optimal search on spatial paths with recall, part ii: computational procedures and examples

Papers Reg. Sci. 79, 293–305 (2000)

c© RSAI 2000

Optimal search on spatial paths with recall,Part II: Computational procedures and examples

Mitchell Harwitz1, Barry Lentnek2, Peter Rogerson2, T. E. Smith3

1 Department of Economics, State University of New York at Buffalo, 415 Fronczak Hall, Buffalo,NY 14260-1520, USA (e-mail: [email protected])2 Department of Geography, State University of New York at Buffalo, Wilkeson Quad, Buffalo,NY 14261, USA (e-mail: [email protected])3 Department of Systems Engineering, University of Pennsylvania, 220 South 33rd Street,Philadelphia, PA 19104-6315, USA (e-mail: [email protected])

Received: 10 October 1996 / Accepted: 3 May 1999

Abstract. This is the second part of a two-part analysis of optimal spatial searchbegun in Harwitz et al. (1998). In the present article, two explicit computationalprocedures are developed for the optimal spatial search problem studied in PartI. The first uses reservation prices with continuous known distributions of pricesand is illustrated for three stores. The second does not use reservation prices butassumes known discrete distributions. It is a numerical approximation to the firstand also a tool for examining examples with larger numbers of stores.

JEL classification: D83, R10

Key words: Search, spatial search, spatial economics

1 Introduction

This article continues the analysis begun in Harwitz et al. (1998), here designatedas Part I. With the general theoretical results in Part I in hand, we turn now to themore practical issue of computing optimal stopping rules for search situations.Initially, we focus on optimal stopping rules for standard search situations, whichwere shown in Part I to be characterized by reservation prices. In this contextwe begin in Sect. 2 below with a sharper characterization of reservation pricesthat is more useful for computational purposes than was the characterization inPart I. These results are then applied, and they yield an operational procedurefor the computation of reservation prices and the identification of optimal searchpaths. It is illustrated by a three-store example, which is small enough to al-low a representative reservation price function to be computed in detail. Thedetailed examination reveals a property of optimal search not noted heretofore,the ’stay close to home’ property. The qualitative properties of optimal stoppingrules are further examined by a numerical procedure based upon discrete price

294 M. Harwitz et al.

distributions which is described in Sect. 3. Detailed calculations again illustratea characteristic of optimal search not emphasized before now, namely that opti-mal search will be extended to more stores as price variation is more importantrelative to incremental travel cost. The article concludes in Sect. 4 with a briefdiscussion of possible extensions of the analysis.

To avoid unnecessary repetition, we take the definitions and results of Part Ias given. In particular we use the symbol ’I’ to denote all references to Part I.For example, ’Theorem I.2.1’ refers to Theorem 2.1 in Part I and ’(I.2.1)’ refersto expression (2.1) in Part I.

2 Computation of reservation prices

The development of an operational procedure for computing reservation prices re-quires a more detailed analysis of their properties than has been done heretofore.In this section, we identify four distinct cases that can arise in the computation ofreservation prices. To do so, we first recall from Theorem I.2.1 that all optimalstopping rules are defined in terms of the family of minimum cost functions,{C0, C1, . . . , Cn}, which are in turn defined by backward recursion. Hence it isclear that all stopping-rule calculations must begin with the final stopping costfunction, Cn in (I.2.7), and proceed backward by successive calculation of thesequence of minimum cost functions,Ci , i = n − 1, . . . , 0, in terms of the asso-ciated sequence of expected minimum cost functions,E (Ci+1| ·), i = n −1, . . . , 0as in (I.2.8). For all standard search situations, (c, P , ρ), these expected min-imum costs were shown in Theorems I.2.3 and I.3.1 to have the explicit form(I.2.35) for each realized price sequence (p0, . . . , pi ), where the expectation in(I.2.35) makes use of ¯pi+1(p0, . . . , pi ) given by (I.3.5). Hence, for purposes ofcomputation, we shall always take this explicit form of expected minimum coststo be understood, and shall simply writeE (Ci+1|p0, . . . , pi ) for notational con-venience. For each given realized price sequence (p0, . . . , pi−1 ) at stagei of thecalculation, it then follows from Theorem I.2.4 together with (I.2.7) and (I.2.8)that the explicit form of the minimum cost function,Ci (p0, . . . , pi−1, ·), on thedomain of pricepi depends critically on the location of the reservation priceri (p0, . . . , pi−1), inside (or outside) this domain. For computational purposes, wenow identify four (mutually exclusive and collectively exhaustive) cases for eachstagei = n −1, . . . , 1, as depicted in Fig. 1 below. Here it is assumed that a real-ized price sequence (p0, . . . , pi−1) is given, and that the minimum cost function,Ci+1, has been previously calculated. In this context, the four cases are distin-guished by the relation of the stopping-cost function,yi (pi ) ≡ yi (p0, . . . , pi−1, pi ),to the continuation-cost function,ci (pi ) ≡ ci ,i+1+E (Ci+1|p0, . . . , pi−1, pi ). In par-ticular, for a given continuation cost function,ci , each casek is represented by acorresponding stopping-cost functionyk

i , k = 1, 2, 3, 4. In addition, the maximumprice levels in (I.2.18) and (I.2.19) are denoted in Fig. 1 by ˆpi = pi (p0, . . . , pi−1 )and pi = pi (p0, . . . , pi−1 ), respectively. The specifics of each case are detailedbelow. For computational purposes, it is convenient to begin with the cases that

Spatial search: computational procedures 295

Fig. 1. Illustration of the four cases

are most easily checked, namely, the cases of negative and infinite reservationprices, respectively.

Case 1. Negative reservation pricesTo determine whether or not storei has stopping power for any given realizedprice sequence (p0, . . . , pi−1), it follows at once from Definition I.3.3 that we needonly check condition (I.3.24). If this condition fails to hold, then by definitionri (p0, . . . , pi−1 ) = −1. Geometrically, this situation corresponds to a stoppingcost function,yi , which is everywhere above the continuation-cost function,ci ,such as the functiony1

i shown in Fig. 1. On the other hand, if this conditionholds, then by Theorem I.3.2 there must exist a nonnegative reservation price,which is possibly infinite.

Case 2. Infinite reservation pricesTo check for the possibility that storei is a terminal point given price information(p0, . . . , pi−1), it is convenient to begin by computing the maximum price levelpi (p0, . . . , pi−1) in (I.3.5), together with the associated expected minimum costlevel:

Ei+1(p0, . . . , pi−1) = E [Ci+1|p0, . . . , pi−1, pi (p0, . . . , pi−1)] (2.1)

In terms of this expected minimum cost level, we have the following simplecomputational criterion for terminal points (see Appendix A, which is availableon request from the authors, for proof):

Lemma 2.1. For any standard search situation (c, P , ρ) and i ∈ {1, . . . , n − 1},point i is a terminal point in ρ given (p0, . . . , pi−1 ) iff:

yi (p0, . . . , pi−1) ≤ ci ,i+1 + E (p0, . . . , pi−1) (2.2)


Hence the case of infinite reservation costs corresponds to a stopping-costfunction, yi , everywhere below the continuation-cost function,ci , such as thestopping-cost function,y2

i , shown in Fig. 1.

Case 3. Proper reservation prices of explicit typeIf (I.3.24) and (2.2) both fail to hold, then (by the proof of Theorem I.3.6)it follows that there must exist a proper reservation price,ri (p0, . . . , pi−1), ati given price information (p0, . . . , pi−1). In this case, observe that if we nowcompute the maximum price level, ˆpi (p0, . . . , pi−1), in (I.2.18), together with theassociated minimum cost level,

Ei+1(p0, . . . , pi−1) = E [Ci+1|p0, . . . , pi−1, pi (p0, . . . , pi−1)] (2.3)

then we obtain the following condition for an explicit solution of this reservationprice level (see Appendix A for proof):

Lemma 2.2. For any standard search situation (c, P , ρ) and i ∈ {1, . . . , n − 1},if condition (2.2) fails to hold, but:

yi [p0, . . . , pi−1, pi (p0, . . . , pi−1 ) ] ≤ ci ,i+1 + Ei+1(p0, . . . , pi−1 ) (2.4)

then there exists a unique proper reservation price, ri (p0, . . . , pi−1 ), given by:

ri (p0, . . . , pi−1 ) = [ci ,i+1 − ci (i )] + Ei+1(p0, . . . , pi−1) . (2.5)

Hence under these conditions, the stopping-cost function,yi , must intersect thecontinuation-cost function,ci , between ˆpi andpi , as illustrated by the functiony3

iin Fig. 1 with corresponding reservation priceri . In particular, it is the constancyof ci in this region that allows the explicit solution forri in (2.5) to be obtained.

Case 4. Proper reservation prices of implicit typeFinally, if conditions (I.3.24) and (2.4) both fail to hold, the proper reservationprice can only be obtained in implicit form, and by Theorem I.2.4 must be givenby the solution of the equation (I.2.41). This equation can be slightly simplifiedunder these conditions as follows (see Appendix A for proof):

Lemma 2.3. For any standard search situation (c, P , ρ) and i ∈ {1, . . . , n −1}, if conditions (I.3.24) and (2.4) both fail to hold, then ri (p0, . . . , pi−1) ∈[0, pi (p0, . . . , pi−1)) and is given by the unique nonnegative solution of the fixed-point problem:

ri = [ci ,i+1 − ci (i )] + Ei+1(Ci |p0, . . . , pi−1, ri ), ri ∈ R+ (2.6)

Hence in this final case it is necessary to solve the fixed-point problem in (2.6).To do so, one can utilize condition (I.3.10) of Lemma I.3.3 as follows. If wenow let:

Di (p0, . . . , pi−1 ) = {pi ∈ R+ : pi + ci (i ) − [ci ,i+1

+E (Ci+1|p0, . . . , pi−1, pi )] ≤ 0} (2.7)

and observe from (I.3.8) together with the proof of Lemma 2.3 that in the presentcase,Pi (p0, . . . , pi−1 ) = Di (p0, . . . , pi−1 ), it follows that the solution to (2.6) canbe equivalently written as:


ri = maxDi (p0, . . . , pi−1) (2.8)

In particular, since (I.3.10) also implies thatDi (p0, . . . , pi−1) = [0, maxDi (p0, . . . , pi−1)], we see thatri is obtainable by a simple one-dimensionalsearch, proceeding from zero up to the point where the quantity,pi + ci (i ) −[ci ,i+1 + E (Ci+1|p0, . . . , pi−1, pi )], first reaches zero. This final case is illustratedby the stopping-cost function,y4

i in Fig. 1, where by definitiony4i (pi ) − ci (pi ) =

pi + ci (i ) − [ci ,i+1 + E (Ci+1|p0, . . . , pi−1, pi )]. Hence in this case the reservationprice,r ′

i , is obtainable by a simple line search in the setD(p0, . . . , pi−1) = [0, r ′1]

By way of summary, the reservation price functions,ri : Ri+ → R∞, can

be computed for each realized price sequence (p0, . . . , pi−1 ) ∈ Ri+, by checking

Cases 1 through 4, respectively, until the appropriate case for (p0, . . . , pi−1) isfound. If we now denote the relevant price domains inR

i+ corresponding to

conditions (I.3.24), (2.2), and (2.4) respectively, by:

P0i = {(p0, . . . , pi−1) ∈ R

i+ : yi (p0, . . . , pi−1, 0)

> ci ,i+1 + E (Ci+1|p0, . . . , pi−1, 0)} (2.9)

Pi = {(p0, . . . , pi−1 ) ∈ Ri+ : yi (p0, . . . , pi−1)

≤ ci ,i+1 + Ei+1(Ci+1|p0, . . . , pi−1)} (2.10)

Pi = {(p0, . . . , pi−1) ∈ Ri+ : yi [p0, . . . , pi−1, pi (p0, . . . , pi−1 )]

≤ ci ,i+1 + Ei+1(p0, . . . , pi−1)} (2.11)

then this computational procedure can be formalized in terms of the followingmore explicit definition of the reservation price functions,ri : R

i+ → R∞, in

(I.3.15) for each (p0, . . . , pi−1 ) ∈ Ri+ and i = 1, . . . , n − 1,

ri (p0, . . . , pi−1 )

=

−1 , (p0, . . . , pi−1 ) ∈ P0i

∞ , (p0, . . . , pi−1 ) ∈ Pi

[ci ,i+1 − ci (i )] + Ei+1(p0, . . . , pi−1) , (p0, . . . , pi−1 ) ∈ Pi − Pi

maxDi (p0, . . . , pi−1) , otherwise (2.12)

Example 1. Reservation price functions and optimal stopping in a three-storeexample. Consider a simple search situation (c, P , ρ) with n = 3, ρ = (0, 1, 2, 3),zero shopping cost (si ≡ 0) and Pi uniformly distributed on [0, 1], for eachi = 1, 2, 3, and with travel costs corresponding to those in Fig. 2 below (where thehome location 0 is denoted by the small circle). Here we let the initial price equalthe upper bound,p0 = 1, to ensure that search will take place, and in addition,that the ‘return home without a purchase’ option is never relevant. To completelyspecify the optimal stopping rule,S ∗(p0, ·) for the initial price conditionp0 = 1,we first evaluate the reservation price functionr2(1, p1). This reservation price issolely a function of the realized price at store 1, p1. Because (c, P , ρ) is simple,Corollary I.3.6. establishes that Case 1 (negative reservation prices) never holds,


Fig. 2. Travel costs for examples 1 and 2

and hence that the reservation pricer2 is everywhere nonnegative. However,all other cases are seen to hold in certain ranges of the realized pricep1. Thecalculations are described in full detail in Appendix B (available on request fromthe authors). In the range of realizations where store 2 is reached, store 2 can bea terminal point and Case 2 may hold. Thenr2 is infinite if p1 ≤ p′

1, which isknown from (I.3.32) to be the solution to the equation:

y2(p0, p1) = c23 + E3(p0, p1) . (2.13)

It can be shown that the unique value ofp1 satisfying this equation isp′1 = 0.657.

The corresponding reservation price,r ′2, is 0.807, which can be shown to be

a proper reservation price of implicit type (Case 4). For valuesp1 > p′1, the

reservation price is always proper, therefore everywhere non-decreasing inp1.The boundary realization between Cases 3 and 4 is the value ˜p1 = 0.844, whichis the unique solution to the equation:

p2(1, p1) + c2(2) = c23 + E [C3|1, p1, p2(1, p1)] (2.14)

where p2(1, p1) is defined using (I.2.17) and (I.2.18). The corresponding reser-vation price level is ˜r2 = r2(1, p) = 0.844. The relation ofp1 to these values ofthe reservation price is depicted in Fig. 3.

With these computations in hand, we have the reservation price function,r1(1, p1) for eachp1 ∈ [0, 1], so we may proceed to calculate the minimumcost functionC2(1, p1). Thereby we may determine the unique reservation pricelevel r1[= r1(p0) = r1(1)] by solving the equation:

r1 + c1(1) = c12 + E (C2 | 1, r1) (2.15)

This yields a valuer1 = 0.625, which completes the specification of the optimalstopping rule,S ∗(p0, ·), for the initial price condition,p0 = 1. This optimalstopping rule is depicted in Fig. 4 below, where each regionSk , k = 1, 2, 3,denotes the set of price pairs (p1, p2) for which stopping occurs at storek underS ∗. In particular,Sk = {(p1, p2) ∈ [0, 1]2 : S ∗(1, p1, p2, 0) = k} for k = 1, 2, andS 3 = [0, 1]2 − (S 1 ∪ S 2) (sinceS ∗(1, 0, 0, 0) > 0 in this example). Notice also


Fig. 3. Reservation price function

Fig. 4. Optimal stopping rule

that the reservation price function,r2, in Fig. 3 above now is seen to define theboundary betweenS 2 andS 3. In particular, observe that sincer1 < p′

1, there isa range ofp1 values, (r1, p′

1], for which store 2 will be reached and will be aterminal point.

An application of the procedure for computing reservation prices leads di-rectly to a computational procedure for determining the expected minimum cost,E (ZS ∗ |p0), of the optimal stopping rule,S ∗ ∈ Sρ, for each possible pathρ.The recursive calculations yield successive values for the minimum cost func-tions,{Cn−1, . . . , C1}, and hence yield the expected minimum cost,E (ZS ∗ |p0) =C0(p0) = min{p0, c01 + E (C1|p0)}, as in (I.2.10). We may denote the expectedminimum cost on search pathρ by EMC (ρ) and designate the search paths,ρ,with the smallest values of EMC as optimal search paths.

For p0 = 1, the search path,ρ = (0, 1, 2, 3), just analyzed for the three-storecase, turns out to be an optimal search path (along with its ’twin’ path, (0,2,1,3)),and yields anEMC value,E (ZS ∗ |p0) = c01 + E (C1|p0) = 0.634. It is interesting


to observe here that the path circuits with minimum total travel cost (i.e., thetraveling-salesman paths) are given by the twin paths, (0,1,3,2) and (0,2,3,1),with identical EMC values of 0.668 > 0.634. Hence even though the pricesat each store are independently and identically distributed, traveling-salesmanpaths are not generally optimal search paths. A closer inspection of this exampleshows that, unlike the traveling-salesman paths, the optimal search paths tend to‘stay close to home’ as long as possible in the initial phase of the search. Wedo not believe that this feature has been noticed before now. It is a reflection ofthe stopping option that is relevant in search situations, but not in the traveling-salesman problem. The stay-close-to-home property is even more evident in thefour-store case below.

3 A discrete computational method, a four-store example,and a further observation

In this section we shall demonstrate for a four-store example a method of cal-culating optimal search paths when price observations are drawn from discreteprobability distributions. In addition to their independent interest, these calcu-lations enable us to make another observation about optimal search not madebefore. We learn that if price variation is large relative to travel cost, optimalsearch is likely to lead to travel paths that visit all stores, but that are still nottravelling salesman paths.

The calculation of optimal search paths based on discrete probability dis-tributions of price observations provides greater convenience and flexibility incomputations than calculations based on continuous distributions. For search situ-ations (c, P , ρ) in which c satisfies IA1, IA2, and IA3 and in whichP is made upof independent discrete distributions, the function defined in I.2.9 is an optimalstopping rule. In particular, for the family of search situationsS = {(c, P , ρ′) : ρ′

is a permutation of (0, 1, .., n) beginning with 0}, we may designate the optimalstopping rule on each path asS ∗(ρ′). The optimal search path then minimizesthe quantity,E (ZS ∗(ρ′)|p0) [= C0(p0)], over S. The theory of Section I.2 can beapplied directly to complete tabulations of possible price realizations, withoutthe need to compute reservation prices. Thus, these calculations constitute anindependent method of characterizing optimal search. We now illustrate such acomputation, and then offer an insight afforded by the study.

It is important to observe that while simple three-store examples with contin-uous probability distributions of prices can be computed with a high degree ofprecision using numerical integration algorithms, such computational proceduresare notoriously time consuming in higher dimensions, and rapidly become infea-sible as the number of stores increases. If we wish to consider larger numbers ofstores by using a discrete method we should note the level of approximation thatcan be attained. In the present case, it was found that a discrete version usingfive equiprobable price levels{0.1, 0.3, 0.5, 0.7, 0.9} achieved a good level ofapproximation (for example,EMC = 0.635 on the optimal search paths). In terms


of this discrete set of prices, it can also be verified that the stopping rule picturedin Fig. 4 is approximated by the simpler form: (i) stop at 1 unlessp1 ∈ {0.7, 0.9}and (ii) stop at 2 unlessp2 ∈ {0.7, 0.9} andp1 = 0.9.

The determination of the optimal search path proceeds as before in two steps:(i) the calculation ofEMC along each path; (ii) the comparison ofEMC valuesto determine the least. In examples with a small number of possible values, step(i) can be accomplished by a complete tabulation of all outcomes, based on therecursion described in (I.2.7) through (I.2.10).

Example 1. Characterizing optimal search paths in a three-store example. Con-sider a three-store example with five possible non-zero price realizations,{0.1, 0.3,0.5, 0.7, 0.9}. For the sake of clarity, it is important to distinguish between thenames of locations (0 = ‘home’, 1 = ‘store 1’, 2 = ‘store 2’, etc.) as givenin Fig. 2, and their position (order) in a given search path. To do so, we nowdesignate the value (location label) of thek -th location visited in search pathρ as ρ[k ]. For example, the search pathρ = (0, 1, 3, 2) is now represented by(ρ[1] = 0, ρ[2] = 1, ρ[3] = 3, ρ[4] = 2); recall that in the present modelρ[1] = 0for all possible search pathsρ. For each store location,i = 2, 3, 4 we may definethe relevant (common) price vector and probability vector, respectively, as:

pi = (pi1, . . . , pi5) = (0.1, 0.3, 0.5, 0.7, 0.9) i = 2, 3, 4 (3.1)

πi = (πi1, . . . , πi5) = (0.2, 0.2, 0.2, 0.2, 0.2) i = 2, 3, 4 (3.2)

so that for any search path,ρ, the price and probability vectors attached to eachstoreρ[k ] are denoted bypρ[k ] andπρ[k ] .1 The matrix of movement costs,cij , isalso needed. From it can be derived the matrix of “recall costs”,ci (j ). In thisexample they are given respectively by:

C =

c00 c01 c02 c03

c10 c11 c12 c13

c20 . . . . . . . c23

c30 . . . . . . . c33

=

0.0 0.1 0.25 0.10.1 0.0 0.20 0.150.25 0.2 0.0 0.20.1 0.15 0.2 0.0

(3.3)

and

R =

r00 r01 r02 r03

r10 r11 r12 r13

r20. . . . . . .r23

r30. . . . . . .r33

=

0.0 0.2 0.5 0.20.1 0.1 0.45 0.250.25 0.3 0.25 0.30.1 0.25 0.45 0.1

(3.4)

where by definition,rij = cij + cj0 for all i , j = 0, 1, 2, 3. For a search pathρ withρ[2] = 1 andρ[3] = 3 , the symbolrρ[2]ρ[3] denotes the entryr13 in the matrixR.

We now recursively define the stopping cost functionsyi (p0, . . . , pi ) of (I.2.1).The functiony1(p0, p1), with p0 = 1, can be viewed as a ‘two way’ table (ormatrix), Y 2, of dimension 3× 5, where each component has the form:

Y 2k ,j = min{p0 + rρ[k ]ρ[1] , pρ[2]j + rρ[k ]ρ[2]} (3.5)

1 More generally, these prices and probabilities can be given any values, provided that the prob-abilities sum to one and are independent between stores.


for k = 2, 3, 4 andj = 1, 2, . . . , 5, and wherepρ[k ]j denotes a particular realizationof the random price variablePρ[k ] . In particular, one should notice here that thesecond and third rows ofY 2 represent acquisition costs whenρ[k ] takes thevaluesρ[3] and ρ[4], that is, when either the home location or the first visitedstore is recalled from a later position in the search path. It should be noted that(3.5) gives only a rule (function) for constructing the tableY 2. In the languageof computational algebra, the function remains unevaluated until it is called withspecific values ofk and j . The ‘three way’ table,Y 3, is of dimension 2× 5× 5,wherek refers only to the second and third visited stores (i.e.,k = 3, 4), and theprice realizations refer toρ[3] andρ[4]. A typical entry ofY 3 can be written as:

Y 3k ,j ,m = min{Y 2k ,j , pρ[3]m + rρ[k ]ρ[3]} (3.6)

where k = 3, 4, j = 1, 2, . . . , 5, and m = 1, 2, . . . , 5. It is to be expected, ofcourse, that the stopping cost function at the third visited store refers to theoption of recalling either the first or second store visited. The tableY 4 dependson the realizations of all three store prices, and has components of the form:

Y 4k ,j ,m,n = min{Y 3k ,j ,m , pρ[4]n + rρ[4]ρ[k ]} (3.7)

wherek = 4 and wherej , m, n = 1, 2, . . . , 5. Here the subscript ‘k ’ is includedsimply to maintain a parallel with the previous definitions.

According to (I.2.7) and (I.2.8), when the value:

Cρ[4] (p0, pρ[2]j , pρ[3]m , pρ[4]n ) = yρ[4] (p0, pρ[2]j , pρ[3]m , pρ[4]n ) = Y 44,j ,m,n (3.8)

is computed for any realization, (pρ[2]j , pρ[3]m , pρ[4]n ), of the random price vector,(Pρ[2] , Pρ[3] , Pρ[4] ), we may recursively define the cost functions,Cρ[3] , Cρ[2] , andCρ[1] as follows. First, it can be seen from (I.2.8) thatCρ[3] can be constructedin terms ofY 3, Y 4, R andp as a 5× 5 table with components of the form:

Cρ[3] (p0, pρ[2]j , pρ[3]m ) = min{Y 33,j ,m , rρ[3]ρ[4] + Σ5n=1(πρ[4]n )(Y 44,j ,m,n )} (3.9)

for eachj , m = 1, 2, . . . , 5. In a similar manner, it also follows from (I.2.8) thatthe components ofCρ[2] are given by:

Cρ[2] (p0, pρ[2]j ) = min{Y 22,j , rρ[2]ρ[3]

+Σ5m=1(πρ[3]m )Cρ[3] (pρ[1] , pρ[2]j , pρ[3]m )} (3.10)

for eachj = 1, . . . , 5, and that the final minimal expected cost,Cρ[1] , is givenby:

Cρ[1] (p0) = min{p0, rρ[1]ρ[2] + Σ5j=1(πρ[2]j )Cρ[2] (p0, pρ[2]j )} (3.11)

For each search path,ρ, the expected minimum cost,EMC (ρ), is given by:

EMC (ρ) = E (ZS ∗ |p0) = Cρ[1] (p0) (3.12)

These cost calculations for any given search path,ρ, have been programmed inboth MATHEMATICA c© and MAPLEc©, and are available from the authors.2

2 These programs were written to analyze the properties of small problems (three and four stores)and are not suitable for much larger problems. They were written for the case of identically dis-tributed price variables, but have been successfully modified to allow for differing but independentdistributions by taking care to associate each distribution with a specific store.


For small problems, one can enumerate all possible search paths and computeEMC (ρ) for each. The least of these numbers defines the optimal path(s).

Example 2. Characterizing optimal search paths in a four-store example. To fa-cilitate comparison with the three-store case above, this example differs only bythe addition of a fourth store, as shown in Fig. 5 below. It is again assumedthat shopping costs are identically zero (so that all search situations are sim-ple) and that all prices are independently uniformly distributed on the price set{0.1, 0.3, 0.5, 0.7, 0.9}. With the addition of the fourth store, a continuation (0,1, 2, 3, 4) of the optimal path (0, 1, 2, 3) in the three-store case of Sect. 3 yieldsthe same value,EMC = 0.635 (indicating that the presence of the fourth storeon this search path is irrelevant and that store 3 is a terminal point on this path).

The optimal search paths in this case, however, are given by the twin paths(0, 1, 2, 4, 3) and (0, 2, 1, 4, 3) withEMC = 0.620< 0.635. The addition of afourth store is indeed relevant. Moreover, an examination of Fig. 5 shows moreclearly than ever that the optimal paths exhibit the stay-close-to-home propertyin the initial search phase.

The present four-store example yields an additional insight. An examinationof the optimal search paths shows that the value,EMC = 0.620, is the same as thatobtained for the shorter search paths in which store 3 is omitted, i.e., that store 4 isa terminal point for this optimal path. The added cost of going to store 3 (c43 =0.35) is sufficiently large to outweigh any expected savings at this additionalstore. That there is a tradeoff between additional travel and added information canbe seen more clearly by considering a modification of the present example. Letall prices be increased by a factor of ten, so thatp0 = 10 andpi ∈ {1, 3, 5, 7, 9}for i = 1, 2, 3. It can then be verified that the optimal search path is given byρ2 = (0, 4, 1, 2, 3) with EMC = 2.77, versusEMC = 2.82 for the previouslyoptimal path,ρ1 = (0, 1, 2, 4, 3). Notice that both of these paths exhibit the stay-close-to-home property (since the first three destinations in both are the storesclosest to home). But there is a critical difference between these paths, which canbe seen by examining the successive store-to-store trip costs incurred along eachpath in Fig. 5, namely (0.10, 0.15, 0.18, 0.35) for ρ1 and (0.10, 0.18, 0.15, 0.20)for ρ2. Observe that if travel costs are very significant relative to prices (as inthe original example), then the large cost involved in visiting the last store (store3) in each path effectively removes it from consideration. We have noted thatthe third store visited is a terminal point on the pathρ1 in our original example.Hence only the first three costs are relevant. But (given the statistically identicalnature of all stores) it is clear that pathρ1 has an advantage overρ2, in theoriginal example, in that the smaller travel cost (0.15 versus 0.18) is incurredfirst in ρ1. If, instead, price variations are much more significant than travel costs(as in the modified example), there is a stronger incentive to visit all four stores.In this context, pathρ2 is now seen to have the advantage of incurring a muchsmaller travel cost to visit the last store (0.20 versus 0.35). Hence when pricevariations are large enough to ensure that the last store is always visited (in the


given travel-cost configuration), then pathρ2 now has an advantage overρ1. Itmay be noted that neither path is a travelling-salesman path.

4 Directions for further research

In this second article, we have developed two explicit procedures for computingminimum costs along spatial search paths, in one of which we used continuousknown price distributions and reservation price concepts, and in the other ofwhich we used discrete known price distributions. However, it should be empha-sized that our use of small examples to illustrate the procedures was not simply toease the presentation. When the number of search options is increased, say to tenstores, the computational task of determining the expected minimum cost evenon a single path becomes enormous. Indeed, it can be shown that in the discretecase, the computation time for the optimal stopping rule on each path is exponen-tial in the number of stores, and in particular increases by a factor approximatelyequal to the number of price levels used (in our example, five). Moreover, theproblem of determining an optimal path, even without recall, is well known tobe NP-hard (see, e.g., Maier 1991). Indeed, the number of potentially relevantsearch paths in ann-store example can in principle be of ordern!. Hence, evenwith very crude discrete approximations, the computation of optimal search pathsrapidly becomes infeasible as the number of stores increases. Thus, to analyzesearch problems involving large numbers of spatial alternatives, a future taskis to develop a range of computational heuristics for approximating solutions.Along these lines, the approximation method developed by Miller (1993, 1994)and Miller and Finco (1995) appears to be promising. Here simulation is used toestimate the probability that a given search path (and implicit stopping rule) isoptimal under very general conditions.

It should be emphasized, however, that the ultimate goal of this research isnot to solve optimal search problems, but rather to provide analytical tools forthe study of actual search behavior. In particular, it should be clear from theabove analysis that no shopper can be expected to compute an “optimal” searchpath through the set of relevant stores. Rather, the notion of an optimal searchpath (or more generally, an optimal search strategy) is most usefully viewed as abenchmark against which actual search heuristics may be gauged. For example,some shoppers may employ a fixed reservation cost level throughout the search,or may simply minimize search effort by purchasing the good at the closestavailable store. Hence it is of interest to ask under what conditions these simpleheuristics might be most appropriate, i.e., might be “close to optimal”.3 Moregenerally, there exist a number of more sophisticated heuristics in the stoppingrule literature that might prove to be very effective in certain spatial search con-texts. For example, the popular one-step look ahead (OLA) rule which in ourcontext defines reservation prices solely on the basis of expected price levels

3 Here the simulation estimates of ‘optimal paths’ employed by Miller (1993, 1994) and Millerand Finco (1995) may be useful.


at the next store, is well known to be optimal in certain types of stopping-ruleproblems (Ross 1970; Yasuda 1988). Hence it would be of interest to computethe expected total cost of the OLA-rule for a selected range of spatial search sit-uations, and to compare the efficiency of this rule with the optimal stopping rule.In the context of economic analysis, the task is to determine the heuristics thatsurvive as search procedures in market situations with limits on available com-putational resources. These types of applications will be explored in subsequentpapers.

References

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242–251Maier G (1995)Spatial search: structure, complexity, and implications. Physica-Verlag, HeidelbergMiller H (1993) Modeling strategies for the spatial search problem.Papers in Regional Science 72:

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optimal search on spatial paths with recall, part ii: computational procedures and examples

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