ESTIMATING MARKET SHARES FOR NEW. BANK LOCATIONS,: -1? THE APPLICABILITY OF THE HUFF MODEL by 'c \ Lorman L. Lundsten & Martin R. Warshaw " 9-~c 4~~ J d I - I6L Vobrklrv fper "'fjS

-2 - Lorman L. Lundsten Vice President, Lundsten Plastics Corporation Formerly Financial Economist, Office of the Comptroller of the Currency Martin R. Warshaw C Ar AndAA Professor of Marketing and head-of-tre Marketing Faculty, the University of Michigan

% -3 -ABSTRACT Data from a study of consumer behavior in one retail banking market are used to determine whether six critical assumptions of a model of retail gravitation would be met in the context of its use as a predictor of bank patronage. The assumptions are generally upheld..'

ESTIMATING MARKET SHARES FOR NEW BANK LOCATIONS: THE APPLICABILITY OF THE HUFF MODEL R-by_ Lorman L. Lundsten & Martin R. Warshaw Marketing research has provided considerable evidence that convenience is the most important factor in a consumer's decision to patronize a specific banking location (Bennet-1975). It is thereforetnot surprising that bank managementiseekp those locations for their main offices and branches which will afford maximum convenience for potential customers. Experience has taught these executives that conveniently located banks attract more deposits and hwae-more loan activity than do less well-located banks. What bankers have needed for some time is a way of- predicting the perope0 '/i!formance of a proposed location prior to its construction and opera-t-i-on. First, such a prediction would be invaluable in convincing the regulatory authorities of the need for a new bank location and second, such predictions would enable management to make better selections of locations from the-a-r-ay available to them. tk;s -- ^ <r The ideal solution to -the(estimation problem/would be some type of algorithm which mni-h.4 utilize the "convenience" effect of a given location to estimate the share of market a given bank might gain if 44-t e-o-.beroql - lett s oc-& r'(5 i built on that location. Because of the close analogy between thejbank 4-ea-4-i.on-preb-effl and the more general problem of-leeat4e-i-e- a retail store for the sale of convenience goodsit is reasonable to start the search for a banking model -4fqir —atrlO.n- those models which have been developed for the location of non-bank retail outlets.

-5 - RETAIL GRAVITATION MODELS The oldest and best-known approach to estimating the extent of retail activity in a market area was developed by Reilly (1929). His model was the first of a series to be known as retail gravitation models and was originally.. devfieped to predicthow retail patronage would be divided between adjacent communities] Even today its use by bank managers is occasionally recommended (Kramer,1972). A second-generation model developed by Huff appears to overcome many of the problems associated with the application of the Reilly model to the location of individual retail outlets (Huff and Batsell 1972). The Huff model is based on estimates of individual probabilities of patronage for a specific retail establishment and appears to provide a logically consistent _ _ approach to the(retail location problem. ~ ~ ~ ~ ----------- __ _-" — i ---- -------- I —~~~~~~~~~~~~~~~~~~- - _ _ _ _ _ --- — - ~ ~ — ~ --- —-- _ ~ ~ ~ - - - - - - `- - - - _ _ _ _ _- _- ___ _- - - - -- -- _-~ _ _ _ __ _ _. _ _ _ _ _ _ _ ---- - ~ -— ~ --- —--------— I --- —-- - ---- ---------------- - ------------- - __ - - _ - - - - ---- ---- " --- — - ~ --- -- ~-~ —~ --- —— ' --- —-- -I-I ---` - -1 1_1 ---_ —1 ---— ______ _____ ~ _II- C -I_ ----L_ --- _Illl_ — —~IXI -- 1.1 --- —-~ 1 — CIII-~ll-l --- —— X ----C --- —--— l —~~

Huff's statement of the share of market model follows the axiomatic approach set out by Luce (1959). It is possible to derive models similar to Huff's model-u-stg several different starting points. Kotler (1971) cites work by Urban, Krishnan and Gupta, and Weiss in developing models of this type and, although he avoids the formal statement of axioms, he presents a discussion that leads to a multiple-variable gravitation model very similar to that suggested by Huff. Bell, Keene;\and Little,(1975) approach the problem of determina-i-o-n-o — market share by offering a set of four axioms that can be developed into a model similar to that of Huff. The axiomatic structure of BKL centers ae~-d the properties Oh of the firm, while that of Luce centers -a.wA4 individual choice. If one assumes homogeneous choice behavior by individuals, the BKL model and the Luce model differ on no major point. The literaturesof mathematical psychology (Coombs et al. 1964,\1975), marketing (BKL et seq'1975) and sociology (Zipf, Co v^,Stewart 1947,1949) w many instances of the formulation and testing of models of interaction and influence. The models oaf ( rf 5 nr I Af- itCCi) ~shorWsome- -dbf-Pe-es-, but it is unlikely that any empirical data —wlt3-al-l-oAthae-.accapana-o — any one model over the others on the grounds of-goodess of fit. Table 1 traces the history of-some models-of this type. -Here as elsewheret the final choice of a model rests with the researcher. Because of its long period of acceptance and use, its grounding in individual behavior and its orientation to marketing, the authors prefer the Huff model. This work will-oe-e-r-ae test o:f several important assumptions of this model in the context of bank marketing.

HUFF'S MODEL ______ --- —------— __ The Huff model is based on an adaptation of Luce's choice axiom to the retail patronage problem. Huff assumes that a measure of size is available for the relevant retail facility and that distance from the consumer to the location is a reasonable surrogate for convenience. The Huff model is stated below: V(j) Pij V-(1T) where: Pij = the probability of consumer i patronizing facility j n V(j) = AADD' and V(all) = z V(j) for n competitors-andj=l Aj = an attraction index for retail facility j' ' 3

-6 - Dij = the accessibility of a retail facility j to a consumer located at i j-ad —Q.. y & X = empirically determined parameters (Huff 1962)b\ Originally rythe exponent of the attraction measure (mass or size in ta-iv oc gravitation t-Befi)swas set at unity for ease of computation. /The similarity to Reilly's model is striking. Both set the mass exponent at unity and both allow empirical determination of the exponent on distance. Unfortunately, the application of the Huff model to problems of market share estimation for proposed retail facilities has been fraught with difficulty. Huff and his associate Batsell have noted that unless certain conditions are met the model may be misapplied or results emanating from its application may be misleading. They have identified six key problem areas which will be considered below in some detail ( Huff and Batsell/1975). PROBLEM AREAS Trip-type The Huff model assumes that the set of retail alternatives specified as choice alternatives -a-e most likely to be associated with single-purpose shopping trips. If choice alternatives are associated with multiple-purpose tripsj i'the proximity of a retail facility to a consumer's place of origin may not be nearly as important as the proximity of a retail facility to other retail facilities in which purchases are intended, or to those non-shopping activity places that the individual intends to visit " (Huff and Batsell 1975). ~( Product-type "It is important that the attraction values that are specified for those retail facilities compr-is-ngi the set of choice alternatives be in keeping K *-K ~* s~ 9 *

-7 - with the product purchase intentions of the customer" (Huff and Batsell, 1975). Thus in those cases where the specific product is not specified or w0l -hu 0/1E -?he Co,-5Vtiq iAu^ -t PVucv'(,45 if specified is ts_~4e -eft, results of model application will be less dependable. @3) Spatial Equilibrium The Huff model assumes that the consumer will share his purchases among feasible alternatives in keeping with a constant probability vector (e.g.,.6 Probability of purchasing from store A and.4 probability of purchasing from store B). The model provides an equilibrium solution without any guarantee that the individual consumer is in equilibrium. i-~ Choice alternatives If the subset of choice alternatives is not defined correctly, two types of problem, can occur. First, ' those aris.ing from nef-4re-i-s4n-ef a consumer choice alternative and second, those which arise from including an alternative which is not in the consumer's set of feasible alternatives. The former error results in overstatement of estimated values understated - the latter error results in expected values which depend on the alternatives which were included in the model. _ Group Behavior The model is based on a choice axiom of individual behavior. If group ('+ I's behavior is being analyzed asiin a market share estimation problem, great care must be exercised to make certain that behaviors in small segments of the market can be averaged out to predictcorrectly behavior in the total market area. XL) Choice Determinants The model requires only two variables (size and distance) to compute a

-8 - probability measure. However, the consumer's h oeiee.Aedu-4tA-i4 —i'yof a retailix facility may be a function of several variables other than the two noted — rtc_ previously. If the variables used in the model are not-t-ru- surrogates for other important factors the resulting measures may be misleading. Given the availability of the Huff model and a recognition of the problems associated with its application, the question arises as to how useful it might be when applied to the problem of market share estimation for a new banking office. To answer this question a telephone survey of 600 families in Farmington, Michigan was tbatei-n. Parmi iurl, Michigan.is a. e suburb of Detroit and -p — -- t-a ns-^r^aoMdre-p]es^~iRadixmtof Detroit rrmajor financial insta &-r 'io ore etC re pTC rr,^ -,),ld(x,\' c. W eThe survey, undertaken in December 1975, was based on a sampling frame of the homes listed in the most recent city directory.' Approxi!ately oj e-oui of eu32 hoi es was contacted. The remainder of this article will examine the Huff assumptions in ti light of the survey data to determine whether or not the Huff model may be usefully applied to the banking problem. () EMPIRICAL EXAMINATION OF HUFF AND BATSELL'S AREAS OF CONCERN Trip and Product Type The attraction function in the Huff model applies for a oiven single^ J^*^ wstV ot\ak- ci ^ri^SU.t'Ac.KstSI J purpose shopping trip. Also, the distance function is plcable ~' ~ the consumer's present location am4 a single destination. If several stores are visited a consumer might seek to maximize some joint utility and to minimize total distance traveled. Hilliard (Hilliard, Vaughnand ReynoldsY 1975) reports some results that indicate that this effect is mixed and may be negligible. ouc w hp c'the of sO h- Qv vi't g waiod c t so te irpe'rigto i g) O n a 2.Sf\<C^e.'-P rpoS: rt ^P) Wa>s ~ e al Adts trC vI l, When questioned about when the household banking was done, Farinington consumers gave the-pe-reae.f shown in Table 2.

-9 - The most frequent response was "special trips to bank." Of the twelve different responses to this question, only three clearly indicate multiplepurpose trips. These account for 43 percent of the responses; however, a clear majority of the responses do not involve multiple-purpose trips. Thte-pjt4i'-ea tn-of-ga-rvTty-mod od e —Tu pbrop-e-irariaeio n a =s1 n'g'elputpoe- b ki n g-trl'pTps'e~~e^- b~a r e-v-ies. The survey results for Farmington show that most families use only one bank for checking services (the question was not asked for savings services). Table 3 shows these figures. X i~ Spatial Equilibrium The model does not specify consumer loyalty and laek.s consider-a-t4on of-temporal and spatial dynamics. In considering any well-established product, it may be reasonable to assume that the survey sample used to specify the model is a reasonable cross-section of the steady-state universe of buyers. Golledge indicates that, when this is not true, model errors may be significant (1970). -~F3l-l.-e-F a.rfirgtan —e-nsume-,s Tables 4 and 5 show the distribution of 'the length of-time consumers have utilized the major retail services. It is obvious that banking services have very loyal patrons and that any predicted change in market share may take a long time to materialize. For this reason, most users of gravity models speak of the prediction of equilibrium market share (BennettT 1975), which is defined as the market share that would result if the modeled conditions were to persist for an indefinite period of time.

-10 - ~ Choice Alternatives Systematic exclusion of establishments that are patronized by the public for the type of good in question can cause-over-p-r-ed.ilction. of the market share of the included establishments. Similarly, inclusion of alternatives not actually considered by the purchaser will result in an understatement of the predicted market share of the establishments actually used. Table 6 shows the usage of each of the major types of competitive alternatives for the primary consumer services. The major conclusion to be +ko-r +taLt. a0 \^.( ti - drawn from this table is the-n-eed to include commercial banks and thrift institutions in any banking market share model. O- Group Behavior The use of a model of individual choice (Luce's choice axiom [Luce;1959]') in a model of group behavior requires assumptions about the homogeneity of the choice process which may not be acceptable to some. A fully equivalent set of assumptions which deal only with group behavior is offered by Bell, Keeney/and Little (Bell, Keeney and Little' 1975). These assumptions may be used to derive Huff's model directly. ur \ As a practical matter, models of the gravity type have been used for some time to-suc-eessfti- y predict human behavior (Olsson 1965). We offer no empirical 'tests of this assumption, as none are appropriate.,f Choice Determinants Huff's final comment deals with the need for a multidimensional model of *consumer utility. The original model requires only two variables to compute the utility measure, distance (either actual or subjective)\and size. Two exponents serve as sensitivity measures and mediate the effects of those measures.

-11 -Huff states that those variables are surrogates for a number of other correlated measures and that two variables were chosen to make parameter estimation possible. Obviously, many variables might enter the consumer choice process, and a method of deciding upon importantvariables and specifying coefficients for the resultant model would be desirable. Huff notes the recent work of Nakanishi and Cooper in showing that least squares regression is an appropriate means of specifying such multivariate multiplicative models (Nakanishi and Cooper 1974). Several authors have reported,*, that such models are effective (Nakanishi and Cooper 1974; Eilon and Fowkes 1972; Lundsten 1976). ---- ~I-~-~ — ~ ~ -'-" I- --- I-~`- ~ =7= 7 — --

* ) SUMMARY AND CONCLUSIONS Data from a study of consumer behavior in one retail banking market were used to determine whether or not six critical assumptions of a model of retail gravitation would be met in the context of its use as a predictor of bank patronage. It appears that five assumptions are realistic inllight of reported consumer behavior in the use of banking services. The assumption of a single-purpose shopping trip is, however, less supportable given survey, — P-a- 2-v" data which indicate that over 400 of households consider their banking to be part ofrmulti-purpose shopping trip. The authors believe that this last finding should not eliminate the use of retail gravitation models such as the one developed by Huff in estimating market shares for new bank locations. The fact tha over half of the respondents did make a single-purpose trip to do their banking, and the findings of Hilliard (Hilliard 1975)A that the single trip assumption was probably not - '-criticalAindicate that the retail gravitation model is a useful tool for those persons in bank management or bank regulation who have a need to predict patronage levels of bank locations before they are built.

FOOTNOTE 1. Strictly speaking, the use of retail gravitation models in this fashion for new banking offices requires an assumption (treated briefly earlier) that the predictions made for an office not yet opened may be meaningfully interpreted as equilibrium market shares. The market share for all banking offices, both actual and proposed, may be computed and the results assumed to describe the market after the new branch has undergone a period of growth. The duration of this period and the rate of growth are not specified. Kramer (1971) has shown that the deposits at a bank branch grow in a regular fashion, typical of a given market, so once the equilibrium market share is known, it is possible to estimate deposit levels for prior years.

-13 -Works Cited David E. Bell, Ralph L. Keeney and John D. C. Little)"A Market Share Theorem." Journal of Marketing Research 12, May 1975, pp. 136-41. Rex 0. Bennetti Bank Location Analysis: Techniques and Methodologym Washi, D. C.'American Bankers Assoc., 1975, pp. 183-~88. Samuel Eilon and Terence R. Fowkes.Applications of Management Science to Bankingsr Essex, England; 1972, p. 215-k24. R. G. Golledge(Q "Some Equilibrium Models of Consumer Behaviore" Economic im grampy 46,970ii R -23. LVaughand Fred D. Re ds"A al Jimmy E. Hilliard, Ronald L. Vaughr and Fred D. Reynoldso"A Generalized Utility Model of Shopping Behavior." In Advances in Consumer Research. Edited by M. J. Schlinger(yChicago: Association for Consumer Research, 1975, XVol. 2.pp. 165-~72. David L. Huffd A Probabalistic Analysis of Consumer Spatial Behavior." I, Emerging Concepts in Marketing. Edited by W. S. Decker. Chicago: American Marketing Association, 1962,pp. 443-61. David L. Huff and Richard R. Batsell. "Conceptual and Operational Problems with Market Share Models of Consumer Spatial Behavior." In Advances in Consumer Research. Edited by M. J. Schlinger) Chicago: Association for Consumer Research, 1975, Aol. 2, p. 165-'72. Robert L. KramerD"Modern Methods for Locating Branches are Discussed," American Banker, March 24, 1972. R. D. Luce Individual, Choice Behavior~^New York: John Wiley and Sons, 1959. Lorman L. Lundsten, A Model to Improve the Quality of the Retail Share of Market Forecast for Banking Offices, Ann Arbor! Xerox Univ. Microfilms, 1976.

-14 - Maseo Nakanishi and Lee G. Coopers "Parameter Estimation for a Multiplicative Competitive Interactions Model - Least Squares Approach." Journalof Marketing Research 2, Augu-s- 1974,pp. 303-11. Gunnar Olsso~n "Distance and Human Interaction: A Review and Bibliography(' Bibliography Series No. 2~ Philadelphia: Regional Science Research Institute (1965). William J. Reilly "Methods for Study of Retail Relationships," Studies in Mark eti ng, No. ~)% 1-9S^r-Bs~r —fB-us:-R-. —U n.i v.wtiy-ofTexa,-Austn Aus'iQ 4 w,. oClup j^i^ < %^e..oC L l~o^^.L-i/tl rl I T^ ^.L~a Ij "''

Table I. Some Important Dominance Models (See Note 1) Individual Models (See Note 2) Date Researcher(s) Comments 1927 Thurstone Led to the robit model! 1952 Bradley & Terry Led to the Logit model 1959 Luce Equivalent to Bradley & -N\ 1961 Restle 1964 Coombs Terry's model )~ Set theoretic adaptation 'o Luce's model Probabflistic unfolding 6t,^4U6 "^dg A, Group Date 1858 1885 1924 1929 1940 1947 1949 1962 Models Res earcher(s) Carey Ravenstein Young Riley Stouffer Stewart Zipf Huff Comments "Social gravitation" model Migration model Migration model "Retail gravitation" model Spatial interaction model Spatial interaction model Social interaction model Market share, based on Luce's i2axioms Similar to Huff's model Market share model Multiplicative.ompetitive / Interaction model ( -^i 1969 Hlavic & Little Nakanishi 1970 Nakanishi & Cooper

k- k Table L. (continued) 1971 Kotler 1975 Bell, Keeney & Little 1976 Barnett Several related models of 'L rket share aV CL-c An axiomatic model for 7jouped data Qx. A refinement of Bell, Keeney ~&Little s Note 1: Many of these models measure absolute phenomena like migration. Share-of-market models, however, measure-a relative phenomenon. This distinction is not material, as any absolute model may be converted into a relative one by simply dividing by the total of the phenomenon under study. (tI.e. / model of sales may be converted into a model of market share by dividing by total sales.) Note 2: The distinction between individual and group models is not as important as it might seem, as nearly all individual models were tested using aggregate group data.

TAN E'E QUESTION 5: WHEN DO YOU BANK?* Do you do your Code banking... Value 1 2 3 Total Percentage_ When shopping for groceries 1 124 - -- 124 16.4 While doing nongrocery shopping 2 36 47 3 86 11.4 While travelling to or from work 3. 104 8 4 116 15.4 During working hours 4 55 13 1 69 9.2 Special trips to bank 5 233 43 16 292 38.7 Whenever handy 6 9 1 4 14 1.9 Bank by mail 7 16 4 1 21 2.8 On pay day during work 8 3 1 1 5.7 On Fridays or Saturdays 9 8 4 -- 12 1.6 When pension check comes 70 5. --- --- 5.7 Not ascertained 10 1 477 568 - First of month 11 4 1 2 7.9 Other times 12 2 1 --- 3.4 Total 600 600 600 Total 599 123 32 Base for/ercentages 754 100.0 75O.O IZ * Up to three mentions were coded.

p A ' TkNU USED3 NUMBER OF BANKS USED QUESTION 7: How many different banks do you and other members of your household use for checking account services? Code Value n Percentac-e 1 2 3 4 5 or more 1 475 74 83.8 13.1 2 3 13 2.3 4 2.4 5 3.5 Refused to answer.Don't use any 11 10 6 27 Total 600 Base for Percentages 567 100.0

QUESTION lOb: LENGTH OF TIME AT THIS OFFICE FOR CHECKING Approximately how many years have you used this particular office for checking account Code services? Would you say.. Value n Percento,A_ 1 or less years 1 69 12.1 About 2 years 2 69 12.1 About 3 years 3 55 9.6 4 to 5 years 4 70 12.2 6 to 10 years 5 135 23.6 More than 10 years 6 174 30.4 Don't use checking 10 27 -- Refused 11-12 1 --- Total 600 Based for /ercentages. 572 100.0 Median: 5.9 years

QUESTIONS 13c AND b: LENGTH OF TIME AT THIS OFFICE QUESTIONS 13c AND i8b: LENGTH OF TIME AT THIS OFFICE ', I --- And how long have you used the savings services at this particular office? Would you say...* Save at Checking Bank PercentiS.e Code Value n Save Elsewhere. PercentcQaS All Savers Percentc f n n Z/ 1 year or,less About 2 years About 3 years I About 3 years About 4 or 5 years 6 to 10 years More than 10 years Refused 1 2. 3 4 5 6 41 34 34 38 72 120 12.1 10.0 10.0 11.2 21.3 35.4 22 30 22 32 36 79 10.0 13.5 10.0 14.5 16.3 35.7 63 64 56 70 108 199 11.3 11.4 10.0 12.5 19.3 35.5 12 4 4 8 q, Base for/ercentages 339 100.0 221 100.0 560 100.0 Median: 6.7 years \ * The wording on question 18b was: Approximately how many years have you used this particular office A____ for savings account services? Would you say...

TITITTI 6 TYPES OF INSTITUTIONS USED n 336 ut_&-2_-At-,?,.~, Commercial /Bank Only Savings andAYoan ~Asociation )'nly Credit,Uni on.Only Commercial,/ank and S & L,Commercial 'ank and Credit.Uion Commercial /ank, Vredit -J.fion and S & L None orN./A. 6 1 108 126 14 9 Percent ac56.0 1.0.2 18.0 21.0 2.3 1.5 600 100.0

14' U-) CII F 0 + 4 ~f Z, Fi - CU?t -:0 -_2 '3 ("El) K I ^ t r, — 1 I' N.. J*^ I

WORKS CITED David E. Bell, Ralph L. Keeney and John D. C. Little. "A Market Share Theorem." Journal of Marketing Research 12, May 1975, pp. 136-41. Rex O. Bennett. Bank Location Analysis: Techniques and Methodology. Washington, D. C.: American Bankers Assoc., 1975, pp. 183-88. Samuel Eilon and Terence R. Fowkes. Applications of Management Science to Banking. Essex, England: 1972, pp. 215-24. R. G. Golledge. "Some Equilibrium Models of Consumer Behavior." Economic Geography 46, 1970, pp. 417-23. Jimmy E. Hilliard, Ronald L. Vaughn, and Fred D. Reynolds. "A Generalized Utility Model of Shopping Behavior." In Advances in Consumer Research. Edited by M. J. Schlinger. Chicago: Association for Consumer Research, 1975, vol. 2., pp. 165-72. David L. Huff. "A Probabalistic Analysis of Consumer Spatial Behavior." In Emerging Concepts in Marketing. Edited by W. S. Decker. Chicago: American Marketing Association, 1962, pp. 443-61. David L. Huff and Richard R. Batsell. "Conceptual and Operational Problems with Market Share Models of Consumer Spatial Behavior." In Advances in Consumer Research. Edited by M. J. Schlinger. Chicago: Association for Consumer Research, 1975, vol. 2., pp. 165-72. Robert L. Kramer. "Modern Methods for Locating Branches Are Discussed." American Banker, March 24, 1972. R. D. Luce. Individual Choice Behavior. New York: John Wiley and Sons, 1959. Lorman L. Lundsten. A Model to Improve the Quality of the Retail Share of Market Forecast for Banking Offices. Ann Arbor: Xerox Univ. Microfilms, 1976. Masgo Nakanishi and Lee G. Cooper. "Parameter Estimation for a Multiplicative Competitive Interactions Model - Least Squares Approach." Journal of / Marketing Research 2, Aug. 1974, pp. 303-11. Gunnar Olsson. "Distance and Human Interaction: A Review and Bibliography." Bibliography Series No. 2. Philadelphia: Regional Science Research Institute, 1965. William J. Reilly. "Methods for Study of Retail Relationships." Studies in Marketing, No. 4. Austin: Bureau of Business Research, University of Texas, 1929.

Tabl Some Important Dominai Individual Models (See Note 2) Date Researcher(s) 1927 Thurstone 1952 Bradley & Terry 1959 Luce 1961 Restle 1964 Coombs Group Models Date Researcher(s) 1858 Carey 1885 Ravenstein 1924 Young 1929 Riley 1940 Stouffer 1947 Stewart 1962 Huff.e 1. ice Models (See Note 1) Comments Led to the Probit model Led to the Logit model Equivalent to Bradley & Terry's model Set theoretic adaptation of Luce's model Probabilistic unfolding Comments "Social gravitation" model Migration model Migration model "Retail gravitation" model Spatial interaction model Socidll interaction model Market share, based on Luce's axioms Similar to Huff's model Market share model Multiplicative competitive interaction model 1969 ( 70 1970 Hlavic & Little Nakanishi Nakanishi & Cooper