An agent who lacks preferences and instead makes decisions using criteria
that are costly to create should select efficient sets of criteria, where the
cost of making a given number of choice distinctions is minimized. Under mild
conditions, efficiency requires that binary criteria with only two categories
per criterion are chosen. When applied to the problem of determining the
optimal number of digits in an information storage device, this result implies
that binary digits (bits) are the efficient solution, even when the marginal
cost of using additional digits declines rapidly to 0. This short paper pays
particular attention to the symmetry conditions entailed when sets of criteria
are efficient.Comment: In Proceedings TARK 2015, arXiv:1606.0729

Mandler, Michael

Electronic Proceedings in Theoretical Computer Science

English

arXiv

Michael Mandler

Crossref

The optimality of coarse categories in decision-making and information storage

An agent who lacks preferences and instead makes decisions using criteria that are costly to create should select efficient sets of criteria, where the cost of making a given number of choice distinctions is minimized. Under mild conditions, efficiency requires that binary criteria with only two categories per criterion are chosen. When applied to the problem of determining the optimal number of digits in an information storage device, this result implies that binary digits (bits) are the efficient solution, even when the marginal cost of using additional digits declines rapidly to 0. This short paper pays particular attention to the symmetry conditions entailed when sets of criteria are efficient

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R. Ramanujam (Ed.): TARK 2015EPTCS 215, 2016, pp. 227–230, doi:10.4204/EPTCS.215.16c© M. MandlerThe optimality of coarse categories in decision-making andinformation storageMichael MandlerDepartment of Economics Royal Holloway College, University of London Egham, UnitedKingdomAn agent who lacks preferences and instead makes decisions using criteria that are costly to createshould select efficient sets of criteria, where the cost of making a given number of choice distinctionsis minimized. Under mild conditions, efficiency requires that binary criteria with only two categoriesper criterion are chosen. When applied to the problem of determining the optimal number of digitsin an information storage device, this result implies that binary digits (bits) are the efficient solution,even when the marginal cost of using additional digits declin s rapidly to 0. This short paper paysparticular attention to the symmetry conditions entailed when sets of criteria are efficient.1 IntroductionSuppose that agents, rather than forming a separate preference judgment for each pair of alternatives,make decisions usingcriteria. A criterion orders a small number of categories, each of which consistsof many alternatives. The potential of a criterion to order alternatives within another criterion’s cate-gories allow sets of criteria to generate large numbers of chice distinctions. If an agent has objectivepreferences that can be inferred from a large set of sufficiently discriminating criteria, the agent will bebetter off if more of the criterion orderings are discovered: the agent will then be able to determine theoptimal allocation from more choice sets. The uncovering ofm re criterion discriminations is costly,however, and we therefore considerefficientpoints where the cost of making a given number of choicedistinctions is minimized.This optimization problem seems to lead to a trade-off. Given a number of choice distinctions, anagent could either use a large set of coarse criteria (criteria with only a small number of categories)or a small set of finer, more discriminating criteria. We showunder mild conditions that large setsof coarse criteria always lead to reductions in decision-making costs. Binary criteria with only twocategories per criterion therefore provide the only efficient arrangement. Under mild restrictions on howcriteria are aggregated into decisions, binary criteria lead torational choice functions, where decisionsare determined by a complete and transitive binary relation.We apply our model to the problem of determining the optimal number of digits in an informationstorage device. We show that, even if the marginal cost of additional digits declines rapidly to 0, binarydigits (bits) offer the efficient solution.In this short paper, we pay particular attention to the symmetry conditions that are entailed when setsof criteria are efficient. A full working paper [2] is available on-line.2 An outline of the modelA criterion Ci is an asymmetric binary relation on a domain of alternativesX and aset of criteriais de-notedC = {C1, ...,CN}. Two alternativesx andy are deemedCi-equivalent ifx andy share the same set228 The optimality of coarse categories in decision-making andinformation storageof Ci-superior alternatives and the same set ofCi-inferior alternatives (see [1]). ACi-categoryis a maxi-mal set ofCi-equivalent alternatives ande(Ci) denotes the number ofCi-categories. Thediscriminationvectorof C is (e(C1), ...,e(CN)). A Ci is coarserthanC′i if e(Ci)< e(C′i ).Let c be a choice function on a domain of finite subsets ofX. Two alternativesx andy are in thesamechoice classof c if c treats them interchangeably: first, whenx is chosen andy is available thenyis chosen too, and second, ifx but noty is available thenx is chosen if and only if, wheny is availableand notx, y is chosen.A choice functionc usesC , denoted(C ,c), if c does not make distinctions that are not alreadypresent in the criteria: for each set of alternativesA that contains only alternatives that are in the sameCi-category,i = 1, ...,N, there is a choice class ofc that containsA.Let κ(Ci) denote thecost of criterion Ci. We assumeκ(Ci) is determined by the number ofCi-categories and therefore also writeκ(e) to denote the cost of aCi with e categories. We assume thatthecost of a set of criteria, κ [C ], equals the sum of the costs of the criteria inC . Letting n(c) be thenumber of choice classes inc, a pair(C ,c) is more efficientthan the pair(C ′,c′) if n(c) ≥ n(c′) andκ [C ] ≤ κ [C ′], with at least one strict inequality, and(C ,c) is efficient if there does not exist a(C ′,c′)that is more efficient than(C ,c).The fundamental advantage of criteria is that each criterion can discriminate within the categories ofother criteria. Given constraints that specify that criterionCi can have no more thanei categories (and as-suming that|X| is sufficiently large), we can find a(C ,c) such that (i) there is a partition ofX with ∏Ni=1 eicells such thatx andy are in distinct cells if and only if they lie in differentCi -categories for at least oneiand (ii) each cell of this partition forms a choice class ofc. Subject to theei constraints, this(C ,c) max-imizesn(c) and accordingly we define(C ,c) to maximally discriminateif n(c) = min[∏Ni=1e(Ci), |X|].3 Main results(1) Since criteria with only one category make no discriminations and require no decisions, we assumethey are costless. To compare a(C ,c) and(C ′,c′) that have the same number of costly categories, sup-pose that∑Ni=1 (e(Ci)−1) = ∑N′i=1(e(C′i )−1). Assume also that either (i) the marginal cost of categoriesis increasing and the smaller ofn(c) andn(c′) is less than the cardinality ofX or (ii) the marginal costsof categories is strictly increasing. We show that ifC has greater proportions of coarser criteria thandoesC ′ and if (C ,c) maximally discriminates, then(C ,c) is more efficient than(C ′,c′).(2) Fix a set of domains that, for each finitem, contains aX with m elements and call a domainadmissibleif it is drawn from this set. Then, every efficient(C ,c) where the domain is admissible has aC that contains only binary criteria if and only ifκ(e)> κ(2)⌈log2e⌉ for all integerse> 2.Thus the cost ofecategories can rise as slowly as log2 e– in which case the marginal cost of categoriesdescends to 0 – and still the only efficient arrangement is forall criteria to be binary.(3) We apply the result in (2) to information storage. Suppose we wish to store some integer between1 andn usingN k-ary digits and that the cost of storage equalsκ(k)N. We show that, for all positiveintegersn, binary digits are the minimum-cost storage method if and only if κ(k)> κ(2)⌈log2 k⌉ for allintegersk> 2.(4) We specify axioms for how to aggregate sets of criteria into choice functions that generalizeweighted voting. Suppose that the choice functionc i the pair(C ,c) satisfies these axioms, thatCcontains only binary criteria, and thatc satisfies the following Condorcet rule: if there is ax in a choicesetA that is chosen byc from all {x,y} with y∈ A thenx is chosen fromA too. Thenc makes selectionsthat maximize a complete and transitive binary relation. Given (2), we conclude that in a broad range ofM. Mandler 229cases, efficient decision-making is rational.4 Symmetry and maximal categorizationMaximal discrimination is necessary for decision-making effici ncy since otherwisen(c) could be in-creased without an increase in costs. The key feature required fo a(C ,c) to maximally discriminate isthat the following property ofC , calledmaximal categorization, is satisfied: thediscrimination partitionP of X that placesx andy in distinct cells if and only ifx andy lie in differentCi -categories for at leastonei must have∏Ni=1e(Ci) cells.We will now see that ifX is a product of attributes and each criterion orders a distinct attribute, thenmaximal categorization is satisfied and conversely if maximl categorization is satisfied then we can labelalternatives so thatX becomes a product of attributes. By joining this conclusionto result (2), that onlybinary criteria are efficient, we can describe efficient decision-making concisely: to be efficient agentsmust be able to describe the alternatives inX so that they form a product of attributes and each criterionmust divide a distinct attribute into exactly two categories.The simplest way to achieve maximal categorization is forX to be formed by a product of attributesand for eachCi to divide X into categories based only on attributei. The domainX might be a set ofcars, and the attributes might be colors, top speeds, and prices. A ‘speed’Ci would then order cars basedon the ranges of top speeds thatCi deems to be equivalent.Formally, anattribute is a setXi andN attributes define the domain of alternativesX = ∏Ni=1Xi. Wewill say that a set of criteriaC is based on a product of attributesif for eachCi there is a setXi and apartition of{X1i , ...,Xe(Ci )i } of Xi such that the categories ofCi are the setsXji ×(∏k6=i Xk), j = 1, ...,e(Ci).So, ifCi is an ordering of cars by color then eachXji would represent a color andx andy would be placedinto distinctCi-categories if and only if theith coordinates ofx andy indicate different colors:xi ∈ Xjiandyi ∈ Xki where j 6= k. The cells of the discrimination partitionP would then be the∏Ni=1e(Ci) setsX j11 × ·· · ×XjNN where, for eachi, j i is an integer between 1 ande(Ci). Maximal categorization thusobtains.This treatment assumes thatX is a product space: for each possible combination of attribues (eachpossible color-speed-price combination), there is a corresponding element ofX. But for maximal catego-rization it is enough that there issomealternative inX for each combination of attribute ranges specifiedby the criteria, that is, it is sufficient forX to be a subset of∏Ni=1Xi such that eachXj11 × ·· · ×XjNNintersectsX.A set of criteriaC that is based on a product of attributes enjoys a wide-ranging symmetry property.Fix someCi in C , and consider a setE−i formed by an arbitrary union of the categories of the remainingcriteriaCj , j 6= i. Given the product structure ofC , any suchE−i must intersect each of theCi-categories.To continue the car example, the set of carsE−i defined by a certain range of top speeds and prices canbe partitioned into all the possible color subsets, say blue, red, and yellow. If we useCi to order the cellsof the color partition of the cars inE−i, the ordering will have the same ‘shape’ as – be order isomorphicto – the original color orderingCi of X. If, for example,Ci onX is a cycle – blue is better than red whichis better than yellow which is better than blue – then theCi ordering of any set of cars defined by a rangeof speeds and prices will also form a cycle. We conclude that any two sets of carsY andZ defined byselections of non-color attributes will be order isomorphic to each other when each is endowed with thecolor orderingCi (or rather the restrictions ofCi to Y andZ).This symmetry property may seem to be of limited value since it appears to apply only to products ofattributes. But in fact the symmetry property characterizes anyC that maximally categorizes. If for an230 The optimality of coarse categories in decision-making andinformation storagearbitrary (possibly nonproduct)C , we applyCi to someE−i and it defines fewer thane(Ci) Ci-categorysubsets then the discrimination partitionP would have to contain fewer than∏Ni=1 e(Ci) cells. And theonly way thatE−i andE ′−i can each define(Ci)Ci-category subsets is for theCi ordering of these subsetsto be order isomorphic.Moreover, if an arbitrary (possibly nonproduct)C enjoys the symmetry property we can relabel theelements of the domainX so thatC is then based on a product of attributes. To do this, we associateeachCi with an attribute (e.g., color) and identify eachCi-category with an arbitrary valueXji for thatattribute (e.g., blue): each cell ofP is thus identified with a vector of attribute values. So, although aproduct of attributes looks special, it provides a model forany set of criteria that maximally categorizes.The following definitions and theorem make these claims precise. We useE1i , ...,Ee(Ci)i to denote thecategories of criterionCi .Given the set of criteria{C1, ...,CN}, E−i is aunion of C−i-categoriesif E−i =⋃j Ejk for some collec-tion of criterion categories{E jk} such thatk 6= i for each j. Let CE−ii denote the binary relation definedby ECE−ii E′ if and only if there areCi-categoriesEi andE′i such thatE = Ei ∩ E−i, E′ = E′i ∩ E−i, andxCi y for x∈ E andy∈ E′. We then sayC satisfies theorder-isomorphism propertyif for any i and anytwo unions ofC−i-categories,E−i andE ′−i, the binary relationsCE−ii andCE ′−ii are order-isomorphic.The set of criteriaC has aproduct representationif (i) for each i, there is a nonempty setYi and apartition{Y1i , ...,Ye(Ci)i } of Yi , (ii) there is a set of criteriaĈ = {Ĉ1, ...,ĈN} defined onY = ∏Ni=1Yi where the categoriesof eachĈi are the setsYji ×(∏k6=i Yk), and (iii) for eachi, there is a order-preserving bijectionf betweenthe categories ofCi andĈi, that is,Eji CiEj ′i if and only if f(E ji)Ĉi f(E j′i).We then have the following result.Theorem. For a set of criteriaC , the following statements are equivalent: (i)C maximally categorizes,(ii) C satisfies the order-isomorphism property, (iii)C has a product representation.5 BibliographyReferences[1] Mandler, M., 2015, Rational agents are the quickest. J. Econ. Theory 155: 206-233, doi:10.1016/j.jet.2014.10.003.[2] Mandler, M., 2014, Coarse, efficient decision-making. Available at SSRNhttp://ssrn.com/abstract=2494600.

The optimality of coarse categories in decision-making and information storage

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