A Game Theoretical Study of Cooperative Advertising with Multiple Retailers in a Distribution Channel

Transcription

1 A Game Theorecal udy of ooperave Adversg wh Mulple ealers a sruo hael Jhua Zhag, Jxg Xe, eparme of Mahemacal ceces, Tsghua Uversy. Bejg 8, ha Asrac Exa sudes of cooperave adversg maly cosder a sgle-maufacurer-sgle-realer chael srucure. Ths ca provde lmed sghs, ecause a maufacurer, real pracces, would frequely deal wh mulple realers a he same me. I order o exame he mpac of he realer s mulplcy o chael memers decsos ad oal chael effceces, hs paper develops a mulple-realer model. I hs model, he maufacurer ad he realers play he acelerg game o mae opmal adversg decsos. Based o he quaave resuls, s oserved ha: Whe here are mulple symmerc realers, as he umer of realers scales up, he maufacurer s aoal adversg vesme corues creasgly o add o chael memers profs, u he oal chael effcecy deeroraes; Whe here are mulple asymmerc realers, he dsruo chael suffers from he maufacurer s uform parcpao sraegy due o he realer s free-rde, ad eefs wh he maufacurer s realer-specfc parcpao sraegy. Ths sudy derves equlrum soluos for all games suded closed form, ad explcly measures he gas/losses of chael effceces uder dffere game segs. Keywords: cooperave adversg, game heory, mulple realers, scalg effec, free-rdg. Iroduco I a ypcal dsruo chael, he maufacurer usually maes s adversg va he aoal wde meda so as o uld up s log-erm rad mage ad elarge s poeal cle ases. I coras, he realer eds o mae local adversg campags whch focus more o he shor-erm sales effec. The dscrepacy of wo ypes of adversg s presumed of a magude suffcecy o creae a role for cooperave co-op adversg. o-op adversg s, praccally, a eracve relaoshp ewee a maufacurer ad a realer whch he maufacurer pays a poro of he realer s local adversg coss; The fraco shared y he maufacurer s commoly referred o as he maufacurer s parcpao rae. o-op adversg offers cosumers he formao eeded whe hey move hrough he fal sages of purchase ad a cogruece of formao ad formao eeds ha would e mpossle f he maufacurer uses aoal adversg oly Huag & L,. o-op adversg prevals oday s mareg pracces. I s repored ha GE s udges for local adversg are hree mes as hgh as s aoal adversg udge Youg & Greyser, 98. Iel s expedure o co-op adversg grows from $8 mllo 999 o $.5 llo El, 999. I 987, he esmaed co-op adversg expedure spe y he U compaes amoued o $ llo omers e al., 99. I 99, llo s used for co-op adversg avs, 99. Accordg o Nagler 6, he oal U expedure of co-op adversg s esmaed a $5 llo, early a four-fold crease real erms comparso wh $9 mllo 97. Icreasg spedg volumes ad hgh levels of parcpao rae co-op adversg have movaed heorecal sudes o explore a mos effecve scheme for chael memers o parcpae. Lyo s 9 dscuss of he adversg allowaces moey pad y a maufacurer o a dealer o cover a par of dealer s orrespodg auhor. Tel.: ; Fax: ; E-mal address: zhag-jh@mals.sghua.edu.c

2 adversg expeses provdes wha may have ee he frs formal, ale qualave reame of he sujec of co-op adversg. Followg Lyo s semal wor, sudes of co-op adversg, cludg oh emprcal vesgaos ad aalycal modelg wors, ever sop, hereafer, for he pas egh decades. Our sudy follows he aalycal modelg approach. egardg aalycal modelg echques, game heorecal models have ee mos wdely used co-op adversg researches. I parcular, hey are used o model he here er-depedece ad coflcs amog chael memers, ad how he co-op adversg decsos, as well as chael performace s flueced y he varous facors, such as adversg effors a aoal ad local levels, he parcpao rae, sales volumes, rad ad sore susuos, prcg, amog ohers. For a exesve survey of he game heorecal models co-op adversg, oe ca refer o Taou & Zaccour 5. We refly revew he co-op adversg leraure relaed o he dyamc ad sac games, respecvely. A dyamc game aes a log-erm perspecve over several or fe me perods e.g., Km & eal, 999; Jorgese e al., ; Taou & Zaccour, ; Jorgese & Zaccour, ; He e al., 9. usomer demad s usually assumed o e ased o a goodwll soc ha s flueced y adversg he curre ad prevous perods Nerlove & Arrow, 96; hagua & Ja, 99. Ivesmes o he goodwll soc are herefore cosdered o have he carryover effec from oe perod o he ex. However, a commo prolem wh he dyamc game models s ha closed-form soluos are usually o avalale whou sgfca model smplfcaos. O he oher had, sac game aalyzes he co-op adversg a sgle-perod e.g., Berger, 97; a & Berger, 996; Berger & Maglozz, 99; Berge & Joh, 997; Huag & L, ; Yue e al., 6; zmereovsy & Zhag, 9; Xe & We, 9. Oe commo assumpo s ha he game s played oly oce, ad he rad-ame vesmes, he local adversg spedg, as well as player s parcpao sraeges are all assumed o e me-vara. ac game models are covee ools o develop aalycal soluos ad o geerae sghs relaed o he ey elemes of he game srucure, cludg adversg expedures, parcpao raes, ad chael effceces, so we choose o use sac game models hs paper. We oe ha all sac aalycal models meoed aove cosder a sgle-realer chael srucure, whch has several lmaos: I s uale o sudy he realer s mulplcy ad s mpac, e.g., scalg effec; I does o capure he uversal mpac of he maufacurer s rad-ame adversg, whch, oce mplemeed, wll eef all resellers equally who sell producs uder he rad-ame; I fals o capure he mpac of dffere realers arues.e., prof margs, mare szes, ec. o her co-op adversg decsos. Alhough mareg pracces sugges ha he maufacurer would ofe mes deal wh mulple realers, exsg leraure of co-op adversg wh mulple realers s sparse. Karray & Zaccour 7 cosders a dsruo chael cossg of wo maufacurers ad wo realers. The realers choose her levels of mareg effors e.g., local adversg, dsplays, ec. ad he maufacurers corol her parcpao raes for he realers mareg effors f he co-op adversg program s a opo. However, he aoal adversg effors are specfed as exogeous parameers sead of maufacurer s decso varales. To he es of our owledge, hs sudy s amog he frs o develop a game heorecal model wh mulple realers. The ma coruo s ha explcly derves ad quaavely measures he mpacs of he realer s mulplcy o chael memers opmal decsos ad o oal chael effceces. Maageral sghs ca e summarzed he followg wo perspecves: Whe here are mulple symmerc realers, we defy scalg effecs o chael memers profs ad he oal chael effceces. pecfcally, as he umer of realers scales up, he maufacurer s aoal adversg vesme corues creasgly o add o chael memers profs. Alhough he oal chael effcecy may crease wh he local adversg effecveess ad/or he maufacurer s relave chael power, deeroraes qucly as he umer of realer scales up. Whe he realers are asymmerc, we sudy how he dffere local mare arues ad he maufacurer s uform parcpao cosra affec he maufacurer s opmal parcpao sraegy ad he sysem s effcecy. I s oserved ha: The maufacurer s uform parcpao sraegy ca e effce due o he realer s free-rdg. pecfcally, he maufacurer s uform susdy eds o e suffce wh realers of larger mare sze ad lower prof margs, ad excessve wh realers of smaller mare sze ad hgher prof marg, gvg he laer a free-rde; I order o preve he realer s free-rde, he maufacurer ca use he realer-specfc parcpao sraegy, whch suggess he maufacurer s parcpao rae wh realers of smaller mare sze ad hgher prof marg should e less geerous ha ha of larger mare sze ad lower prof marg.

3 Ths paper s orgazed as follows: eco preses he model framewor. eco - vesgaes mpacs of he realer s mulplcy whe here are mulple symmerc/asymmerc realers. eco 5 summarzes he fdgs ad proposes drecos for fuure research.. Model framewor osder a dsruo chael wh oe maufacurer M sellg a cera produc hrough realers,,,. Assume ha dffere realers are geographcally separaed ad ha cross-order producs rasferrg s prohed, so here are o ra-rad compeos. Ths assumpo capures he real suao whe a maufacurer s mareg chaels are wdespread, wh each reseller eg auhorzed as he sole represeave of s local mareg area ad mag adversg ad prcg decsos depedely. As far as we ow, some hese maufacurers of Elecroc Isrume follow hs way. They desgae a frs-class sales age who s, headquarered some frs-er cy, fully charge of he sales ad serves he cy ad he surroudg areas e.g., Bejg for he Norh-ha area, hagha/guagzhou for he Eas-ha area, ec.. Alhough geographcally separaed, he realers eef equally from he aoal adversg e.g., rad-mage adversg o aoal TV chaels ecause hey carry he same producs uder he maufacurer s rad-ame. Our sudy maly focuses o chael memers adversg decsos. o we do o ae drecly prcg decsos o accou, sead, we assume a fxed gross u marg adversg coss excluded for chael memers. Le M s gross prof marg e ρ ad s e ρ, =,,,. Fxed margs ca e jusfed wh a shor-ru plag horzo. mlar assumpos are also used y Jorgese e al. ad gue & hagua 8. Le a ad a, =,,,, e M s ad s adversg spedgs, respecvely. We assume he resulg cusomer demad for realer, = a, a, ofe called he sales respose fuco, s joly deermed y oh he aoal adversg spedg a ad local adversg spedg a. There s a susaal leraure o he esmao of he sales respose fuco wh respec o adversg vesmes. Followg Xe & We, 9 we use where a, a a a, =,,,, s he demad ase wh respec o zero adversg pu, s he relave effecveess of he local adversg compared wh he aoal adversg, ad s he mare sze. To oa a eer udersadg of a, a, we sugges ae a a as he average demad of a ypcal cusomer wh s mareg area ad he populao sze. We assume furher... ad... o mply ha a ypcal cusomer s average demad level s much he same all local mares. The he local mares are prmarly dffereaed y he populao sze wh he mareg areas. Techcally, here are o dffcules o exed he model y cosderg ad eg depede wh he realer s dex. However, he assumpo of a uform ad ca help us o eer focus o he ey parameer of, whch we wll use hs paper o dsgush large realers from small realers. Fally, for smplcy of our dscusso, we ormalze. Noe ha s o he realer s decso varale; Isead, s oly a cosa parameer, o couplg wh ay decso varales. o ormalzg wll o affec he equlrum oucome regardg chael memers opmal decsos. The s sales respose fuco s specfed as a, a a a. Ovously, a, a s couously dffereale, srcly creasg, ad srcly jo cocave wh respec o a, a. The square roo formulao also reflecs he commoly oserved adversg saurao effec,.e., addoal adversg spedg geeraes couously dmshg reurs, whch has ee verfed ad suppored y mo & Ard 98 ased o her revew of over case sudes. To mpleme co-op adversg, le M shares a poro [,] of s local adversg cos a.

4 Le a : a, a,..., a ad :,,...,, he profs for M,, =,,,, ad he whole dsruo chael are: a, a, a a a a,, a a a a a, =,,...,. a, a a a a a. 5 All hroughou hs paper, we use suscrps, =,,,, ad o mar he parameers correspodg o M,, ad he whole chael sysem.. calg effec wh mulple homogeeous realers uppose he realers he dsruo chael are homogeeous. Whou loss of geeraly, le all realers e decal wh, he,, for all =,,,... Bechmar case ceralzed chael Followg Equao 5, he oal chael prof ca e wre as, a a a a a a. 6 Please oe ha a, a s a src joly cocave fuco wh respec o a, a. To see hs po, oe ca smply chec he Hessa of a, a eg egave-defed. By defo,, where A / a a, j, l,,...,. If j l, A ; Oherwse, f j=l=, A jl A jl jl a s egave, so a, a j l a, f j l, A a. Thus, jl jl j has a uque gloal opmal po, whch ca e solved from he frs order codo. eoe he opmal po y a, a := a, a,, a. Proposo. Wh symmerc realers he dsruo chael he ceralzed soluo ha maxmzes he oal chael prof s gve y a, 7, =,,,. 8 a Hereafer hs paper, we wll use o mar ceralzed soluos, ad he upper-ar equlrum oucomes correspodg o he -symmerc-realer case. o mar Noe ha represes he whole chael s gross marg for each u sales. The, Equaos 7-8 sugges ha he opmal levels for aoal/local adversg expedures should e proporoal o he squared value of he chael s gross marg scaled y he mare sze,.e.,. Wha s dffere, M s opmal adversg level a s creasg, whereas, s opmal adversg level a says cosa wh. usue a, he opmal chael prof y a o Equao 6, we oa

5 . 9 eferrg o Equaos 7-8, we ca acually erpre Equao 9 he followg way: a +, whch reveals = ha, he opmal ceralzed chael prof adversg expese expeses a. ce a a / a coruo o oal chael prof creases wh. cosss of wo pars: oe s M s opmal aoal, he oher s oaed y addg up all realers opmal local adversg = / creases, we coclude ha he aoal adversg s relave Isgh. I a ceralzed chael, as he realers umer scales up, he aoal adversg plays a creasgly prome role erms of geerag profs for he whole dsruo chael... eceralzed chael For he deceralzed chael, we model chael memers decso process as a wo-sage acelerg game. I he frs sage, M decdes s aoal adversg expedure a ad parcpao raes, =,,,, ad he secod sage, smulaeously chooses s local adversg level a, =,,,, accordgly. ue o he symmery of he model, le M s parcpao raes wh, =,,, all e decal wh, he players prof fucos are reduced o a,, a a a a a, a a, =,,,., a, a a Noe ha a he secod sage of he game, gve a ad, s smulaeous move geeraes a Nash Equlrum. To deal wh he Nash Equlrum, oe should always dscuss s exsece ad uqueess. Foruaely, for he curre model suded, oh he exsece ad he uqueess of he Nash Equlrum are guaraeed. For exsece, we use Tops s 979 argume of he supermodular game as he suffce codo Because / a a for all j, =,,,, he he game s always a j supermodular game. For uqueess, we refer o Moul s 986 saeme, whch requres ha he slope of each player s es resposes ever exceed asolue value. I our model, he codo / aa j = <, j, =,,,, s always vald. / a Lemma. Gve M s decso of a ad, he realers smulaeous move geeraes a uque Nash equlrum, whch ca e explcly expressed as a a,... a, a..., a a, a, =,,,. Because a /, he a hgher level of M s parcpao rae duces a hgher level of s local adversg pu equlrum. Proposo. O codo ha, he uque su-game perfec equlrum soluo for he acelerg game s gve y Whe, M s opmal parcpao rae wll e zero,.e., M chooses o o parcpao o he realers local adversg. mlar argumes apply o Proposo ad 5 aalogously. 5

6 a,, =,,,, 6 a. 5 Hereafer hs paper, we wll use he superscrp o ofy all equlrum soluos ad he assocaed ems wh a deceralzed chael. Equao 5 reveals ha M s opmal parcpao rae [,] s oly depede wh chael memers gross margs: I s creasg ad decreasg. Please oe ha he rao / represes M s relave chael power agas, ad he po / s he realer s local adversg-demad elascy asolue value log / a / a, herefore, codo s equvale o sayg ha M s relave chael power s larger ha or equals o he realer s adversg-demad elascy asolue value. We assume hs codo hold all hrough hs seco. ompare Equaos - wh 7-8, we oa smlar oservaos as eco.: s opmal adversg cos a s depede wh ; M s opmal adversg vesme a s proporoal o, herefore, wll crease qucly as he umer of realers scales up. usue - o -, we ge M s ad s profs as: 6, 6, =,,,. 7 8 = 6 Equaos 6 ca e reformed as a a suggess us o cou M s opmal prof y wo pars: oe s M s ow adversg vesme aoal level, he oher s he sum of all s realers adversg expeses, whch a a he a a he local level. ce a / a creases, we coclude ha he aoal adversg s relave coruo o M s prof creases wh. mlarly, Equao 7 ca e reformed as a a, =,,,, eg a lear comao of s opmal local adversg spedg a ad he average spedg of M s opmal aoal adversg, a, where /, /. Whle a s cosa wh, a s creasg wh. Therefore, M s aoal adversg spedg creasgly corues o add o s prof as he realer s umer scales up. Isgh. The maufacurer s aoal adversg coducs a uversal mpac geerag chael memers profs: I helps geerae sales all local mares ha carry producs uder s rad-ame. As he realer s umer scales up, corues creasgly o add profs o oh he maufacurer ad he realers. I geeral, he chael prof ased o deceralzed decsos -5 wll e less ha ha ased o ceralzed decsos 7-8. To quaavely measure he sysem effcecy uder he curre deceralzed sysem, we roduce he rao [,] as he opmal chael prof uder he deceralzed game seg over ha uder he ceralzed game seg. For example, we wll use : / o measure he chael effcecy for he deceralzed game seg wh mulple symmerc realers, whch s oaed y addg all memers profs: 6

7 . 8 6 Proposo. The sysem effcecy of he curre deceralzed game seg s whch has he followg properes: For ay gve, lm /, 9 ad, s a covex decreasg fuco of, ad For ay gve, creases wh ad /. The effcecy oud s esmaed y 5 9,. Whe = ad, ges s lower oud 5/9; whe, ges s upper oud. ce he rao /, =, represes realer s relave chael power. Proposo saes ha a src Pareo mproveme laeral parcpao scheme s guaraeed as log as he realer s average chael power s greaer or equals o he maufacurer s. ecall ha g maufacurers used o have more chael power ha realers efore he early 98 s. Maufacurers such as Procer&Gamle P&G lmed he quaes of hgh-demad producs hey would delver o a gve supermare cha ad ssed ha he supermare carry all szes of a cera produc Huag e al.. However, hgs chaged ever sce he mddle 98 s wh a mare red whch he realers ega o rea equal or eve more powers Acheaum ad Mchel 987; Buzzell e al. 99; Kumar 996;. Accordg o of Proposo, creases as he local adversg s relave effecveess ges hgher ad/or he maufacurer s relave chael power / ges larger. However, as show y of Proposo, he chael effcecy suffers from a creasg umer of realers. Fgure shows how he chael effcecy chages wh he realer s umer. I parcular, f =,. 5 ad, he chael effcecy s elow 6% whe here are more ha 7 realers he dsruo chael. Isgh. Wh mulple symmerc realers he dsruo chael, he chael effcecy creases wh he local adversg effecveess ad/or he maufacurer s relave chael power, however, deeroraes qucly as he umer of realer scales up.. Opmal parcpao sraeges wh asymmerc realers Accordg o Berger 97, co-op adversg veures pracce are frequely specfed o a 5-5 ass, wh each parcpa payg half of he expeses. Nagler 6 coduced a large-scale sudy of,86 rads he U, fdg 6.5% of he realers ejoy a parcpao rae of 5% from he maufacurers. oes mae sese for dffere realers o ejoy a uform level of parcpao rae for sace, 5%? Iuvely, f he maufacurer s uform parcpao rae s derved from a sese of faress raher ha sysemac prof maxmzao, ca e effce. Oe mpora reaso ca e he realer s free-rdg. I s commoly eleved ha a well-desged chael corac should preve horzoal free-rdg amog realers Lal, 99. I hs seco, we cosder co-op adversg wh mulple asymmerc realers. We wll exame how dffere parameers affec he maufacurer s co-op adversg decsos, ad wll quafy he effcecy losses due o realers free-rdg uder he maufacurer s uform parcpao sraegy. Whou loss of geeraly, le =... The realer s free-rdg uder he maufacurer s uform parcpao sraegy osder M, he acelerg game leader, offers a uform co-op adversg parcpao raes o 7

8 ad. The chael memers profs are: a, a, a, a a a a, a, a, a a a, a, a ; a a a. The game sequece s he same as eco. Whou loss of geeraly, we assume all hrough hs paper. Before movg o, le s defe :, where. By defo,, =,, ad have he followg properes: Lemma. [,], =,,. creases ad decreases j, j, =,. Whe /, we ge, ad ; oversely, whe, we ge, ad. / Accordg o Lemma, s a covex comao of ad, weghed y ad, so. We call he realer s average prof marg. Proposo. O codo ha acelerg game wll e gve y, he uque su-game perfec equlrum soluo for he aˆ ˆ, a, 6 The maufacurer s opmal parcpao rae wll e aˆ. 5 6 ˆ. 6 Hereafer hs paper, we wll use he upper-cap o ofy all he equlrum soluos ad he assocaed ems wh M s uform parcpao sraegy. Wh Proposo we are ready o oserve he realer s free-rdg whe M uses a uform parcpao rae ˆ wh dffereaed realers. Accordg o Equao 6, M s opmal parcpao rae creases ad decreases. Le s cosder he exreme case whe s mare sze s /. By propery of Lemma, we have,, he M s opmal parcpao rae wll e rased up o larger ha s mare sze y far,.e., ad cosequely goes dow o /, as f s he oly dowsream reseller. For hs suao, we say ha ges a free-rde, ecause ejoys he same hgh level of parcpao rae as. Exreme suaos le / are uusual real scearos, u suaos wh ca e reasoaly expeced 8

9 gve ; o log as, oe ca use smlar argume o show ha he realer wh smaller mare sze ges he free-rde. Isgh. Wh mulple asymmerc realers, he maufacurer s uform parcpao sraegy resuls he realer s free-rdg. The maufacurer s opmal uform parcpao rae eds o e favor of realers wh larger mare szes; herefore, realers wh smaller mare szes usually accompaed y hgher prof margs ge he free-rde... The maufacurer s realer-specfc parcpao sraegy Because of he realers free-rdg, he maufacurer s uform parcpao sraegy ca e effce. I order o quafy he effceces ad dsclose he fac ha M s uform parcpao sraegy deed mposes approprae ceves co-op adversg, we roduce he maufacurer s realer-specfc parcpao sraegy. Uder he realer-specfc parcpao sraegy, M s allowed o use dffereaed parcpao raes owards dffere realers. eoe M s parcpao rae wh y, =,. The chael memers prof fucos are a, a, a,, a a a a, 7 a, a, a a a, 8 a, a, a a a. 9 The game sequece s he same as ee specfed he uform parcpao case. Proposo 5. If, =,, he uder he maufacurer s realer-specfc parcpao sraegy, chael memers equlrum adversg levels are, a The maufacurer s opmal parcpao raes are a 6, a 6.,. Proof of Proposo 5 s almos he same as Proposo. o we om here o save space. Hereafer hs paper, we use he lde o represe equlrum soluos ad he assocaed ems for cases wh M s realer-specfc parcpao sraegy. Mareg pracces sugges ha a realer who possesses a ga mare sze usually ses s u margs eg comparavely low, for example, he Wal-Mar, eg famous for s Everyday Low Prces, operaes wh a low gross marg of o more ha % over decades. A rage of sudes has foud ha prces a Wal-Mar are aywhere from 8 o 9 perce less ha s major compeors, ad ha eve a very small crease s coss, whou a correspodg crease reveues, would wpe ou all Wal-Mar s profs erely Furma, 5. I coras, a realer wh lmed mare sze would ofe mes mpose a hgh u marg. Ths s well explaed y he old mareg dscple: low profs go alog wh good sales. 9

10 Mag comparsos ewee Equaos - ad 6, ad also y Lemma, we ow ha ˆ. Ths oservao ca e uvely explaed as Isgh 5. I coras wh he opmal realer-specfc parcpao sraegy, M s opmal parcpao rae uder he uform parcpag sraegy mposes approprae ceves o he realers: I s suffce wh he realer of larger mare szes.e., our model, ad s excessve wh he realer of smaller mare szes.e., our model. egardg how o mae a dfferece ewee asymmerc realers, Equaos - sugges ha M s opmal parcpao rae owards he realer wh smaller mare szes should e less ha ha wh larger szes. The followg proposo, alog wh Fgure, quaavely compares M s uform parcpao sraegy wh M s realer-specfc parcpao sraegy. esuls of Proposo 6 are aural cosequeces of Proposo ad 5, so we om he proof o save space. Proposo 6. If, he M s opmal polcy s o ae o parcpaos he realers local adversg, eher uder he uform parcpao sraegy or uder he realer-specfc parcpao sraegy.e. ˆ. If, he M wll parcpae o s local adversg wh rae ad parcpae o s local adversg wh rae uder he realer-specfc parcpao sraegy; Bu wll ae o parcpaos wh eher realers uder he uform parcpao sraegy.e. ˆ. If, he M wll parcpae o s local adversg wh rae ad parcpae o s wh rae uder he realer-specfc parcpao sraegy; Ad wll parcpao o oh realers local adversg wh rae ˆ uder he uform parcpao sraegy. v If, he M wll parcpae o s local adversg wh rae ad parcpae o s local adversg wh rae uder he realer-specfc parcpao sraegy; Ad wll parcpaos o oh realers local adversg wh rae ˆ uder he uform parcpao sraegy... ysem effcecy esmaos For all cases dscussed aove, we ca measure he chael effceces y comparg he opmal chael prof uder a specfc game seg agas he opmal prof of he ceralzed chael. We use wo measuremes: Oe s he asolue measure, deoed y, whch descres he dsace of he equlrum chael prof uder deceralzed soluo ad he opmal oal chael prof uder ceralzed soluo. For example, whe M s allowed o use he realer-specfc parcpao raes, we defe : ; whe M s cofed o he uform parcpao sraegy, ˆ : ˆ ; The oher s he relave measure, deoed y [,], whch s he rao of he equlrum chael prof uder deceralzed soluo over he opmal oal chael prof uder ceralzed soluo has ee roduced eco.. For example, whe M s allowed o use he realer-specfc parcpao raes, we defe : / ; whe M s cofed o he uform parcpao sraegy, ˆ : ˆ /. By defo, he asolue measure ad he relave measure are cosse; a lower hgher value of

11 dcaes a hgher level of sysem effcecy. We ls all equlrum soluos Tale for a referece, where he oal chael prof uder wh ceralzed soluo s. The rows a he oom gves he effcecy esmao y oh asolue measure ad relave measure. We frs focus o he asolue measure. ompare ˆ wh, we quafy he effcecy losses due o M s uform parcpao sraegy y ˆ ˆ, whch s greaer ha zero as log as. The uos here are que clear: M s uform parcpao sraegy curs effcecy losses as log as he realers gross margs do o cocde; The sysem s effcecy ges worse whe he dsace of ad elarges. ce he uform parcpao sraegy ca e ae as a specal case of he realer-specfc parcpao sraegy wh, so formally, we may cosder he em ˆ as he added value o he sysem y swchg o he realer-specfc parcpao sraegy. We he focus o he relave measures for chael effceces. We do o fully dsplay s orgal form, sead we esmae s upper oud ad lower oud ealed dervaos he appedx. All oud esmaos are coduced wh, ad, =,. The lower ouds upper ouds are correspodg o he wors es suao of he chael effceces. For example, he lower oud 5/9 of suggess ha, whe he realer-specfc parcpao sraegy s adoped, he dsruo chael s, heorecally speag, a rs of aag a low effcecy of mere 55.6%. For aoher example, he lower oud of ˆ suggess ha he uform parcpao sraegy ca e exremely effce f ad. The explaao s que sragh forward. Whe he aove codos are me, s reasoale o posulae ha: s gross marg s smaller ha y far, ad s mare sze s larger ha y far. Accordg o propery of Lemma, M s uform parcpao rae ˆ eds o e oe, whch wll gve a g free-rde, ecause M s parcpao rae wh would have ee zero uder M s realer-specfc parcpao sraegy. I s, deed, he egave mpac of s free-rdg ha drves he sysem s effcecy o zero. 5. ocludg remars Ths sudy s srogly movaed y he scarcy of he modelg wors whch explcly cosder co-op adversg a mulple-realer s evrome. o hs paper, as our ma coruo, a game heorecal model wh mulple realers s developed o explore he mpacs of he realer s mulplcy o chael memers opmal decsos ad o he oal chael effcecy. We exame ad quafy he mpacs wo dffere aspecs. I s oserved ha: Wh mulple symmerc realers, whe he umer of realers scales up, he maufacurer s aoal adversg vesme corues creasgly o add o chael memers profs, u he oal chael effcecy goes dow qucly ad coverges o a cera value; Wh mulple asymmerc realers dffereaed y gross margs ad mare szes, he maufacurer s uform parcpao sraegy ca e effce due o he realer s free-rdg. We show ha he maufacurer s opmal decso uder he uform parcpao cosra eds o mpose approprae ceves o realers, ad he dscuss how he dermeal mpacs of hs dsoro ca e mgaed ad correced y adopg he realer-specfc parcpao sraegy. For oh he symmerc ad asymmerc-realer cases, we solve he games all closed form ad explcly esmae effcecy ouds. For fuure researches, we would frs le o exed he curre sudy o cosder realers adversg/prcg compeos. Aoher eresg aemp would e o desg some mechasm o coordae he dsruo chael. Fally, all models proposed hs paper are deermsc aure. I would also e eresg o vesgae how uceraes ad dyamcs could chage he coclusos of hs paper.

12 Fgures ad Tales Fgure. calg effec o he sysem s effcecy wh mulple symmerc realers whe parameers are specfed as. 5,, ad = % =7 Fgure. omparso of M s opmal parcpao raes wh ad uder oh he uform ad realer-specfc parcpao sraeges, ˆ Tale. hael performace ad sysem effcecy esmao wh wo asymmerc realers Parcpao raes Local adversg ealer-specfc parcpao sraegy 6 a 6 a aˆ aˆ Uform parcpao sraegy ˆ 6 6

13 Naoal adversg a ˆ a Effcecy y asolue measure 6 6 ˆ Effcecy y relave measure,] 9 5 [ ] ˆ [,

14 Appedx Proof of Proposo. The frs order codo a, a s equvale o a a a A a, =,,,. A olvg he aove wo Equaos, we oa 7-8 as he uque ceralzed soluo. Proof of Lemma. Gve M s decso of a ad, he realers wll move smulaeously o choose her local adversg levels ha opmze her dvdual profs, =,,,. Accordg o Equao, s depede wh a j, j, so doma sraegy for. a s also uque ecause uque Nash equlrum as show y Equao. a / whch maxmzes should e he s srcly cocave wh a, so we oa he Proof of Proposo. Accordg o Lemma, he realers Nash equlrum s gve y a a,... a, a..., a /, =,,,. usue a, =,,,, o M s prof fuco, we ge, A a, a a The frs order codos of a, A o a ad are, respecvely, a a, A, A5 olvg A-A5, we ge ad 5. Ovously, Equao A s srcly cocave wh a for ay gve, so a s he uque opmal aoal adversg level ha maxmzes M s prof. Accordg Equao A5, f, he wll crease wh whe ad decreases wh whe ; However, f, he wll decrease wh all. o M s opmal parcpao rae wll e f ad f. Here we are oly eresed he former case where M s opmal parcpao rae aes posve values. Whe Proof of Proposo., y susug 5 o we ge Equao. Tae he frs order ad secod order dervaves of wh respec o, we have, A6. A7

15 o, for ay gve, lm / ad, s a covex decreasg fuco of. Le goes o fy, we have For ay gve, ad, we have. Le r /, he for ay gve ad, we also have. A8. A9 r ce creases wh for all, ad, so aas s lower oud whe reaches dow o, ha s /. O, whch has a mmum value of 5/9 whe he oher had, creases wh r for all ad, so whe, ouches s upper oud. Proof of Proposo. We use acward-duco. A he secod sage of he game, for ay gve a ad, s prof fuco s srcly cocave wh a, so s opmal respose a a, s uquely oaed as: a a a, usug a,, =,, o Equao, we ge M s prof fuco as, =,. A a, a a,, a a,, = a a. A Tae frs order codos of A o a ad, respecvely, we have a a, A. A olvg A-A, we ge ad 6. Nex, we wll cofrm ha ad 6 gves he uque soluo ha maxmzes M s prof. Equao A s addve separale wh respec o a ad, so we may deal wh he wo varales separaely. Ovously, Equao A s srcly cocave wh a for ay gve, so a Equao s he uque opmal aoal adversg level. The we chec how chages wh. Accordg Equao A, f, he wll crease wh whe ad decreases wh whe ; However, f, he wll decrease wh all. o M s uque opmal parcpao rae wll e ˆ f ad f. Here we are oly eresed he former case where M s opmal ˆ parcpao rae aes posve values. Whe -5. Boud esmao for chael effceces y relave measure Tale Whou loss of geeraly, we assume, y susug 6 o A we oa. The he codo, =, s equvale o 5

16 6, ad he parameer rages wll e,, ad, =,. Frs of all, we cosder he case whe M adops he realer-specfc parcpao sraegy. The chael effcecy y relave measure s /. A ce /,, A5 so creases for all,, ad. The he upper oud of mus e, ecause. O he oher had, he lower oud of mus e aaed a, where Equao A wll e reduced o A6 Tae he frs order dervave of A6 wh, we fd ha wll e creasg wh for all ad,, he we ow ha he lower oud of mus e aaed a ad =, where Equao A6 wll e furher reduced o, = 5. A7 Aga, Equao A7 s creasg for all ad, so we ca smply ae dow o zero o oa he lower oud of A7 eg 5/9, whch should e mmal possle value of wh, ad,. o he effcecy ouds are esmaed as ], [ 9 5. The we cosder he case whe M adops he uform parcpao sraegy. The chael effcecy y relave measure s preseed as / ˆ. A8 Oe ca chec ha whe ad, ˆ ca reach dow o ; Whe ad, ˆ ca rse up o. o he effcecy oud s esmaed y ] ˆ [,.

17 efereces [] Berge, M. & Joh, G Udersadg cooperave adversg parcpao raes coveoal chaels. Joural of Mareg esearch, : [] Berger, P Vercal cooperave adversg veures. Joural of Mareg esearch, 9: 9- [] Berger, P.. & Maglozz, T. 99. Opmal co-operave adversg decsos drec-mal operaos. Joural of he Operaoal esearch ocey, : [] hagua, P.K. & Ja,. 99. A dyamc model of chael memer sraeges for mareg expedures. Mareg cece, : [5] a,.p. & Berger, P Modelg cooperave adversg decsos frachsg. Joural of he Operaoal esearch ocey, 79: -6 [6] avs,.a. 99. ealers ope doors wde for co-op. Adversg Age, Augus [7] El, T o-op crossroads. Adversg Age 7: ad -6 [8] Furma, J. 5. Wal-Mar: A Progressve uccess ory. Washgo : eer for Amerca Progress, hp:// [9] He, X., Prasad, A. & eh,.p. 9. ooperave ad adversg a dyamc sochasc supply cha: feedac acelerg sraeges. Produco ad Operaos Maageme, 8: 78-9 [] Huag, Z. & L,.X.. o-op adversg models a maufacurer-realer supply cha: a game heory approach. Europea Joural of Operaoal esearch, 5: 57-5 [] Jorgese,., gue,.p. & Zaccour, G.. yamc cooperave adversg a chael. Joural of ealg, 76: 7-9 [] Jorgese,. & Zaccour, G.. A dffereal game of realer promoos. Auomaca, 97: 5-55 [] Karray,. & Zaccour, G. 7. Effecveess of coop adversg programs compeve dsruo chaels. Ieraoal Game Theory evew, 9: 5-67 [] Km,.Y. & ael, Maufacurer allowaces ad realer pass-hrough raes a compeve evrome. Maageme cece, 8: [5] Lal,. 99. Improvg chael coordao rough frachsg. Mareg cece, 9: 99-8 [6] Lyo, L. 9. Adversg Allowaces. Washgo : The Broogs Isuo [7] Moul, H Game Theory for he ocal ceces. New Yor: New Yor Uversy Press [8] Nagler, M.G. 6. A exploraory aalyss of he deermas of cooperave adversg parcpao raes. Mareg Leers, 7: 9- [9] Nerlove, M. & Arrow, K.J. 96. Opmal adversg polcy uder dyamc codos. Ecoomca, 9: 9- [] mo, J.L. & Ard, J. 98. The shape of he adversg fuco. Joural of Adversg esearch, : -8 [] gue,.p. & hagua, P. 8. Adversg sraeges a frachse sysem. Europea Joural of Operaoal esearch, 98: [] omers, T.M., Gupa, Y.P. & Herro, Aalyss of cooperave adversg expedures: a rasfer-fuco modelg approach. Joural of Adversg esearch, 5: 5-5 [] zmereovsy, J.G. & Zhag, J. 9. Prcg ad wo-er adversg wh oe maufacurer ad oe realer. Europea Joural of Operaoal esearch, 9: 9-97 [] Taou,. & Zaccour, G.. Impac of realer s myopa o chael s sraeges. Opmal 7

18 orol ad ffereal Games: Essays Hoor of effe Jorgese, G. Zaccour ed., Kluwer Academc Pulshers: 79 9 [5] Taou,. & Zaccour, G. 5. oordao Mechasms Mareg haels: a urvey of Game Theory Models. Les ahers du GEA, G-5-6 [6] Tops,.M Equlrum pos ozero-sum -perso sumodular games. IAM Joural o orol ad Opmzao, 76: [7] Xe, J. & We, J.. 9. oordag adversg ad prcg a maufacurer-realer chael. Europea Joural of Operaoal esearch, 97: [8] Youg,.F. & Greyser,.A. 98. Maagg ooperave Adversg: A sraegc Approach. MA: Lexgo Boos [9] Yue, J., Aus, J., Wag, M.. & Huag, Z. 6. oordao of cooperave adversg a wo-level supply cha whe maufacurer offers dscou. Europea Joural of Operaoal esearch, 68: