Economic Theory 16, 295–312 (2000) Choice under complete uncertainty: axiomatic characterizations of some decision rules⋆ Walter Bossert1 , Prasanta K. Pattanaik2 , and Yongsheng Xu3,4 1 2 3 4 Department of Economics, University of Nottingham, Nottingham NG7 2RD, UK (e-mail: lezwb@unix.ccc.nottingham.ac.uk) Department of Economics, University of California, Riverside, CA 92521, USA (e-mail: ppat@ucrac1.ucr.edu) Department of Economics, Andrew Young School of Economics, Georgia State University, Atlanta, GA 30303, USA (e-mail: yxu3@gsu.edu) Department of Economics, University of Nottingham, Nottingham NG7 2RD, UK Received: August 20, 1998; revised version: November 3, 1999 Summary. We provide characterizations of four new rules for individual decision-making under complete uncertainty. They are what we call the minmax rule, the max-min rule, the lexicographic min-max rule and the lexicographic max-min rule. These rules provide orderings of the sets of possible outcomes associated with uncertain prospects. They provide significant alternatives to commonly-used rules that focus on worst outcomes or best outcomes only, and lexicographic versions of those rules. Keywords and Phrases: Complete uncertainty, Nonprobabilistic decision rules. JEL Classification Number: D81. 1 Introduction Consider an agent choosing an action in a situation of ‘complete’ uncertainty, where the agent knows the set of possible outcomes for each action, but has no information about the probabilities of those outcomes or about their likelihood ranking. Thus, each action is associated with an uncertain prospect represented by the set of possible outcomes corresponding to that action. How does such an agent make her choice? A number of authors have discussed this ⋆ We thank Steven Humphrey and two referees for comments and suggestions. Correspondence to: P.K. Pattanaik 296 W. Bossert et al. and related problems; see, among others, Arrow and Hurwicz (1972), Bandyopadhyay (1988), Barberá, Barrett, and Pattanaik (1984), Barberá and Pattanaik (1984), Bossert (1989a,b, 1995, 1997), Cohen and Jaffray (1980), Fishburn (1984, 1992), Heiner and Packard (1984), Holzman (1984a,b), Kannai and Peleg (1984), Maskin (1979), Milnor (1954), Nitzan and Pattanaik (1984), and Pattanaik and Peleg (1984). There are several different approaches to address this issue in the literature. For example, this choice situation can be modelled by taking into account a set of possible states of the world, and every pair of a feasible action and a state of the world leads to a specific outcome. Actions are then ranked on the basis of the vectors of contingent outcomes that they generate. An alternative approach consists of ranking actions on the basis of the sets of possible outcomes. Clearly, the latter approach involves some loss of information. For a given outcome and a given action, it only matters whether or not this outcome results in at least one state from choosing that action, whereas the former model allows the decisionmaker to take into account in how many states an action leads to this outcome. In spite of this loss of information when employing a set-based decision rule as compared to a vector-based one, there are circumstances where a ranking of sets can be considered an appropriate way to model choice under complete uncertainty. For example, if there is a large number of possible states of the world, a decision-maker may find it too complex to take into account the entire vector of possible outcomes, and restricting attention to the set of possible outcomes may allow her or him to represent the available information in a more compressed and tractable form. Furthermore, some applications of the vector-based model require the set of all possible contingencies to be partitioned in a way that can be considered quite arbitrary. Another scenario where the set-based model can be considered appropriate emerges in the context of a Rawlsian (Rawls, 1971) veilof-ignorance framework, where it may not be obvious how states and outcomes can be distinguished in a satisfactory fashion. Even in ordinary contexts, one can think of plausible situations where the decision maker may not have any notion of the possible states of the world. A student may feel that, if he takes an examination this week, he will get either a C or a B but certainly not an A. At the same time, the student may not have any idea about the states of the world under which he will get a C and the states of the world under which he will get a B . Unless one defines the states of the world to be ‘getting C ’ and ‘getting B ,’ the problem in this case cannot be posed in terms of the states of the world. On the other hand, it is not clear what will be gained if, in this fashion, one defines the states of the world in terms of the outcomes themselves. Thus, while the type of ‘complete uncertainty’ that we are modelling is even more ‘complete’ than the type of ‘complete uncertainty’ that Arrow and Hurwicz model, it does succeed in modelling some empirically interesting situations which either cannot be captured in the Arrow-Hurwicz framework or can be captured in this framework only by resorting to a somewhat artifial conception of a state of the world. For a more detailed discussion regarding the relative merits of set-based and vector-based choice models, see Pattanaik and Peleg (1984). Choice under complete uncertainty 297 This paper is a contribution to the literature on set-based rankings. In addition to a number of impossibility results, the earlier literature on the subject also contains axiomatic characterizations of several rules for decision-making under complete uncertainty. We extend this analysis by characterizing four decision rules which do not seem to have attracted much attention in the previous literature. They are what we call the min-max rule, the max-min rule, the lexicographic min-max rule and the lexicographic max-min rule. While the names of the decision rules that we characterize may seem somewhat familiar, there are important substantive differences between our rules and rules with similar names discussed in the earlier literature. For example, under our min-max rule, in comparing two uncertain prospects A and B , the agent first considers the worst possible outcome of each uncertain prospect (recall that an uncertain prospect is represented by a set of possible outcomes). If the worst outcome of one uncertain prospect is better than that of the other, then the first uncertain prospect is considered better than the second. But, if the two worst outcomes happen to be identical (we do not permit indifference between distinct outcomes), then the agent does not necessarily declare the two uncertain prospects indifferent (as in the standard ‘min’ rule) nor does she necessarily go on to consider the next-to-worst outcomes in the two sets (as in the usual ‘leximin’ rule). Instead, if the worst outcomes of A and B are identical, then the agent proceeds to compare the best outcomes of the two sets (it is, of course, possible that the best element of a set may coincide with the worst element in that set). Two uncertain prospects A and B are indifferent under the min-max rule if and only if the worst outcome of A is the same as the worst outcome of B and the best outcomes of A and B are the same. Thus, our min-max decision rule is different from the familiar rule based exclusively on the ‘min’ as well as from its lexicographic extension. Similarly, our lexicographic min-max rule differs from the ‘lexi-min’ rule. Under our lexicographic min-max rule, the agent first proceeds as under the min-max rule. However, if the worst outcomes of the two sets turn out to be identical and so do the best outcomes of the two sets, then the agent eliminates the best and the worst outcomes of each of the two sets and applies the min-max rule to the two reduced sets. Again, if the worst outcomes of the reduced sets are identical and so are the best outcomes, then the agent eliminates these outcomes and considers the remaining subsets of the two sets, and so on. If, in the process of this successive reduction, one of the original sets gets eliminated completely while some elements of the other set remain, then the latter is considered to be better than the former. Though, as far as we know, these rules have not been discussed earlier in the axiomatic literature, they have intuitive plausibility when one recognises that often people are capable of processing only a limited amount of information at a time, and, therefore, may tend to concentrate on certain ‘focal’ features of a decision situation, ignoring the rest of the information available. In Sections 3 and 4, we comment in somewhat greater detail on the intuitive justifications for our rules in terms of ‘limited rationality’ and the consequent tendency to focus 298 W. Bossert et al. on a few conspicuous features of a decision situation to the exclusion of other information available in such situations. The plan of the paper is as follows. In Section 2, we introduce our basic notation. In Section 3, we lay down some axioms and use them to characterize our min-max and max-min rules. In Section 4, we introduce some further axioms and characterize our lexicographic min-max and lexicographic max-min rules. We conclude in Section 5. 2 The basic notation Let X denote the universal set of alternatives. We assume that 1 < #X < ∞, where #X denotes the cardinality of X . K denotes the set of all non-empty subsets of X . A subset of X which has more than one element is interpreted as an uncertain prospect where the agent does not have a probability distribution (or even a likelihood ranking) over the possible outcomes. Thus, for example, {x , y, z } ∈ K is to be interpreted as an uncertain situation where the agent knows that x , y and z are the only possible outcomes but the agent does not not assign probabilities or a likelihood ranking to x , y and z . A singleton set {x } ∈ K is interpreted as a ‘trivial’ uncertain prospect with only one possible outcome, that is, as a certain prospect of having the outcome x ∈ X . K2 denotes the class {A ∈ K | #A ≤ 2}. Thus, K2 is the class of all singleton and doubleton sets in K . R is a given linear preference ordering (‘at least as good as’) over X , which is assumed to be fixed throughout our discussion (a linear preference ordering is a reflexive, transitive, complete, and antisymmetric binary relation). P denotes the asymmetric factor of R. For all x ∈ X and all A ∈ K , we write xPA to denote that xPa for all a ∈ A, and we write APx to denote that aPx for all a ∈ A. For all A ∈ K , min A and max A denote, respectively, the worst and best elements of A according to R. Note that, owing to the assumption that X is finite and R is a linear ordering, worst and best elements are well-defined and unique for all A∈K. Let  be an ordering over K (an ordering is a reflexive, transitive, and complete binary relation). This ordering is interpreted as the agent’s preference ordering over the uncertain prospects represented by the elements of K . ≻ and ∼ denote, respectively, the asymmetric and symmetric factors of . 3 Min-max and max-min rules As a preliminary result, we show that the worst and best outcomes must play an important role in the ordering , given some plausible axioms. In particular, the two axioms defined below imply that every uncertain prospect A is indifferent to the uncertain prospect where the only outcomes are the best and worst elements in A. Choice under complete uncertainty 299 Definition 1  satisfies (1.1) simple monotonicity (SM) if, for all x , y ∈ X such that xPy, {x } ≻ {x , y} ≻ {y}; (1.2) independence (IND) if, for all A, B ∈ K and all x ∈ X − (A ∪ B ), A ≻ B implies A ∪ {x }  B ∪ {x }. SM was introduced by Barberá (1977)—see also Barberá and Pattanaik (1984). This axiom requires that, if an outcome x is strictly preferred to another outcome y, then the certain prospect of having x is better than the uncertain prospect with the two possible outcomes x and y which, in turn, is better than the certain prospect of having y. Clearly, SM is a very plausible condition. IND was introduced by Kannai and Peleg (1984). It requires that, if a set of possible outcomes A is better than another set B , then adding a new outcome x to both A and B does not reverse this strict ranking. Note that the addition of x does not require the augmented sets to be strictly ranked in the same way as A and B are ranked. Thus, this formulation, by itself, does not rule out rankings based on worst elements only (maximin) or best elements only (maximax). We obtain Theorem 1 If  satisfies SM and IND, then, for all A ∈ K , A ∼ {min A, max A}. Proof. We proceed by induction. Clearly, A ∼ {min A, max A} for all A ∈ K such that #A ≤ 2. Now suppose m > 2 and A ∼ {min A, max A} for all A ∈ K such that #A < m. (1) Let #B = m and B = {b1 , . . . , bm } with b1 Pb2 P . . . Pbm . By SM, {b1 } ≻ {b1 , bm−1 }. (2) {b1 , bm−1 } ∼ {b1 , . . . , bm−1 }. (3) By (1), By (2) and (3), noting the transitivity of , we have {b1 } ≻ {b1 , . . . , bm−1 }. (4) {b1 , bm }  {b1 , . . . , bm }. (5) {b2 , . . . , bm } ∼ {b2 , bm }. (6) {b2 , bm } ≻ {bm }. (7) By (4) and IND, By (1), By SM, From (6) and (7), noting the transitivity of , we have {b2 , . . . , bm } ≻ {bm }. (8) 300 W. Bossert et al. From (8) and IND, we have {b1 , . . . , bm }  {b1 , bm }. (9) (5) and (9) together imply B ∼ {min B , max B }, which completes the proof. While the axioms SM and IND are consistent (see the main characterization result of this section), SM is incompatible with the strengthening of IND that requires a strict preference between A ∪ {x } and B ∪ {x } as a consequence of a strict preference between A and B —see Barberá and Pattanaik (1984). Because SM appears to be a very mild axiom (note that it applies to comparisons of singletons and doubletons only), we avoid this impossibility by choosing our version of the independence axiom rather than the strengthening mentioned above, as is done in most of the literature on the subject. Theorem 1 is a strengthening of Kannai and Peleg’s (1984) lemma, where they use IND together with a stronger axiom than SM to conclude that any set of uncertain prospects is indifferent to the set consisting of its best and worst elements. Bossert (1989a) shows that Kannai and Peleg’s result remains valid if the transitivity of  is weakened to quasi-transitivity. See also Barberá, Barrett, and Pattanaik (1984) for a related observation. Nitzan and Pattanaik (1984) characterize decision rules with the property that they declare any set of uncertain prospects indifferent to its set of median alternatives. Note that Theorem 1 is also reminiscent of the central theorem of Arrow and Hurwicz (1972). However, the formal framework of Arrow and Hurwicz (1972) is very different from ours insofar as they use an informationally richer structure where an uncertain prospect is a function from a set of states of the world to the universal set of outcomes—see also the discussion in the Introduction. Structurally, our Theorem 1 comes very close to the analysis of Shackle (1952, 1954). Like us, Shackle also considered the problem of decision-making under ‘complete uncertainty’ where there is no information about the states of the world. By Theorem 1, in considering alternative actions, an agent, who behaves according to a rule satisfying SM and IND, basically does not look at anything other than the best and the worst outcomes of each action. Similarly, in Shackle (1952, 1954), the agent ignores all information about an action excepting the information about the best and the worst outcomes corresponding to the action. There are however, significant differences between our result and Shackle’s decision rule insofar as, in our model, we do not have anything resembling the ‘potential surprise’ of Shackle. The behaviour that follows (by Theorem 1) from SM and IND is consistent with the notion of ‘limited rationality’ which is familiar in the theories of organization and bounded rationality (see, for example, March, 1988; March and Simon, 1958), and which suggests that, given a complex decision problem, the agent often seeks to simplify the problem by focusing on only a few salient features of the complex situation. If the agent’s ability to process information is rather limited, then it seems reasonable to assume that, in looking at an action, Choice under complete uncertainty 301 the agent will judge the attraction of that action on the basis of only a few conspicuous aspects of the information available about the action. The question then arises about the exact features of a decision situation that are likely to be conspicuous for the agent. The best and worst outcomes are certainly some prominent focal points of an action. What our Theorem 1 tells us is that a rule satisfying SM and IND will concentrate only on these two focal points to the complete exclusion of all other information regarding the action(s) under consideration. The rules introduced in this paper are, to a large extent, motivated by the result of Theorem 1 and related observations in the literature see some of the above references). As illustrated in those results, various combinations of plausible axioms imply that best and worst elements play a central role in designing set-based rankings. In this paper, we propose rules that go beyond the mere comparison of best or worst elements while retaining the importance of those elements. Especially the lexicographic versions of our rules (which will be introduced formally in the next section) can be considered refinements of such rankings: the prominent roles of the best and worst elements are preserved, but in the case of a ‘tie,’ we move on to a comparison of second-worst and second-best elements etc. rather than immediately imposing indifference. The following definition introduces two distinguished members of the class of all orderings under which every uncertain prospect A is indifferent to the set consisting of the best and worst elements of A. Definition 2 (2.1) The min-max relation mnx is defined by letting, for all A, B ∈ K , A mnx B :⇔ [(min A P min B ) or (min A = min B and max A R max B )]. (2.2) The max-min relation mxn is defined by letting, for all A, B ∈ K , A mxn B :⇔ [(max A P max B ) or (max A = max B and min A R min B )]. Under the min-max rule, when comparing two uncertain prospects A and B , the agent first compares the worst alternative in A with the worst alternative in B . If the worst alternative in one set is better than the worst alternative in the other set, then the former set is considered to be better than the latter set. However, if the two worst alternatives happen to be identical, then the agent does not necessarily declare A and B to be indifferent (as he would under the usual maximin rule); nor does he then necessarily compare the worst-but-one alternative in A with the worst-but-one alternative in B (as he would under the usual leximin rule). Instead, if the worst alternatives in A and B turn out to be identical, then he goes by the comparison of the best alternatives in the two sets. It is only when the worst alternatives of A and B are identical and so are the best alternatives of A and B, that the agent is indifferent between A and B. Thus, our min-max rule is different from the usual maximin rule and also from the usual leximin rule. Our max-min rule is the ‘dual’ of our min-max rule. Here the agent first compares the best alternatives in the two sets; if these best alternatives turn out to be identical, 302 W. Bossert et al. then the agent proceeds to compare the worst alternatives in the two sets. Again, our max-min rule is different from the usual maximax and leximax rules. The intuition of the two decision rules (namely, the max-min rule and the min-max rule) introduced in Definition 2 hinges crucially on a particular version of limited rationality that leads the agent to concentrate on some focal points in the range of outcomes corresponding to any given action, and to exclude, once and for all, from consideration all outcomes other than these focal points. However, each of these two rules combines this intuition with specific types of uncertainty attitude: from an intuitive point of view, each of these two rules appeals to the principle of limited rationality in a particular fashion, and this is then followed by an appeal to a specific type of uncertainty attitude. An example may be helpful in clarifying the sequence in which these two intuitive ingredients are brought into play in the two decision rules. Consider two sets of possible outcomes: A = {a1 , a2 , a3 } and B = {b1 , b2 , b3 }. Assume that a1 and a3 are, respectively, the best and the worst outcomes in A. Similarly for b1 and b3 in B . Suppose the principle of limited rationality, which comes into operation first, takes a form such that the agent’s focal points in a set of possible outcomes are always the best and the worst outcomes in that set to the exclusion of all other alternatives (our Theorem 1 shows that SM and IND, together, imply this specific type of limited rationality). It is then intuitively obvious that, if a1 = b1 and a3 = b3 , then the individual will be indifferent between A and B . What happens if the focus of the agent is on the best and the worst, and the set of best and worst outcomes in A is not identical with the set of best and worst outcomes in B ? In this case, under our decision rules, the uncertainty attitude of the agent becomes the crucial factor in determining the ranking of A and B . If the individual is uncertainty averse then he will compare the two worst outcomes a3 and b3 , and will rank A and B on the basis of that comparison provided a3 and b3 are distinct so that they are strictly ranked. However, if a3 = b3 , then he will proceed to compare a1 and b1 . Given our assumption that the agent is uncertainty averse, it may be asked why, when a3 = b3 , the individual does not proceed to compare a2 (the worst-but-one outcome in A) with b2 (the worst-butone outcome in B ) instead of proceeding directly to compare the best outcomes in the two sets. The intuition of proceeding straight to the comparison of the best outcomes, when the two worst outcomes turn out to be identical, lies in the very starting point of our sequence, namely, in the principle of limited rationality which led the individual to focus exclusively on the best and the worst outcomes only. Given that the focus is only on the best and the worst, it is natural to assume that when the two worst outcomes turn out to be identical, the agent proceeds to compare the remaining focal points for the two sets, namely the two best outcomes: when a3 is found to be identical to b3 , the individual does not proceed to compare the worst-but-one outcomes, a2 and b2 , in the two sets simply because, the operation of the principle of limited rationality has removed, right at the beginning, these alternatives from the scope of the agent’s attention. We would like to note that, not only do min-max and max-min rules have intuitive plausibility, but rules based on best and worst elements have also been studied Choice under complete uncertainty 303 in applied problems. For example, the selection of actions on the basis of the best and/or worst possible outcomes associated with those actions are some of the heuristics that are employed in some simulation models where, in addition to expected payoffs, criteria such as complexity consideration may play a role (see, for example, Johnson and Payne, 1985; Thorngate, 1980). It is easy to check that mnx and mxn are orderings over K and that the restrictions of mnx and mxn to K2 are linear orderings. Our next definition introduces four properties, which, together with SM and IND, will be used to characterize mnx and mxn . Definition 3  satisfies (3.1) type 1 simple dominance (SD1) if, for all x , y, z ∈ X such that xPyPz , {x , z } ≻ {y, z }; (3.2) type 2 simple dominance (SD2) if, for all x , y, z ∈ X such that xPyPz , {x , y} ≻ {x , z }; (3.3) simple uncertainty aversion (SUA) if, for all x , y, z ∈ X such that xPyPz , {y} ≻ {x , z }; (3.4) simple uncertainty appeal (SUP) if, for all x , y, z ∈ X such that xPyPz , {x , z } ≻ {y}. The intuition underlying SD1 and SD2 is straightforward. While SD1 and SD2 can be regarded as ‘rationality conditions’ for choice under complete uncertainty, SUA and SUP are alternative empirical assumptions regarding the agent’s attitude towards uncertainty in comparing ‘simple’ uncertain prospects. SUA says that, if the agent strictly prefers x to y and y to z (and, hence, x to z ), then he strictly prefers the certainty of the middle-ranking outcome y to an uncertain prospect where he may get outcome x which is better than y or outcome z which is worse than y. Thus, SUA reflects ‘uncertainty aversion’ in situations involving comparisons of such ‘simple’ prospects. Similarly, SUP reflects a liking for uncertainty in situations involving comparisons of ‘simple’ prospects. See Bossert (1997) for a more detailed discussion and characterization of uncertainty aversion and uncertainty appeal in nonprobabilistic decision models. Lemma 2 (2.1)  satisfies SM, SD1 and SUA if and only if, for all A, B ∈ K2 , [A  B ⇔ A mnx B ]. (2.2)  satisfies SM, SD2 and SUP if and only if, for all A, B ∈ K2 , [A  B ⇔ A mxn B ]. Proof. First, we prove Lemma 2.1. It can be seen easily that SM, SD1 and SUA are satisfied if  agrees with mnx on K2 . To prove the converse implication, suppose  satisfies SM, SD1 and SUA, let A, B ∈ K2 , and let a, a, b, and b denote, respectively, min A, max A, min B , and max B . Because the restriction of  to K2 is an ordering, it is sufficient to show that A ∼mnx B ⇒ A ∼ B and (10) 304 W. Bossert et al. A ≻mnx B ⇒ A ≻ B . (11) Suppose A ∼mnx B . Then, because the restriction of mnx to K2 is linear, it follows that A = B . In that case, by the reflexivity of , A ∼ B . This proves (10). Now suppose A ≻mnx B . Then there are four possibilities: (i) a = a and / a and b = b; (iii) a = a and b = / b; (iv) a = / a and b = / b. b = b; (ii) a = Suppose (i) holds. Then, given A ≻mnx B , we must have aP b. Hence, by SM, {a} ≻ {b}. Hence, given (i), {a, a} ≻ {b, b}. Suppose (ii) holds. Then, given A ≻mnx B , we have either aP aP b = b or aP a = b = b. In the former case, SM and the transitivity of  imply {a, a} ≻ {a} ≻ {b, b}. In the latter case, again using SM, it follows that {a, a} ≻ {a} = {b, b}. Suppose (iii) holds. Then, given A ≻mnx B , we must have a = aP b. Since / b, we have two possibilities: a = aP bP b or bP a = aP b. In the former case, b= by SM and the transitivity of , we have {a, a} ≻ {b} ≻ {b, b}. In the latter case, SUA implies {a, a} ≻ {b, b}. Suppose (iv) holds. Given A ≻mnx B and (iv), it can be checked that exactly one of the following must be true: aP aP bP b; (12) aP a = bP b; (13) aP a and bP aP b; (14) aP bP a = b. (15) Suppose (12) holds. Then, by SM and the transitivity of , {a, a} ≻ {a} ≻ {b} ≻ {b, b}. Suppose (13) holds. Again using SM and the transitivity of , it follows that {a, a} ≻ {a} = {b} ≻ {b, b}. Now suppose (14) holds. By SUA, {a} ≻ {b, b}, and by SM, {a, a} ≻ {a}. Thus, by the transitivity of , {a, a} ≻ {b, b}. Lastly, suppose (15) holds. Then, by SD1, {a, a} ≻ {b, b}. This completes the proof of (11) and thus of Lemma 2.1. The proof of Lemma 2.2 is similar to the proof of Lemma 2.1. We now obtain Theorem 3 (3.1)  satisfies SM, IND, SD1 and SUA if and only if =mnx ; (3.2)  satisfies SM, IND, SD2 and SUP if and only if =mxn . Proof. To prove Theorem 3.1, note first that it is straightforward to show that mnx satisfies SM, IND, SD1 and SUA. Conversely, suppose  satisfies SM, IND, SD1 and SUA. By Theorem 1, A ∼ {min A, max A} for all A ∈ K , (16) Choice under complete uncertainty 305 and by Lemma 2, A  B ⇔ A mnx B for all A, B ∈ K2 . (17) Combining (16) and (17) with the transitivity of , we obtain =mnx . The proof of Theorem 3.2 is analogous to the proof of Theorem 3.1. The following examples establish the independence of the axioms used in Theorem 3.1, provided that X contains at least three elements (clearly, if #X = 2, the ordering =mnx is uniquely determined by SM and no further axioms are required). Let X = {x , y, z } and xPyPz . (i) Let {x } ≻ {x , y} ≻ {x , z } ∼ {x , y, z } ≻ {y} ≻ {y, z } ≻ {z }. Then  satisfies SM, IND and SD1 but violates SUA. (ii) Let {x } ≻ {x , y} ≻ {y} ≻ {x , z } ∼ {x , y, z } ∼ {y, z } ≻ {z }. Then  satisfies SM, IND and SUA but violates SD1. (iii) Let {x } ≻ {x , y} ≻ {y} ≻ {x , z } ≻ {x , y, z } ≻ {y, z } ≻ {z }. Then  satisfies SM, SD1 and SUA but violates IND. (iv) Let {x } ∼ {x , y} ≻ {y} ≻ {x , z } ∼ {x , y, z } ≻ {y, z } ≻ {z }. Then  satisfies IND, SD1 and SUA but violates SM. That the properties SM, IND, SD2, and SUP are independent can be shown by employing the ‘duals’ of the above examples. 4 Lexicographic min-max and max-min rules In this section, we introduce lexicographic versions of the min-max and the maxmin rules and provide axiomatic characterizations of these rules. A few additional definitions are required in order to do so. For all A ∈ K , let A0 := A and  #A/2 if #A is even; nA := (#A − 1)/2 if #A is odd. If nA > 0, let, for all t = 1, . . . , nA , At := At−1 − {min At−1 , max At−1 }. For all A, B ∈ K , let nAB := min{nA , nB }. Definition 4 (4.1) The lexicographic min-max relation Lmnx is defined by letting, for all A, B ∈ K , A Lmnx B :⇔ ∃t ∈ {0, . . . , nAB } such that [As ∼mnx Bs for all s < t] and [At ≻mnx Bt or Bt = ∅]. (4.2) The lexicographic max-min relation Lmxn is defined by letting, for all A, B ∈ K , A Lmxn B :⇔ ∃t ∈ {0, . . . , nAB } such that [As ∼mxn Bs for all s < t] and [At ≻mxn Bt or At = ∅]. 306 W. Bossert et al. Under the lexicographic min-max rule, when comparing two sets A and B , the agent first compares A and B in terms of the min-max rule. If, according to this criterion, one of the sets is better than the other, then the former is considered to be better than the latter according to Lmnx as well. However, if, in terms of the min-max rule, A is indifferent to B , then the agent removes the best and the worst elements from both sets so that she is left with A1 and B1 . A1 and B1 are then compared using the min-max rule and if one of these two sets turns out to better than the other set according to this criterion, then A is regarded as better than B . If A1 and B1 are indifferent in terms of the min-max rule, then the agent again removes the best and the worst elements from A1 and B1 , thus obtaining A2 and B2 , and so on. If, at any stage in this process of successive reductions of the two original sets, the set of alternatives surviving from one set is non-empty while the set of alternatives surviving from the other is empty, then the former is considered better than the latter. The lexicographic max-min rule involves a similar multi-stage process, where the agent uses the max-min rule rather than the min-max rule to compare the two relevant sets in each stage. If, at some stage in this process, one of the reduced sets is empty and the other is not, the former is considered better than the latter. It can be checked easily that Lmnx and Lmxn are linear orderings. Like the min-max and max-min rules discussed in Section 3, the lexicographic rules introduced in Definition 4 are also based on an interplay of limited rationality and certain types of uncertainty attitudes, but limited rationality manifests here in a somewhat different form. In the case of min-max and max-min rules, the agent picks the focal points (the best and worst outcomes) right at the beginning, and once that happens, the rest of the outcomes never enter his deliberations. In contrast, in the case of our lexicographic rules, while the agent’s limited ability for calculation and deliberation leads him to focus on a few focal outcomes at any given stage, there can be successive layers of focal outcomes in the following sense. In comparing two sets of possible outcomes, A and B, the agent first looks at the two focal subsets, {min A , max A} and {min B , max B }. However, if the consideration of these two focal subsets does not enable the agent to discriminate between A and B , then she ‘discards’ these two subsets from the original sets and shifts her attention to new focal points, namely the best-but-one and worst-but-one outcomes in A and B . Thus, given her limited ability for absorbing complex information, at each stage the agent concentrates only on a few ‘distinguished’ or ‘focal’ outcomes. However, unlike in the case of min-max and max-min rules, now there can be different layers of focal points, the transition from one layer to the next being made if an examination of the focal points in the earlier layer does not help the agent to discriminate between the two sets of outcomes under consideration. Note that, under the lexicographic min-max rule, as well as under the lexicographic max-min rule, the agent is assumed to have the same uncertainty attitude in every successive stage of this process. Thus, under the lexicographic min-max rule, the agent is pessimistic in each stage of his deliberation. However, in any particular stage, the exercise of this pessimism is confined only to the layer of focal outcomes appropriate for that stage. Simi- Choice under complete uncertainty 307 larly, optimism characterizes the uncertainty attitude in each stage of deliberation under the lexicographic max-min rule. The following definition introduces some additional properties of . Definition 5  satisfies (5.1) type 1 dominance (D1) if, for all A ∈ K and all x , y ∈ X such that xPAPy, {x , y} ≻ A ∪ {y}; (5.2) type 2 dominance (D2) if, for all A ∈ K and all x , y ∈ X such that xPAPy, A ∪ {x } ≻ {x , y}; (5.3) type 1 extension principle (EP1) if, for all A ∈ K and all x , y ∈ X − A, ([{a} ≻ {x , y} for all a ∈ A] and [A ≻ {x , y}]) implies A ∪ {x , y} ≻ {x , y}; (5.4) type 2 extension principle (EP2) if, for all A ∈ K and all x , y ∈ X − A, ([{x , y} ≻ {a} for all a ∈ A] and [{x , y} ≻ A]) implies {x , y} ≻ A ∪ {x , y}; (5.5) type 1 monotonicity (MON1) if, for all x ∈ X and all A, B ∈ K , ({x } ≻ A and {x } ≻ B ) implies {x } ≻ A ∪ B ; (5.6) type 2 monotonicity (MON2) if, for all x ∈ X and all A, B ∈ K , (A ≻ {x } and B ≻ {x }) implies A ∪ B ≻ {x }; (5.7) extension independence (EIND) if, for all A, B ∈ K and all x , y ∈ X − (A ∪ B ) such that xPA ∪ BPy, [A  B ⇔ A ∪ {x , y}  B ∪ {x , y}]. D1 requires that, if x is preferred to every alternative in A and every alternative in A is preferred to y, then {x , y} is preferred to A ∪ {y}. Analogously, D2 requires that if x is preferred to every alternative in A and every alternative in A is preferred to y, then A ∪ {x } is preferred to {x , y}. The intuitive appeal of these conditions, which are stronger versions of SD1 and SD2, respectively, is immediate. EP1 stipulates that, if x and y do not belong to A, A is preferred to {x , y}, and, for every element a of A, the certainty of having a is better than the uncertain prospect {x , y}, then A ∪ {x , y} is better than {x , y}. Again, EP2 is the ‘dual’ requiring that if neither x nor y belongs to A, {x , y} is better than A, and, for every element a of A, the uncertain prospect {x , y} is better than the certainty of having a, then {x , y} is better than A ∪ {x , y}. MON1 requires that, if each of the uncertain prospects A and B is worse than the certainty of having x , then A ∪ B is worse than receiving x with certainty. MON2 is the ‘dual’ of MON1. EIND has intuitive similarities with IND, but neither of these conditions implies the other. If every alternative in A ∪ B is worse than x and better than y, then the relative ranking of A ∪ {x , y} and B ∪ {x , y} is the same as the relative ranking of A and B . Note that the axiom only applies to situations where x and y are neither in A nor in B and, thus, no restriction is imposed by this axiom when new elements are to be added to one of the sets A and B only. The following lemma will be used in the proofs of the characterization results in this section. 308 W. Bossert et al. Lemma 4 (4.1) If  satisfies SM, SUA, and EP1, then, for all A ∈ K and all x , y ∈ X − A such that xPAPy, A ∪ {x , y} ≻ {x , y}. (4.2) If  satisfies SM, SUP, and EP2, then, for all A ∈ K and all x , y ∈ X − A such that xPAPy, {x , y} ≻ A ∪ {x , y}. Proof. We prove only Lemma 4.1; the proof of Lemma 4.2 is analogous to that of Lemma 4.1. Suppose  satisfies SM, SUA, and EP1. Let A = {a1 , . . . , am } ∈ K with a1 Pa2 P . . . Pam , and let a0 , am+1 ∈ X be such that a0 PAPam+1 . We have to show that A ∪ {a0 , am+1 } ≻ {a0 , am+1 }. By SUA, {aj } ≻ {ai , ak } (18) for all i , j , k ∈ {0, . . . , m + 1} such that i < j < k . Suppose m is odd. Let m = 2g + 1 where g is a non-negative integer. By (18), {ag+1 } ≻ {ag , ag+2 }. By EP1, {ag , ag+1 , ag+2 } ≻ {ag , ag+2 }. (19) Using (18) again, we obtain {ag+2 } ≻ {ag−1 , ag+3 }. Therefore, by SM and the transitivity of , {ag , ag+2 } ≻ {ag−1 , ag+3 } which, together with (19) and transitivity, implies {ag , ag+1 , ag+2 } ≻ {ag−1 , ag+3 }. Hence, by (18) and EP1, {ag−1 , ag , ag+1 , ag+2 , ag+3 } ≻ {ag−1 , ag+3 }. Continuing in this fashion, we eventually obtain {a0 , . . . , am+1 } ≻ {a0 , am+1 }. Now suppose m is even. Let m = 2h where h is a positive integer. By (18), {ah+1 } ≻ {ah−1 , ah+2 }. Hence, by SM and the transitivity of , {ah , ah+1 } ≻ {ah−1 , ah+2 }. By EP1, {ah−1 , ah , ah+1 , ah+2 } ≻ {ah−1 , ah+2 }. Again using (18), SM and the transitivity of , we obtain {ah−1 , ah+2 } ≻ {ah−2 , ah+3 } and, using transitivity, it follows that {ah−1 , ah , ah+1 , ah+2 } ≻ {ah−2 , ah+3 }. Therefore, by (18) and EP1, {ah−2 , ah−1 , ah , ah+1 , ah+2 , ah+3 } ≻ {ah−1 , ah+2 }. Continuing this procedure, we eventually obtain A ∪ {a0 , am+1 } ≻ {a0 , am+1 }. Choice under complete uncertainty 309 The following theorem is the main result of this section. It provides characterizations of Lmnx and Lmxn by means of some of the axioms introduced earlier. Note that, even though the axioms SUA and SUP are used in these characterizations, one may not want to consider these rules uncertainty averse or uncertainty appealing in a strong sense because, rather than proceeding from worst to best or best to worst element, they make comparisons in a different order. Instead, they can be considered uncertainty averse or uncertainty appealing in a much weaker sense—namely, in the sense reflected by the simple versions of these axioms that apply to comparisons of singleton and doubleton sets only. Theorem 5 (5.1)  satisfies SM, SUA, D1, EP1, MON1 and EIND if and only if =Lmnx ; (5.2)  satisfies SM, SUP, D2, EP2, MON2 and EIND if and only if =Lmxn . Proof. The proof of Theorem 5.2 is similar to the proof of Theorem 5.1. Therefore, we only prove Theorem 5.1. It can be checked easily that Lmnx satisfies SM, SUA, D1, EP1, MON1 and EIND. Now suppose  satisfies these properties. It is sufficient to prove that A ∼Lmnx B ⇒ A ∼ B (20) A ≻Lmnx B ⇒ A ≻ B (21) and for all A, B ∈ K . (20) follows immediately from the linearity of Lmnx and the reflexivity of . To prove (21), suppose A, B ∈ K are such that A ≻Lmnx B . Let a = min A and a = max A. There are three possibilities: A ≻mnx B ; (22) ∃t ∈ {1, . . . , nAB } such that As ∼mnx Bs for all s < t and At ≻mnx Bt ; (23) / ∅ = Bt . (24) ∃t ∈ {1, . . . , nAB } such that As ∼mnx Bs for all s < t and At = First, consider case (22). There are four possible subcases: (i) #A ≤ 2 and #B ≤ 2; (ii) #A > 2 and #B ≤ 2; (iii) #A ≤ 2 and #B > 2; (iv) #A > 2 and #B > 2. In subcase (i), Lemma 2 implies A ≻ B . In subcase (ii), Lemma 4 implies A ≻ {a, a}. By Lemma 2, we have {a, a} ≻ B , and transitivity implies A ≻ B . Now consider subcase (iii). Let B = {b1 , . . . , bm }, where b1 Pb2 P . . . Pbm . Since #A ≤ 2 and #B > 2, A = {a, a} and m ≥ 3. Given A ≻mnx B , either (aPbi for all i ∈ {1, . . . , m}) or (b1 P aPbm ) or (a = bm and aPb1 ). If (aPbi for all i ∈ {1, . . . , m}), then, by repeated application of MON1, we have {a} ≻ B and, hence, using SM and the transitivity of , it follows that {a, a} ≻ B . If (b1 P aPbm ), then, by SUA, ({a} ≻ {bi , bm } or {a} ≻ {b1 , bi }) for all i ∈ {1, . . . m} and, hence, by repeated application of MON1, we have {a} ≻ B . By 310 W. Bossert et al. SM and the transitivity of , {a, a} ≻ B . Finally, suppose a = bm and aPb1 . Then, by D1, {a, bm } ≻ {b1 , . . . , bm−1 } ∪ {bm } = B , that is, A = {a, a} ≻ B . Finally, consider subcase (iv). Then, as in the proof for case (iii), it follows that {a, a} ≻ B . By Lemma 4, A ≻ {a, a}, and transitivity implies A ≻ B . Now consider case (23). Then, by (possibly repeated) application of EIND, it follows that A  B ⇔ At  Bt . As in case (22), it follows that At ≻ Bt and thus A ≻ B . Finally, consider case (24). By (possibly repeated) application of EIND, we have A  B ⇔ At−1  Bt−1 . Since Bt = ∅, Bt−1 = {min At−1 , max At−1 }. / ∅, Lemma 4 implies At−1 ≻ {min At−1 , max At−1 } = Bt−1 . Hence, Because At = A ≻ B. If #X = 2,  is determined by SM alone. For #X = 3, EIND is trivially satisfied and, furthermore, D1 is implied by SM and SUA, and MON1 is implied by SM. The following examples show that SM, SUA, D1, EP1, MON1 and EIND are independent if X contains at least four elements. Suppose X = {x , y, z , w} and xPyPzP w. (i) Define  by replacing {x , y, z , w} ≻Lmnx {x , z , w} with {x , y, z , w} ∼ {x , z , w} and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except EIND. (ii) Define  by replacing {y} ≻Lmnx {x , y, z } with {y} ∼ {x , y, z } and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except MON1. (iii) Define  by replacing {x , y, z } ≻Lmnx {x , z } with {x , y, z } ∼ {x , z } and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except EP1. (iv) Define  by replacing {x , w} ≻Lmnx {y, z , w} with {x , w} ∼ {y, z , w} and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except D1. (v) Define  by replacing {y} ≻Lmnx {x , y, z } ≻Lmnx {x , z } with {y} ∼ {x , y, z } ∼ {x , z } and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except SUA. (vi) Define  by replacing {x } ≻Lmnx {x , y} with {x } ∼ {x , y} and letting all other relative rankings in  be the same as in Lmnx . The resulting ordering satisfies all of the above axioms except SM. Again, the ‘duals’ of these examples can be used to establish the independence of the axioms used in Theorem 5.2. 5 Conclusion We have provided axiomatic characterizations of four decision rules for choice under complete uncertainty, namely, the min-max rule, the max-min rule, the lexicographic min-max rule and the lexicographic max-min rule. To the best of our knowledge, these results are the first axiomatizations of these decision rules. Choice under complete uncertainty 311 All these rules share an important feature. Each of them makes use of both the min and the max, either in one round (as in the case of min-max and max-min rules) or, possibly, in several rounds (as in the case of lexicographic versions of min-max and max-min rules). In this respect, the rules of choice under complete uncertainty, which we have considered in this paper, provide some plausible ways of going beyond the exclusive concern with the min alone or the max alone, which marks most of the familiar rules in the existing literature. A related issue that remains to be addressed is the question whether attractive lexicographic extensions of the median-based rules can be found. Furthermore, it would be interesting to conduct empirical investigations in order to examine to what extent the rules introduced in this paper can be used to approximate observed choice behaviour under complete uncertainty. References Arrow, K.J., Hurwicz, L.: An optimality criterion for decision-making under ignorance. In: Carter, C.F., Ford, J.L. (eds.) Uncertainty and expectations in economics: Essays in honour of G.L.S. Shackle, pp. 1–11. Oxford: Basil Blackwell 1972 Bandyopadhyay, T.: Extensions of an order on a set to the power set: some further observations. Mathematical Social Sciences 15, 81–85 (1988) Barberá, S.: The manipulation of social choice mechanisms that do not leave “too much” to chance. Econometrica 45, 1573–1588 (1977) Barberá, S., Barrett, C.R.,, Pattanaik, P.K.: On some axioms for ranking sets of alternatives. Journal of Economic Theory 33, 301–308 (1984) Barberá, S., Pattanaik, P.K.: Extending an order on a set to the power set: some remarks on Kannai and Peleg’s approach. Journal of Economic Theory 32, 185–191 (1984) Bossert, W.: On the extension of preferences over a set to the power set: an axiomatic characterization of a quasi-ordering. Journal of Economic Theory 49, 84–92 (1989a) Bossert, W.: On the extension of preferences over a set to the power set: a generalization. Mimeo, Universität Karlsruhe (1989b) Bossert, W.: Preference extension rules for ranking sets of alternatives with a fixed cardinality. Theory and Decision 39, 301–317 (1995) Bossert, W.: Uncertainty aversion in nonprobabilistic decision models. Mathematical Social Sciences 34, 191–203 (1997) Cohen, M., Jaffray, J.: Rational behavior under complete ignorance. Econometrica 48, 1281–1299 (1980) Fishburn, P.C.: Comment on the Kannai-Peleg impossibility theorem for extending orders. Journal of Economic Theory 32, 176–179 (1984) Fishburn, P.C.: Signed orders and power set extensions. Journal of Economic Theory 56, 1–19 (1992) Heiner, R., Packard, D.: A uniqueness result for extending orders; with applications to collective choice as inconsistency resolution. Journal of Economic Theory 32, 180–184 (1984) Holzman, R.: A note on the redundancy of an axiom in the Pattanaik-Peleg characterization of the lexicographic maximin extension. Social Choice and Welfare 1, 123–125 (1984a) Holzman, R.: An extension of Fishburn’s theorem on extending orders. Journal of Economic Theory 32, 192–196 (1984b) Johnson, E., Payne, J.: Effort and accuracy in choice. Management Science 31, 395–414 (1985) Kannai, Y., Peleg, B.: A note on the extension of an order on a set to the power set. Journal of Economic Theory 32, 172–175 (1984) March, J.G.: Rationality, ambiguity, and the engineering of choice. In: Bell, D., Raiffa, H., Tversky, A. (eds.) Decision making: descriptive, normative and prescriptive interactions, pp. 33–57. Cambridge: Cambridge University Press 1988 March, J.G., Simon, H.A.: Organizations. New York: Wiley 1958 312 W. Bossert et al. Maskin, E.: Decision-making under ignorance with implications for social choice. Theory and Decision 11, 319–337 (1979) Milnor, J.: Games against nature. In: Thrall, R., Coombs, C., Davis, R. (eds.) Decision processes. pp. 49–59. New York: Wiley 1954 Nitzan, S., Pattanaik, P.K.: Median-based extensions of an ordering over a set to the power set: an axiomatic characterization. Journal of Economic Theory 34, 252–261 (1984) Pattanaik, P.K., Peleg, B.: An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set. Social Choice and Welfare 1, 113–122 (1984) Rawls, J.: A theory of justice. Cambridge: Harvard University Press 1971 Shackle, G.L.S.: Expectation in economics, 2nd edn. London: Cambridge University Press 1952 Shackle, G.L.S.: Expectation in economics. In: Carter, C.F., Meredith, G.P., Ford, J.L. (eds.) Uncertainty and business decisions, pp. 90–97. Liverpool: Liverpool University Press 1954 Thorngate, W.: Efficient decision heuristics. Behavioral Science 25, 219–225 (1980)