A TWO-PLAYER MODEL FOR THE SIMULTANEOUS LOCATION OF FRANCHISING SERVICES WITH PREFERENTIAL RIGHTS Pedro Godnho and oana Das Faculdade de Economa and GEMF Unversdade de Combra Av. Das da Slva 65 3004-5 Combra Portugal Faculdade de Economa and Inesc-Combra Unversdade de Combra Av. Das da Slva 65 3004-5 Combra Portugal {pgodnho joana}@fe.uc.pt Keywords: Abstract: Locaton problem Competton Smultaneous decsons Game theory Nash equlbrum. We consder the dscrete locaton problems faced by two decson-makers franchsees that wll have to smultaneously decde where to locate ther own servces (unsure ab the decsons of one another). All servces compete among themselves. At most one servce can be located at each potental locaton. We consder that one of the decson-makers has preferental rghts meanng that f both decson-makers are nterested n the same locaton only to ths decson-maker wll be gven the permsson to open the servce. We present a mathematcal formulaton and some conclusons based on computatonal results. INTRODUCTION Compettve locaton problems consder the stuaton where t s not suffcent for a gven decson maker to consder only hs own facltes when faced wth a locaton decson (through the paper we wll refer to faclty and servce nterchangeably). Most of the tmes these facltes wll compete wth smlar facltes n the market so that the customers share that wll be assgned to the decson-maker s facltes depends on hs own choces as much as on the compettors decsons. In ths paper we work wth a compettve dscrete locaton problem where two decson makers (players) wll have to decde smultaneously where to locate ther own facltes unsure ab the decsons of one another. Several authors have studed compettve locaton problems (for a revew see for nstance Plastra 00). Dobson and Karmarkar 987 study a dscrete compettve locaton problem n whch prce and demand are fxed and consderng an exstng frm a compettor and clents that want to mnmze the dstance traveled. Labbé and Hakm 99 study the problem n whch two frms have to make decsons regardng the locaton of facltes and also the quanttes of a gven commodty they wll make avalable. Vetta 00 also proposes a locaton game where multple decson makers (servce provders) start by decdng where to locate ther facltes and then defne how much they charge ther customers. Hande et al 0 study a sequental compettve locaton problem where the follower can react to the decsons made by the leader adjustng the attractveness of ther own servces. Among the compettve locaton lnear programmng problems most approaches ether consder that the frms already present n the market wll not be able to react to the decson-maker s new chosen locatons or consder a Stackelberg problem where there s a follower that wll react to a leader knowng what the leader has decded. These types of problems dffer sgnfcantly from the problem tackled here. We consder a stuaton where a franchser ntends to open new facltes n a gven area. There are two potental nvestors and the facltes to be open wll compete among themselves. They provde the same type of commodtes to consumers at the same prces and t s assumed that customers patronze the closest avalable faclty. The franchser defnes the fnte of potental locatons for facltes but he s not famlar wth the demand patterns of the area. So he wll defne more potental locatons than he expects the nvestors to choose and leave the choces among them to the nvestors who are better acquanted wth the area. The franchser payoff wll be a percentage calculated
over the total demand assgned to the new facltes. Each nvestor s nterested n maxmzng the total demand that s assgned to hs own facltes. Each nvestor s aware of the fxed costs ncurred by openng each and every faclty whch can be dfferent for both nvestors. Each nvestor has a budget constrant. They are also aware of the demand assocated wth each customer. Ths demand wll not ncrease wth dstance meanng that the closer the assgned faclty s of a gven customer the greater the demand from the customer. At each locaton at most one faclty can be opened. If the decson-makers were to decde sequentally ths problem would be a sequental problem that could be formulated as a blevel lnear programmng problem. But we consder that both decson-makers wll have to decde smultaneously. In ths stuaton t wll be necessary to clarfy what happens f both nvestors apply for the same locaton. As at most one servce can be opened the franchser wll have to decde what to do n ths case. We consder that the franchser patronzes one nvestor n detrment of the other. For the sake of smplcty consder that the franchser always chooses nvestor. Ths means that f both apply for the same locaton then the franchser wll allow nvestor to open the faclty and nvestor wll not be able to do so. We can say that nvestor has preferental rghts whch s known by both decson-makers. Ths problem can also be nterpreted as a full nformaton game (because each player knows the payoffs and strateges of the other) wth a fnte number of players (the two decson-makers) and a fnte number of pure strateges (for each player a pure strategy can be defned as a partcular combnaton of locatons of the of potental new locatons where the player chooses to open facltes). That s why we wll not dstngush between nvestor decson-maker and player and wll use these terms nterchangeably as havng the same meanng. We approach ths smultaneous decson problem from a mathematcal programmng pont of vew (a prelmnary mathematcal formulaton appeared n the research report Das and Godnho 0) and from a game theory pont of vew. The game wll have at least one Nash equlbrum possbly wth mxed strateges that can be calculated algorthmcally. Some computatonal results are presented and conclusons drawn. PROBLEM FORMULATION In ths problem we are consderng that both decson-makers wll decde smultaneously wth knowng the decson made by the other. We wll also accommodate the exstence of already opened servces. Let us consder that these servces belong to nvestor. Consder the followng defntons: F of pre-exstng facltes that belong to nvestor ; G of potental locatons for new facltes; of customers; dj demand assocated wth customer j when he s assgned to a faclty located at ; c dstance between customer j and locaton ; j fp fxed cost assocated wth nvestor p openng a faclty at locaton (and such that f 0 F ) p p percentage over the demand captured to be pad to the franchser by nvestor p; O maxmum budget avalable to nvestor p. p We consder that demand wll not ncrease wth dstance. We wll addtonally assume that potental locatons at the same dstance wll capture the same demand. Ths means that: cj ckj d j d kj k F G j () cj ckj d j d kj k F G j () Let us defne the followng decson varables: f nvestor ether opens a faclty at or has a y F G pre-exstng faclty at 0 otherwse f nvestor bds for w openng a faclty at FG 0 otherwse f nvestor opens a faclty at z FG 0 otherwse f clent j s assgned to faclty that belongs to xj FG j nvestor 0 otherwse f clent j s assgned to faclty that belongs to mj FG j nvestor 0 otherwse
In most locaton problems only bnary varables smlar to y and z are needed. For ths problem however another of varables w s essental to allow the dstncton between two dfferent stuatons: of bddng for and of beng able to open a faclty. Ths dstncton s not needed for nvestor : he wll open every faclty that he bds for because he has preferental rghts. But nvestor can bd for a locaton and stll not be able to open a faclty there f nvestor has also shown nterest for the same locaton. In ths problem dfferent s of varables are controlled by dfferent stakeholders: nvestor controls varables y G ; nvestor controls varables w G ; the franchser controls varables z G (accordng to a predefned rule known by both decson makers); customers control varables x m FG j (also accordng j j to known rules n ths case resortng to the mnmum dstance crtera). Decsons made by the franchser and by customers are not controlled by the two decson-makers despte the fact that these decsons play a crucal role n the determnaton of each players payoff. An mportant pont to make s that despte not beng under ther control decson makers are both fully aware of how these decsons are made. As a matter of fact once varables y and w are fxed t s possble to mmedately compute the correspondng values for x j m j and z. Each nvestor wll make hs own decsons condtoned only by hs own constrants. A of connecton constrants s then consdered that wll determne the values of the remanng varables accordng to the pre-establshed rules. Let us now formulate the problem followng the representaton ntroduced n Godnho and Das 00: Decson-Maker Max Subject to: FG j ( ) d j x j f y O G (3) (4) y F (5) Decson-Maker Max ( ) d j m j Subject to: (6) FG j f w O G (7) w 0 F (8) Connecton restrctons y z F G (9) m j z F G j (0) x j y F G j () ( x j m j ) j () FG mj xj zk yk G j k Tj (3) z w F G (4) z w y 0 F G x 0 F G j j m 0 F G j j (5) Regardng decson-maker he wll maxmze hs payoff subject to the restrcton that he has to afford to open all the facltes he bds for (constrant (4)). Constrant (5) guarantees that the exstng facltes wll stay open. A smlar objectve functon s consdered by decson-maker and smlar constrants: a budget constrant (7) and a constrant that does not allow hm to bd for already opened servces (8). Constrant (9) guarantees that at most one servce s opened at each locaton. Customers can only be assgned to opened facltes (constrants (0) and()) and have to be assgned to exactly one faclty (). Each customer s assgned to the closest opened faclty (3). We are not consderng the stuaton such that a customer s equally dstant from two or more opened facltes. Ths possblty can easly be consdered assumng that the demand of a customer s equally splt by two or more opened facltes (see Godnho and Das 00). Constrants (4) state that nvestor can only open facltes he has bd for. Each soluton to ths problem s composed of a of y varable values whch we wll denote as vector y and a of w varables values whch we wll denote as vector w. Interpretng ths problem as a game y s a strategy for player and w s a strategy for player. An admssble soluton s a Nash equlbrum soluton. In the case of a Nash equlbrum wth pure strateges ths means that y w s admssble f y s a best response to w and vce-versa. 3 COMPUTATIONAL RESULTS There s no obvous procedure for solvng the two-player smultaneous decson problem presented n the prevous secton. Therefore n order to
calculate the game equlbra we resorted to an algorthm based on the best responses of each player to the other one's strategy proposed by Godnho and Das (00). The algorthm was mplemented n C programmng language usng LP Solve rnes for solvng the lnear programmng problems (source: http://lpsolve.sourceforge.net). For each nstance we appled the algorthm twce for the game n whch player has preferental rghts. The frst tme we chose a null strategy for player (openng no locatons) as the startng pont; the second tme we chose a null strategy for player as the startng pont. In fact n a model wth preferental rghts the algorthm wll often fnd solutons that are more favorable to the player whose best response s consdered frst (the algorthm wll only fnd one equlbrum and the game may have several equlbra so the results may be somewhat based by the choce of the startng pont as shown n Godnho and Das 00). However n the problem here addressed the equlbrum soluton that s found s usually ndependent of the startng pont of the algorthm; moreover when dfferent startng ponts lead to dfferent equlbra the dfferences n the players payoffs n the two equlbra are small. was used as a reference the parameters of the remanng test s beng defned as changes over the parameters of ths test. For test we defned a network wth 00 nodes (that s 00 possble locatons for the customers) wth both players beng able to open facltes at 48 of these locatons. The budget for each player was to 000 and the average cost of openng a faclty was to 350. s -4 were desgned to allow us to analyze the mpact of smultaneously changng the number of potental locatons for both players facltes. The number of potental locatons for the players facltes was to 36 4 and n test s 3 and 4 respectvely and the other parameters values were dentcal to the ones used n test. The results obtaned wth test s -4 are summarzed n Table. As expected the average payoffs of both players ncrease as the number of potental faclty locatons ncrease but ths ncrease takes place at a decreasng rate. Ths behavor occurs both when there are preferental rghts and when they do not exst and t s consstent wth the results of Godnho and Das (00). Both the beneft that player gets from havng preferental rghts and the loss player ncurs when player has such rghts seem farly stable n absolute terms. Snce payoffs ncrease wth the number of potental locatons ths means that the relatve gan of player and the relatve loss of player become less sgnfcant as the number of potental locatons ncrease. Ths makes sense because an ncrease n the number of potental locatons leaves player wth more places n whch he can avod player and provdes player wth more nterestng locatons so he has a relatvely smaller ncentve to try to choose the same locatons as player. s 5-7 allow us to analyze the consequences of changng the potental locatons avalable to just one of the players. Player has 48 potental locatons and the number of potental locatons for player s facltes s 48 36 4 and n test s 5 6 and 7 respectvely. Ths s done by randomly choosng a sub of G and consderng f for all facltes n ths sub. The other parameters values were dentcal to the ones used n test. The results are summarzed n Table. As the number of locatons avalable to player ncreases player s payoff ncreases and player s payoff tends to decrease. The relatve loss of player from the preferental rghts of player s farly stable. In the case of player both the absolute and the relatve gan ncrease wth the number of potental locatons for player. Ths means that as player gets more Potental locatons Average return (wth preferental rghts) wth wth Table : Summary of the results obtaned wth test s -4. Average return (wth preferental rghts) Player beneft from preferental rghts Relatve wth w/ / wth w/ Player loss from player rghts wth Relatve / wth w/ 48 47.8 95.8 97.6 96.9 30. 9.% 44. 0.4% 36 46.9 93.7 79. 80.6 37.7 0.% 47.9.0% 3 4 30.8 83.4 084. 060.7 6.6 0.9% 47.4 3.3% 4 089.9 633. 840.3 866.9 49.6 9.7% 33.7 7.0% wth wth : average payoffs for player and player respectvely when player has preferental rghts; : average payoffs for player and player respectvely when there are no preferental rghts.
Potental locatons for player Table : Summary of the results obtaned wth test s (repeated for easer reference) and 5-7. Average return (wth preferental rghts) wth wth Average return (wth preferental rghts) Player beneft from preferental rghts Relatve wth w/ / wth w/ Player loss from player rghts wth Relatve / wth w/ 48 47.8 95.8 97.6 96.9 30. 9.% 44. 0.4% 5 36 49. 96.8 43.7 5.0 75.4 4.% 35. 0.4% 6 4 59.5 88.5 365.7 09.5 53.7.3% 0.0 9.% 7 543.6 74.3 4.8 93.4.8 8.6% 90. 0.4% wth wth : average payoffs for player and player respectvely when player has preferental rghts; Player s budget : average payoffs for player and player respectvely when there are no preferental rghts. Table 3: Summary of the results obtaned wth test s (repeated for easer reference) and 8-0. Average return (wth preferental rghts) wth wth Average return (wth preferental rghts) Player beneft from preferental rghts Relatve wth w/ / wth w/ Player loss from player rghts wth Relatve / wth w/ 000 47.8 95.8 97.6 96.9 30. 9.% 44. 0.4% 8 750 567.4 763.8 340.8 000.9 6.6 6.9% 37. 3.7% 9 500 576.7 505.6 45. 675.4 5.4 0.6% 69.8 5.% 0 50 678.4 44.4 576.4 348.0 0.0 6.5% 03.7 9.8% wth wth : average payoffs for player and player respectvely when player has preferental rghts; Average fxed faclty cost : average payoffs for player and player respectvely when there are no preferental rghts. Table 4: Summary of the results obtaned wth test s (repeated for easer reference) and -3. Average return (wth preferental rghts) wth wth Average return (wth preferental rghts) Player beneft from preferental rghts Relatve wth w/ / wth w/ Player loss from player rghts wth Relatve / wth w/ 75 96. 50. 590.0 580.0 37. 3.3% 49.9 7.% 6.5 709.6 089.9 408.5 49.0 30..4% 339. 3.7% 350 47.8 95.8 97.6 96.9 30. 9.% 44. 0.4% 3 55 09.6 83.4 965.0 989.0 44.6 5.0% 56.7 5.8% wth wth : average payoffs for player and player respectvely when player has preferental rghts; : average payoffs for player and player respectvely when there are no preferental rghts. potental locatons t becomes more mportant to player to get preferental rghts n order to secure exclusve benefts from the most nterestng locatons. s 8-0 consdered along wth test allow us to analyze the consequences of changng the budget of a player whle keepng the other player s budget constant. We defned that player s budget s 000 and player s budget to 000 750 500 and 50 n test s 8 9 and 0 respectvely wth all other parameters values held constant. The results are summarzed n Table 3. As expected player s payoff ncreases when hs budget ncreases and player s payoff decreases n that stuaton. The beneft from havng preferental rghts becomes more sgnfcant for player as player s budget ncreases. Ths means that as player s able to buld more facltes t becomes more mportant for player to secure exclusve benefts from the best locatons. As for player the absolute loss from player s rghts ncreases wth hs budget but the relatve payoff reducton becomes less sgnfcant for hgher budgets. s -3 consdered along wth test allow us to analyze what happens when the average fxed cost of each faclty changes and the players budgets are kept constant. We the average cost of each faclty to 75 6.5 350 and 55 n test s and 3 respectvely. The other parameters values were dentcal to the ones used n test. The results are summarzed n Table 4. The payoffs
of both players decrease as the average cost of each faclty ncreases. When the average cost ncreases players are able to open less facltes thus reducng ther payoffs. At the same tme the ncrease n average faclty cost reduces the absolute and relatve beneft player gets from preferental rghts and t also reduces the absolute and relatve loss ncurred by player. In fact wth the ncrease n average faclty cost and the consequent reducton n the number of facltes the level of competton between players decreases reducng the mpact of preferental rghts. 4 CONCLUSIONS We have ntroduced a smultaneous dscrete locaton problem wth two decson-makers n a franchsng envronment where one of the players has preferental rghts. Ths model has several dstngushng features namely the fact of consderng explctly smultaneous decsons nstead of sequental decsons. We have formulated the problem as a lnear programmng problem and have defned as admssble solutons those that are Nash equlbrum solutons. The computatonal results show us that f the level of competton ncreases then the mportance of havng preferental rghts also ncreases. The level of competton s hgher when there are fewer potental locatons for openng facltes when fxed openng costs decrease keepng the budget constant or when the budget szes are smlar. The developed work rases other questons namely what happens f t s gven to the player wth preferental rghts the possblty of bddng for more facltes than the ones he can afford. Ths wll be the subject of further research. Dobson G. and U. S. Karmarkar. (987). Compettve Locaton on a Network. Operatons Research 35 565-574. Godnho P. and. Das. (00). A two-player compettve dscrete locaton model wth smultaneous decsons. European ournal of Operatonal Research 07 49-43 Hande K. Aras and N. Altnel. (0). Compettve faclty locaton problem wth attractveness adjustment of the follower: A blevel programmng model and ts soluton. European ournal of Operatonal Research 08 06-0. Labbé M. and S. L. Hakm. (99). Market and Locatonal Equlbrum for Two Compettors. Operatons Research 39 749-756. Plastra F. (00). Statc Compettve Faclty Locaton: An Overvew of Optmzaton Approaches. European ournal of Operatonal Research 9 46-470. Vetta A. (00). Nash Equlbra n Compettve Socetes wth Applcatons to Faclty Locaton Traffc Rng and Auctons. Proceedngs of the 43rd Annual IEEE Symposum on Foundatons of Computer Scence (FOCS 0) IEEE Computer Socety Press. ACKNOWLEDGEMENTS Ths research was partally supported by research project PEst-C/EEI/UI0308/0. REFERENCES Das oana and P. Godnho. (0). Some thoughts ab the smultaneous locaton of franchsng servces wth preferental rghts. Inesc-Combra Research Report 3-0