aa r X i v : . [ ec on . T H ] N ov Intensity-Efficient Allocations
Georgios Gerasimou ∗ November 10, 2020
Abstract
We study the problem of allocating n indivisible objects to n agents when the lattercan express strict and purely ordinal preferences and preference intensities. We suggest arank-based criterion to make ordinal interpersonal comparisons of preference intensities insuch an environment without assuming interpersonally comparable utilities. We then definean allocation to be intensity-efficient if it is Pareto efficient and also such that, wheneveranother allocation assigns the same pairs of objects to the same pairs of agents but in a“flipped” way, then the former assigns the commonly preferred alternative within every suchpair to the agent who prefers it more. We show that an intensity-efficient allocation existsfor all 1,728 profiles when n = 3. ∗ University of St Andrews. Email address: [email protected]. The main concepts in this paper were firstpresented as part of other work at the June 2019 meetings of the
Risk, Uncertainty and Decision conference atthe Paris School of Economics. Acknowledgements to be added.
Introduction
In this paper we consider the social allocation problem when agents express theirordinal preferences as well as their ordinal preference intensities over the availablealternatives. In particular, similar to the way in which the agents’ preferencesare elicited, we assume that information about their intensities can be obtainedby asking them to respond to simple questions such as “do you prefer a to b more than you prefer c to d ?” . Crucially, however, we don’t assume that thesecomparisons are necessarily quantifiable/cardinalizable. Operating within such aninformational setting, our focus is on the assignment of n indivisible items to n agents when monetary transfers are infeasible or prohibited, as is the case, forexample, in several matching markets. Thus, we rule out the agents’ willingnessto pay for the different items from being a potential source of information abouttheir generally differing preference intensities.This non-cardinal, non-utilitarian framework where preference and preference-intensity information is nevertheless still available to the social planner/matchingplatform raises the question of how it might be used in the problem at hand to arriveat some intuitive refinement of Pareto efficiency that would also reflect differencesin the agents’ preference intensities. Similar to utilitarian or other cardinal-utilitynotions of efficiency, such a refinement seems to require that one be able to makesome kind of interpersonal comparisons. Unlike those notions, however, in our caseinterpersonal comparisons cannot be based on the agents’ utilities and must relyon the information contained in the above ordinal intensity rankings.To this end, we assume that such comparisons can be made when all agents’preferences and intensities are strict by contrasting the rank/position of pairs ofalternatives ( a, b ) in the different agents’ intensity orderings. In particular, whenboth agents i and j prefer a to b but the pair ( a, b ) lies higher in i ’s intensity rankingthan in j ’s, then we assume that i prefers it more . Building on these ordinal andunweighted interpersonal intensity comparisons, we say that an allocation x is intensity-efficient if it is Pareto efficient and also such that, whenever anotherallocation y assigns the same pairs of objects to the same pairs of agents but in a“flipped” way, i.e. when ( x i , x j ) = ( y j , y i ) = ( a, b ) for agents i , j and alternatives a , b , then x assigns the commonly preferred alternative in each such pair to theagent who prefers it more.We show that an intensity-efficient allocation exists in all 1 ,
728 strict intensityprofiles that correspond to the case where n = 3. When n ≥
5, however, theexistence of intensity-efficient allocations is not guaranteed because the underlyingdominance relation may be cyclic if additional restrictions are not imposed. Model
There is a finite set X of n ≥ n individuals.Agent i ≤ n is assumed to have a preference intensity relation ˙ % i on X . This canbe thought of as a binary relation over pairs in X × X or as a quaternary relationover elements of X . We assume that ˙ % i is representable by a preference intensityfunction (Gerasimou, 2020), i.e. a mapping s i : X × X → R that satisfies( a, b ) ˙ % i ( c, d ) ⇐⇒ s i ( a, b ) ≥ s i ( c, d ) (1a) s i ( a, b ) = − s i ( b, a ) (1b) s i ( a, b ) ≥ , s i ( b, c ) ≥ ⇒ s i ( a, c ) ≥ max { s i ( a, b ) , s i ( b, c ) } . (1c)A preference intensity function for ˙ % i is unique up to an odd and strictly increasingtransformation in the sense that t i : X × X → R also represents ˙ % i in this way ifand only if t i = f ◦ s i for some f : R → R that is odd [ f ( − z ) = − f ( z )] and strictlyincreasing in the range s i ( X × X ) of s i . Oddness ensures that the skew-symmetrycondition (1b) is preserved. This condition is without loss of generality and isassumed for convenience, as it implies that a % i b ⇔ s i ( a, b ) ≥
0, where % i is agent i ’s preference relation that is induced by ˙ % i .A notable special case of (1a)–(1c) is obtained when ˙ % i admits a utility-difference representation whereby ( a, b ) ˙ % i ( c, d ) ⇔ u i ( a ) − u i ( b ) ≥ u i ( c ) − u i ( d )for some function u i : X → R + . This special case allows one to write s i ( a, b ) ≡ u i ( a ) − u i ( b ). It is possible if and only if (1c) is strengthened to the additivity condition s i ( a, c ) = s i ( a, b ) + s i ( b, c ) for all a, b, c ∈ X (see Gerasimou (2020) andreferences therein). As is well-known, when such a utility-difference representa-tion exists it is unique neither up to a positive affine transformation nor up toan arbitrary strictly increasing transformation. Instead, it is invariant to certainnon-cardinal and strictly increasing transformations that are ˙ % i -dependent. Animplication of this fact is that these utility indices, even if normalized so thatthey have the same range for all agents, cannot be interpreted as precise units ofmeasurement.In addition to being representable as in (1a)–(1c) we will also assume thateach ˙ % i is strict in the sense that s i ( a, b ) = s i ( c, d ) if and only if ( a, b ) = ( c, d )or ( a, c ) = ( b, d ). This condition implies that no agent perceives an intensity-equivalence between distinct pairs of distinct alternatives. It is therefore analogousto the strictness condition of preferences. Under such uniform strictness we canassume without loss of generality (see Gerasimou, 2020; Section 4.1) that each ˙ % i s represented by a canonical s i in the sense that s i ( X × X ) = {− k, . . . , − , − , , , , . . . , k } , where k ≡ (cid:0) n (cid:1) is the number of distinct pairs of alternatives in X . That is,every agent’s intensity function is onto the same set of consecutive integers thatis symmetric around zero. Focusing on pairs ( a, b ) where a is preferred to b , thiscanonical normalization implies that the value of the agents’ canonical intensityfunctions at such a pair reflects the rank/position of that pair in the respectiveagents’ strict intensity rankings. It allows in turn for a novel kind of meaningful ordinal interpersonal comparisons of preference intensities to be assumed, withoutalso assuming interpersonally comparable utilities, cardinal, pseudo-cardinal, orotherwise.An intensity profile is an n -tuple ( ˙ % , . . . , ˙ % n ) of preference intensity relations,one for each agent i ≤ n . Every such profile corresponds to a unique canonicalpreference intensity function profile s = ( s , . . . , s n ). The second column of Table1 lists the number of distinct strict intensity relations when n = 3 , ,
5. Thesenumbers have been computed by distinct constraint-solving programs which areavailable from the author upon request. The third column lists the correspondingnumber of strict intensity profiles for such n . Table 1: The number of strict intensity relations and profiles for small values of n . n Strict intensity relations Strict intensity profiles3 12 12 , We are now in position to state the central assumption of this paper.
Ordinal Unweighted Interpersonal Comparisons of Intensities
Given a strict preference intensity profile ˙ % = ( ˙ % , . . . , ˙ % n ) that is canonicallyrepresented by s = ( s , . . . , s n ) , the statement s i ( a, b ) > s j ( a, b ) > is assumed to imply that agent i prefers a to b more than j does. Towards motivating this assumption, let us first recall that our underlying modelof preference intensities at the level of the individual decision maker effectivelyassumes that no intensity comparison of any agent can be quantified with any recision beyond the level of an ordinal ranking. The question now is whether,under the maintained assumption of such simple/non-quantifiable intensities, thedifferent agents’ intensity-difference rankings should be treated equally by the so-cial planner/matching platform or not. In particular, is it the case that, in theabsence of information regarding the degree to which agents i and j would suf-fer if they received b instead of a , the planner should declare that i prefers a to b more than j does if all that she knows is that the former intensity differencelies higher in i ’s ranking than the latter does in j ’s? Since the agents’ separateintensity orderings convey all the available welfare-relevant information, treatingthem in any way other than equal would call for a justification that appears elu-sive. Thus, the above assumption might be thought of as a reasonable benchmarkfor interpersonal comparisons in such an informational environment. We note, fi-nally, that the canonical interpersonal comparisons of preference intensities thatare postulated in the above assumption can be thought of as an ordinal analogueto the standard “relative utilitarianism” assumption that rests on interpersonalcomparisons of normalized von Neumann-Morgenstern utilities whose range is theunit interval for every agent. Fleurbaey and Zuber (2021) discuss the applicationof this normalization over the past 70 years and propose a new an alternative one,also in a cardinal-utility world, where the marginal utilities at the poverty line areequalized instead. Building on the equally-weighted interpersonally comparable intensities assump-tion of the previous section, we can now introduce the following novel notions ofdominance and efficiency over allocations.
Definition 1 (Intensity Dominance and Efficiency)
Given a strict intensity profile (cid:0) ˙ % , . . . , ˙ % n (cid:1) that is represented canonically by ( s , . . . , s n ) , and given two allocations x and y that are Pareto efficient with respectto the induced strict preference profile ( ≻ , . . . , ≻ n ) , x intensity-dominates y if ( x i , x j ) = ( y j , y i ) = ⇒ s i ( x i , x j ) ≥ s j ( y j , y i ) (2) for every such pair of agents ( i, j ) , and at least one inequality is strict. A Paretoefficient allocation x is intensity-efficient if it is not intensity-dominated. Thus, a Pareto efficient allocation x intensity-dominates y if, in every pair of agentsthat is “flipped” by x and y in the sense that both allocations assign the same twoalternatives a and b to the two agents in that pair but do so in opposite ways, the gent receiving a (which, under the postulated Pareto efficiency, is the mutuallypreferred one) under x prefers it to b more than the agent receiving it under y .Therefore, if allocations x and y are Pareto efficient and x intensity-dominates y ,then the interpersonal preference trade-offs in all pairs of agents that receive thesame two alternatives under x and y but in reverse order are always resolved by x in favour of the agent in the pair who prefers the relevant alternative more. Thus,the concept of intensity efficiency conforms with intuitive principles of distributivejustice. Moreover, it appears to be the first refinement of Pareto efficiency that isoperational in an environment where neither the agents’ utilities are required tobe inter- and intra-personally comparable nor monetary transfers between agentsare assumed to be feasible. Proposition 1
An intensity-efficient allocation exists under every strict intensity profile (cid:0) ˙ % , ˙ % , ˙ % (cid:1) on X = { a, b, c } . Proof of Proposition 1 . Let D be the intensity-dominance relation over allocations that is defined by (2).Suppose to the contrary that w Dw D . . . Dw k Dw (3)for Pareto efficient allocations w , . . . , w k on X := { a, b, c } . Notice that, because n = 3, for any two allocations w i , w i +1 such that w i Dw i +1 we must have w il = w i +1 l for exactly one agent l ∈ { , , } and ( w ij , w ik ) = ( w i +1 k , w i +1 j ) for j, k = l . Noticealso that n = 3 implies k ≤ k = 2 because D is asymmetric by construc-tion. Suppose k = 3. Given the remark in the previous paragraph, we can start bywriting, without loss of generality, w = ( a, b, c ), w = ( b, a, c ) and w = ( b, c, a ).Since w and w are D -incomparable by construction, the postulate w Dw iscontradicted.Now suppose k = 4. By (3) and the above implications, we may take w , w , w to be as in the k = 3 case, from which it then follows that allocation w must satisfy w = ( c, b, a ). Together with the postulated Pareto efficiency, w Dw implies s ( a, b ) > s ( a, b ) > w Dw implies s ( a, c ) > s ( a, c ); w Dw implies s ( b, c ) > s ( b, c ); and w Dw implies s ( a, c ) > s ( a, c ). Since all s i are canonical, s ( a, c ) > s ( a, c ) > s ( a, c ) implies s ( a, c ) = 3 and s ( a, c ) = 1. Given the latter,however, it is impossible for s ( a, b ) > s ( a, b ) and s ( b, c ) > s ( b, c ) to both betrue. Thus, the postulate w Dw is contradicted. ext, suppose k = 5. Arguing as above, allocations w , . . . , w in (3) must beas in the previous two cases, while allocation w must satisfy w = ( c, a, b ). Thisis D -incomparable to w by construction, which contradicts the postulate w Dw .Finally, suppose k = 6. With allocations w , . . . , w in (3) being as above, w must satisfy w = ( a, c, b ). It follows from w Dw that s ( a, c ) > s ( a, c ), while s ( a, b ) > s ( a, b ) follows from w Dw and s ( b, c ) > s ( b, c ) is implied by w Dw .Recalling again that each s i is canonical, from these inequalities and the onesimplied by w Dw Dw Dw (cf the case of k = 4 above) we deduce the following:(i) s ( b, c ) = 1 is implied by s ( b, c ) > s ( b, c ) and s ( b, c ) > s ( b, c ); (ii) s ( a, b ) = 3and s ( a, b ) = 2 is implied by s ( a, b ) > s ( a, b ) and s ( b, c ) = 1; (iii) s ( a, c ) = 3follows from (i) and (ii); (iv) s ( b, c ) = 2 follows from s ( b, c ) > s ( b, c ) = 1and s ( a, b ) = 3; (v) s ( a, c ) = 1 follows from s ( a, b ) = 3 and s ( b, c ) = 2; (vi) s ( a, b ) = 3 is implied by s ( a, b ) > s ( a, b ) = 2; (vii) s ( b, c ) = 2 is implied by s ( b, c ) > s ( b, c ) = 1; (viii) s ( a, c ) = 1 follows from (vi) and (vii). Therefore, wehave s = s = s , where a ≻ b ≻ c and a ≻ i c ≻ i b for i = 1 , w = ( a, c, b ) is Pareto dominated by w = ( a, b, c ).Since D is always acyclic when n = 3, and because the set of allocations is fi-nite, we conclude that an intensity-efficient allocation always exists in that case. (cid:4) Example
Suppose X = { a, b, c } and consider the canonical intensity function profile ( s , s , s )where s ( a, c ) = 3 , s ( a, c ) = 3 , s ( a, c ) = 3 ,s ( a, b ) = 2 , s ( b, c ) = 2 , s ( a, b ) = 2 ,s ( b, c ) = 1 , s ( a, b ) = 1 , s ( b, c ) = 1 . This induces the homogeneous strict preference profile ( ≻ , ≻ , ≻ ) where a ≻ i b ≻ i c for i = 1 , ,
3. Since the three agents’ preferences coincide, all six allocations( a, b, c ) , ( a, c, b ) , ( b, a, c ) , ( b, c, a ) , ( c, a, b ) , ( c, b, a )are Pareto efficient. Now, because s ( a, b ) = s ( a, b ) = 2 > s ( a, b ), allocations( b, a, c ) and ( c, a, b ) are intensity-dominated by ( a, b, c ) and ( c, b, a ), respectively.Moreover, since s ( b, c ) = 2 > s ( b, c ) = s ( b, c ) = 1, ( a, c, b ) and ( b, c, a ) are lso intensity-dominated by ( a, b, c ) and ( c, b, a ), respectively. The latter two al-locations, finally, are incomparable by intensity-dominance. Thus, these are theintensity-efficient allocations under this profile. Therefore, even in this exampleof identical preferences and almost identical intensities, the refinement of intensityefficiency allows for discarding 2/3 of Pareto efficient allocations. ♦ n ≥ The example strict intensity profile shown in Table 2 on X = { a, b, c, d, e } clarifiesthat an intensity-efficient allocation does not always exist when n ≥
5. Therefore,additional restrictions must be imposed to guarantee existence in such cases.
Table 2: An intensity profile with n = 5, for which no intensity-efficient allocation exists. ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ s i ( a ′ , b ′ ) = Agent i = 1 2 3 4 510 ( a, e ) ( a, e ) ( a, e ) ( d, · ) ( c, · )9 ( a, d ) ( a, d ) ( a, d )8 ( b, e ) ( b, e ) ( a, c )7 ( b, d ) ( a, c ) ( b, e )6 ( a, c ) ( b, d ) ( b, d )5 ( c, e ) ( c, e ) ( c, e )4 ( a, b ) ( c, d ) ( c, d )3 ( b, c ) ( a, b ) ( d, e )2 ( c, d ) ( b, c ) ( a, b )1 ( d, e ) ( d, e ) ( b, c ) In particular, Table 2 shows that agents 1 – 3 have identical preferences whichare captured by a ≻ i b ≻ i c ≻ i d ≻ i e for i = 1 , ,
3, but their intensity orderings are distinct. On the other hand,the last two columns in Table 2 are meant to be read as suggesting that agents4 and 5 prefer d and c over everything else, respectively. Other than that, thepreferences and intensities of these two agents are inconsequential for the argument.Therefore, Pareto-efficient allocations that correspond to the preferences inducedby ( s , . . . , s ) are( c, a, b, d, e ) | {z } s , ( c, b, a, d, e ) | {z } t , ( a, b, c, d, e ) | {z } x , ( a, c, b, d, e ) | {z } y , ( b, c, a, d, e ) | {z } w , ( b, a, c, d, e ) | {z } z . Now let D stand for the intensity-dominance relation over allocations. Start by onsidering allocation s . We have s ( b, c ) > s ( b, c ) . Thus, zDs . Now consider allocation z . We have s ( a, c ) > s ( a, c )Thus, wDz . Now consider allocation w . We have s ( a, b ) > s ( a, b ) . Thus, yDw . Now consider allocation y . We have s ( b, c ) > s ( b, c )Thus, xDy . Now consider allocation x . We have s ( a, c ) > s ( a, c ) . Thus, tDx . Finally, allocation t . We have s ( a, b ) > s ( a, b ) . Thus, sDt . We have therefore arrived at the intensity-dominance cycle sDtDxDyDwDzDs over the all Pareto efficient allocations. Hence, no intensity-efficient allocationexists for this profile.
Our approach to the assignment problem is to our knowledge the first that focuseson a refinement of Pareto efficiency that incorporates information on the agents’preference intensities without assuming existence (or requiring the elicitation of)cardinal utility functions, e.g. via expected-utility preferences over lotteries overthe alternatives of interest that are, in addition, quasi-linear in money. Yet, ourunderlying motivation of Pareto efficiency being a weak normative requirementin welfare economics and matching theory has also been raised explicitly in thisliterature, for example in Che, Gale, and Kim (2013), Lee and Yariv (2018) andPycia and ¨Unver (2020). A key methodological difference in this regard between hese authors’ approach and ours is that we do not employ any social welfareaggregation method that builds on interpersonally comparable utilities –cardinalor otherwise– in order to define a social ranking over allocations/matchings. In-stead, our approach assumes interpersonally comparability of the agents’ ordinalstrict preference intensity relations –which is operationalized through the canonicalnormalization of their preference intensity functions– and puts this interpersonalcomparability to use by means of the intensity efficiency refinement of the Paretocriterion. This requires a different kind of information from the agents in additionto their ordinal preferences, and is based on a partial dominance relation. It istherefore logically distinct from any efficiency refinement that is derived from anynotion of social welfare aggregation. References
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Fleurbaey, M. and S. Zuber (2021): “Fair Utilitarianism,”
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