Partially Ordered Preferences in Decision Trees: Computing Strategies with Imprecision in Probabilities

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1 Partially Ordered Preferences in Decision Trees: Comuting trategies with Imrecision in Probabilities Daniel Kikuti scola Politécnica University of ão Paulo Fabio G. Cozman scola Politécnica University of ão Paulo Cassio P. de Camos Pontifícia Universidade Católica PUC-P Abstract Partially ordered references generally lead to choices that do not abide by standard exected utility guidelines; often such references are revealed by imrecision in robability values. We investigate five criteria for strategy selection in decision trees with imrecision in robabilities: extensive Γ-maximin and Γ-maximax, interval dominance, maximality and -admissibility. We resent algorithms that generate strategies for all these criteria; our main contribution is an algorithm for - admissibility that runs over admissible strategies rather than over sets of robability distributions. Introduction A rational agent is often exected to comly with strict guidelines concerning decisions: acts are encoded as functions from states to consequences, consequences are measured by utilities, and utilities are weighted by robabilities. The rational agent is then assumed to have a comlete order that ranks all acts: any two decisions can be comared, and either one is better than the other, or the two are equivalent. Preferences are then revealed by the agent s consistent attern of choice among acts [amuelson, 948]. In this aer we wish to exlore situations where references are artially ordered: given two acts, the agent may refer one to the other, or find them to be equivalent, or find them to be incomarable. In this aer we want to restrict attention to models that assume a unique utility function (u to a linear transformation) but that contemlate nonunique robability values as a source of artially ordered references [Jaffray, 999; Machina, 989; eidenfeld, 24; Walley, 99]. Imrecise beliefs may arise from an incomlete understanding of a decision situation, from lack of rior knowledge or emirical data, from disagreements between exerts, or from lack of resources for a comlete elicitation rocedure [Walley, 99]. A gradual assessment of references may in fact create intermediate models that are incomlete but that are still useful for decisions [Wang and Boutilier, 23]. Whatever may be its origin, imrecise beliefs reresented by a set of robability measures lead to artially ordered references because each robability measure in the set may create a different comlete ordering among acts: the intersection of these comlete ordering is the agent s artial ordering. Thus our agents exress their artially ordered references by recise utilities and imrecise beliefs. Consider first the static scenario where the agent must select a single act. The agent may select an act that maximizes the minimum exected utility. This solution is called a Γ-maximin one [Berger, 985; Gilboa and chmeidler, 989]. An alternative solution is to find a set of acts such that any act in the set is an otimal act with resect to at least a robability measure in the set of ossible robability measures. uch acts are called -admissible [Levi, 98]. Other solutions, such as maximality and interval dominance, can be found in the literature, and there is considerable debate on which solution should be adoted in ractice [eidenfeld, 24; Troffaes, 24]. In this aer we focus on the more comlex dynamic situation, where a decision tree reresents a sequential decision roblem with imrecise robabilities. We resent algorithms for comuting strategies under extensive Γ-maximin and Γ-maximax, interval dominance, maximality and - admissibility we develo these algorithms within a multilinear rogramming framework. Our main contribution is an algorithm for -admissibility whose comlexity deends essentially on the number of admissible strategies thus avoiding a direct deendency on the otentially high comlexity of the underlying set of robabilities. ection 2 briefly reviews the basics of decision trees and sets of robabilities. ection 3 resents several algorithms that handle decision trees from different ersectives. ection 4 resents examles that illustrate our methods, and ection 5 concludes the aer. 2 Decision trees and credal sets A decision tree reresents a sequential decision roblem using nodes (choice, chance and value nodes) and arcs between nodes [Raiffa, 968]. Arcs indicate ossible decisions (when coming out of choice nodes) or ossible states (when coming out of chance nodes). A chance node is associated with robability values, and value nodes are associated with utility values. An obvious solution method for solving a standard decision tree is by comlete enumeration of all strategies (equivalent to the reresentation of games in normal form [Luce

2 D 2.a 2.b 3 C C C q q q q q q D 2 D a 3.2b a 3.2b Figure : Decision tree for xamle. and Raiffa, 957]). xlicit enumeration clearly becomes infeasible for large roblems. A better aroach is to take the rincile of dynamic feasibility : to assess sub-strategy s at a choice node n, one has to anticiate how one will choose at (otential) future choice nodes n, and declare infeasible all future alternatives under s which are inadmissible at n [eidenfeld, 988]. xamle Figure deicts a sequential decision roblem, adated from [eidenfeld, 24]. The chance node C is the toss of a fair coin with robability of.5 for heads or tails (q = q =.5). uose that is a recise robability value, say.25. At the initial choice node, there are four otions. We must ay.4 utiles to take, 2a or 2b, and we ay.35 utiles to take decision 3. At D we have twelve strategies to evaluate. Using dynamic feasibility, we only have to evaluate four strategies (, 2a, 2b,(3, 3.b, 3.a)). In this aer we only consider methods that emloy dynamic feasibility. ome solutions, such as Γ-maximin, roduce strategies under dynamic feasibility that are not identical to strategies roduced in the normal form (that is, when all strategies are enumerated) [eidenfeld, 24]. We focus on dynamic feasibility solutions for comutational reasons: it is imractical to enumerate all strategies as decision trees grow larger. That is, we always work with the extensive form of decision trees. This rincile is also known as backward induction or is simly taken as the basis for dynamic rogramming In this aer we wish to study situations where a chance node is not associated with a single robability measure, but rather with intervals or sets of measures. uch reresentations encode artially-ordered references [Walley, 99; eidenfeld, 995]; here we briefly review essential concets concerning sets of robabilities. A credal set K(X) is a set of robability distributions (or measures) for random variable X [Levi, 98]. A credal set catures imrecision in robability values; given a credal set and a function f(x), one may comute lower exectations [f(x)] = inf [f(x)] and uer exectations [f(x)] = su [f(x)], where [f(x)] denotes standard exectation. Lower and uer robabilities are defined similarly [Giron and Rios, 98; Walley, 99]. A conditional credal set is obtained by alying Bayes rule to every distribution in a credal set. We adot the following definition of indeendence, usually referred to as strong indeendence: two variables X and Y are strongly indeendent when the credal set K(X, Y ) has all vertices satisfying stochastic indeendence of X and Y (that is, all vertices factorize as P (X) P (Y )) [Couso et al., 2; Cozman, 2]. In this aer we assume that all variables are categorical. We also assume that credal sets are closed and convex with finitely many vertices. Finally, we assume that any conditioning event has lower robability strictly larger than zero. 3 Algorithms for decision trees associated with credal sets In this section we resent algorithms that roduce one or several strategies for a given decision tree associated with imrecise robabilities that is, chance nodes are associated with credal sets. We start with the relatively simle extensive Γ-maximin and Γ-maximax solutions (here extensive indicates that solutions are not necessarily valid for the normal form, but they are valid in an extensive form emloying dynamic feasibility). We then consider interval dominance, maximality and -admissibility solutions all of which tyically roduce sets of strategies [Troffaes, 24]. Given our reliance on dynamic feasibility, all solutions follow a iterative lan: start with the leaves of the tree, and gradually build the artial strategies that are admissible from a certain oint on. The skeleton of the rocedure is as follows: DCIIONTR(decisiontree) for each decision node D from deth N to do 2 Aux null; 3 for each branch i of D do 4 if i links D to a choice node then 5 Aux Aux TRATGI(i); 6 else if i links D to a chance node then 7 Aux Aux COMBINATION(i); 8 else //value node 9 Aux Aux i; endif endfor 2 X.Adm CRITRION X(Aux); 3 endfor 4 return TRATGI(root of the decision tree); The function TRATGI receives a decision node and returns a list of admissible strategies. The function COM-

3 BINATION receives a chance node and makes recursively the combination of all admissible strategies available on decision nodes and value nodes. These functions simly build lists of admissible strategies as the algorithm roceeds. The function CRITRION X is a generic function that must be roerly imlemented to select all valid strategies in an array of strategies. This function is imlemented in several forms in the remainder of this section, relacing X by the aroriated criteria. Variable Aux is an array of strategies. ach strategy is defined by an array of choices; in xamle, we have strategies (3, 3.b, 3.b) and (3, 3.a, 3.b), among others. It is imortant to understand that a strategy defines a multilinear constraint as long as the robabilities that influence the strategy are imrecise. Consider for instance the strategy (3, 3.a, 3.b) in xamle. If robabilities q and are interval-valued, then the value of the strategy is the multilinear exression q + ( )( q). All algorithms resented in this aer require the comutation of uer and lower exectations for strategies or algebraic oerations on strategies; these uer and lower exectations are obtained by multilinear rogramming. To comute an uer exectation, it is necessary to maximize a multilinear function subject to whatever constraints are imosed on robability values. In some roblems it may be the case that the necessary robabilities are not directly secified and must be generated through Bayes rule. For examle, it may be necessary to maniulate P (X Y ) for variables X and Y, but the decision tree may be associated with robabilities P (Y X) and P (X) this is articularly common in influence diagrams. If P (Y X) and P (X) are recisely secified, direct alication of Bayes rule yields P (X Y ). If P (Y X) or P (X) are only secified u to constraints, then it is necessary to maniulate values of P (Y X) as unknowns and to introduce the multilinear constraints P (Y X) P (X) = P (X Y ) P (Y ). Note that this constraint corresonds to Bayes rule. To solve multilinear rograms, we have used herali and Tuncbilek s Reformulation-Linearization (RL) method [herali, 992] in our imlementation. The RL method substitutes each roduct of variables j J rt θ j by a new artificial variable ϑ Jrt for all terms t in the roblem, thus obtaining a linear rogram. The solution of each linear roblem gives an uer bound to the solution of the multilinear roblem. The method iterates over the variables by branching over their ranges whenever necessary, until each ϑ Jrt is close enough to j J rt θ j. We have used the same method for comutation of uer and lower conditional robabilities in multivariate models with remarkable success [Camos and Cozman, 24]. 3. Γ-Maximin and Γ-Maximax The Γ-maximin criterion selects the strategy with highest lower exected value a essimistic solution [Berger, 985; Gilboa and chmeidler, 989]. In a sequential setting, the extensive Γ-maximin solution is to take, at each choice node, the Γ-maximin solution at that oint (this may be different from the normal form) [Jaffray, 999; eidenfeld, 995]. The Γ-maximax criterion selects the strategy with highest uer exected value certainly a very otimistic solution [atia and Lave, 973]. This extensive form of Γ-maximin leads to a rocedure that selects a single strategy for any given set of strategies. Note that there may be several strategies with the same highest lower exected value, but these are all equivalent for this criterion. The resulting algorithm is comutationally simle and similar to solution of standard decision trees; the only difference is that the comutation of a lower exectation requires multilinear rogramming. CRITRION Γ-MAXIMIN(Aux) null; M ; 2 for each s in Aux do 3 if [s] > M then 4 s; M []; 5 endif 6 endfor 7 return M; The Γ-maximax solution has the same structure, but instead of comaring lower exectations in line 3, we must comare uer exectations. Both the Γ-maximin and the Γ- maximax lead to a single strategy, even though the underlying references are artially ordered. 3.2 Interval dominance Interval dominance classifies admissible choices according to a strict artial ordering. The ordering is generated by airwise comarison. Given two strategies r and s, if [r] > [s], then s is inadmissible. The set of admissible strategies consists of those strategies not classified as inadmissible [Troffaes, 24]. The algorithm is quite simle, again requiring multilinear rogramming. In this algorithm we associate an attribute admissible to each strategy. CRITRION INTRVAL DOMINANC (Aux) M CRITRION GAMMA-MAXIMIN(Aux); 2 for i running over every strategy in Aux do 3 if [M] > [Aux[i]] then 4 Aux[i].admissible false; 5 endif 6 endfor 7 return All alternatives not marked as false; This algorithm avoids unnecessary comutations of uer and lower exected values: instead of conducting exlicit airwise comarisons, it uses the Γ-maximin solution to yield a linear number of multilinear rograms (linear on the number of ossible strategies). To show that the algorithm is correct, note that the choice with maximum lower exectation is always admissible according to interval dominance criterion, and the comarison of all strategies with the Γ-maximin strategy is sufficient for determine the admissible ones. 3.3 Maximality The maximality criterion is also based on airwise comarisons between strategies. Consider that a credal set reresents the imrecise beliefs of a articular roblem. A strategy r is maximal rovided that there is no strategy s such that, for each robability measure P in the credal set, the exected value P [s] is larger than P [r]. The maximality criterion

4 rescribes that any maximal strategy can be selected by a rational agent; the comutational roblem is to generate the set of maximal strategies. CRITRION MAXIMALITY(Aux) N Number of strategies in Aux; 2 for i =... N- do 3 for j = i+... N do 4 if [Aux[i] Aux[j]] > then 5 Aux[j].admissible = false; 6 else if [Aux[i] Aux[j]] < then 7 Aux[i].admissible = false; 8 endif 9 endfor endfor return All alternatives not marked as false; The algorithm CRITRION MAXIMALITY comares any air of strategies only once. If we know the uer/lower value for [s i s j ], we also know whether one dominates the other and we do not need to evaluate [s j s i ]. The term (s i s j ) refers to a multilinear exression, obtained by subtracting the multilinear exression of s j from the multilinear exression of s i, and of course retaining all constraints on robability values in these exressions. To verify whether N alternatives are admissible, the algorithm runs through at most O(N 2 ) multilinear roblems Admissibility The criterion of -Admissibility restricts the decision maker s admissible choices to those that are Bayes for at least one robability measure P in the relevant credal sets. That is, given a choice set of feasible strategies and a credal set K reresenting imrecise beliefs, the strategy s is - admissible when, for at least one P K, s maximizes exected utility [chervish et al., 23]: = arg max([s]) K If neither otion s i or s j is -admissible, then their convex combination αs i ( α)s j is not -admissible. The following fact, similar to the usual decision tree construction, is also true. Let be the set of -admissible strategies and the set of -inadmissible alternatives in a subtree D of a decision tree D. A strategy s i in D cannot be a substrategy of an -admissible strategy in D; that is, if we detect that a substrategy is not -admissible in a subtree, we can discard any strategy that contains it. At first one may think that -admissibility is qualitatively different from the revious criteria, because it does not directly comare strategies. Rather, -admissibility looks at distributions on the underlying credal set, and comares strategies for all distributions. Thus one might think that - admissibility is much more difficult to handle than the revious criteria; in fact this seems to be the existing consensus on the issue [Troffaes, 24]. However we wish to demonstrate that -admissibility can also be exressed using airwise comarisons, when one works in our multilinear rogramming framework. For each strategy s, we are interested in finding a robability distribution for which s is otimal in the standard exected utility sense. If this robability distribution exists, then s is -admissible. That is, strategy s i is -admissible if there exists a P K such that for all s j, s j s i, we have [s i s j ]. These (multilinear) constraints must all be satisfied to show that s i is -admissible; if the constraints cannot be satisfied, then s i is not -admissible. We thus obtain the following algorithm, where LR is a list of constraints roduced by airs of strategies: CRITRION -ADMIIBILITY(Aux) N Number of strategies in Aux; 2 for i =... N do 3 LR null; 4 for j =... N do 5 if i j then 6 LR LR [Aux[i] Aux[j]] ; 7 endif 8 endfor 9 Q set of all constraints on robability values lus LR; P arg max [s i] s.t. constraints on Q; if P is non-null then 2 Aux[i].admissible = true; 3 else 4 Aux[i].admissible = false; 5 endif 6 endfor 7 return All alternatives not marked as false; Lines 3 to 8 generate all constraints that are required to satisfy -admissibility of a strategy s i, and line 9 collects constraints on robabilities. Line requires the solution of a multilinear rogram. We emhasize that the whole algorithm deends on N, the number of strategies, and not directly on the number distributions in the credal sets. ven though the roerties of the credal sets certainly affect the solution of the relevant multilinear rograms, there is no need to reresent the credal sets exlicitly, or to enumerate their vertices stes that are necessary in existing methods [Troffaes, 24]. In a sense, the comlexity of credal sets is hidden within the multilinear rograms. This raises the question of how efficient can be multilinear rogramming; we mention that revious work has indicated that state-of-the-art multilinear rogramming methods can handle roblems containing hundreds of variables [Camos and Cozman, 24]. 4 xamles In this section we aly the various decision criteria to two roblems where beliefs are reresented by credal sets. We start by analyzing the examle resented in ection 2, with a small change: instead of adoting recise robability values, we take q [.4,.5] and [.25,.75]. Algorithm DCIIONTR begins evaluation at D 3. The array Aux contains three strategies (3.), (3.2a) and (3.2b). Function COMBINATION is called twice, roducing exressions: (3.2a) = +( ) and (3.2b) = ( ) +. The choice node D 2 is analogous to D 3. At D, the function COMBINATION generates the following multilinear exressions for strategies 2.a and 2.b: [2.a] = q ( + ( ) ) + ( q) ( + ( ) ) and [2.b] = q ( +( ) ) +( q) ( +( ) ). Note that uer and lower exectations are obtained through

5 the maximization and minimization of these exressions subject to and.4 q.5. At sequential otion 3, the function COMBINATION combines all admissible strategies (those returned by CRITRION X in the revious stes). The function CRITRION X is called once more and finally, the function TRATGI returns the admissible strategies. The evaluation by Γ-Maximin in our examle yields strategy (2.a) or (2.b) or (3, (3.,3.)) with a ayoff of.5 units each. This examle shows that sometimes the Γ-Maximin criterion may dislay somewhat strange behavior: while choices (3.2a) and (3.2b) are inadmissible at D 2 and D 3, their combination is the same as (2a) and (2b) which are admissible at D [eidenfeld, 24]. Using Γ-Maximax at D 2 and D 3 we have two admissible alternatives: (3.2a) and (3.2b). Using interval dominance at decision nodes D 2 and D 3, we have that all strategies are admissible, thus, at decision node D we have nine ossible combinations for the fourth decision lus the three first strategies. At node D, we obtain that strategies () and (3,(3.,3.)) are dominated. From the remaining strategies, four of them have [s] [.4,.26] (combinations between the decision (3.) and the otions (3.2a) or (3.2b)), two have [s] = [.,.4] (same choice at D 2 and D 3 ) and the last two have [s] [.,.2]. According to the maximality criterion, we also have nine ossible combinations at D for the fourth decision. The dominated strategies are (), (2a), (2b) and (3, (3.,3.)). The other eight strategies are admissible. Finally, if we use -admissibility in the sequential otion 3, the first of these (3.) is -inadmissible. Thus, at the initial node we have the first three alternatives lus the four ossible combinations for the fourth otion. Alying -admissibility on these strategies we have the first three inadmissible. Two have [s] [.,.4] (same choice at D 2 and D 3 ) and the other two have [s] [.,.2]. Consider now a second examle, the classic oil wildcatter roblem, but with robability intervals. The roblem is as follows. An oil wildcatter must decide either to or not to. The cost of ing is $7,. If the decision is to, the hole may be, or with a return of $2,, $, and $27,, resectively. At the cost of $,, the oil wildcatter could decide to take seismic soundings of the geological structure at the site. The soundings will disclose whether the terrain has no structure (almost no hoe for oil), closed structure (indication for much oil) or an oen structure (indication for some oil). Table shows conditional robabilities (as interval-valued robabilities); take that rior robabilities of the test on no structure, oen structure and closed structure are interval-valued as [.8,.222], [.333,.363] and [.444,.454]. Table : Conditional robabilities for the oil wildcatter roblem. T no [.5,.666] [.222,.272] [.25,.8] oen [.222,.333] [.363,.444] [.25,.363] closed [.,.66] [.333,.363] [.454,.625] Figure 2 shows the decision tree for this roblem. no sounding D sounding $, D2 T $7, no oen D3 D4 closed D5 $7, $7, $7, $ $27, $2, $ $ $27, $2, $ $ $27, $2, $ $ $27, $2, Figure 2: Decision tree for the oil wildcatter roblem. We solve this roblem using a criterion that roduces a single strategy (Γ-maximin) and a criterion that roduces several strategies (-Admissibility). Using Γ-maximin we start by finding the uer and lower exectations at D 2, D 3, D 4, D 5, that is, max ( i i ) and min ( i i ) subject to i =, i. The admissible strategies in these nodes are resectively: (), (), () and (). At decision node D we have just two strategies (two multilinear rograms to solve): s = {ns, d} and s 2 = {s, (d, n, d, d)}. Choosing s we obtain the exectation [2,, 32, ] and, choosing s 2 the exectation is [25, 537.9, 42, ]. Thus, according to Γ-maximin, the best otion is take the strategy s 2. According to -admissibility, the admissible strategies at D 2, D 3, D 4, D 5 are: (), () or (), () and (). At D the strategies are s = {ns, d}, s 2 = {s, (d, n, d, d)} and s 3 = {s, (d, d, d, d)}. All three are admissible, that is, these strategies can roduce a maximal exectation for some robability. 5 Conclusion In this aer we have resented algorithms for strategy generation in decision trees associated with imrecise beliefs. As such decision trees reresent artially ordered references, there are several criteria that can be used to generate strategies. The aer contributes in two ways:. It resents a multilinear rogramming framework for strategy generation. We emhasize that existing techniques for multilinear rogramming can handle roblems with hundreds of variables [Camos and Cozman, 24], thus guaranteeing that our algorithms can be alied to large roblems. 2. It resents an algorithm for -admissibility that deends essentially on the number of strategies to be comared, and not so much on the underlying credal set. Given the diversity of criteria, one may wonder whether there is a best criterion in the field. The following comments may be relevant to this question. It seems that Γ-maximin is $

6 aealing concetually, and relatively simle from a comutational oint of view but extensive Γ-maximin solutions can be incoherent in a sequential manner [eidenfeld, 24]. The Γ-maximax criterion seems too otimistic, even though it may be aroriate in some situations [atia and Lave, 973]. The other three criteria, interval dominance, maximality and -admissibility, roduce sets of strategies with increasing selectivity that is, they are rogressively more faithful to the artial order of references. In articular, -admissibiliy does reveal the artial order of references in its sets of admissible strategies. Our results show that interval dominance is linear while maximality and -admissibility are quadratic (here linear and quadratic are used informally to refer to the number and size of multilinear rograms). This is a significantly simler icture than reviously believed [Troffaes, 24]. Altogether, -admissibility emerges as a concetually elegant and comutationally feasible criterion for decision trees with imrecise robabilities. Acknowledgments The first author has a scholarshi from FAPP, Brazil. The second author is artially suorted by CNPq, Brazil. This work has also received generous suort from HP Brazil R&D. References [Berger, 985] J. O. Berger. tatistical Decision Theory and Bayesian Analysis. ringer-verlag, New York, 985. [Camos and Cozman, 24] C. P. de Camos and F. G. Cozman. Inference in credal networks using multilinear rogramming. Proc. of the 2nd tarting AI Researcher ym.,. 5 6, Valencia, 24. IO Press. [Couso et al., 2] I. Couso,. Moral, and P. Walley. A survey of concets of indeendence for imrecise robabilities. Risk, Decision and Policy, 5:65 8, 2. [Cozman, 2] F. G. Cozman. earation roerties of sets of robabilities. In Conf. on Uncertainty in Artificial Intelligence, ages 7 5, an Francisco, 2. Morgan Kaufmann. [Gilboa and chmeidler, 989] I. Gilboa and D. chmeidler. Maxmin exected utility with non-unique rior. Journal of Mathematical conomics, 8(2):4 53, 989 [Giron and Rios, 98] F. J. Giron and. Rios. Quasi- Bayesian behaviour: A more realistic aroach to decision making? Bayesian tatistics, ages University Press, Valencia, ain, 98. [Jaffray, 999] J.-Y. Jaffray. Rational decision making with imrecise robabilities. In Proceedings of st Int. ym. on Imrecise Probabilities and Their Alications, Ghent, Belgium, 999. [Levi, 98] I. Levi. The nterrise of Knowledge. MIT Press, Massachusetts, 98. [Luce and Raiffa, 957] R. Duncan Luce and H. Raiffa. Games and Decisions. Wiley, New York, 957. [Machina, 989] M. J. Machina. Dynamic consistency and non-exected utility models of choice under uncertainty. Journal of conomic Literature, 27: , 989. [Raiffa, 968] H. Raiffa. Decision Analysis: Introductory Lectures on Choices under Uncertainty. Addison-Welsey, Massachusetts, 968. [amuelson, 948] P. amuelson. Are counterfactual decisions relevant for dynamically consistent udating under nonexexted utility? conometrica, 5: , 948. [atia and Lave, 973] J. K. atia and R.. Lave, Jr. Markovian decision rocesses with uncertain transition robabilities. Oerations Research, 2(3):728 74, 973. [chervish et al., 23] M. J. chervish, T. eidenfeld, J. B. Kadane, and I. Levi. xtensions of exected utility theory and some limitations of airwise comarisons. In Proc. of 3rd Int. ym. on Imrecise Probabilities and Their Alications, ages 496 5, Lugano, witzerland, 23. [eidenfeld, 988] T. eidenfeld. Decision theory without indeendence or without ordering: What is the difference. conomics and Philosohy, 4:267 29, 988. [eidenfeld, 995] T. eidenfeld, M. J. chervish, and J. B. Kadane. A reresentation of artially ordered references. Annals of tatistics, 23(6): , 995. [eidenfeld, 24] T. eidenfeld. A contrast between two decision rules for use with (convex) sets of robabilities: Gamma-maximin versus -admissibility. ynthese, 4(3):69 88, June 24. [herali, 992] H. D. herali and C. H. Tuncbilek. A global otimization algorithm for olynomial rogramming roblems using a reformulation-linearization technique. J. of Global Otimization, 2: 2, 992. [Troffaes, 24] M. Troffaes. Decision making with imrecise robabilities: A short review. IPTA Newsletter, ages 4 7, ociety for Imrecise Probability Theory and Alications, Manno, witzerland, December 24. [Walley, 99] P. Walley. tatistical reasoning with imrecise robabilities. Chaman and Hall, Londom, 99. [Wang and Boutilier, 23] T. Wang and C. Boutilier. Incremental utility elicitation with the minimax regret decision criterion. Proc. of Int. Joint Conf. on Artificial Intelligence, ages 39 36, 23.