Decision - Making under Ignorance 321

A new investigation is launched into the problem of decision-making in the face of 'complete ignorance', and linked to the problem of social choice. In the first section the author introduces a set of properties which might characterize a criterion for decision-making under complete ignorance. Two of these properties are novel: 'independence of non-discriminating states', and 'weak pessimism'. The second section provides a new characterization of the so-called principle of insufficient reason. In the third part, lexicographic maximin and maximax criteria are characterized. Finally, the author's results are linked to the problem of social choice. Several authors, [2], [3], [7], and [9] have dealt with the problem of an individual who must choose from a set of alternatives when he cannot associate a probability distribution with the possible outcomes of each alternative. The problem has been called that of decision-making under complete ignorance; presumably 'complete ignorance' captures the notion that the axioms of subjective probability cannot be fulfilled. This paper is a further investigation in that tradition. In the first section we suggest a set of properties which might characterize a criterion for decision-making under ignorance. Most of these properties are familiar, but, in particular, ' independence of non-discriminating states' and 'weak pessimism' are new in this context. The second section recapitulates some of the important decision criteria in the literature and suggests a new characterization of the so-called principle of insufficient reason. In the third part, we drop the assumption invariably made by previous authors that preferences for consequences satisfy the von Neumann-Morgenstern axioms, and characterize the so-called lexicographic maximin and maximax criteria. We also provide a new axiomatization of the ordinary maximin principle. Finally, we show that several of our results translate quite easily into the theory of social choice. THE P R O P E R T I E S Let C be a consequence or 'outcome' space. C contains a subset C* of 'sure' or 'certain' outcome as well as all finite lotteries 1 with outcomes in C*, We Theory and Decision 11 (1979) 319-337. 0040-5833/79/0113-0319~01.90. Copyright 9 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A. 320 E R I C M A S K I N assume that the decision-maker has a preference ordering ~ on C which satisfies the yon Neumann-Morganstern axioms for decision-making under uncertainty. Let ~ * be the family of von Neumann-Morgenstern utility representations of ~ . Define a decision d as a map d : ~ -~ C where ~ is an exhaustive list of possible states of nature. Obviously there are many ways in which nature can be described, and therefore many conceivable ~'s could apply to the same world. Following Arrow-Hurwicz [2] we shall define a decision problem P as a set of decisions which share a common domain ~2(P). The decision-maker solves a non-empty decision problem P by choosing a non-empty subset/~_c p. 16 is interpreted as the 'choice' or 'optimal' subset of P. Let g be the class of all non-empty decision problems P such that P and ~(P) are finite.2 A decision criterion f is a mapping f : G ' ~ f f such that VP E if, f(p)c p, f(P)4:O and such that d(w)~d'(w) 3, for all w E ~2, implies that d ~f(P) if and only if d' E f(P). The following are conditions that have, at various times, been deemed reasonable properties for a decision criterion f to satisfy. PROPERTY (1). VP,, P2 E if , d E P, c_P 2 ~ [ d E f(P~) ~ d E f(P0l Property (1) is Sen's Property a of rationality [10]. PROPERTY (2). VP1, P2E D ~ [d, dlCf(P1) and Plc__P2] ~ [dEf(P2) o d' ~ f(P9]. Property (2) is Sen's Property/3 and Milnor's 'row adjunction' [9]. Together (1) and (2) constitute the Arrow-Hurwicz Property A, and, as Herzberger [6] has shown, imply that, for every ~2, f induces an ordering ~ on D~ = {d I domain of d = fZ} such that for any P with ~(P) = ~2, d* Ef(P)od* E P ana a ~ a for all d E P. PROPERTY (3).VP1, P2 E if,, d E -Pl c--P2 ~[dEf(P1),d ~ f(P2) ~ f(P~)\PI --/:0]. Properties (1) and (3) together are quivalent to Luce's and Raiffa's Axiom 7' and Chernoff's [3] Postulate 4. (1) and (3) combined are somewhat weaker than the combination of (1) and (2)~ PROPERTY(4). VPE g d , dl@P, if dEf(P) and dl(w)~d(w) for all w E ~2(P), then d 1 ~ f(p). Property (4) is the weakest form of the domination principle. It is ArrowHurwicz Property D. DECISION-MAKING UNDER IGNORANCE 321 PROPERTY (5). VPC ~ V d l E P VdEf (P) , if d(w)~'d~(w) for all w E ~2(P), then d 1 q~ f(P). Property (5) is another rather weak version of domination. It is Milnor's 'Strong Domination' property. PROPERTY (6). V P E . ~ Vd, d l E P if VwEFZ(P), d (w)~d l (w) and 3Wo E ~2(P) such that d(wo)>-d l(wo), then d 1 q~f(P). Property (6) is the usual admissibility condition. It is obviously stronger than property (5). Combined with continuity (see below), it is also stronger than property (4). PROPERTY (7). VP1, /~ C ~ such that g2(P1) = gZ(P2) = ~ , if, for some u E f t * , there exist k E ~r Wo E g2, and bijection h :P1 -;P2 such that (u(cl(w)) + k , w = Wo, ' V d ~ e ~ , u ( h ( d ) ( w ) ) = ~ l u (d(w)) , w--/=Wo. then, d E f(Pl) if and only if h(d) E f(P2). J Property (7) is Milnor's column linearity condition. It amot/nts to demanding that if two decision problems are isomorphic except that in one, the utility derived from any decision if a certain state of nature Wo prevails is uniformly higher than the utility from the corresponding decision in the other problem when Wo arises, then if a given decision is optimal in the other. PROPERTY (8). VP~, P2E ~r that f](P1) = fZ(P2) and IPI[ a_ Ie21, if for some u E f f * , there exists a bijection g:Pa 4p~ such that for some a>0, b E ~ , u(g(cO(w)) = au(d(w)) + b for all d EP1 and w E ~2, then d El(P1) e, g(d) E f(P2). Property (8) is Milnor's linearity condition. PROPERTY (9). ~'P~r b'dl, d2, d EP, if 3u E ~'* such that uod = 89 489 then dl, d2 Ef(P) =~ d El(P). Property (9) is Milnor's convexity condition. PROPERTY (10). Consider a sequence {Pi} g g a n d P E ~ . Suppose that for all i, ~2(P 0 = g2(P) and IP~I--= IPI n. Write P = {d~, . ' . ,dn} ,P i = {d~, " ' . , d~}. Then, if 3u E~ '* such that WVwEg2(P) ifirn u(dj(w))---u(dj(w)), 322 E R I C M A S K I N d~(P i) E f (P i) for all/implies dj f(P). Property (10) is Milnors's continuity axion. Up to this point, all of the stated properties are arguably reasonable, but none embodies the idea of ignorance. Properties 11-13 are an attempt to capture this notion. PROPTERTY (11). Suppose there exists a bijection h :~21 ~ ~2. For P with fZ(P) = ~2, define p1 with g2(P t) = ~21 as p1 = { d l l d 1 = d o h f o r d E P } . Then, d El(P) if and only i fdoh Ef(P1). Property (11) is the Arrow-Hurwicz Property B and the Milnor Symmetry condition. It insists that the labelling of states and decisions be irrelevant for the decision criterion. Consider PI~ P2 E ~r Following Arrow-Hurwicz, P2 is said to be derived from P1 by deletion of repetitious states (/'1 ~ P2) if ~(P2)--~(Pt) and if there exists a bijection h :P1 -->/'2 such that V w E ~2(P2) h(d)(w) = d(w) and such that V w E fZ(PI)/fZ(P2), 3w ~ E fZ(P2) with d(w) = d(w ~) for all dEP1. PROPERTY (12). VP1, P2~ ~,, if P1 ~P2 via bijection h, then h(d) E f(P2) "~ d f(el). Property (12) is the Arrow-Hurwicz Property C and the Milnor 'Deletion of Repetitious States'. More than any other property, it captures the idea of complete ignorance, for, in effect, it asserts that dividing a state into several substates should have no effect on the chosen decision. The next condition is just a weakened version of Property (12). PROPERTY (13). VP1, P2 E ~ , if P1 ~/ '2 via bijection h and if for all dl, d2 @ P1 with dl --b d2 , V w , w 1 E fZ(P1) , not dl (w) ~ d 2 ( wl ) , then h( d) E f (P2 ) r d f((eO. PROPERTY (14). Consider Px, P2 C ~ with ~"~(P2)D..D_~~(P1) and a surjection g:PI~P2 such that V d E P l V w E F t ( P 1 ) g ( d ) ( w ) = d ( w ) . Then, if for d, d~E~f(Px), g (d ) (w)~g(d ' ) (w) for all wEa(P2)/gt(P1), g(d)Ef(P2)~" g(a') f(e2). This last property requires that adding additional states for which all DECISION-MAKING UNDER IGNORANCE 323 decisions are equivalent does not affect the choice of optimal decisions. In effect this requirement is the strong separability axiom of Debreu [5]. Although it has not previously appeared in discussions of decision-making under ignorance, entirely analogous properties have been used recently in the social choice literature under the names of 'elimination of indifferent individuals' [4] and 'unanimity' [10]. II. THE DECISION CRITERIA We may now state the results for the case where preferences obey the yon Neumann-Morgenstern axioms. THEOREM 1. (Arrow-Hurwicz): A decision criterion f satisfies properties (1), (2), (4), (11), (12) if and only if for each u E ~'* there exists a weak ordering~* in the space of real ordered pairs (M, m) with m < M such that (a) MI >~M2 and ml >~ m2 implies that (MI, ml)~u (M2, m2), (b) vP f(P) = {d EPl(max u(d(w)), min u(d(w))~* (max u(d'(w)), min u(d'(w))) for all d'EP}. DEFINITION. A criterion f is the Hurwicz a-criterion for a E [0, 1 ] if VP E ~r Vu E 7Z, d* E f(P) if and only if a max w u(d*(w)) + (1 L--a) rain w u(d*(w)) >~ maxwu(d(w)) + (1 -a) minw u(d(w)) for all d EP. THEOREM 2. A decision criterion fsatisfies properties (1), (2), (4), (5), (8), (11), (12) if and only if 3a ~ [0, 1] such that VP ~ ~,f(p)c_fa(p) where f a is the Hurwicz a-criterion. Remark. It should be noted that this theorem does not require continuity (property (10)). If, however, continuity is also stipulated, we obtain Theorem 3 (see below). Proof. If f satisfies the stipulated properties, we may apply Theorem 1 and, for choice of u E X/*, define an ' * ordermg~u as above. Following Milnor's argument, let a~ be the supremum of all a~E R such that (1, 0 )~u (a , a'). By property (5), 0~-~(a ,a ) if a ' < a u , and