Arrow, Hausdorff, and Ambiguities in the Choice of Preferred States in Complex Systems
AArrow, Hausdorff, and Ambiguities in the Choice of PreferredStates in Complex Systems
T. Erber † and M. J. Frank ‡ Department of Physics Illinois Institute of Technology, Chicago, USA † Department of Applied Mathematics Illinois Institute of Technology, Chicago, USA † ‡
Department of Physics University of Chicago, Chicago, USA † Abstract
Arrow’s ‘impossibility’ theorem asserts that there are no satisfactory methods of aggregatingindividual preferences into collective preferences in many complex situations. This result hasramifications in economics, politics, i.e., the theory of voting, and the structure of tournaments. Byidentifying the objects of choice with mathematical sets, and preferences with Hausdorff measuresof the distances between sets, it is possible to extend Arrow’s arguments from a sociological to amathematical setting. One consequence is that notions of reversibility can be expressed in termsof the relative configurations of patterns of sets.
PACS numbers: 01.55,+b,01.90+g,01.70+w a r X i v : . [ ec on . T H ] S e p . 1. ARROW’S IMPOSSIBILITY THEOREM One of the most significant ’no-go’ results discovered in the twentieth century is Ar-row’s theorem concerning the impossibility of devising satisfactory methods for aggregatingindividual preferences into collective preferences in many complex situations [1]. Arrow’sarguments have given rise to a voluminous literature with wide ranging applications in eco-nomics, politics, and the organization of tournaments. The feasibility of further extendingthese results to areas of mathematics and physics depends on finding appropriate counter-parts to Arrow’s objects of choice, i.e., various social states, and reinterpreting the conceptof preference in quantitative terms. One possible approach is to identify the objects of choicewith mathematical sets, and to relate the associated preferences to Hausdorff’s asymmetricdistances between sets.To be specific, let
A, B, C, etc. denote the objects of choice, and represent the relationsof preference by means of the symbols (cid:31) and ≺ : consequently a formula such as A (cid:31) B signifies that A is preferred over B , and similarly the expressions B (cid:31) A and A ≺ B bothmean that B is preferred to A . Suppose now that there are three individuals labeled 1, 2, 3,each of which can choose between the two alternatives A and B . Then, irrespective of whichcombination of preferences - - - out of a total of eight possibilities - - - occurs, there isa reasonable way of aggregating the individual preferences into a collective preference. Forinstance, the pattern 1 : A (cid:31) B ; 2 : A (cid:31) B ; 3 : B (cid:31) A (1)indicates the overall preference for the choice of A in virtue of majority rule.Arrow’s arguments become relevant in the slightly more complicated situation where thereare three individuals who can choose among three alternatives A, B, and C . Although inthis case there are many combinations of preferences that can be aggregated into collec-tive preferences with the help of plausible schemes such as majority rule, there are somerecalcitrant outliers whose antecedents date back more than two centuries [2]. Suppose, forinstance that the first individual orders the objects of choice in the sequence1 : A (cid:31) B (cid:31) C (cid:31) A . (2)This is an instance where the preferences of A over B and B over C do not imply that A is preferred over C . Sequences of this kind are said to be intransitive [3]. The ambiguity is2eightened still further if the remaining two individuals order their preferences in a cyclicversion of Eq.(2), viz. 2 : B (cid:31) C (cid:31) A (cid:31) B , (3)3 : C (cid:31) A (cid:31) B (cid:31) C , (4)Clearly majority rule can’t achieve a consensus in this situation. Experience shows thateven more elaborate voting schemes fail to extract a fair and favored choice from suchsets of intransitive preferences. Arrow’s point is that this is not a matter of ingenuity butrather an impossibility: in some complex situations there is simply no reasonable methodfor aggregating individual preferences into a collective preference. The reasoning is basedon making precise the notions of ‘fair’ and ‘reasonable’.Arrow lists four criteria that ought to be satisfied by any acceptable voting procedure:1. Local and global harmony. Suppose that in a bloc of voters every individual has thesame preference, say, A over B ( A (cid:31) B ), then the collective preference of the entiregroup is also A over B .2. All choices are possible. Every individual can in principle choose among all availablealternatives and these can be ordered in every possible sequence of preferences.3. Independence of irrelevant alternatives. The set of choices, A, B, etc., available to everyindividual constitutes an environment of admissible options. All methods of aggregat-ing individual preferences into a collective preference then must be independent of anychoice that lies outside of the environment.4. Non-dictatorship. The collective preferences of a group of individuals are not to bedetermined solely by the preferences of a single individual.At first sight these innocuous propositions appear to be part of any reasonable votingscheme, but more than sixty years ago Arrow discovered an astonishing twist - - - thesepropositions are actually incompatible! This basic flaw is the reason that it is impossible todevise a general method for aggregating individual preferences into a collective preferenceapplicable under all circumstances. The technical details of the proof are given in [1]. Amore mathematically oriented treatment is presented in [4].3
IG. 1: Arrangement of the point sets A and B over a grid of unit squares
2. 2. THE HAUSDORFF DISTANCE BETWEEN SETS
Let S be a planar set composed of the points s i , ≤ i ≤ N S , and T be another set inthe same plane composed of the points t j , ≤ j ≤ N T . Further, let d ( s i , t j ) denote theordinary Euclidean distance between s i and t j . Then the Hausdorff distance from the set S to the set T is given by the radius of the smallest disk centered at any point of S that alsoincludes at least one point of T [5]. This definition corresponds to the expression δ H ( S → T ) = sup s i ∈ S inf t j ∈ T d ( s i , t j ) . (5)In a similar fashion the Hausdorff distance from the set T to the set S is given by the radiusof the smallest disk centered at any point of T that also includes at least one point of S .This definition corresponds to a formula analogous to (5), δ H ( T → S ) = sup t j ∈ T inf s i ∈ S d ( s i , t j ) . (6)In general, these distances depend both on the configurations as well as the relative positionsof the sets S and T . Consequently, the directed Hausdorff distances δ H ( S → T ) and δ H ( T → S ) may be unequal, even though the underlying Euclidian metric d ( s i , t j ) is symmetric. Thisis the essential property that furnishes a link between the concepts of choice and preferenceand the mathematical notion of a distance between setsThe transition from Arrow to Hausdorff can be illustrated with the aid of several verysimple examples. Suppose that A and B are both two-point sets with the elements a , a ∈ A , and b , b ∈ B , arranged over a grid of unit squares as shown in Figure 1. In this case4he basic distance definition (5) reduces to the simpler form δ H ( A → B ) = max (cid:8) min i d ( a , b i ) , min i d ( a , b i ) (cid:9) . (7)The four Euclidean distances between the points in the two sets can then be displayed inthe form of a 2 × b b a √ √ a √ (8)Inserting these numbers into (7) one obtains δ H ( A → B ) = max (cid:8) min( √ , √ , min(1 , √ (cid:9) , (9)or δ H ( A → B ) = max (cid:8) √ , (cid:9) = √ . (10)The reverse Hausdorff distance δ H ( B → A ) can then be computed from the analog of (7),viz. δ H ( B → A ) = max (cid:8) min i d ( b , a i ) , min i d ( b , a i ) (cid:9) . (11)Again, inserting the numbers, the result is δ H ( B → A ) = max (cid:8) min( √ , , min( √ , √ (cid:9) , (12)or δ H ( B → A ) = max (cid:8) , √ (cid:9) = √ . (13)Clearly, the numerical differences between the two Hausdorff distances are due to the factthat (10) corresponds to a ‘max - min by rows’ algorithm whereas (13) corresponds a ‘max- min by columns’ algorithm. The two parallel interpretations of the expression A (cid:31) B are now complete: in Arrow’s language this means that the social state A is preferred overthe social state B ; in Hausdorff’s terminology A and B are sets whose configurations andrelative positions imply that the Hausdorff distance from A to B is greater than the distancefrom B to A .The next increment of complexity is the Condorcet triplet Eq.(2). This intransitivesequence of preferences can also be associated with a pattern of sets. Figure 2 shows onepossible arrangement of three seta A, B, C whose relative Hausdorff distances mirror the5reference rankings in Eq.(2). It is easy to confirm this numerically since the A and B setsare exactly the same as in Figure 1; and it merely remains to evaluate the new Hausdorffdistances δ H ( B → C ) and δ H ( C → A ). Following the steps of the prior calculations inEqs.(7)-(8), we first list the Euclidian distances between the points B and C in an array c c b √ b √ (14)The corresponding Hausdorff distance is then δ H ( B → C ) = max (cid:8) min(1 , √ , min(2 , √ (cid:9) , (15)or δ H ( B → C ) = max (cid:8) , (cid:9) = 2 . (16)The reverse Hausdorff distance is given by δ H ( C → B ) = max (cid:8) min(1 , , min( √ , √ (cid:9) , (17)or δ H ( C → B ) = max (cid:8) , √ (cid:9) = √ . (18)Similarly the distance between C and A can be inferred from the array c c a √ √ a √ √ (19)Specifically, δ H ( A → C ) = max (cid:8) min( √ , √ , min( √ , √ (cid:9) , (20)or δ H ( A → C ) = max (cid:8) √ , √ (cid:9) = √ . (21)The last distance is given by δ H ( C → A ) = max (cid:8) min( √ , √ , min( √ , √ (cid:9) , (22)or δ H ( C → A ) = max (cid:8) √ , √ (cid:9) = √ . (23)6he equivalences between the preference rankings in Eq.(2) and the distance inequalitiesimplicit in Figure 2 can now be listed in a unified form: A (cid:31) B ⇔ δ H ( A → B ) = √ > √ δ H ( B → A ) B (cid:31) C ⇔ δ H ( B → C ) = 2 > √ δ H ( C → B ) (24) C (cid:31) A ⇔ δ H ( C → A ) = √ > √ δ H ( A → C ) FIG. 2: Arrangement of three point sets
A, B, C whose mutual Hausdorff distances satisfy theinequalities in Eq.(24).
3. 3. REVERSIONS
In ordinary particle dynamics a reversal of motion is usually effected by a reversal ofall velocity components. This typographical device is adequate for systems governed bydifferential equations, but in more general situations where the dynamical evolution is de-scribed by mathematical flows the replacement of t by − t does not necessarily correspondto a physical time reversal [6]. The disconnect between reversions and sign changes is evenmore drastic in Arrow’s scheme of preference rankings. In this situation it seems reasonableto associate reversions with an interchange of preferences: instead of A (cid:31) B we presume B (cid:31) A . In this sense, the reverse of the Condorcet triplet Eq.(2) is A (cid:31) C (cid:31) B (cid:31) A (25)Switching preferences also implies a reversal of the inequalities in the corresponding Haus-7 IG. 3: Arrangement of three point sets
A, B, C , corresponding to the Hausdorff inequalties inEq.(26). dorff distances : and this, in turn, requires that the pattern of sets in Figure 2 has to beshifted. A minimal rearrangement consistent with the sequence of preferences in Eq.(25) isshown in Figure 3. Comparing with Figure 2, it is clear that all of the individual pointsretain their positions, just the labeling of the sets has changed. The resulting collection ofpreferences and Hausdorff distances can then be summarized in a form similar to Eq.(24): A (cid:31) C ⇔ δ H ( A → C ) = √ > √ δ H ( C → A ) C (cid:31) B ⇔ δ H ( C → B ) = 2 > √ δ H ( B → C ) (26) B (cid:31) A ⇔ δ H ( B → A ) = √ > √ δ H ( A → B )Evidently, this perspective on reversions has no relation to the introduction of minus signs. [1] Arrow, K. J. : Social Choice and Individual Values, 2nd ed., Wiley, New York (1963).[2] Condorcet, M. : ´Essai sur l’application de l’analyse `a la probabilit´e des d´ecisions redues `a lapluralit´e des voix. Paris (1785).[3] Tarski, A. : Introduction to Logic. Oxford University Press, New York (1941).[4] Kelly, J. S. : Arrow Impossibility Theorems, Academic Press, New York (1978).[5] Hausdorff, F. : Mengenlehre, de Gruyter, Berlin (1927).[6] Bernstein, B., Erber, T. : Reversibility, Irreversibility : Restorabiity, Nonrestorabiity. J. Phys.A : Math. Gen. , 7581 (1999)., 7581 (1999).