A choice theory of spatial interaction

Regional Science and Urban Economics 5 (1975) t37- 176. 0 North-Holland

A CHOICE

THEOltY

OF SPATIAL INTERACTION” Tony E. SMITH

A general probabilistic theory is developed for modeling a variety of spatial interaction choices by individual behaving units. This theory, designated as the Choice Theory, is shown to be consistent with a wide class of empirical spatial interaction hypotheses, designated as Gravity Hymtheses. Moreover, it is shown that under certain structural conditions the Choice Theory completely char~ctcrizes this class of hypotheses. Finally, a number of explicit statistical procedures for testing the Choice Theory ate developed and discussed.

1. rntroduction

One of the central problems of interest to regional scientists is the explanation and prediction of patterns of human spatial interaction. At the micro level, this spatial interaction behavior can be characterized in terms of a choice paradigm for a single behaving unit among a number of possible interaction opportunities distributed over space. A common approach to modeling this paradigm has been to characterize choice behavior in terms of certain interaction probabilities, which may be taken to reflect either the relative likelihood of specific choices in a given situation or the relative frequencies of choices over a series of similar situations.’ In this context the decision process of the nc involving a trade off between the positive behaving unit has been characterized cI., .&, attraction of each relevant opportunity and the negative deterrewe of their distances from the behaving unit. ’ In particular, the choice probability associated

with any given opportunity is assumed to vary directly with some measure of the attractiveness of that opportunity and inversely with some measure of the distance to that opportunity. The classical form of this type of spatial interaction hypothesis is the Gravity Hypothesis, first s!Jggested by the work of Stewart (1948), which asserts that the probability of interacting with any given opportunity should be proportional to the ratio of these two quantities. Sinie the original introduction of this hypothesis some twenty-five years ago, a wealth of literature has immergcd attempting to refine, extend, and modify the hypothesis. These efforts have focussed mainly on the particular forr,l in *‘I%~ rtscmli ws ~ttpportcc! by the National Science Foundation under grant no. GS-35640X. ‘A recent survey of such choice models in tltc cantcut of forxasting trawl demand is ?.iLrn in Nrand (1’373). 2&e, for example, the classical survey in lsrtrd (1960, ch. 111, A more recent survey is aiven in Ruiter (1973).

138

T.E. Smith, Chr,ice theory of spatial interaction

which distance enters the rclatiqn. While various theories have been advanced in support of particular distance relations, 3 the conclusions of these theories are in fact more notable for their similarities than for their differences. As we thaw (in section 2.3.2 below) if one adopts a broad behavioral approach to distance, each of these theories- can be viewed as yielding specific forms of the original Gravity Hypothesis in which physical distance is replaced by some measure of ‘behavioral’ distance. Hence the central purpose of this paper is to ask what behavioral content can be given to this collection of hypotheses as a wholen To do so, we begin ty developing a broad class of spatial interaction systems within which to study such hypotheses. We then formulate a rather general definition of the Gravity Hypothesis within this context. Finally we develop a Choice Theory of spatial interaction behavior within such systems which is consistent with the Gravity Hypothesis. Our main result is then to show that for systems in which all pairwise choice situations are possible, i.e., which are pairwise complete, this Choice Theory completely characterizes the Gravity Hypothesis in the sense that a behaving unit’s spatial interactions are consistent with the Gravity Hypothesis if ,and only if they are consistent with the Choice Theory. In particular we show that this theory implies the existence of an attraction function over opportunities and a behavioral distance function for the behaving unit such that all interaction probabilities are representable by these two functions within the classical gravity model. Moreover, we show that these two functions are essentially unique for the behaving unit, in that like physical dist:.nce for example, they are indeterminant only up to a choice of uni,s. These results su~~.z.t that the Choice Theory may well provide a unified foundation upon w&h a number of more specific theories of spatial interaction might be constructed. For example, it is of interest to ask what additional behavioral assumptions are necessary in order to imply the specific forms of attraction and behavioral distance f’unctions currently in use. We return to this question in the concluding remarks of this paper {section 5.1 below). Because of the technical nature of the analysis to be developed, we choose to present a self-contained informal discussion of the theory and its implications in section 2 below. In section 3, we then formalize the theory and analyze its consequences in detail. ln section 4, we attempt an initial evaluation of the behavioral appropriateness of the theory and outline a number of explicit methods for testing its behavioral assumptions. The paper conclurtes in section %Iost notable among thesethcocicsarc (I) the ‘search thcoricsof Schwidcr (1959). Harris (1964) and others, (2) the ‘entropy’ theories of Wilson (1967) and others;. and (3) ths ‘utility’ theories of Niedercornand Bcchdolt (1969), Bcckmann, Gustafson and Golob (1973) and others. ‘The possibility of unifying these hypotheses has been rccognizcd by a number of authors. Most notable among these is the recent work of Choukroun (1975) in which a unification of these hypothesesis achieved at the macro level by employing the statistical theory of ‘mixture distributions to aggregate individual interaction prob:lbilitics.

5 with a bribf discussion of certain possible extensions of both the theory and the methods of‘ealibrating and testing the theory. 2. Overview of the theory To fix the general setting for the analysis, we first outline our notion of spatial interaction systems more explicitly. In this context, the behavioral assumptions of the Choice Theory are then motivated and discussed. Finally, we develop a general definition of the Gravity Hypothesis for spatial interaction systems, and summarize the major consequences of the Choice Theory for this hypothesis. 2.1. General self ing Consider a behaving unit. or spatial actor, a situated in some space ,Y of locations s. These locations may be physical locations, or more generally, may represent positions within any relevant space of attributes. We next suppose that there exists some universe S of potential opportunities i, j, k, . . . for a, and that each of these opportunities may occupy various locations in A’. These potential opportunities for a mry include other behaving units, collections of behaving units, physical facilities, or in general, any definable entities which can meaningfully be said to occupy positions in A’. The pair of sets X and Q will be called a spatial interactiort context for a. 2.1.1. Spatial interaction structures

Within any spatial interaction context for c1there will generally exist a broad spectrum of possible spatial interaction situations which a may confront. Each such situation is characterized by the specific set /of opportunities in R which a cor,siders to be relevant at the moment, together with the particular configuration of those opportunities over the space A’. To be more precise we now designate a conflgtrration c for / and a in X to be any assigr,ment of the opporrunitics i E I to specific locations cI E Xtogether with an assignment of a himself to some locatkn c, E A’. While all possible combinations of opportunity sets I ;;nd configurations c could theoretically constitute choice situations for a, we shall assume that in any relevant situation a can only consider finitely many clpportunities explicitly. Hence we now designate each pair (I, c) consisting rf a non-empty finite subset I of s1 together with a configuration c for I and a in Xas a spatial iuteroctim situutiov for a. The set I will be called the op/>outunillp \rt for that situation, In each spatial interaction context, the set of spatial interaction situations which are potentially relevant for a will bc called a .v,srjul interaction itnrcture Y for a. This structure will usually not contain all conceivable spatial interaction situations within the given context. For example. if the relevant universe of opportunities s? is very large (possibly infinite), then Y’will generally only involve those subsets Iof 52which are sufficiently small to warrant practical consideration.

140

TX. Smith, Choice theory of spatial inleraction

2.12. iiiustrativtiPxamples As a first illustration of the above concepts suppose that a is an economic consumer and that we are interested in modeling a’s leisure trip behavior. Then

as an appropriate spatiai interaction context we might take X to be some represeniation of physical space ar.d take L? to include the set of potential recreational opportunities (museums, movie theaters, parks, etc.) which a might consider to be relevant in various situations. Within this general contiut we may imagine a specific situation for example in which a wishes to see a Then on the basis of available information (newspaper listings, elc.) we may suppose that a considers some finite set I of available movies in $2and

movie.

makes a choice among these. Such a choice problem would thus be characterized by the spatial interaction situation (1, c) where c denotes the configuration in X consisting of a’s location together with the locations of all theaters in I. The collection of all such situations which might be relevant for a would thus constitute the appropriate spatial interaction structure 9’ for a. As a second illustration we consider a case which need not involve physical space at all. Suppose that a is a registered voter and that we are interested in modeling a’s voting behavior. Here we may imagine that X is some space of possible positions on a number of reievant political issues for a, and that R includes the set of potential political candidates who might immerge and take stands on these issues. In this context a configuration c for any set I of relevant political candidates would consist of a’s own position on these issues together with the positions of all political candidates in I. Hence each possible voting situation (1, c) for a would be characterized by the set I of candidates in R whom a explicitly considers voting for, together with the relevant configuration c of positions in X. The collection of all such potentially relevant voting situations in this context would thus constitute the appropriate spati;, interaction structure 4r 1;7ra. 2.1.3. Spatial interaction systems Given any spatial interaction structure 9’ for a, we now wish to model a’s choice behavior for each possible situation (I, c> in 9 by means of a probability distribution over his choices in I. To do so we must of course assume that in each such situation a confronts a set of mutually exclusive and collectively exhaustive choice alternatives. Hence we first assume that in any situation (I. c} in 9, a chooses at most one opportunity in I to interact with.’ SA number of situations which appear tti involve mulriplc interaction chuiccs can noncthelc.ss be treated within the context of this assumption. Consider for cxamplc a multiple-purpose shopping trip for a. WMc such a trip may involve a joint decision to visit several stores, a in fact visits only one store at n time. Henceif the first store on a’s shopping list is i then we may take the relevant observable outcome of this choice situation to be an interaction with i. Moreover, assuming that a is free to reconsider his list of remaining stores after having shopped at i, we may treat his decision to visit a second store on the list as a new spatial interaction situation in which a is now located at i. and so on.

T. E. Smith. Ckv’ce theory of spatial interaction

141

MoreoT;er, we must assume that a U/KXS chooses some opportunity from I. However, it is quite possible that a may choose not to interact with any of the opportunities he explicitly considers. For example, in the theater illusrration cisection 2. I .2,cr may find no movie to his liking and hence chooses to stay home. Thus we wish to allow the possibility of such a ‘no interaction’ choice for a. One simple way to do so is to include a himself among the potential interaction opportunities in R and to interpret CI’Schoice of the opportunity ‘a’ in any situation as effectively exercising his ‘no interaction’ option (i.e., interacting with himself). With this objecti\:e in mind; we henceforth assume that the universe of opportunities R alwa_r~i~rclir&s the elemetlt u. If in any situation (1. c} the spatial actor a considers his ‘no interaction’ option to be relevant, this will be reflected by the inclusion of the opportunity a in 1. On the other hand, if a considers himself to be committed to some overt interaction (such as an essential shopping trip, or a moral obligation to vote), then this commitment will be I dlected by the absence of a from I. With these assumptron: the set of choice alternatives I in any spatial interaction situation clearly satisfies both mutual exclusiveness and collective exhaustiveness. Hence within each such situation , or alternatively, as a measure of the relative frequency with which a will choose i in a series of (I, c) situations. We make one final assumption regarding these choice probabilities. In ‘particular, since the opportunity set I in each situation (I, c> is taken to consiqt oniy of those opportunities which u explicitly considers interacting with, it is natural to suppose that each such opportunity has a definite chance of being chosen, i.e., that the interaction probabilities P<,,=,(i) are positice for each i E I. Alternatively, one may view this positivity assumption as effectively defining the relevant opportunity set I, i.e., in each situation (I, c) the set I is taken to consist of precisely those opportunities in f2 which have some positive probability of being chosen. With these conventions, we now designate each possible assignment P of positive probability distributions PCr,,> to every situation (I, c) in .Y as an itttcrtlctiorr probahilit~* SCJW~Wfor 9. The structure 9 togcthcr with any specific interaction probability scheme P for 9’ will be called a .r~~nf,+~l intc~zvfiort sysfem (.V’. P} for a.

Given the above notion of a spatial interaction system, WCnow develop n theory of O’S choice behavior within such systems. This theory, which we designate as tfre Choice Theory, focusses on the specific nature of a’s choice

142

TX.

Smith, Choice theory

of spotid

ittteructi~rr

0 l-r;kin arly given spatial interaction situation. Hence we begin by probabiBiaiecl *vJ positing a given spatial interaction system (.Y, P} for a and motivate the behavioral assumptions of the theory in telms of the interaction probabilities for some arbitrary spatial interaction situation (/, c> within that system. The three behavioral assumptions of the Choice Theory are developed respectively in the three sections below. Th!: emphasis of this development is on the general motivation and explanation of the assumptions. A formal statement of each assumption is given in section 3.5 below. Moreover, a full evaiuation of the b+aviora! appropriateness of each assumption is postpc ned tinti sectio.7 4. I bdm, where their implications may be seen in perspective. 2.2. I. Independence axiom Our first assumption focusses on a’s attitude toward the opportunities in $2. To motivate the main idea, suppose that in the movie example of section 2.1.2 above, a starts aith a list of ten possible movies, and succeeds in narrowing his list down to two alternatives. In this context, we assume that a’s probable choice among the remaining pair of alternatives is the same as it would have been if he had started with only these two initially. In other words, a’s probable choice among the remaining pair is assumed to be illdependent of those alternatives already eliminated. Similarly, if in a voting situation a reduces his list of candidates down to only two, then we assume that his probable choice among these two is the same as it would have been if they were the only candidates running. More generally, we adopt the following assumption: Independence Axiom. 6 If a given spatial interactiortsituation (I, c> is reduced by eliminating all but two alternatkes irj I, saJ*i aud j, then the relutive probability of choosing i overj irrthe new situatiortremains the same.

This t.ype of behavioral assumption was fitst proposed by Lute (19591, and has since been applied to a variety of choice contexts.’ The behavioral implications of this axiom will be developed in sections 3.6. I and 4.1. I below. 2.2.2. Separability axiom Our second assumption relates more directly to N’S spdtia! attitudes. 7 3 motivate the main &a, suppose for example that a considers a possible shopping trip to one of two stores i and j located in the same shopping center. In this context we assume that a’s clloice between these stores does not depend on where “The formal statement of this axiom (section 3.5) in\olvcs no temporal ordering, and hcncc is symmetric with respect to additions and deletions of opportunities. See the IlISt paragraph ofsection 4.2.1 below for further discussion of this point. ‘See, for example, the applications cited in Lute (1959) and Luccand Galanter (I 963) and the more recent applications to travel behavior cited in Brand (1973).

thL shopping center itself is located. In other words, if a has to make the same trip regardless of which opportunity he chooses, then we assume that his trip considerations can be ‘separated’ from his considerations of opportunities. Similarly, if a is considering a choice between two pol.itical candidates i and j with identical positions on all relevant issues, then we assume that a’s choice will he based on considerations other than the issues. More generally, we adopt the following assumption : Separabilit_v Axiom. Given arly spatial interaction situation ((i j ;, c) in which i and j are located together, if the corlfguration c is altered in any way such that i andj remain located together. then a’s interaction probabilities remain the same. The behavioral implications 3.62 and 4. I.2 below.

of this axiom

will be developed

in sections

2.2.3. Accessibility axiom Our third and final assumption is also spatial in nature, and in particular. relates to a’s perception of opportunity accessibility in the space X. In essence we assume that increased accessibility to an opportunity does not inhibit a’s potential interactions with that opportunity. For example, suppose that a is observed to frequent two grocery stores i and j. In this context we assume that if store i is moved next door to u, then a is as likely to shop at i after the move as before. Similarly, suppose that a is considering a choice between two political candidates i and j. Then we again assume that if i’s position were coincident with a’s own position on all the ‘ssues, then Q would be at least as likely to vote, for i in this situation as otherwise. More generally, we adopt the following assumption : Accessibility Axiom. If in a given spatial interaction siiuatioit ((i j). c> rite conjiguration c is altered only b)y moving opportunity i to a’s location, then a’s probability ofinteracting with i is not decreased. The behavioral implications of this axiom will be dcvelcped in sections 3.6.3 ar1.i 4.1 3 below. Given these three axioms, we now say that a spatial interaction system (,V’, P) is consistent with the Choice T/KwI’~ if and only if the Independence, Separability, and Accessibility Axioms arc satisfied by (.U, P). 2.3. Gravity hypothesis Having developed the behavioral axioms of the Choice Theory, we now wish to consider certain consequences of the theory. Our main interest focusses on

T.E. Smith, Choice theory of spatial interaction

144

the implications of this theory for the class of spatial interaction hypotheses which have come to be called ‘gravity hypotheses’. To do so, we must fist formulate an appropriate definition of such hypotheses within the context of spatial interaction systems. We begin by considering the claAca1 formulation of the Gravity Hypothesis, and then proceed to broaden this hypothesis in section 2.3.2 below by the introduction of ‘behavioral distance functions*. Finally we formulate in section 2.3.3 a general definition of the Gravity Hypothesis for spatial interaction systems. 2.3.1. Classical gravity hypothesis Recall that aside from the spatial actor himself, the Choice Theory essentially involves three primitive concepts: namely locations, opportunities, and interaction probabilifies. In contrast to this theory, the typical gravity-type models of spatial interaction add two new primitives to this list: namely distance and attraction. In particuiar it is generally hypothesized that a’s probability of interacting with any opportunity is inhibited by a’s perceived distance from the opportunity and efianced by the degree of attractiveness of the opportunity as perceived by a. More precisely, if we let a(i) denote some measure of the attraction of opportunity i for a, and let d(c,, CJ denote some measure of a’s perceived distance to i under any given configuration c, then this hypothesis asserts that in any spatial interaction situation (I, c) for a, the interaction probabilities P cr,,1(i) tend to vary directly with oc(i)and inversely with d(c,. ci). The simplest explicit formulation of this hypothesis is the Classical Graait} Hypothesis, first introduced by Stewart (1948), which asserts that P(,,,>(i) should be proportional to the ratio of these measures, i.e., that for some positive scale factor A,

holds for all

i E I. *

While this hypothesis has obvious appeal in terms of its simplicity, it possesses certain equally obvious shortcomings in terms of physical distance measurements. In particular, if we attempt to consider opportunities i situated at a’s location (i.e., with c, = ci) and if we take the distance from any location to itself to be zero, then expression (1) clearly entails division by zero. WC could of course avoid this problem by simply assuming that no opportunities can occupy a’s location (other than a himself). But if c: considers his relevant ‘location’ in a given spatial interaction situation to be his own neighborhood, ‘This expression broadly interprets Stewart’s original ‘demographic potential’ formulation, but the essential.form of the hypothesis is the same.

T.E. Srrith. Chcice theory of sf dtial interacrim

145

or his city, or his position on political issues, then clearly many .*elevant opportunities may occupy the same location as u. Hence it would appear that a more satisfactory solution to this problem would be to relax the requirement that the ‘distance’ from a location to itself be zero. In fact, this is precisely the approach commonly ‘employed by more recent formulations of gravity hypotheses. In these formulations distance enters the gravity relation by means of a transformation which renders zero quantities positive. To be more precise, if we let t denote *somegiven non-negative measure of distance (such as physical distance, travel time, travel cost, or even some type of ‘social’ or ‘political’ distance), then with respect to t we may obtain three types of gravity relations currently employed by making each of the following substitutions into expression (I), respectively,

where c, j? and 0 are all positive constants. These specifications of the gravity relation, which have each been studied by a number of authors,9 all have the property that the resulting d function is positive even when the t function is zero. Hence in each of these specifications the problem of division by zero is avoided. But having made this transformation from t to d, one may equally well interpret d to be the relevant distance measure in terms of expression (1). In other words, if any of the above specifications accurately represents a’s interaction probabilities, then we may view that particular d function as an appropriate measure of behavioral distance’ for representing a’s spatial interactions within the classical gravity framework. 2.3.2. Behavioral distance funcrions With this in mind, we now wish to reformulate the Classical Gravity Hypothesis in terms of th;* ‘behavioral’ distance functions exemplified by expressions (2) through (4). To develop a general characterization of such functions, we first observe that any meaningful notion of distance should at least satisfy the condition that no two locations be closer to one another than they are to themselves. Hence it is reasonable to require that for all locations X, J E X, any appropriate measure 9An excellent survey of these formulations (among others) is given in Choukroun (1975) and Kuiter (1973). See also the basic theoretical papers of Schneider (1959). Harris (1964) and Wilson ( :967) among others.

146

T.E. Smith, Choice theory of spatial interaction

of distance nshould satisfy d(x, x) 5 d(x, y). For example, if the distance function t employed in (2) through (4) satisfies this minimal condition, then the positive monotonicity” of the transformations in each of these expressions implies that the resulting d functions also satisfy this condition. Moreover, each of these d functions is also seen to satisfy the reasonable condition that the distance from any location to itself is the same for all locations, i.e., that for all X, J’ E X, d(s, A-) = d(y, y).

(6)

Hence the only essential difference between d 2nd r is that this constant distance from a location to itself is posirive for d. With these observations in mind, we now designate each positive function d satisfying (5) and (6) as a behavioral distancefunction on the space A’.” The d functions in expressions (2) through (4) are thus seen to be instances of behavioral distance functions under this definitional2 2.3.3. General gravity hypothesis Given this notion of behavioral distance we may formulate a general statement of the Gravity Hypothesis for spatial interaction systems as follows. First observe that the positivity of behavioral distances together with the nonnegativity of probabilities implies that the only measures Qf attractiveness r(i) which can be consistent with (1) are themselves non-negative. Moreover, if ci(i) = 0 for any opportunity i, then (I) must imply that Pf,,rl(i) = 0 for l0A transformationfon the real numbers is positive monotone if and only if for all numbers x and y, x P y impliesf(x) z i(y). “It is also of interest to note that the additional postulates which usually characterize physical distance (i.e., the ‘symmetry’ and the ‘triangle inequality’ conditions) may well bc inappropriate in many behavioral contexts. Cansidcr, for example, situations in which the spatial actor perceives travel cost to be the relevant measure of distance. Then the ‘symmetry’ condition may well fail for this distance measure since, for example, the cost of going down hill or down stream is not the same as the cost of going back up. Also the ‘triangle inequality’ condition may Fail For such distance measures since, for example, a sequence of short trips may be less costly than one long trip covering the same ground (a five-milt run is more exhausting than five one-mile runs). “It is of interest to note that the transformations in each of thcsc expressions arc in fact general methods ofconvcrtingarty distance Functioninto a behavioral distance function. Among the other types of transformations ;vhich also have thi; property arc the Be.~selfi~nrtiorrsstudied by Schneider and Choukroun [see choukroun (i975, section 4.6)]. Among those transformations which Fail to have this property are the power jmctions [Le., d(:.,y) = atb, VP; LX,/I > 01 employed in classical gravity formulations, and the gamma functions [i.e., d(x, y) = 0, ~9”exp [MS, JJ)]; j&O r 01 studied by Tanner (1961) and others.

7: E. Smith, Choice theory of y &al intvactior

147

every spatial interaction sitliation (I, c} in which i is involved. Hence, without loss of generality, we may (gnore all such opportunities and include in Q only those potential interaction Dpportunities which have pcuitiw attraction for a. With this convention, if we designate any positive function r on 52as an attractioufhction. then we now take the appropriate notion of opportunity attractiveness for u to be representable by an attraction function a on Q. Next observe from the positivity of a and n that we may ehminate the unknown scale factor I in expression (I) as follows. First, since all intetacticn probability distributions must sum to unity. it follows from (1) that for any spatial interaction situation (/, c}. (7) =

C i.r(i)!d(c,.

ri3

iel

Hence the positivity of a and d ailow us to solve for 1 in (7) and thus to write ( 1) explicitly in terms of a and d as

With these observations, we inay now state our general definition of the Gravity Hypothesis for any spatial interaction system (.V, P} for a. There e.lists u behavioral distance fumtion d on X and cm1attraction funcrion B 011R~such that for every spatial interachon situation ’ (1. c> in 9, a’s interactio!,probabilitiesare reprcscntable by eq. (8). Gravity Hypothesis,

__

if such a behavioral distance function and attraction function exist for system (9. P}. then we shall say that (.V, P> is consistent bith the Gravity Hypothesis.

WC are now ready to summarize the main consequences of the Choice Theory for the class of spatial interaction hypotheses encompassed by oui definition of the Gravity Hypothesis. These results, designated respectiveiy ir.s the Ccnsistency Theorem, the Equivalence Theorem, and the Uniqueness Theorem, will be developed in the three sections below.

148

T.E. Smith, Choice theory of spatial intcractiort

2.4.1. Consistency theorem Our first result is to show (in Theorem 2 below) that the Choice Theory is in fact entailed by the Gravity Hypothesis. In particular we establish the following consequence : Consistency Theorem. Every spatial interaction system consistent with the Gravity Hypothesis is also consistent with the Choice Theory. Hence if the spatial interaction behavior of a beh;dving unit is to be representable by any form of the Gravity Hypothesis [such as expressions (2) through (4)], it is essential that this behavior be consistent with the Choice Theory. In other words, our three axioms constitute a set of prerequisites which must be satisfied by any behavioral theory of spatial interaction systems which leads to gravity-type representations of interaction probabilities. In this sense, the Choice Theory might usefully be viewed as a general foundation upon which a number of more explicit behavioral theories can be built. (See section 5.1 below for further discussion of this point.) 2.4.2. Equivalence theorem While these three axioms are entailed by the Gravity h’ypothesis, it is important to observe that individually they are entailed by any number of different hypotheses. Hence, a far more interesting question is whether these three axioms taken together entail the Gravity Hypothesis itself. The main result of our analysis is to establish precisely the sense in which this is true. To motivate the structural conditions under which this result obtains, it is instructive to consider the special case of a spatial interaction system (9, P} in which only one spatial interaction situation (I, c) is possible. In such a restrictive case the Choice Theory tells us very little about o’s behavior. !n particular, since there is only one possible opportunity set, I, the Independence Axiom has no real meaning for this system. Moreo-:zt, since only one configuration, c, is possible, it follows that the Separability and Accessibility Axioms have no real meaning either. Hence it should be clear that the Choice Theory is only meaningful for those systems in which a variety of spatial interaction situations is possible. With this in mind, it is of interest to ask how ‘rich’ a variety of spatial interaction situations is necessary? One way to give substance to this question is to ask more specifically what range of spatial interaction situations would be required to construct a complete test of the Choice Theory for a given spatial interaction context? Since the axioms themselves are stated for the most part in terms of pairwise choice situations, it is readily apparent that such a test would require the observation of every possible pairwise choice situation in the given context. Hence for purposes of testing the Choice Theory, it is natural

T.E. Stnirh, Choice rheory

of spatial inleracliotr

119

to suppose that the relevant spatial interaction structure 9 for the given context contains all possible pairwise choice situations far 17. Within this setting. our main result (Theorem 6 below) is to show that for spatial interaction systems which are rich enough in potential interaction situations to permit a full test of the Choice Theory, the validity of the Choice Theory for such systems is equivalent to the validity of the Gravity Hypothesis. More precisely, if we now designate systems in which al! pairwise choice situations are possible as pairwise complete systems,’ 3 then our main result can be stated as follows: Equimience Theorem. A pairwisc complete spatial interaction system is consistent with tlte Gravity Hypotltesis if and only tyit is consistent witit the Choice Theory.

This result has several important theoretical consequences. First of all, since the Choice Theory makes no mention of the primitive concepts of distance and attraction, the Equivalence Theorem shows that the Gravity Hypothesis can be derived within a theo’ry in which neither of these primitives is postulated. Hence if one regards the notions of ‘locations’, ’ opportunities’, and ‘interaction probabilities’ as the externul primitives for the Choice Theory, then the behaviorai notions of ‘distance deterrence’ and ‘opportunity attraction’ now take on the role of internal primitives derivable within the theory itself. This internalization of distance and attraction implies one additional interesting consequence for the theoretical status of the Gravity Hypothesis itself. In particular it suggests a reversal in the role of this hypothesis relative to the distance and attraction measures by which it is defined. Traditionally, researchers have tended to start with specific measures of distance and attraction as given data (such as travel costs and population levels, respectively) and have entertained the Classical Gravity Hypothesis as one among many possible methods of combining such data to describe various types of spatial interaction behavior. However, the Equivalence Theorem suggests a strong rationale for starting with this hypothesis itself as the given mathematical structure and seeking to determine appropriate distance and attraction measures to ‘fill in’ this structure. More spec,fically, if empirical tests’* suggest that the Choice Theory is valid within a given (pairwisc complete) spatial interaction system, then this theorem implies that the Gravity Hypothesis must hold for some behavioral distance and attraction measures yet to bc determined. Hence the task of the researcher in this process is shifted from seeking the relevant mathematical structure for a I JA precise dcfinifion of pairwise cornplrrcncss is given in parallels the traditional assumption of individual choice theory are (al lcagt potentially) definable oker clcry pair of choice individual IGJV ncvcr be obsened to mahe explicit choices alternatives, ii is assumed that all such choice situatk 4~ are for the individual.

l JSre sections 4.2 and 5.2 glow

section 3.2 below. This notion that an individual’s preff:re*ces alternatives. Hence, while the

between each possible pair Of at least potentially

for a discussion of ;uch testing procedures.

meaningful

.

150

T.E. Smith, Choice theory of spatial interaction

of data to seeking the relevant data set for a given structure. In this sense, the Equivalence Theorem is perhaps tnost usefully viewed as a representation theorem (see Theorem 5 below) which specifies certain behaviorn! conditions (the ‘Choice Theory) under which individual spatial interaction behavior is representable within a given mathematical structure (the Classical Gravity Hypothesis).’ ’ given set

2.4.3. Uniqueness theorem In the above context we have one final result which considerably enhances the empirical status of these internally derived concepts of distance and attraction. In particular we establish (in Theorem 1 below) the following uniqueness ie3llt:

Uniqueness Tlteorent. For any pairwise complete spatial interaction system ronsistent with the Gravity Hypothesis, the behavioral distance and attraction functions fo: this system are borh unique up to a choice of units.

Hence it follows that ifa’s behavior is consistent with the Choice Theory in any pairwise complete system, then there must exist a measure of behavioral distance d as unique as physical dista’nce itself, and a measure of opportunity attraction a as unique physical forze itself, which together represent a’s interaction behavior in terms of the Classical Gravity Hypothesis.’ 6 This uniqueness result thus lends strong behavior-theoretic impetus to the estimation of such functions. For if such functions can be obtained, then we can effectively transform the relevant space X of locations into a ‘behavioral dirtance’ space for a which shares all the uniqueness properties of physical distance ISSuch representation theorems are not new in the behavioml sciences. In particular, it is of interest to point out a strong parallel between our treatment of ihe Classical Gravity Hypothesis and the modern treatment of the Expected Utility I-!ypothesis in statistical decision theory. This hypothesis, which was originally postulated as a rather natural way of combining given utility and probability measures into a summary evaluation of risky alternatives, has more recently undergone a similar transformation [first proposed by Savage (1954)]. In the modern view, one takes this hypothesis as a given representational scheme and asks whether there exist ‘subjective* probability and utility measures such that the individual’s preferences among risky alternatives are representable within this scheme. Hence the notions of subjective probability and utility become the internally defined primitives for this theory of behavior. For a complctc discussion of this hypothesis together with some of its more recent representation theorems, see Krantzet al. (1971, ch. 8). ‘% view of footnote 15 it is of interest to compare this uniqucncss result with the corrcsponding results for the subjective probability and urility measures of the Expected Utility Hypothesis. The uniqueness of the subjective probabilities obtained is essentially identical with our uniqueness result (i.e., unique up to a positive scalar transformation) except that the usual normalizing constraint on probabilities dictates a natural choice of units. The uniqueness of the utility measures obtained is in fact weaker than our result since the utilities obtained are only determined up to a positive linear transformation. See Krantz et al. (1971, Theorem I, p. 381) for further details.

T.E. Smith, Choice theory of spatial interaction

151

space. Moreover, each of the relevant opportunities in R which may assume locations in X can now be said to generate probabilistic ‘attraction fields’ for a which share all the uniqueness properties of physical force fields. These results suggest, for example, that the Choice Theory might serve as 2 behavioral foundation at the micro level for certain of the ‘field theory’ approaches which have been employed to study patterns of human interaction at the macro level.” In this context it would be of interest to study the relation between the micro concepts of spatial interaction situations and the macro concepts of these descriptive models. ’ * Moreover, to unify both approaches, it would be of considerable interest to determine the extent to which the individual behavioral distance and attraction functions entailed by the Choice Theory can be aggregated into corresponding functions at the macro level. We return to this question briefly in section 5.2.1 below.

3. Formal developmentof the theory In this section we present a self-contained formal deveiopment of the theory outlined above. To do so we begin with certain preliminary concepts which will facilitate this development, and then proceed to formalize spatial interaction structures, spatial interaction systems, the Gravity Hypothesis and the Choice Theory for such systems, respectively. The formal development culminates in a number of representation theorems (section 3.6) which constitute the major results of our analysis. A final section on conditional probabilities is included m order to facilitate the discussion of testing procedures in section 4.2 below.

3. I. Preliminary concepts

For any non-empty sets X and I’, we denote the Cartesianproduct of X and Y by X x Y, and denote each map (or function) f from X to Y by f: x’ + Y. The set of all maps from X to Y is denoted by Yx. For finite sets X = {x,, . . ., x,> we sometimes write each map fi X + Y as f = VI, . . .,I;,) where fi =f(xI), i= 1 ,...* n. If P(X) = {SlS E X} denotes the power set of X, then for each mapJc YK and each non-empty set SE P(X), the mapflS:S --) Y defined for all .y E S by f($(~> = $(-Y) is called the restriction off to S. Special sets of interest are the empty-set 0, the set R of real numbers, and the set R”’ = {x E RI x > 0) of positire real numbers. For any set X we designate the cardinaiit~~ of X(i.e.. the number ofelements in X) by 1x1. “Of particular rcic\ance the many extensions of their

hcrc is the work of StewarI (1948) and ZipT (1949~ together with work by Warntz (1957) and others. “For example, it would be of interest to determine the extent to which the macro concepts of ‘demographic potential’. ‘income potential’, etc. each reflect micro behavior within different types of spatial interaction contexts. Conversely, it would be of interest to determine whether macro concepts such as ‘total potential’ have any behavioral significance at the micro level.

T.E. Smith, Choice th-ory of spatial interaction

152

Finally, we record the following result on separable maps which will prov: useful for oulr purposes : For any non-empty sets X and Y and any mq f: Xx Y + R*, if

Lemma I.

(9) holds foral! x1, x2 E X and yl, y2 E Y, then there exist maps t : X 4 R+ and s: Y 3 R” si& thatfor all (x, y) E XX Y,

m

Y) = t(x) SW

(10)

l

Proof. U’ we choose any fixed elements x0 E:X and y. E Y and let t : X + R’ and s: Y + R+ be defined respectively for all x E X and y E Y by t(x) =. J(x, J’,,) and s(y) = ,f(~ nr y)lP
.fk Y)

f&9 Y)

t(x)

J-(x,Yo)

-=-z-m

f(xo* Y) J-(x0,Yo) =

so!)v

(11)

and hence that (10) holds identically.

3.2. Spaticjl interaction structures

Given any non-empty sets Q and X together with a distinguished element a E Q called a spatial actor, the triple (a, Q, X) will be designated as a spatial interaction context for a with opportunities i E D and locations x E X. Within any such context each pair (1, c) consisting of a non-empty finite subset I s Q and a map c:lu {a> 3 X will be called a spatial interaction situation in (a, C?, X) with opportunity set I and configuratio,l c. Each non-empty set ~7 of spatial interaction situations in context (a, $2, X) will be called a spatial interaction structure for a. If

denotes the family of subsets of Q containing at least two opportunities and at most two opportunities distinct from a, I’ then a spatial interaction structure Lf is said to bepairwise closed iff for all S E 9 and (I, c> E .Y’, S C I *

(S, clS u (a}) E 9.

“Equivalently, for each pair of distinct opportunities (I)} E atile sets (ai}, {aj}, (ii}, and {a#).

(13) (a}, the family f contains

T.E. Smith, Choice theory of spatial interactiorr

153

Structure Y is said to be p&wise emplete iff every opportunity set in 3 together with each of its possible configurations in X is a situation in -SF,i.e., iff for all I E 9(f2) and c E X’“t”j, IES=+(l,C)ECE By

w

definition every pairwise complete structure 9 is also pairwise closed.

3.3. Spatial interaction systems If for any spatial interaction structure 9’ and any map P: 9 x 52 3 R we write P( E 9 the following two conditions are satisfied:

Hence each interaction probability scheme P for 9’ associates with every situation E 9’ a positive probability distribution over the elements of I, where P,,,cl(i) is interpreted as the probability that opportunity i is chosen in situation (/. c). Given any (pairwise closed, pairwise complete) spatial interaction structure 27 and any interaction probability scheme P for 9, the pair <9’, P} will be designated as a (pairwise closed, pairwise complete) spatial interaction system hr a.

3.4. Gravity hypothesis Each positive function a:R --, R+ will be called an attraction function on Q, and the class of such functions will be denoted by A(Q). Similarly, if D(X) denotes the set of positive functions d: X2 --I R’ satisfying

d(Y, Y) = 4% x) 2 J(x, u),

(17)

for all (x, J) E X2, then the functions in D(X) will be called behaviorul &stance furxtiorrs on X. For any spatial interaction system (9, P> and any functions II c‘ A(R) and d E D(X), the pair (a, d) will be called a gruvity rept-e.ientafionfor (.Y, P) iff for all (I, c} e .V12’

z”Observe from the positivity of LXand d that every gravity representation satisfies both (15) and (16), and hence is consistent with the definition of an interaction probability scheme.

T.E. Smith, Choicetheoryof sputialinteraction

154

If we designate the class of all gravity representations for (Y, P> by G(Y, P), then the system (9, P> is said to be consistent with the Gravity Hypothesis iff there exists at least one gravity representation for (9, P j,i.e., iff G(Y’, P) # 0. For the special case of pairwise complete spatial interaction systems we may characterize the uniqueness of such representations as follows: Theorem I [Uniqueness Theorem]. For any pairwise complete system (9, P} and gravity representations (a,, d,), (ctz, d2) E G(Y, P> there exist positive constants RI2 andpI z such that a2 = A12a1 andd2 = p1 2d,. Proof. From pairwise completeness it follows that for each i E 9- (a> there exists some ((ai), c) E 9’ with c, = cI. Hence from (I@, each gravity representation (cc,d) E G(9, P) must satisfy

Thus for any (al, d,), (a2, d2) E G{9’, P) we may conclude from (19) that for all i E 52,

adal = at(a) -- =>a2(i) = a2(a) f3(9 al(i)

a Ai)

a l(a)

*

a2C)

=

(20)

~12al(0,

where AI2 = a2(a)/a,(a) > 0. Turning next to d, we observe again from pairwise completeness that for any X, y E X we can choose c in (19) so that c, = x and c i = y. Hence by (19) and (20) it follows that for all x, y E X and all (rl, d,), da2, d2) E GW’, P>,

j

4x, 4 3 dk, x) 4(x, Y) ddx, Y) ’

But since (17) implies that for any fixed location .‘cOE A’,

ddx, x) = dk(xr,,xc,) > 0

(22)

T.E.

Smith,

Choice

tLory

of quzticll irrterrlctiot~

holds for ail x E X and k = I, 2, we may conclude

155

from (21) and (22) that for

(23)

where

PI2

= d2(x0, xO)/dl(xO, x0) > 0.

Theorem 1 thus implies that for any pairwise complete system (9, P), if there exists a gravity representation (a, d) for (9, P) then this representation must be unique up to positiw scalar transformations of a and d. 3.5. Choice theory

Following section 2.2 we may now state the behavioral axioms of our theory in precise terms as follows. Given any spatial interaction system (9, P), the formal equivalents of the Independence, Separability, and Accessibility axioms for (9, P) are easily seen to be: (A.]) [Independence].” c’l{aij} = c,

For all

( {Q}, c),

(I, c’) E 9’

with

{ij} E I aid

(24) (A.2)

[Separability].

Forall({ij},

c), ({ii), c’) E -9,

(A .3)

[Accessibility].

Forall({i/,\,c),({ij},

c’) c .r/iwithcl{aj}

= c’l{ajj,

These three axicms taken together are designated as the C/~oice Tlleor_~ of spatial interaction systems. A system {.‘F, P> is said to be consLtcnt with the Choice Tbory ill’ (.V, P) satisfies axioms (A. I), (A.2) and (A 3). Given these definitions, we now claim that the Gravity Hypothesis for s+tial interaction systems implies the Choice Theory for such systems. zLThis axiom

is a form of the ‘choice axiom’ in Lute (1959, Lemma 3).

T.E. Smith, Choice theory of spatial interactiort

156

Theorem 2 [Consistency Theorem]. Every spatial interact&n system (9, P} consistent will!the Gravity Hypothesis is also consistent withthe Choice Theory. ProoJ If (9, P) is consistent with the Gravity Hypothesis then G(S+‘,P) # 0. Hence to establish that (A.1) Iholds, choose any (a, d) E C(Y, P) and observe that for all ((ij}, c),
implies (27) Next to show (A.2), observe that for any ({ij}, c), {{ij}, c’) E 9 with c1 = ci = XE Xandcf = c> = y E X, (18) implies

(28)

ati) = a(i)+ a(j) = a(W(c,, yl + @>/t-&, y) =

p,,i,,,cm

Hence (A.2) also holds. Finally to show (A.3), consider any ({ij}, c), ({ij), c’) E 9 with cl(aj> = c’I(aj: and set A = a(i)/d(c,,.q)

Then from

> 0,

B = a(i)/&

c;) = a(i)/d(cd, ci) > 0,

C = au)/d(ca,

Cj) = a(j)&:,

(I

Ci) >

0.

7), (18) and (29),

c, = c1 * d(c,, c,) r; d(c,, c:) *AA2 * AC+AB

2 BC+AB

* A(B+ C) 2 B(A t C)

* MA + C) z B/(Bf C) * Hence

(2%

every (9, P>

p<[ilI,c>(i)

2 p({i],,&>(i)-

with G(Y, P} # 0 satisfies (A. I), (A.2) and (A.3).

(30)

T.E. Smith. Choke fheory

3.6.

Reprr.wntGrion

of sparid

irrfernrfion

157

tlm.wnu

In this section we develop the major consequences of our behavioral ax,oms in terms of a series of consecutively stronger representation theorems for

spatial interaction systems. These three theoreins serve to clarify the representational implications ofeach of the axioms of the Choice Theory in turn. 3.6. I. Indepe~~denceaxiom To motivate our first result, we begin by observing that for any spatial interaction system (Y, P} if there exists a family of positive functions (Oi: A” -+ R’li E L?)such that

(31)

holds for all (I, c) E 9, then (.Y. P> clearly satisfies the Independence Axiom since &L,) (i)/P,,,.,u) = O,(cpc,)~O,(c,,ci)holds identically for all (I, c> E 9’ with {ij} E I. In this context our first result shows that for pairwise closed systems the converse is also true, namely that the Independence Axiom implies the existence of a family of positive functions satisfying (3 1). Tlreorenr 3. For Amy pairwise closed system (9, P> satisfying (A.]) there e_vists u family of funcrions (0,: X2 -+ R + Ii E Q> such thatfor all E 9’ and iE I,

b>(i) =

Qikl, c,) x

u,(cu,

(32)

c,).

JEl

Proo,/.

If for each i E Q- {a) we let ‘X1= (c E X@ill({ai}, c> E 9)

and take

0,: .X2 -B R+ to be any map satisfying (33)

for all c = (c,, c,) E %,, then we claim that the family of maps (Ui: x’2 + R’l i E f2) dcfincd by (33) together with 0,(x, JJ) = 1: sntktics (32).

(s, y) E x2 <

To cst(ablish this result first observe that if it can be shown that

(34)

158

T.E. Smith, Choice theory of spatial irtteraction

holds for all (I, c) E 9 and {ij) c f, then by (15) and (16) we must have (36)

i E I. Hence to establish the theorem it suffices to show (35). To do so, we first consider the case in which {ii} c I- {a}. Then since 9 is pairwise closed it follows that for any (I, c} E -9 and any pair (ii} c I-(u) the situations designated respectively by

(37)

must also appear in 9’. Hence it follows from (A. 1) and (33) that

08)

TX.

Smith. Choice tlreory of sjxtial

intemcti0u

159

Finally, if i = a E 1 then again from (A.l) together with (33j..(34) and (37),

I

=-

ej(c&j)

e*(c*cjl =

F&q

A similar argument holds forj = a E I. Hence (35) is valid in all cases, and we are finished. This result, which parallels the representation of choice probabilities by Lute (1959) and the representation of conditional probabilities by Renyi ( !970),2 2 shows that within any pairwix closed system (.V, P} a!! interaction probabilities are representable in terms of a family of positive t’unLii3ns 6i each of which depends only on the location of the opportunity i together with the location of the spatial actor u. Hence for each fixed location of the spatial actor these 0, functions are entirely independent of one another. 3.62

Separability axiom

While the above independence pro,perty serves to simplify the estimation of these functions, we must still estimate a separate function Oi for each possible opportunity i in order to construct interaction probabilities. However, observe that if each 8i function could in turn be represented up to a scale factor by some common function over space, i.e., if there were to exist a function 8: X2 + R+ such that for some set of positive weights {zCi)liE Q) the relation

holds identically for a!! i E R and (x, y) E X2, then the problem of estimating the totality of individual 0, functions reduces to the estimation of a sing!e spatial function 8 together with a ‘weighting’ function 2: Cl -+ R+. Hence it is of considerable interest to establish conditions under which such a simplification is possible. For the case of pairwise complete systems v:e may readily establish one such set of conditions. To motivate these conditions, obsep c”that if such funcrions z and 0 exist for a pairhise complete system (yf, P> iq which the Independence l’Thr: wn~truction of our 0, functiorls parallels Renyi’s Cl970, Theorem 2.2. I ) construct idn.

Lute’s

1135’3, p. 34) construction

and

160

T.E. Smith, Choice theory of spatial i.nterac:ion

Axiom holds, then substituting (40) into (32) and employing an argument identical to that of (ZS), it follows easily that (9, P) must also satisfy the Separability Axiom. With this in mind, our next representation theorem establishes .the co;:verse of this- observation, namely that if a pairwise complete system (9, P> satisfies the Separability Axiom as well ac the Independence Axiom, then (9, P> is always representable by positive functions a and 0 as in (40). Tllecrem 4. For any pairwise complete system (9, f) sarisfiling (A.I) und (A.2) there exist maps a: a + R+ and 0: X2 -+ R+ SK!: that for all (J, c} E 9’ andiE J,

(41)

Proof.

Choose any family of functions {Oi:X2 + R+li E S2) as in Theorem 3 and define the map 1: Szx X2 + R+ for all i E Q and (x, y) E X2 by A(i, X, y) =

Oi(X,

y) > 0.

(42)

Then from (A.2) together with (15) and (32) it follows that for all ({ii>, c). ({ij), c’) E 9, (43)

But fro,m paiGse completeness it follows that for an) locations .v, J, z, w E X and opportunities i, jE R there exist situations ({ij}, c), {{ij), c ) E .!f’ with c&l= _y,ci z C’iZ: 1, ci = r, and c; = $ = w. Hence from (43) we m?y conclude that R(i, .i, .11)=

W x9 Y)

I,(i, z, w)

l.(j, z, w)

(44)

holds for all i, jE R and (x, Y), (z, u.) E X2. By Lemma 1 it then follows that

T. E. Smith, Choice theory of sptial

internctiotl

I61

there must esist maps 3: f2 + R’ and 0: X2 + R L such that ;.(/I I, 1.) = r(i)Q(.r. _I-)

(45)

hold< for all i E Sz and (x, .r_)E X2. Hence the result fo!lows directly from (321, (43) and (45) by simple substitution.

3.63. Accessibility axiom Our final result is to show that if a pairwise complete system also satisfies the Accessibiky Axiom then the spatial function 0 in Theorem 4 may always be taken to be the reciprocal of a belrocioruldistartce function. Theorem 5. For any pairwise complere sysrem (9, P) safi.fyily (A.Z), (A-2), and (A.3), there existfimctio~u r E A(Q) urtd d E D(X) such thof for all (I, c} E 9 djtd i E I,

(46) Tilefurrctiws r and d are both unique up to positive scalar transformaticns. Proof. From Theorem 4 there exists a pair of maps r E A(B) and 0: X2 + R+ such that (41) holds. Hence, if we now define the map d: X2 -+ R+ for all (x, y) E X2 by 0(x, s) > o

c&s, y) = 0(x, y)

’

(47)

then from (4 I), (48)

holds for :I1 (I, c) E Y’ and iE 1. Hence (46) will follow if we can show that ti C D( r\‘#, i.e., that (17) holds for all .v. J; c, X. To do so, observe first from (47) that for all x E A, (49) so that the !‘!rst half of (17) holds identically. Next observe that (A.3) together

162

T.E. Smiih, Choice theory

of spatial

interaction

with (48) implies that for all ie s2 and all ({ai), c}, ((ai), c’) ~9’ with c,= Ci = CL = .xE Xandc; = J*E X, e,t = c1 *

p<{ai,,c>Cil

a(i)/d(xx) * a(i)/d(xx) + a(a)/d(xx) ’ 1 * a(i) + a(u)

GO)

22 p<(ai),cn>li’

2--

a(O/d(xy) s/d(xy) + a(u)/d(xx)

1I4-v~ a(i)/d(xy) + a(a)/d(xx)

* a(i)/d(xy) + a(u)jd(xx) 2 a(i)/d(xy)+ a(a)/d(xy) = l/J(xx) r l/d(xy)

cc-d(xx) I d(xy). But since arbitrarily d E D(X). follows at

the pairwise completeness of 9 implies that x and y may be chosen in (50). it fdlowsthat the inequality in (17) also holds, and hence that Finally, the uniqueness of o[and d up to positive scalar transformations once from (46) together with Theorem 1.

As an immediate consequence of this result we may also conclude that: Tlreorem 6 [Equivalence Theorem]. A pairwise cmplete system is consistent with the Gmcity Hypothesis iflit is consistent with the Choice Theory. ProoJ

Necessity follows from Theorem 2 and sufficiency follows from Theorem 5. 3.1. Conditionalprobabilities For purposes of estimation and testing the Choice Theory it is convenient to reinterpret the Independence Axiom in terms of conditional probabilities. To do so, observe that if for any spatial interaction situation (I. c) and any nonempty subset S c Iwe let

denote the probability that an opportunity from S is chosen in situation (f, c), then for any i E S lthe conditional probabilit? that opportunity i is chosen girrn that some opportunity from S is chosen in (I, c} is defined to be

In this contest. we then ha\-e the folfowinf equivalent form of the Independence Axiom :

Theorent ((ij).

c}.

7. {I,

A spatia!

c’) E 9

interacrion

with {iii

P
sysrent (9,

C I andc’l(aQj

P> saQfim

(A.1)

[ff

forall

= c.

(53)

P,r.,ytillij)).

A-CX$ From (24) and (52) it follows that (‘4.1) holds iff t’or all ({ii). c). (I. c’> E 9~1th (ijt E Inndc’l[aij) = c, (54)

and observing that PCltl,,,Jj) = But letting /i = P 0 and f~,.,.~(il{ijlj = 1 -B > 0. it then follows from the identity [A/(1 -A) = B/(1 -B))e [A-AB = B- BA]e A = B that (54) is equivalent to (53).

Moreover, for the case of pairwise closed systems we have the following stronger equivalence Iparalleling Lute (1959, Lemma 2, p. 7j] ., TIleorem 8. (A.11 i’for

all(S.

A pairwise c>, (I.

P Cs,,:(i)

closed spatial

inferacriorrt s_vstwr (.Y,

c’) E Y’ wirlt S c Iandc’(S

= P~t,,+IS).

iES.

u {a)

P>

sa~isj?cs

== c, (55)

Proof. Since (55) implies (53) and thus (A. l), it .;~J.TIcY~ eo show that (Al) implies (55). But from Theorem 3 it follows that for d’! pairwise closed systems satisfying (A.1) there exist positive functions {Otti c l2) s: :‘I that for all i c- S.

(56)

4. Evaluating and testing the theory Having developed the formal consequences of the Choice Theory, we now turn to the important problems ol’evaluating and testing the theory. TO do SO,

164

T.E. Smith, Choice theury of spatial inreracriotr

we hegin with a general consideration of the appropriatenessof the individual axioms fc;lr a variety of spatial interaction situations. The overall applicability of our theoretical framework is then considered briefly in section 4. I .4. Following this general discussion, we take up the specific question of testing each of the behavioral axioms in specific situations. Likelihood-ratio tests for each axiom are developed and discussed in sections 4.2.1,4.2.2 and 4.2.3, respectively. 4.1. Behavioral evaluation of the axioms Before adopting any behavioral theory of spatial interaction situations, one must naturally consider the plausibility of the theory’s assumptions for that situation. From this viewpoint, the Choice Theory offers one clear advantage in that all of its axioms have direct behavioral interpretations. Hence, in modeling a shopping situation or voting situation for example, one can develop prior judgments based on common experience as to the reasonableness of these axioms for such situations. With this in mind, we now wish !o consider some possible general guidelines for evaluating the three axioms of the Choice Theory in specific situations. 4.1.1. Independence axiom

Since the introduction of the Independence Axiom in Lute (1959). the behavioral implications of this axiom have been studied by a number of authors. As a result of these studies, several convincing counter-examples have been put forth [such as those of Debreu (1960) and of Tversky and Russo (1969)] which suggest that this axiom may be inappropriate for many choice situations. These counterexamples [which have been supported experimentaNy by Becker, DeGroot and Marschak (1963) among others] may be illustrated in a spatial context by the following spatial interaction situation. Suppose that in dining out, a is observed to frequent two restaurants i and .i with equal frequency, where i serves American food and j serves Italian food. If a new American food restaurant k is then added to,& perceived opportunities, he may still frequent i fifty percent of the time (namely every time he feels like Italian food) and now divide the remaining fifty percent equally between i and k. Hence a clearly fails to satisfy the Independence Axiom (even approximately) in this case. More generally, in those situations where a’s opportunities differ markedly in terms of their relative substitutability for one another, one may expect this axiom to fail. While the ultimate appropriateness of the Independence Axiom for any given situation can only be established by direct empirical testing procedures (such as those outlined in section 4.2.1 below), the above examples suggest that this axiom is most likely to be appropriate in those contexts where the array of potential opportunities in &?all exhibit a similar degree oj’ substitutability.23 ’ 3This conjecture is in accord with the conclusions of Tvenky and Russo (I 969, p. 12), and moreover, does not appear to be inconsistent with any empirical findings of which this author is aware.

T.E. Sntith, Choice theory of spatial interaction

165

For example, if in the above restaurant situation American and Italian restaurants were treated as separate contexts for a, then we might expect the Independence Axiom to hold more closely within each of these contexts in view of the relatively higher degree of substitutability among alternatives. At the other extreme, if no pair of opportunities in a are close substitutes (i.e., if all opportunities exhibit a similarly low degree of substitutability), then we may ag4n expect the Independence Axiom to be appropriate. For example, if restaurant k served Chinese food, then a might still patronize i andj with equal frequency in those cases where he doesn’t feel like having Chinese food. However, it should be emphasized that while ‘degree of substitutability’ may serve as an informal guideline in applying the Independence Axiom, the task of incorporating this concept within the formal theory itself is far more difficult. 24 Some initial results along these lines are suggested by the recent work of Tversky (1972a, b) and will be considered more explicitly in a subsequent paper. Having considered the restrictive nature of the Independence Axiom, we also wish to point out some of its desirable features for practical applications. In particular, this axiom has important consequences for the problem of estimating interaction probabilities. First of all, from the viewpoint of an outside observer, the relevant opportunitity set I considered by a in any spatial interaction situation (/, c) is almost never fully revealed. (in fact a himself may not be consciously aware of all the opportunities he implicitly considers.) Hence even if relative frequencies can be constrllcted on the basis of observed interactions, one may never be able to identify precisely the spatial interaction situation being observed - thus resulting in an apparently insoluble estimation plob1em.25 However, if the Independence Axiom can be assumed to hold (at least approximately) then meaningful estimates can be obtained for the case of pairwise clmed systems. In particular, if for any pairwise closed system (.V, P} and any sequence of N observed interaction choices by a we define the relevant spatial interaction situation (I, c} to consist of precisely those opportunities for which at least one interaction is observed,26 then by assuming only that I is at least contained in the actual opportunity set f, for each observation I?= 1,...* N, we can estimate the interaction probabilities for (I, c} as follows. Recall from Thearem 8 that for all i E /and tz = I, . . ., IV,

**The need for such a rormnlbation was recognized by Lute and Suppcs (1966, p. 337) who wrote that any complefely successful choice theory must include ‘some mathematical structure over the set of outcomes that, for cuamplc, permits us to characterize those outd;omcs that are simply substitutable for one another and those that are special casts of one another.’ “For a pertinent Acussion of this point in a Focial choice context see Fishburn (1973. pp. 6 8). z6Mo~-~ precisely, if II is situated at location c,, E Xand is obserkcd to interact zt least once witheach ot thcopportunilie’i,, . . ., it at iocationscl, . . ., ck rcspcctivcly, then f = {il, . . ., ik} is taken to be the relevant opinnrtunity set and c = (e,, c,, . . ., c&j ;hc relevant configuration. C

166

T.E. Smith, Choice theory qf spatial interaction

where c = c,,ll u {a}. Hence the observations from eti(i), i E 1. A second estimation problem in which the Indepradence Axiom provides a powerful tool relates to the ‘no interaction’ option for a. In practice we can rarely observe the exercise of this option. Hence our probability estimates would again appear to be biased by this lack of information. However, if the Independence Axiom holds (at least approximately) then Theorem 8 also implies that for any pairwise closed system (9, P> and any spatial interaction situation (I, c) E 9’ with Q 4 I and (I u (a), c} E 9,

Pdi)

= P<,vc,,,,,(ilO.

(58)

Hence by assuming oniy that (I, c} and (I u (a), c) are each potentially meaningful situations for a, the Independence Axiom permits us to obtain valid relative frequency estimates for the distribution PCr,C) regardless of whether or not the ‘no interaction’ opportunity is present. If it is present then we estimate PCr,,>(i) by means of P Clv(.,,r)(i]Z), and if it is not present then we estimate P,,, ,+j’) directly. In summary, while the Independence Axiom places definite restrictions on the range of spatial interaction situations which can be treated, it would appear to be an almost indispensable assumption for any operational theory of choice which is applicable under the conditions of limited c-bservational information mentioned above. 4. I .2. Sepurability axiom The Separability Atiom may also be viewed as a type of independence axiom in that it asserts a degree of independence between opportunities and locations. In essence it implies that in making comparisons among opportunities the spatial actor a implicitly compares both their ‘spatial’ and ‘non-spatial’ attributes, and that a‘s comparison of non-spatial attributes is independent of any locational considerations. For example, if the quality of a store’s merchandise is a relevant non-spatial attribute, then in essence the Separability Axiom asserts that a’s preferences among the merchandise of any two stores i and j remains the same over space. The comparatiue nature of this independence assumption is crucial. For example, while a lake resort will surely take on an entirely different character when moved away from the lake, it is still reasonable to suppose that Q’Scomparisons among the relevant non-spatial features (quality of service, room sizes, etc.) of two resorts i andj similarly situated will remain the same. However, the Separability Axiom may still fail to be appropriate in many spatial interaction situations. In the resort example above, for instance, if

*T.E. Smith, Choice theory of s.w~‘iai interactim

167

resort i is designed to take maximum advantage of the view of the lake while .j is not SO designed, then resortj may appear relatively more attractive in locations where the view is of little interest. Moreover, while the trip to two opportunities i andj located together may be the same, the aature of this trip itself may have a differential impact on Q’S perception of i and j. For example, if twc, siores i and j are both located across the street from a then the presence or absence of a o&cry service by the stores is of little significance. However, if i and j are both located across town from u and if i has a delivery servke lvhile j does not, then a may consider i to be relatively more attractive in this situation. As one final example in non-physical space, if u is voting for one of two Presidential candidates whose positions on the relevant issues are both identical with his own, then he may well prefer the candidate he believes will be the most effective in influencing legislation. On the other hand, if the positions of both candidates are similarly opposed to his own, then a might well prefer the candidate he believes will have the least influence over legislation. All these examples indicate that the Separability Axiom may have varyicg degrees of appropriateness in specific spatial interaction contexts. However, while the ultimate worth of this axiom can only be determined in each situation by empirical testing procr..?ures (such as those outlined in section 4.2.2), the above examples do suggest certain informal guidelines for applying the axiom. In particular, recall that in the case of the Independence Axiom it was suggested that by defining the relevant opportunity set 0 to include only opportunities with similar degrees of substitutability one might render this axiom more appropriate. A similar argument applies to the Separability Axiom with respect to the definition of the relevant locution spuce A’. More specifically, if the definition of ‘spatial’ attributes is extended to include all those properties of opportunities which 4 perceives to be of differing significance depending on location, then the problems raised in the above examples may be at least partially overcome. In the shopping example. for instance, one might wish to treat ‘level of delivery service’ as a relevant spatial attribute. Moreover, in the voting example, the ‘degree of legislative influence’ of each candidate may bc more properly treated as a spatial attribute for u. In doing so howevsr it must be borne in mind that the resulting location space X is thereby made rzrore complex and difficult to interpret. Moreover. if this process is carried to an extreme and all relevant attributes are included in the definition of X, then ‘cjpportunities’ become indistinguishable from ‘locations’ and the Separability Axiom loses its meaning altogether. Hence if the primitive concepts of locations and opportunities are to retain any distinct meaning in a given context, then there must clearly be some trade off between the degree of integrity which can be maintained for these two concepts and the degree of satisfiability v.hich can be achieved for the Separability Axiom. Having illustrated the limitaticns of the Separability Axiom, it is also of

168

T.E. Smith, Choice theory of spatial interaction

interest to consider certain of the desirable features of this axiom for estimation purposes. II particular, recall from the discussion of section 3.6.2 that if the Separability Axiom can be said to hold (at least approximately), then for the case of p&wise complete systems (9, P} one can z,m?narize the non-spatial ‘attractiveness’ of each opportunity i E L? by a single weightiq factor a(i). Moreover, the effect of space on a’s interaction behavior in (9, P) LLXIbe summarized by a single function 0(x, y) of a’s location x E X together with the location y E X of any opportunity in $2. Hence if there are 100 opportunities in the universe Q, for example, then in terms if expression (40) this axiom allows us to reduce the problem of estimating 100 separate spatial functions (OiliE 0) to the estimation of one spatial function 0 together with a set of 100 simple attraction weights {a(i) 0). Hence from a practical viewpoint, the adoption of thir: axiom as an approximation to u’s interaction behavior yields an enormous reduction in the estimation problem. 4.1.3. Accessibility axiom The third and final axiom of the Choice Theory is in many respects the least essential part of the theory. The inclusion of this Accessibility Axiom is designed to reflect the traditional focus of spatial interaction models on those situations in which some notion of ‘distance deterrence’ is appropriate. (Recall from section 3.6.3 that the only formal role of this axiom is to render the spatial function 8 in Theorem 4 interpretable as the reciprocal of a behavioral distance function d, as in Theorem 5.) Such a ‘distance deterrence’ effect is clearly appropriate for a wide range of spatial interaction situations in which some form of ‘accessibility cost’ tends to impede u’s interactions with opportunities over space. For example, this assumption would certainly appear to be plausible in the shopping and voting situations of section 2.2.3 above. However, there are many situ;, tions in which the reverse may well be true. If, for example, we consider Q’Schoice of vacation trips, there is strong reason to suppose that the vacation opportunities at his own location are among his least likely choices. More generally, there are many spatial interaction situations where trips t%mselves may have positive value for a . 27

As in the case of the Independence and Separability Axioms, the ultimate appropriateness of the Accessibility Axiom for modeling specific situations can only be determined by empirical testing procedures. One such procedure is outlined in section 4.2.3.below. 2’Notice that the failure of the Accessibility Axiom need not rule out the possibility of ‘behavioral distance’ itself. For example, if as in the situation above there is a ‘negative’ accessible effect, i.e., if moving an opportunity to u’s location does not increase his probable in’eraction with that opportunity (other things being equal), then one easily sees from Theorem 4 that 0 itself becomes a behavioral distance function for a. Hence by reversing the Accessibility Axiom one again obtains a behavioral distance representation in which distance now acts as a stimulant rather than a deterrent to interaction.

T-E. S&h,

Choice theory of spatial interaction

169

4.1.4. Overa!! appropriateness of the choice theory Before leaving the question of behavioral appropriateness, it is important to point out certain general properties of the Choice Theory which may limit its applicability in certain situations. First of all, the Choice Theory focusses on individual behavior rather than group behavior, and hence is not well suited for direct application to large scale interaction phenomena. For example, if one were to apply this theory directly to the question of predicting travel patterns within a given metropolitan area, it would be necessary to study the travel behavior of each individual resident. Hence the Choice Theory by itself provi:%s us with no practical method of studying aggregate behavior. In this sense the theory is perhaps best viewed as a possible component within a larger theoretical framework for studying spatial interaction behavior. For example, if prevailing theories of a certain type of aggregate interaction behavior suggest that this behavior is well approximated by a limited number of ‘representative’ i%L viduals, then the Choice Theory could easily be applied in this context to model such representative individuals. We shall return to the question of applying the Choice Theory to aggregate interaction behavior in section 5.2. I below. A second basic property of the Choice Theory which may limit its scope of application relates to the probabilistic nature of the theory. In particular, the theory characterizes spatial interaction situations as involving a number of alternative opportunities, each of which has a definite possibility of being chosen. Hence the theory implicitly focusses on those situations for individual behaving units in which some form of discretionary choice behavior is involved. Shopping trips and recreational trips are good examples of such situations. However, many important types of spatial interaction behavior do not share this discretionary quality. For example, trips to work and to school can hardly be considered as discretionary travel behavior. Hence it should be clear that the Choice Theory is by no means a general theory of spatial interaction. As with any substantive theory, its assumptions impose definite restrictions which must be recognized explicitly if the theory is to be applied successfully. 4.2. Direct tests of the axioms Having considered the general behavioral appropriateness of the Choice Theory, we now set down some initial thoughts on testing the theory. In particular we develop certain methods of direct statistical testing which may be applied to each of the behavioral axioms individually. The possibility of more general methods of indirect testing will be considered briefly in section 5.2.2 below. To begin with, recall that each of the behavioral axioms of the Choice Theory is stated in terms of three primitive concepts: locations, opportunities, and interactton probabilities. Hence if we can identify the relevant locations and

170

T.E. Smith, Choice theory of spatial interaction

opportunities in a given spatial interaction context, and if we are able to observe replications of various spatial interaction situations within that context, then direct tests of these axioms can be constructed in terms of relative frequency estimates of the interaction probabilities. We now consider such tests for tlhe three behavioral axioms of the Choice Theory respectkly. 4.2, I. Indepmdence axiom Numerous direct tests of the Independence Axiom may be constructed. By way of illustration, suppose we consider the restaurant example of section 4.1.1. In particular, suppose that a is observed to frequent two restaurants i and j, and that at some point in time a third restaurant k is pelrceived by o to be a new relevant opportunity. If in this situation a’s behavior is consistent with the Independence Axiom, then we may hypothesize that a”s relative probabilities of dining at i versus j will stay the same after the introduction of k. Hence d test of this hypothesis amounts to a test of the Independence Axiom in this situation. To develop such a test, suppose more generally that in a given spatial interaction system (9, P) we observe n, instances of interaction choices by a in some situation S, = ((ij}, cl) E 9’ (such as n, instances in which a is observed to dine at restaurants i or j in our illustration above). Suppose, moreover, that we also observe n2 instances of choices from the extended situation Sz = ({ijk}, c2) E 9 with c,l{aij) = cl. Then using these observations we wish to test the hypothesis that

From Theorem 7 t&s is clearly equivalent to testing the hypothesis that

Hence if we assume that our observations in both situations S, dnd Sz are independent, and if we let p1 = P,s,(i) and p2 = Ps,(il(ij}), then we may treat our two sets of observations as coming from the independent binomial distributions B(n,,p,) and B(11~,p& respectively. Hence we may then test the hypothesis H,:p, = p2 against the alternative hypothesis H:p, # p2 by employing the standard likelihood rariv testing procedure. In particular, if we let X, and x2 denote the number of interactions with opportunity i observed in situations S, and Sz, respectively, and if we define the appropriate likelihood ratio A by c

T.E. Smith. Choice theory of spatial interaction

where

II = n, +nz

and

s = _Y,+x,.

then

171

the statistic

C = -2 In ). is well known to be assymptotically distributed chi-square with one degree of freedom,

i.e., assymptotically x:. under hypothesis HO [see Wilks (1962, p. 423)j. Hence for any level of statistical significance r we would reject I?, iff C is sufficiently large to fall within the critical region of size 01for the XTdistribution. Generalizations of this testing procedure may easily be constructed for a number of more complex situations [see, for example, Wil ks (1962, p. 427)]. Numerous other testing procedures are also applicable to the Independence Axiom. Of particular interest is the work of Becker, DeGroot and Marschak (1963) together with the many citations in Lute (1959, pp. 27-28). As one final observati._ul on testing the Independence Axiom, it is important to note that this axiom ,zcessarily implies a behavioral symmetry with respect to the addition and deletion of opportunities. In particular, the equality of relative probabilities in (59) must hold whether situation S, is exfettded to situation Sr or situation St is restricted to situation S,. Thus u’s choices between restaurants i and j. for example, are assumed to be independent of whether restaurant k is added to P’S options as a new restaurant or deleted from u’s options, say, by going out of business. Hence to take this symmetry property into account, a well-balanced test of the Independence Axiom for any spatial interaction system must include both situations in which opportunities are added and sitztations in which opportunities are deleted. 4.2.2. Separak:lity axiom

A direct test of the Separability Axiom requires the observation of a series of spatial interaction situations for a involving the same opportunities situated successively at two or more common locations. In particular, suppose that for some pair of opportunities i and j in a given spatial interaction system (.sP. P} we are able to observe n, instances of choices for a within two spatial interaction situations S, = ({ii}, c,) E ,Y where c,~ = c,] E X for a = 1, 2. In this context, a test of the hypothesis

(62) is clearly a test of the Separability Axiom for system (.Y’, P). Hence letting P* = P,=(i) for Q = 1,2, and again assuming independent observations in each situation, we may test the hypothesis H,:p, = pz against the alternative H:p, # pr by employing the likelihood ratio test of (61). As with the Independence Axiom, numerous generalizations of this simple test to more complex situations arc easily constructed. 4.2.3. Accessibility axiom As one possible direct test of the Accessibility Axiom, suppose that we are able to observe ftl, instances of choices for a within two situations S, = ((ij),

172

T.E. Smith, Choice theory of spatial interaction

c,) E 9, a = 1,2, where it is now assumed that c,l{uj) = czI{uj}. clt # cl,. and C2i = ~2~. Then a test of the hypothesis

is easily seen to be a test of the Accessibility Axiom for system (9, P). But since (63) involves an inequa!iry rather than an equality as in (60) and (62),, the problem of constructing a statistic whose distribution is known under this. hypothesis is far more difficult. One approach to this problem is to construct the standard likelihood ratio1 statistic for hypothesis (63) and to associate with this statistic the ‘least favorable’ distribution consistent with the given hypothesis. To construct such a test, observe that if we again let X, denote the number of observed interactions with i in situation S, and let pa = P,(i) as before, then we wish to test the composite hypothesis H0:p2 2 pI against the composite alternative H:p2 < p,. The appropriate likelihood ratio for this test is given by

max [p’;‘( 1 -pl)n*-xtp~z

(1 -p2)nzBX*] (641

An analysis of the numerator of (64) shows that p, = xr/nr and pz = x2/rtr whenever x2/nl 2 x,/nr, and that pl = p2 = (xl +_uJ(n, +nz) whenever 28 Hence if we let C(X,X,) ~2 -2 In i(x,xz) then it follows at x2ln2 < xlh. once that

C(x,x2)

=

I

-2

0,

In 2

if x2/n, 9

Z x,/n,,

if x2/n2 c xl/n,,

*sTo see this observe that the likelihoodfunction

055)

1 defined for all (p, .p2) in the unit square

U={(p,,p~)[O~~~~1;a=1,2}by

I(p,,p,)

= pfl(l

-p,)“‘-X’p;z(

I -p2)Q-*z

is strictly concave on Uand achieves its global maximum at the point p” = (x,/n,, xz/nl) 6 U. Moreover, the set of feasible pels, namely F = {(p, , p2) E U 1y, 3 p,) is a compact convex subset of II which contains p* iff xz/n2 2 x,/n,. Hence if x&r2 < .r,/n,, then 1 achieves its unique maximum in F on the boundary of F. Moreover, thib maximum cannot be on the boundary of CJsince I(p,, p2) = 0 on the boundary of U and I(p, , pl) > 0 interior to U. Hence the maximum oi’Iover Fmust be achieved on the boundary of Fdcfmed byp, = pz = p. But subject to this additional constraint, the maximum of /(p,p)

= p~l+-~l(~_p~l+n~-(x!t~z,

is always at p = ix1 “xzMn1

+nzj.

T.E. S&h,

Choice theory of spatial interaction

173

where A.is given by (61). Intuitively Ho should be rejected if C is sufficiently large. However, the assymptotic distribution of C is not known for composite hypotheses such as ff0.29 One common method of resolving this problem is to consider the assymptotic distribution of C under the ‘least favorable’ parameter values consistent with the hypothesis H. i.e., those parameter values which are as ‘close’ to the alternative hypothesis as possible and hence which tend to favor the alternative hypothesis. 3o The basic idea is thus to adopt a conservative posture by constructing the most stringent test of Ho which is consistent with the supposition that Ho is true. In our case the ‘least favorable’ values of the parameters p1 and p2 are easily seen to be those with p1 = p2. Hence if we consider the distribution of C under the hypothesis that p, = p2 then it can be shown that the assymptotic distribution of C is closely related to the chisquare distribution. in particular, the critical region of size ti for testing Ho is obtained by taking a critical region of size 23 from the xi distribution.” fience the construction of this critical region yields a conservative test of the hypothesis W,:p? 2 pI at the z level c?significance. S. Directions for further research As with any theory of behavior, our development of the Choice Theory

perhaps raises more questions than it answers. In this final section we wish to consider briefly some of the questions which we have already begun to pursue. In particular. we begin by considering certain questions relating to the specification and estimation cf behavioral distance functions. We then consider a number of questions concerning the possibility of developing more operational methods of testing the Choice Theory as a whole. 5.1. Behavioral distancefunctions

Throughout this paper there has been little discussion of the important question of calibrating the Choice Theory. The major reason for this omission is that the relevant parameters, namely attraction and behavioral distance functions, are characterized at a level of generality which for the most part precludes practical estimation procedures. While maximum likelihood estimates for certain attraction and distance values could be extracted from the test statistics of section 4.2, it would be a practica: impossibility to estimate a “*The usual asymptotic results for compasitc likelihood ratio statistics rcqulre that the dimcnsionality of the parameter space under /A, bc strictly smaller than that of thcl parameter space itself. This condition clearly fails for our case [see for example Wilks (1962, section 13.8)]. J0Sec Lehmann (1959, section 3.8) for a full explication of this idea. J1fhe form of this test is easily seen to be equivalent to the ‘minimum discrimination information’ statistic for the two-sample binomial problem developed by Kullback (1968, pp. 132-l 331, where his statistic 2PlH,’ : Hz) is identical with the statistic C of (65).

174

T.E. Smith, Choice theory of spatial interaction

sufficiently large range of values to approximate the overall functions themselves. Hence we require more specific behavioral assumptions in order to render these functions estimable. With this objective in mind, we have begun to seek behavioral characterizations of certain explicit forms of distance functions. Along these lines, a first natural question to ask is whether a given behavioral distance function is dependent on certain readily observable meastires of distance, such as travel time, travel costs, or physical distance itself. In answer to this general question we have obtained necessary and sufficient conditions uuder wl+?h a’s spatial interaction behavior implies such a dependency, i.e., under which the appropriate measure of behavioral distance d is solely a function of some given distance measure f. Within this general framework, we have also established necessary and sufficient conditions under which such a dependency must be of the fo;m (2). Similar results have been obtained for (3). Hence such conditions may serve as behavioral axioms for a number of more explicit theories of spatial interaction behavior. These results will be presented in a subsequent paper. 5.2. Indirect testing procedures A second area of investigation relates to the construction of more operational procedures for testing the Choice Theory. While the methods of direct testing outlined in section 4.2 have the advantage of requiring a minimum of additional assumptions to test the Choice Theory, ” they are very objectionable in terms of data requirements. In particular, these testing procedures generally require a controlled testing environment in which the researcher is able to manipulate both the opportunity set I and the configuration c perceived by the spatial actor a. But since one is hardly able to move stores at will, or change candidates’ positions at will, such experimentation tnust generally be conducted either in terms of a very limited number of observations or by an appeal to hypothetical situations for a. The former is generally ineffective in that without sufficient replications, no significance can be obtained for the estimates of relative frequencies. The latter method suffers from the inherent problem that a’s answers to hypothetical questions may bear little relation to his actual interaction behavior. In short, the practical requirements for conducting meaningful direct tests of the Choice Theory may well be prohibitive. In this context it is thus desirable to seek less restrictive methods of testing. We have thus far obtained results in two areas which will hopefully lead to such testing procedures. The first of these areas concerns the aggregation of individual spatial interaction behavior, and the second relates tnore directly to comprehensive testing methods. ‘*In fact the only major assumptionrequired by these direct testing procedures is that the observed interactions are statistically independent,

T.E. Smith,

Choice

rheory of spntid

irtteroctivrr

175

5.2. I. Aggregution of spatial actors

The most common method of overcoming the problem of limited individual observations is to aggregate individuals into groups which are on the one hand suflkiently homogeneous to exhibit similar spatial interaction behavior, and are on the other hand sufficiently large to permit a reasonable frequency distribution of interactions to be observed within each group. If such aggregations are possible then the testing procedures of section 4.2 become more feasible. For example, if one wishes to test the Separability Axiom for a homogeneous agflegate of shoppers with respect to the stores in a given shopping center, then one can now compare the interaction frequencies of shoppers at different locations from the center rather than moving the shopping center from location to location. However, the construction of such aggregates requires a host of pdditionai assumptions as to what constitutes ‘homogeneity’, and, inoreover, must inevitably involve a trade off between the size of the aggregates and the degree of homogeneity obtainable. Along these lines we have been able to establish certain homogeneity conditions which permit inferences to be drawn about individual behavior from group behavior. We have also obtained conditions under which group interaction behavior constitutes a simple averaging of individual interaction behavior. These aggregation results will be reported in a subsequent paper. 5.2.2. Comprehensive tests qf the Choice Theory While the possibility of aggregating individuals into homogeneous groups renders the testing procedures of section 4.2 more feasible, the task of testing each axiom of the Choice Theory individually still creates many problems. For example, in testing the Independence Axiom one may still be unable to observe the addition or removal of opportunities from a given configuration. Hence it is of interest to seek less restrictive methods of testing. One procedure which is suggested by the representation theorems of section 3.6 involves an extension of the likelihood ratio tests of section 4.2 to uncontrolled observations from homogeneous aggregates of individuals. In particular, using the results of Ford (1957) one can readily obtain maximum likelihood estimates of the 0 function in Theorem 3 on the basis of such observations. Given these estimates it is a simple matter to construct likelihood ratio te& of the Indepcndence Axiom which do not require any manipulatron of opportunity sets. More generally, the estimation procedure of Ford can bc cxtcndcJ to the functional form of the Gravity I-Iypothesis itself, i.e., expression (lb). Hence by means of Theorem 5 we may thereby utilize uncontrolled observations from homogeneous populations to construct comprehensive likelihood ratio tests of the Choice Theory itself. These testing results will be presented in a subsequent paper.

176

T.E. Smith, Choice theory of spatial interaction

References Beckmann, M.J., R.L. Gustafson and T.F. Golob, 1973, An economic utility theory approach to spatial interaction, Papers of the Regional Science Association 30. Becker, GM., M.H. DeGroot and J. Marschak, 1963, Stochastic models of choice behavior, Behavioral Science 8. Brand, D., 1973, Travel demand forecasting resource paper, in: Urban travel demand forcasting, Special Report no. 143 (National Highway Research Board, National Academy of Sciences, Washington, DC.). Choukroun, J.M., 1975, A general framework for the development of gravity-type trip distribution models, this issue. Debreu, G., 1960, Review of R.D. Lute, Individual choice behavior: A theoretical analysis, American Economic Review 50,180-l 88. Fishbum, PC., 1973, The theory of social choice (Princeton University Press, Princeton, N.J.). Ford, L.R., Jr., 1957, Solution of a ranking problem from binary choices, Herbert E. Slaught Memorial Papers, American Mathematical Monthly, 28-33. Harris, B., 1964, A note on the probability of interaction at a distance, Journal of Regional Science 5, no. 2,31-35. Isard, W., 1960, Methods of regional analysis (M.I.T. Press, Cambridge, Mass.). Krantz, D.H. et al., 1971, Foundations of measurement, vol. I (Academic Press, New York). Kullback, S., 1968, Information theory and statistics (Dover, New York). Lehmann, E.L., 1959,Testing statistical hypotheses (Wiley, New York). Lute, R.D., 1959, Individual choice behavior: A theoretical analysis (Wiley, New York). Lute, R.D. and E. Galanter, 1963, Discrimination, in: R.D. Lute et al., eds., Handbook of mathematical psychology, vol. I (Wiley, New York) 191-243. Lute, R.D. and P. Suppes, 1966, Pretxence, utility, and subjective probability, in: R.D. Lute et al., eds., Handbook of mathematical psychology, vol. III (Wiley, New York). Niedercorn, J.H. and B.V. Bechdolt, Jr., 1969, An economic deriva?ion of the ‘gravity law’ of spatial interaction, Journal of Regional Science 9, no. 2,273-282. Renyi, A., 1970, Foundations of probability (Holden Day, San Francisco, Calif.). Ruiter, E.R., 1973, Analytical structures resource paper, in: Urban traveldemand forecasting. Special Report no. 143 (National Highway Research Board, National Academy of Sciences, Washington, D.C.). Savage, L., 1954, The foundations of statistics (Wiley, New York). Schneider, M., 1959, Gravity models and trip distribution thmry, Papers of the Regional Science Association 5,51-58. Stewart, J.Q., 1948, Demographic gravitation: Evidence and applications, Sociometry Il. Tanner, J.C., 1961, Factors affecting the amount of travel, Road Research Technical Paper no. 51 (H.M. Stationary Office, London). Tversky, A. and J.E. Russo, 1969, Similarity and substitutability in binary choices, Journal of Mathematical Psychology 6, l-l 2. Tversky, A., 1972a, Elimination by aspects: A theory of choice, Psychological Rebiiew 79, no. 4,281-299. Tversky, A., 1972b, Choice by elimination, Journal of Mathematical Psychology 9. Warntz, W., 1957, Geography of prices and spatial interaction, Papers of the Regional Science Association 3. Wilson, A.G., 1967, A statistical theory of spatial distributiun models, Transportation Research 1, no. 3.253-269. Wilks, S.S., 1962, Mathematical statistics (Wiley, New York). Zipf, G.K., 1949, Human behavior and the principal of least action (Addison Wesley. Cambridge, Mass.).