeking. Such more complex ambiguity attitudes can be reflected by other representations of interval-valued expectations. A convenient and flexible choice is a linear combination (compare with Ellsberg (1961, p. 664), Jaffray (1989), Weichselberger &: Augustin (1998), Weichselberger (2001, Ch. 2.6)) which represents E~u()~) by U. LIE~u(A) + (1-77). UE~u(A). (18) The parameter ~, called caution by Weichselberger (2001), reflects the degree of ambiguity aversion: the closer 77 comes to 1, the higher is the ambiguity aversion, y = 1 corresponds to strict ambiguity aversion, 71 = 0 expresses maximal ambiguity seeking attitudes. Remark 4 This criterion again contains two criteria from classical decision theory as border cases (compare with Remark 2): If there is no ambiguity, then //(-) = lr(.) for a classical probability It(-), and optimality based on (18) again coincides with Bayes optimality with respect to lr(.). In the case of completely lacking information, (18) leads to the Hurwicz criterion with optimism parameter 1 - y.

14 18 T h o m a s Augustin 5.2 Calculating Optimal Actions To obtain actions maximizing generalized expected utility with respect to the representation (18), one has to maximize min ~-~ (u(as;v~j)a(as)).tr({v~j}) + +(1 - r/) 9 ( max s \ ' ( ) ~ M j=l under the constraints (14). Trying to make this problem linear, the s u m m a n d corresponding to LIE~u(A) can be handled as in the previous section. Though looking quite similar, the second summand, which corresponds to UlE~u(A), needs some additional care. Now, instead of the maximization of a minimum, two connected maximizations have to be performed: one has 'there exists a 7r(-) E ~ / ' instead of 'for all 7r(.) E A/I'. The direct way to deal with this is to put UIE~u(A) into the objective function. Then one introduces new variables #({z91}),..., #({0m}) describing t h a t element #(.) of the structure under which UlE~u(A) becomes maximal. The condition #(-) G A4 can be rewritten as U( U {OJ}) -> Z#({'O./})_> L( U {0i}), V,.7C_{1,...,m}. (19) iej jej iej All in all one obtains L e m m a 2 Consider a basic decision problem (IA, O, u(.) ), an F-probability 11(.) on (0, 7~o(0)) with structure A4, the caution 77, and the programming problem ~?.g + (1-7/). (u(as; Oj)A(as)). #({Og}) ---* max (20) j=l under the constraints (14), (16), and (19). A randomized action A*(.) G A(IA) is maximizing generalized expected utility with respect to the representation (18) and the prior 1-I(.), iff it corresponds to an optimal solution (x* ({al }), 99 9 ~* ({a,, }), g*, h', ~-" ({Ol } ),..., ~* ( { o,, })). Note that in (20), for 7/ < 1, A(.) and 7r(.) vary simultaneously. As a consequence, the objective function is no longer linear but bilinear. The resulting optimization problem can be solved by viewing it as a special case of a quadratic optimization problem. (See, e.g., Fromm (1975) or Bazaraa & Shetty (1979) for quadratic programming.)

15 Generalized Expected Utility 19 Surprisingly, one can go a step further: it is possible to reformulate even this problem in terms of a single linear programming problem, which again is helpful for efficient calculation by standard software as well as for deriving properties of the optimal solution. Inspired by (16) one can introduce an auxiliary variable h with the additional constraints j=l and consider the objective function -< h, e E(M), (21),7 -g + (1 - ri). h --, max. (22) Of course, this does not immediately yield an optimal action. The fact that one has 'less or equal h' in (21) lets, together with the maximization in (22), h become unbounded; every increase in h also results in a higher value of the objective function. Now, the major idea is to additionally bound h by a penalty variable z which guarantees that h satisfying (21) becomes, ceteris paribus, as small as possible. Then, the value h* of h in the optimal solution will be the smallest value being greater than or equal to ~-~.jm l ()-~sn l(u(as, ~j). )~(as)))" ~r({~j}), for all lr(.) E E(A~t). This means nothing else but that h* is the maximum of )'~jm=l()-~,n l(u(a,, ~j). A(as))))- 7r((#j )), for all Ir E ~(~4). By the reduction lemma (Corollary 1), h* is even the maximum over all 7r E rid. Combining these arguments with the considerations preceding Theorem 1 yields the following result: Theorem 2 Consider a basic decision problem (IA, O, u(.)), an F-probability II(.) on (0, go(o)) with structure rid, a caution ~1 < 1, and the programming problem under the constraints (14) and (7 "g + (1-7/). h) - z --* max z>o. a) A randomized action ~*(.) E A(IA) is maximizing generalized expected utility with respect to the representation (18) and the prior H(.), iff it corresponds to an optimal solution (~*((al)),..., ~* ({a~)), g*, h*, 0). b) Mutatis mutandis, Corollary ~ on maximizing Choquet expected utility as well as the remaining parts of Theorem 1 still hold.

16 20 Thomas Augustin 6 Concluding Remarks The paper extended expected utility theory to interval probability, a generalized probabilistic setting which has the power to take ambiguity into account and to allow for a more appropriate modeling of the decision situation. While classical decision theory can only deal with the unrealistic border cases of perfect probabilistic information or of complete ignorance, the framework developed here allows to express the decision maker's state of knowledge in a flexible way and without the need of introducing artificial assumptions. A general algorithm was derived to calculate optimal actions and to describe their properties. As a side product, applying them to the special case of two-monotone capacities, the results also provide a new point of view how to handle Choquet expected utility efficiently. The mathematical main arguments are based on the fact that the structure associated with the interval-valued prior is a convex polyhedron. Hence, most of the results presented should carry over to those approaches which work with more general types of sets of classical probabilities (in particluar imprecise probabilities (with a finite domain) in the sense of Walley (1991) or linear partial information (cf. Kofler (1989)). Beside this aspect three topics of further research appear to be of special interest. Firstly, the framework developed has to be extended to incorporate sampling information. This can be done, just as in classical decision theory, by considering data-dependent decision functions. Some first results on this topic can be found in Augnstin (2001B). Secondly, duality theory should be studied carefully. In the very special decision problem of hypotheses testing, duality theory for linear programming provides valuable insights leading to an extension of the generalized Neyman Pearson lemma (Augnstin (1998, Ch. 5.1f)). Applying similar techniques here promises to be very helpful. One will obtain a deeper understanding of the decision problem, mainly by characterizing least favorable elements of the structure of the interval-valued prior. Finally, because one source of ambiguity is the aggregation of (expert) opinions, further work has to clarify the implications of the results presented here for group decision making. Acknowledgements I am indebted to two anonymous referees, Dieter Denneberg and Kurt Weichselberger for many very valuable comments. I am also grateful to participants of the Workshop on Risk, Uncertainty and Decision (Paris, 2000) and to members of the research group 'Making Choices' (ZIF, Bielefeld, 1999/2000) for inspiring discussions.

17 Generalized Expected Utility 21 References 1. Augustin, T. (1998): Optimale Tests bei lntervallwahrscheinlichkeit. Vandenhoeck & Ruprecht, GSttingen. (In German, with an English summary on pages ) 2. Augustin, T. (2001A): Neyman-Pearson testing under interval probability by globally least favorable pairs - reviewing Huber-Strassen theory and extending it to general interval probability. To appear in: Journal of Statistical Planning and Inference. 3. Augustin, T. (2001B): On decision making under ambiguous prior and sampiing information. In: decooman, G.; Fine, T.L.,; Moral, S., Seidenfeld, T. (eds.): ISIPTA 01: Proceedings of the Second International Symposium on Imprecise Probabilities and their Apphcations. Cornell University, Ithaca, New York; c.f. also htzp://decsai.ugr.es/~smc/isiptaol/proceedings/index. html 4. Bazaraa, M.S.; Sherry, C.M. (1979): Nonlinear Programming. Theory and Algorithms Wiley. New York. 5. Berger, J.O. (1984): Statistical Decision Theory and Bayesian Analysis. (2nd edition). Springer. New York. 6. Chateauneuf, A.; Cohen, M.; Meilijson, I. (1997): New tools to better model behavior under risk and uncertainty: An overview. Finance. 18, decooman, G; Fine, T.L.; Moral, S.; Seidenfeld, T. (eds.): ISIPTA 01: Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications. Cornell University, Ithaca, New York; c.f. also http ://decsai. ugr. es/- smc/isiptao i/proceedings/index, html 8. decooman, G.; Walley, P. (eds.) (2001): The Imprecise Probability Project. http ://ippserv. rug. ac. be/ 9. Dempster, A.P. (1967): Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics. 37; Denneberg, D. (1994): Non-Additive Measure and Integral. Kluwer, Dordrecht. 11. Ellsberg, D. (1961): Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics. 75, Fromm, A. (1975): Nichtlineare Optimierungsmodelle. Ausgew~hlte AnMitze, Kritik and Anwendungen. Harri Deutsch, Frankfurt/M. 13. Gilboa, I.; Sehmeidler, D. (1989): Maxmin expected utility with nonunique prior. Journal of Mathematical Economics. 18, Grabisch, M.; Nguyen, H.T.; Walker, E.A. (1995): Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer, Dordrecht. 15. Huber, P.J.; Strassen, V. (1973): Minimax tests and the Neyman-Pearson lemma for capacities. Annals of Statistics. 1, ; Correction: 2, Jaffray, J.Y. (1989): Linear utility theory and belief functions. Operations Research Letters. 8, Kofler, E. (1989): Prognosen and Stabilitiit bei unvollstiindiger Information. Campus, Frankfurt/Main. 18. Kofler, E.; Menges, G. (1976): Entscheidungen bei unvollst~ndiger Information. Springer, Berlin (Lecture Notes in Economics and Mathematical Systems, 136). 19. Papamareou, A.; Fine, T.L. (1991): Unstable collectives and envelopes of probability measures. Annals of Probability. 19,

18 22 Thomas Augustin 20. Schmeidler, D. (1989): Subjective probability and expected utility without additivity. Econometrica. 57, Shafer, G. (1976): A Mathematical Theory of Evidence. Princeton University Press. Princeton. 22. Shapley, S. (1971): Cores of convex games. International Journal of Game Theory. 1, Smithson, M. (1999): Human judgement research on imprecise probabilities -- a reference list distributed at ISIPTA99. Division of Psychology, The Australian National University. 24. Vidakovic, B. (2000): F-minimax: A paradigm for conservative robust Bayesians. In: Insua, D.R., and Ruggeri, F. (eds.) Robust Bayesian Analysis. Springer. 25. Walley, P. (1991): Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London. 26. Walley, P.; Fine, T.L. (1982): Towards a frequentist theory of upper and lower probability. The Annals of Statistics. 10, Weichselberger, K. (1995): Axiomatic foundations of the theory of intervalprobability. In: Mammitzsch, V., Schneeweifl, H. (eds.) Symposia Gaussiana Conference B. de Gruyter, Berlin; Weichselberger, K. (1996): Interval-probability on finite sample-spaces. In: Rieder, H. (ed.) Robust Statistics, Data Analysis and Computer Intensive Methods. Springer, New York; Weichselberger, K. (2000): The theory of interval-probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning. 24, Weichselberger, K. (2001): Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung L IntervaUwahrscheinlichkeit als umfassendes Konzept. Physika, Heidelberg. 31. Weichselberger, K.; Augustin, T. (1998): Analysing Ellsberg's Paradox by means of interval-probability. In: Galata, R.; Kiichenhof, H. (eds.): Econometrics in Theory and Practice. (Festschrift for Hans Schneewei]3.) Physika. Heidelberg; Yager, R.R.; Fedrizzi, M.; Kaeprzyk, J. (eds.) (1994): Advances in the Dempster-Shafer Theory of Evidence. Wiley. New York.