Competitive Multi-period Pricing with Fixed Inventories

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1 Compettve Mult-perod Prcng wth Fxed Inventores Georga Peraks y and Anshul Sood z y MIT Sloan School, Cambrdge, MA, USA z MIT Operatons Research Center, Cambrdge, MA, USA Abstract Ths paper studes the problem of mult-perod prcng for pershable products n a compettve (olgopolstc) market. We study non cooperatve Nash equlbrum polces for sellers. At the begnnng of the tme horzon, the total nventores are gven and addtonal producton s not an avalable opton. The analyss for perodc producton-revew models, where producton decsons can be made at the end of each perod at some producton cost after ncurrng holdng or backorder costs, does not extend to ths model. Usng results from game theory and varatonal nequaltes we study the exstence and unqueness of equlbrum polces. We also study convergence results for an algorthm that computes the equlbrum polces. The model n ths paper can be used n a number of applcaton areas ncludng the arlne, servce and retal ndustres. We llustrate our results through some numercal examples. I. INTRODUCTION The am of ths paper s to propose and analyze a model of compettve non-collusve olgopoly for sellers competng to sell gven nventores of a pershable product over a fnte mult-perod tme horzon. We start by descrbng the termnology used and gvng example of practcal stuatons where such a model would be more applcable than other proposed models n the exstng lterature. A pershable product s defned as a product that has a fnte lfe, or equvalently, loses ts value f not sold before a preset deadlne. A market s an olgopoly f there are more than one partcpatng seller competng wth each other. Each of the partcpatng sellers affects but does not control the market. The competton s non-collusve f the sellers do not (or, by law, are not allowed to) enter collusve agreements wth each other, but nstead compete to capture demand. A mult-perod tme horzon mples that the sellers have a fxed tmetable for changng ther prces and we characterze the behavor of demand wthn these dscrete ntervals. The followng examples help explan these concepts. Consder the problem of prcng advance reservatons for Hotel StayHere n Atlanta for a partcular weekend n October. For smplcty, assume that there s only one type of room offered by the hotel. The hotel starts makng reservatons for ths weekend three months n advance but s unsure Georga Peraks s the Sloan Career Development Assocate Professor of Operatons Research, Sloan School of Management and Operatons Research Center, Massachusetts Insttute of Technology, E53-359, Cambrdge, MA Emal: georgap@mt.edu Anshul Sood s a PhD Canddate, Operatons Research Center, Massachusetts Insttute of Technology, E4-13, Cambrdge, MA Emal: anshul@mt.edu of the actual demand that wll materalze. Apart from the demand uncertanty, another reason behnd ths uncertanty s the presence of competng hotels n the same area offerng smlar rooms. In ths paper we wll focus on the latter and wll assume that the demand s determnstc. The hotel competes wth other hotels on the bass of prce and offers ncentves for early reservatons for purposes of customer dfferentaton (revenue management). Whle t wants to make sure that t does not end up wth too many vacances, t also also wants to prevent sellng too many rooms at low rates early on. There are several settngs whch have smlar characterstcs to the one descrbed above. For example, those nvolvng competng arlnes sellng arfares for ther own flghts leavng wthn a small tme wndow wth the same orgn and destnaton, are very smlar to the one descrbed above. Instead of hotel reservatons or arlne fares, problems wth the same structure could nvolve shppng contaners, broadband communcaton lnks or any capacty constraned ndustry product. Several models have been proposed for monopolstc versons of ths problem. McGll and van Ryzn [31] and the references theren also provde a thorough revew of revenue management and prcng models. Btran and Caldentey [6] provde an overvew of prcng models for the monopolstc verson of the revenue management problem n whch a pershable and non-renewable set of resources satsfy stochastc prce-senstve demand processes over a fnte perod of tme. They survey results on determnstc as well as nondetermnstc, sngle as well as mult-product, and statc as well as dynamc prcng cases. Elmaghraby and Kesknocak [2] revew the lterature and current practces n dynamc prcng n ndustres where capacty or nventory s fxed n the short run and pershable. They classfy monopolstc models on the bass of whether nventory can be replenshed or not, whether demand s dependent over tme or not, and whether customers are myopc or strategc optmzers. Yano and Glbert [41] revew models for jont prcng and producton under a monopolstc setup. On the compettve sde, Vves [4] dscusses the development of olgopoly prcng models. A survey by Chan et al [1] summarzes research on jont prcng, nventory control and producton decsons n a supply chan. They also survey lterature on prce and quantty competton n supply chan settngs. Cachon and Netessne [8] also survey the problem of competton from a supply chan perspectve where the problem s characterstcally a perodc producton-revew model. They dscuss both non-cooperatve and cooperatve games n

2 statc and dynamc settngs. Our model dffers from the compettve supply chan models snce we have rgd nventory constrants over the entre horzon and the flexblty to replensh nventory between perods through producton s not an avalable opton. Under these modellng restrctons, we lose the convenent structure of the problem whch would otherwse allow us to analyze equlbrum wth standard technques. In ths paper we take and alternate approach usng deas from varatonal nequaltes. To the best of our knowledge, no general results whch could be used to analyze such a model have been presented. A more detaled dscusson of ths follows n Secton II. The paper s organzed as follows. In Secton II we revew some of the relevant lterature n the feld and dscuss how the results n ths paper could not be acheved under the framework of other papers. In Secton III we descrbe the model for the problem and the notaton used. Sectons III-A and III- B descrbe the best-response polcy problem and the market equlbrum problem respectvely. In Secton IV we gve some theoretcal results for these two problems and the condtons under whch those results hold. In Secton V we analyze an algorthm that can be used to compute the equlbrum polces and provde suffcent condtons for convergence. In Secton VI we llustrate these results through some numercal examples. II. LITERATURE REVIEW Cournot and Bertrand establshed the foundatons for the analyss of olgopolstc competton between sellers wth ther quantty and prce competton models. Cournot [11] proposed a soluton concept for olgopolstc nteracton, stablty of the resultng solutons, phenomena of colluson, and compared perfect competton wth olgopoles. These models were based on quantty competton: The competng sellers controlled ther ndvdual producton and the prevalng market clearng prce was determned by the net producton. Bertrand [3] proposed an alternate prce competton model where the competng sellers controlled prces whle quanttes for each seller were determned by the prevalng prces. In Bertrand s model, each seller was assumed to supply the entre quantty demanded at ther set prce, at an ncreasng producton cost. Edgeworth [17] proposed a prce competton model where no seller s requred to supply all the forthcomng demand at the set prce. In such a case, the resdual demand s splt amongst all sellers on some ratonal rule. We refer the reader to the surveys mentoned n Secton I for a comprehensve overvew of lterature n the area, but we would lke to brng specal attenton to some partcularly relevant papers. Rosen [38] proves exstence and unqueness results for general olgopolstc games. The paper shows exstence under concavty of the payoff to a seller wth respect to t s own strategy space and convexty of the jont strategy space and unqueness under strct dagonal domnance of the payoff functon. Murphy et al [32] analyze equlbrum n a sngle-perod quantty competton model usng mathematcal programmng results. Harker [27] analyze the same model usng varatonal nequaltes. Elashberg and Jeuland [19] model a two stage problem. The market n the frst stage s a monopoly and becomes a duopoly n the second stage wth the entry of a second seller. The sellers dynamcally prce ther product. The paper analyzes the prcng behavor under the cases that the ncumbent seller foresees or does not foresee the entrant. Prcng models n tradtonal revenue management research can be classfed nto two broad categores: statc and dynamc. Statc prcng models are based on aggregated demand dstrbutons and can be seen as a specal case of the multproduct newsvendor problem wth fxed producton costs and pershable product wth no salvage. The extenson of the newsvendor problem wth prce as a decson varable was studed by Zabel [43], Young [42], Dada and Petruzz [14], etc. Other relevant research ncludes Zabel [44], Thomas [39], Dada and Petruzz [13] and Federgruen and Hechng [22] who study the sngle-product, mult-perod combned prcng and nventory control problem that s typcally solved by dynamc programmng. Dynamc prcng models represent demand as a controllable stochastc pont process wth prce dependent ntensty. Gallego and van Ryzn [24] and Zhao and Zheng [45] consder the problem of optmally prcng a gven nventory of a sngle product over a fnte plannng perod before t pershes or s sold at salvage value. There s no reorderng. Gallego and van Ryzn [25] and Paschalds and Tstskls [35] extend ths type of model to the dynamc prcng of multple products whose producton draws from a shared supply of resources. Kleywegt [3] gves an optmal control formulaton of the mult-perod dynamc prcng problem. Kachan and Peraks [28] propose a determnstc flud model for dynamc prcng and nventory management for non-pershable products n capactated and compettve make-to-stock manufacturng systems. Some work has been done recently that explctly consders the presence of competton wthn the prcng framework. Dockner and Jørgensen [16] provde a treatment of the optmal prcng strateges for olgopolstc markets from a marketng perspectve but not a computatonal perspectve. Bernsten and Federgruen [2] develop a stochastc general equlbrum nventory model for supply chans n an olgopoly envronment where the polces nvolve prces, servce level targets and nventory control wth lnear models of demand. Bertsmas and Peraks [4] propose an optmzaton approach for jontly learnng the demand as a functon of prce and the compettor response by dynamcally settng prces of products n a duopoly envronment. Prevously publshed results prove exstence and unqueness for equlbrum strateges for prcng games under varous condtons. We found that none of these condtons hold for our model, hence requrng a new approach for analyss. Results for games wth supermodular payoff functons and lattce strategy spaces are well known (See Vves [4]). The problem n ths paper does not fulfll the latter requrement. Other results requre the payoff functon to be concave over a convex strategy space. The problem dscussed n ths paper can be reformulated so that the strategy space s a lattce and ncely convex but the resultng objectve functon to be maxmzed s nether concave nor supermodular. Alternatvely, t can be

3 formulated to have a concave objectve functon, but then the resultng strategy space s no longer convex. We study the unqueness of equlbrum prces usng varatonal nequaltes snce ths analyss does not requre the payoff functons to be concave. To the best of our knowledge, no such analyss for mult-perod prce competton models for pershable products has been done before. III. MODEL FORMULATION Consder a market of a sngle product wth a set of competng sellers. Each seller has a gven nventory of the product. The tme horzon over whch each seller wants to sell her nventory of product s dvded nto dscrete ntervals. We make the followng assumptons regardng the model and polces. 1) Perfect Informaton: We assume that each seller has perfect nformaton about the structure of demand, mpact of ther prces on ther compettors demand and the startng nventory levels of each of her compettors. 2) Consumer Choce: We assume that the demand for each seller s a functon only of the current perod prces (the seller s and her compettors ) and that s the only dstngushng factor between products from dfferent sellers. 3) Demand: We assume that the demand that each seller wll see s a determnstc functon of the prces set by all sellers n that perod. 4) Product: We assume that there s only a sngle product and the nventory that s saleable over all tme perods s pershable at the end of the tme horzon. 5) Objectves: We assume that the sellers are objectvely maxmzng ther respectve revenue over the tme horzon of the problem and practces lke short-squeezng compettors out of the market by short-term prcecuttng, ntroductory dscount prcng to capture market share, etc. are absent. The notaton we use s as follows. A sngle seller s denoted by 2 I, where I s the set of all sellers. For ease of notaton, we denote the set of all compettors of by. The nventory of products belongng to seller s denoted by C. A tme nterval s denoted by t 2 T. A prce s set by seller n every perod t and s denoted by p t. Thus, we denote a prcng polcy for seller by p =(p 1 ;p2 ;:::;pt ) and a set of prcng polces for all sellers by p =(p 1 ; p 2 ;:::;p I ). In perod t, the resultng number of buyers who wsh to purchase from seller s denoted by h t (pt ) (observed demand) and s a functon of the prce levels set by all sellers n that perod. We assume that the current demand s not affected by the prevous hstory of prces. The demand that seller realzes (.e the sale made) n perod t s denoted by d t. Note the mplcaton that d t» h t (pt ) snce the sale made (realzed demand) cannot be greater than the demand (observed demand). The relaton s not an equalty snce the seller mght be restrcted by the actual nventory level avalable. We use the notaton d = (d 1 ;d2 ;:::;dt ) and d = (d 1; d 2 ;:::;d I ) to denote the realzed demand. We denote the strategy of the sellers by the prces set, together wth the maxmum realzed demand usng z = (p ; d ) and z = (z 1 ; z 2 ;:::;z I ). As before, we use the notaton h (p) = (h 1 (p1 );:::;h T (pt )) and h(p) =(h 1 (p);:::;h I (p)). Gven the nventory nformaton C =(C 1 ;:::;C I ), the total payoff to seller, over the entre tme horzon as a functon of the sellers polces p, s denoted by J (p). The correspondng payoff n any sngle perod t s gven by ß t. We denote the best response polcy that maxmzes the payoff of seller over the entre tme horzon gven that her compettors have adopted polces p by BR (p ). Ths wll be obtaned by solvng the best response mult-perod prcng problem. We defne ths problem n secton III-A and formulate t as an optmzaton problem. We denote the resultng best response polcy BR (p ) for seller by p. In the specal case where the prcng polces are at equlbrum, we denote them by p Λ = (p Λ ; pλ ). In what follows, we defne the concept of Nash equlbrum polces: Defnton 3.1: The prcng polces for each seller are Nash equlbrum prcng polces f no sngle seller can ncrease her payoff by unlaterally changng her polcy. Ths defnton mples that each seller sets her equlbrum prcng polcy as the best response to the equlbrum prcng polces of her compettors. Ths set of polces would then, by defnton, be a Nash equlbrum set of polces. See Nash [34] for further detals on the noton of a Nash equlbrum n non-cooperatve games. A. Best Response Polcy The best response prcng polcy for seller s the polcy that maxmzes seller s payoff n response to all others sellers prcng polces. We frst defne the mult-perod prcng problem followed by the formulaton of the best response problem for ths problem. Defnton 3.2: Mult-perod Prcng Problem Consder a set of sellers I wth nventores C and tme horzon T. The strategy of each seller conssts of settng her prce levels p optmally,.e. as best response prces arsng from formulaton (1) below. The demand observed by seller n any perod s equal to the number of buyers who are wllng to buy from her gven the prce levels for all sellers. Seller wll realze that demand f she has enough nventory. The best response polcy p of seller, gven all her compettors polces μp s the soluton of the followng optmzaton problem: P T argmax d;p t=1 dt pt (1) such that d t» ht (pt; μpt P ) 8t 2 T T t=1 dt» C p mn t» pt» pt max 8t 2 T 8t 2 T: In compact notaton, the above can be rewrtten as: d t max z =(d ;p ) J (z )= 1 2 z Qz such that d» h (p ; μp ) 1 d» C p mn» p» pmax d ;

4 I where Q =, I denotes a square dentty matrx of I sutable dmenson. Note that n ths optmzaton problem, gven a μz, seller selects that μz that maxmzes the objectve functon J (d ; P T p )= t=1 dt pt wthn the feasble space K (μz )= f(d ; p ) j d t» P h t (pt ; μpt ); T t=1 dt» C ;p mn t» pt» p t max ;dt ; 8t 2 Tg. In Secton IV-A we show that the soluton z to the best response problem for seller, gvenμz, wll also satsfy the followng varatonal nequalty problem: rj (z ) (z z ) 8z 2K (μz ) (2) B. Market Equlbrum Model The defnton of a Nash equlbrum (Defnton 3.1) mples that, at equlbrum, each seller would select a prcng polcy that optmally solves her own best response problem. Notce that all compettors solve ther best response problems smultaneously. Therefore each of them solve varatonal nequalty (2) gven ther compettors polcy z. Gven a potental canddate for an equlbrum set of prcng polces for her compettors, z, seller sets her equlbrum prcng polcy by solvng varatonal nequalty problem (2). Thus the equlbrum set of polces wll solve the followng set of varatonal nequalty problems: rj (z Λ ) (z z Λ ) 8z 2K (z Λ ) 2 I (3) The next result shows how ths set of varatonal nequalty problems can be combned nto a sngle varatonal nequalty problem. Ths new problem solves the best response problem for each seller smultaneously, hence determnng a set of equlbrum prcng polces. Proposton 3.1: The equlbrum set of prces satsfes the followng varatonal nequalty formulaton: F (z Λ ) (z z Λ ) 8z 2K; (4) where F (z Λ )= rj (z Λ ), 8 2 I and K = fz =(z 1 ; z 2 ;:::;z I )jz 2K (z )8 2 Ig : Proof: See Peraks and Sood [36]. 2 In the remander of the paper, we refer to varatonal nequalty (4) as the market equlbrum model. IV. ANALYSIS OF EQUILIBRIUM In ths secton we examne the analytcal propertes of the best response model (2) and the market equlbrum model (4). These propertes hold under certan condtons. We state these condtons and try to provde some ntuton on when these condtons hold. Frst we mpose a condton that ensures that the space of allowed prces s bounded. One way to acheve ths boundedness property would be to constran the prces between some allowable upper and lower lmts. Under ths condton, we can elmnate strateges nvolvng nfntely hgh prce levels. Note that the lower lmt could be the zero prce level and the hgher lmt could be the prce level at whch the demand functon vanshes. Condton 4.1: There exsts a mnmum and maxmum allowable prce level. We denote ths by p mn and p max for each perod respectvely. The next condton ensures that the demand for a seller s concave n the seller s prce for each perod. Ths condton ensures that the strategy space n the best response problem s convex. The lnear demand model trvally satsfes ths condton. From a demand elastcty pont of vew, ths condton holds for demand functons whch have an ncreasng prce elastcty as prce ncreases. Ths holds for products where demand decreases faster as prce ncreases. Condton 4.2: The demand functon h t (pt; μpt ) s a concave functon of p t over the set of feasble prces for all 2 I;t2 T for a fxed μp t. We also requre that the demand s strctly monotonc n prce n order to ensure that the best response polcy s unquely defned. For a lnear demand case, ths mples that the demand functon s downward slopng wth respect to prce as s true for normal goods. Condton 4.3: For any perod t, for any fxed μp t, the functon h t (pt; μpt ) s strctly decreasng wth respect to pt over the set of feasble prces. Mathematcally, h t (^pt ; μpt )+ht (»pt ; μpt ) (^p t»pt ) > 8(^p t ;»pt ), ^pt 6=»p t, 2 I: The next condton ensures that the market equlbrum model has a unque soluton by requrng strct monotoncty on the demand functon as a whole. For a two seller lnear demand case ths s equvalent to sayng that the senstvty of seller s demand to seller s prce s hgher than the senstvty of seller s demand to seller s prce and the senstvty of seller s demand to seller s prce. Ths makes ntutve sense snce we expect the decrease n demand seen by seller when she rases prces to be more than the resultng ncrease n demand seen by her compettor. Ths can be nterpreted as sayng that upon seeng an ncrease n seller s prce, some of her customers wll prefer to swtch to her compettor and some wll prefer not to buy at all. Condton 4.4: The functon h(p) s strctly monotone wth respect to p, over the set of feasble prcng polces K. That s, ([ h(^p) +h(»p)] (^p»p)) > 8^p;»p 2K, ^p 6=»p: Note that Condton 4.3 refers to the strct monotoncty of h ( ) wth respect to only p, whle Condton 4.4 refers to the strct monotoncty of h( ) wth respect to p and s thus a stronger condton. The followng lemmas are consequences of these condtons: Lemma 4.1: Under Condton 4.2, the constrants d t» ht (pt ; μpt ) 8t 2 T; 2 I defne a convex set for any gven μp t. Lemma 4.2: Under Condton 4.3, for any fxed μp, the functon h (p ; μp ) s strctly monotone wth respect to p,

5 over the set of feasble prcng polces K (μp ). That s, [ h (^p ; μp )+h (»p ; μp )] (^p»p ) > 8^p ;»p 2K (μp ), ^p 6=»p, 2 I: Lemma 4.3: Under Condton 4.4, the functon h t (p t ) s strctly monotone wth respect to p t, over the set of feasble prcng polces. That s, h t (^p t )+h t (»p t ) Λ (^p t»p t ) > 8(^p t ;»p t ), ^p t 6=»p t, t 2 T: A. Best Response Problem We start wth an ntutve result that characterzes a soluton to the best response problem. We use ths result whle provng the unqueness of the soluton to the best response. Lemma 4.4: Gven a compettor strategy ( d μ ; μp ), the soluton z = (d; p ) to varatonal nequalty problem (5) satsfes the followng relaton: d t = h t (p t ; μpt ); f d t > : If d t = then there exsts z that also solves varatonal nequalty problem (5) and satsfes the above relaton. Proof: See Peraks and Sood [36]. 2 1) Exstence and Unqueness of Soluton to the Best Response Problem: To show that the varatonal nequalty (5) has a soluton we frst show that a best response strategy whch solves model (1) exsts. We then show that t satsfes the varatonal nequalty. Proposton 4.1: For any fxed μp, there exsts a soluton z =(d; p ) to the Best Response optmzaton model (1). Proof: It s easy to show that the feasble space s nonempty and compact and the objectve functon s contnuous. Under these condtons the result follows from the well known Weerstrass theorem (See Bazaraa, Sheral and Shetty [1]). 2 Proposton 4.2: Let z = (d ; p ) be a soluton to Best Response optmzaton model (1). Condton 4.2 mples that also solves the followng varatonal nequalty problem: z rj (z ) (z z ) 8z 2K (μz ) (5) Proof: See Peraks and Sood [36]. 2 Theorem 4.1: Under Condton 4.3, for fxed μz, there exsts a unque soluton to best response problem (1). Proof: See Peraks and Sood [36]. 2 B. Market Equlbrum Prces 1) Exstence and Unqueness of Equlbrum Prces: Theorem 4.2: Condton 4.2 mples that there exsts at least one soluton to varatonal nequalty (4). Proof: The result follows from Knderlehrer and Stampaccha (198) snce the feasble set s compact (closed and bounded) and (under Condton 4.2) convex, and functon rj ( ) s contnuous over the feasble set. 2 Theorem 4.3: Under Condtons 4.2 and 4.3, there exsts a unque soluton to the Market Equlbrum Model (3). Proof: See Peraks and Sood [36]. 2 V. COMPUTATION OF MARKET EQUILIBRIUM PRICES A. Iteratve Learnng Algorthm In ths secton we study an algorthm for computng the market equlbrum prces arsng from varatonal nequalty (4). A number of algorthms proposed for solvng varatonal nequaltes exst n lterature. The algorthm we study s based on a smple ntutve process nspred by the concept of fcttous play, frst ntroduced by Brown [7] and Robnson [37]. The tatônnement process descrbed n Vves [4] s very smlar n nature and s shown to converge (tatônnement stablty) for supermodular games. We gve suffcent condtons for convergence of the algorthm for the mult-perod prcng game dscussed n ths paper and dscuss how these condtons can be nterpreted for the lnear demand case. Consder the market we descrbed n Secton III, consstng of several sellers prcng a product n a mult-perod settng. Assume that the process s repeated under the same condtons of ntal nventory and perod-wse demand. The sellers do not start wth the equlbrum polces but rather follow a nave myopc optmzaton approach: They prce usng the best response polcy gven all compettors prces from the prevous nstance of the process. The key queston s that f ths process s repeated suffcently many tmes, under what condtons wll the sellers prces converge to the equlbrum prces, rrespectve of the assumed startng polces? The outlne of the general algorthm s as follows. Start by consderng an ntal estmate for the soluton denoted by z 2Kand set k =1. Compute z k by solvng the followng set of separable varatonal nequalty subproblems for each 2 I: F (z k ) (z z k ) 8z 2K (z k 1 ): (6) For our problem, ths teraton step corresponds to each seller settng the best response polcy to her compettors strateges from the last teraton. We dscuss the detals of ths computaton step n Peraks and Sood [36]. We check for convergence (f the polces from two successve teratons are the same or ffl-close to each other) and stop; otherwse we repeat wth an ncremented value for k. Ths algorthm s formally presented n Algorthm 1 below. B. Convergence of the Iteratve Learnng Algorthm In ths secton we study the convergence of Algorthm 1. We prove that the followng set of condtons are suffcent for convergence. Condton 5.1: For any gven μp, h (μp ; p ) s Lpschtz contnuous wth respect to p wth parameter L. kh (μp ; ^p ) h (μp ;»p )k»lk^p»p k Condton 5.2: For any gven μp, h (p ; μp ) s strongly monotone wth respect to p wth parameter A. ( h (^p ; μp )+h (»p ; μp )) (^p»p ) Ak^p»p k 2 Condton 5.3: A > L where A and L are defned as above. For the two seller, lnear demand case, the above condtons hold when for all 2 I, the mnmum senstvty of seller s demand to seller s prce over all perods, s greater than the

6 Algorthm 1 1: for =1:::N do 2: p t ψ pt ntal 3: end for 4: for =1:::N do 5: 1 p ψbr ( p ) 6: end for 7: k ψ 1 8: whle k p 6= k 1 p do 9: for =1:::N do 1: k+1 p ψbr ( k p ) 11: end for 12: k ψ k +1 13: end whle 14: p Λ ψ k p 15: RETURN p Λ maxmum senstvty of her demand to her compettor s prce over all perods. In partcular, f the demand for a two seller market s gven by h t (pt ;pt )=Dt base ft pt + fft pt then A =mn t (f t ) and L =max t(ff t ). In order to formally prove the convergence of Algorthm 1, we ntroduce the followng reformulaton of the best response problem. We am to move the constrant nvolvng observed demand and realzed demand nto the objectve functon. We frst defne the relaxed strategy space: K μ = fz = P T (p ; d ) j t=1 dt» C ;p t mn» pt» pt max ;dt ; 8t 2 Tg, and μk = fz =(z 1 ; z 2 ;:::;z I ) j z 2 μ K 8 2 Ig: Note that these spaces are dfferent from K n that the constrant nvolvng d t» h t (pt ) s mssng. To move the constrant nto the objectve functon, we ntroduce a dummy varable v n addton to the prce p and allocaton varables d, where v = (v 1 ; v 2 ;:::;v I ) 2 R T I and v = (v 1 ;v2 ;:::;vt ) 2 RT. The complete varable space s thus defned n terms of w = (w 1 ; w 2 ;:::;w I ) where w = (z ; v ) 2 K μ R T. The feasble strategy space n terms of ths varable s K w = Φ w =(z; v)jz 2 μ K; v 2 R I T Ψ : The varatonal nequalty problem can then be stated as: where G(w Λ ) (w w Λ ) 8w 2K w ; (7) G (w) = ( r z J (z ); d h(z ; z )) = ( Qz ; d h(z ; z )): The functon G(w) n varatonal nequalty problem (7) s non-separable (e. dependng on the seller s and her compettors strateges) makng the problem dffcult to solve. Algorthm 1 consders an approxmaton of (7) whch modfes the problem nto a separable varatonal nequalty problem whch s easer to solve. Ths separable problem s actually nothng but the best-response problem for an ndvdual seller. We apply Algorthm 1 by solvng ths separable verson at each step for each ndvdual seller. For a gven μw, solve where G(w ; μw) (w w ) 8w 2K w (8) G (w ; μw) = ( r z J (z ); d h(z ; μz )) = ( Qz ; d h(z ; μz )): Ths varatonal nequalty problem can be separated nto smaller sub-problems (s separable) and for each 2 I, we fnd w satsfyng G (w ; μw) (w w ) ; 8w 2K w : (9) The followng lemma and proposton prove that the varatonal nequalty n w and the varatonal nequalty n z are equvalent. Lemma 5.1: Any w that solves varatonal nequalty (8) satsfes d t = h t (p t ; μpt ) 8 2 I;t2 T Proof: The result follows from the fact that v t 2 R, 8 2 I;t2 T. 2 Remark: From Proposton 5.1 t follows that f vector w Λ = (z Λ ; v Λ ) solves varatonal nequalty (8), v Λ can be any vector n R T I. Wthout loss of generalty, we select v Λ = n our soluton. Proposton 5.1: The Best Response Problem (6) and varatonal nequalty problem (8) are equvalent. Proof: It s easy to see that any soluton to the Best Response Problem (6) s also a soluton to the formulaton (8). To show the converse, we use Lemma The next theorem proves that Condtons 5.1, 5.2 and 5.3 are suffcent condtons for convergence. Theorem 5.1: Under Condtons 5.1, 5.2 and 5.3, Algorthm 1 converges to an equlbrum prcng polcy. Proof: See Peraks and Sood [36]. 2 VI. NUMERICAL EXAMPLES In ths secton, we examne the results presented n ths paper numercally. Note that the results presented hold for general demand functons though n ths secton we use the lnear demand case for llustraton. We study the nature of the resultng equlbrum prcng polces when the ntal nventores fc 1 ;C 2 g and prce senstvtes are vared. The general trends observed are as expected: 1) The hgher the nventory that any seller has avalable for sale over the entre horzon, the lower the prces that she sets. The revenue earned, however, s hgher even though the prces set are lower. 2) Correspondngly, an ncrease n the nventory of a compettor results n lower revenues for the seller snce the compettor reduces prces. 3) Prces are hgher n perods wth lower prce senstvtes. We also examne the convergence behavor of Algorthm 1 numercally as the relatve rato of prce senstvtes s vared

7 and also as the ntal estmate of prces used n the algorthm s vared. In general, numercal experence led us to the followng conclusons regardng the practcal convergence of the algorthm: 1) The algorthm converges to the equlbrum polces rapdly n practce. 2) The numercal results verfy the theoretcal analyss regardng convergence of the algorthm to the unque equlbrum prcng polces when startng from dfferent startng ponts. 3) The number of teratons taken to converge were dependent on the startng pont. Convergence was tested by ntalzng the algorthms wth dfferent ntal prces. In general, numercal experence led us to conclude that the number of teratons requred to converge were smallest for cases where the startng prces were taken close to the equlbrum prces for all sellers. However, the rate of convergence dd not depend on the startng pont. 4) Changng the relatve rato of demand senstvtes to prce affected the rate of convergence n accordance to Theorem 5.1. The prces converged to the equlbrum prces at a geometrc rate roughly proportonal to the theoretcally predcted rate. For llustraton purposes, we consder a two-seller multperod, symmetrc lnear demand example. For ths example, I = f1; 2g and T = f1; 2; ; 1g. The demand s lnear n prces and symmetrc wth respect to both sellers and vares wth tme: 8 2 I, the demand functon h t = D base t f t p t + fft p t. We assume the symmetry of demand for the sake of convenence. Note that the results hold n general for asymmetrc demand. We consder markets where customers wth lower prce senstvtes typcally arrve n later perods. As a result the senstvty of the demand to the seller s prce (and also to her compettor s prce) n the examples decreases towards the end of the tme horzon. In Table I we study the trend n prcng polces wth varyng nventory balances. We consder three cases wth dfferent nventores for each of the two sellers. In the frst case both players are over-nventored wth fc1; C2g = f3; 2g and the optmal equlbrum polcy results n nether of them sellng ther entre nventory. Ths case s effectvely equal to the uncapactated case. Fgure 1 shows the evoluton of the prcng polces as the algorthm terates, the resultng equlbrum prces, the remanng nventory over the tme horzon under the equlbrum prces, and the cumulatve revenue from those sales. In the second case, only one of them s over-nventored (fc1; C2g = f3; 5g). Fgure 2 shows the results from ths case. Note that the seller wth less nventory sets prces hgher than the seller wth hgher nventory. Even though the average prce s lower for the latter, her total revenues are hgher. The prces n general are also hgher than n the prevous case. Fnally, n the thrd case, nether has suffcent nventory (fc1; C2g = f1; 5g) so the demand supply mbalance results n a general prce hke (Fgure 3). In Table II we study the movement of prcng polces as Algorthm 1 terates wth varyng ntal estmates for startng Inventory level Fg. 1. Inventory level Fg. 2. Inventory level Iteratons: Evoluton of prcng strategy Inventory: Intal level & Level at end of perod Cumulatve Revenue Equlbrum prcng polces: Seller 1 & 2 1 x 14 Cumulatve Revenues Both sellers have excess nventory. fc1;c2g = f3; 2g Iteratons: Evoluton of prcng strategy Inventory: Intal level & Level at end of perod Cumulatve Revenue Equlbrum prcng polces: Seller 1 & 2 15 x 14 Cumulatve Revenues One seller has excess nventory. fc1;c2g = f3; 5g Iteratons: Evoluton of prcng strategy Inventory: Intal level & Level at end of perod Cumulatve Revenue Equlbrum prcng polces: Seller 1 & 2 15 x 14 Cumulatve Revenues Fg. 3. Nether of the sellers have excess nventory. fc1;c2g = f1; 5g

8 3 25 Iteratons: Startng prce = 3 25 Iteratons: Startng prce = 15 TABLE I TREND IN PRICING POLICIES WITH VARYING INVENTORY BALANCES Iteratons: Startng prce = Iteratons: Startng prce = 45 Model parameters held constant D base = f11; 1; 1; 1; 9; 9; 1; 1; 8; 6g f = f1:2; 1:2; 1:1; 1:; :9; :8; :7; :6; :5; :4g ff = f1:; 1:1; 1:; :8; :8; :7; :5; :4; :4; :4g Model parameters vared fc1;c2g = f3; 2g; f3; 5g; f1; 5g TABLE II MOVEMENT OF PRICING POLICIES IN ITERATIONS OF ALGORITHM 1 WITH VARYING INITIAL ESTIMATES FOR STARTING PRICES. Fg. 4. Actual trend of prcng polces over successve teratons of Algorthm 1 when startng wth dfferent ntal prces. 1 1 Model parameters held constant D base = f11; 1; 1; 1; 9; 9; 1; 1; 8; 6g f = f1:2; 1:2; 1:1; 1:; :9; :8; :7; :6; :5; :4g ff = f1:; 1:1; 1:; :8; :8; :7; :5; :4; :4; :4g fc 1 ;C 2 g = f1; 5g Model parameters vared Startng estmate of prces 8 2 I and t 2 T p t =; 15; 3; Dstance TABLE III PRACTICAL CONVERGENCE BEHAVIOR OF ALGORITHM 1 WITH VARYING RELATIVE PRICE SENSITIVITIES. Fg Iteraton Example: Convergence startng wth dfferent ff. Model parameters held constant D base = f11; 15; 1; 95; 9; 85; 8; 75; 75; 85g f = f1:2; 1:15; 1:1; 1:5; 1;:95;:9;:85;:8; :75g fc1; C2g = f1; 5g Model parameters vared ff = kf where, k = 2 32 ; ; 1 52 ; 5 32 prces. Fgure 4 shows how Algorthm 1 converges to the equlbrum prcng polcy when startng from four dfferent startng ponts. We consder prces whch are constant over all tme perods as our ntal estmates. We fnd that the convergence occurs fastest when the startng pont s close to the equlbrum prce. In Table IV we look at the same ssue by measurng the 2-norm dstance between the prce polcy vector from successve teratons. In Table III we study the practcal convergence behavor of Algorthm 1 wth varyng relatve prce senstvtes. Fgure 5 shows the 2-norm dstance between the prce vectors p n the current teraton and the prevous teraton of Algorthm 1. The four cases correspond to the choce of dfferent ratos of the senstvty of seller s demand to her own prce and her compettor s prce. The steepest lne occurs for the smallest rato and vce versa. In Table IV we study the practcal convergence behavor of Algorthm 1 wth varyng ntal estmates for startng prces. Fgure 6 shows the 2-norm dstance between the prce vectors p n the current teraton and the prevous teraton of Algorthm 1. The four cases correspond to the choce of dfferent ntal estmates of the p. We observe that the slope of the lne s the same and the dfferent cases just result n parallel dsplaced lnes. Fg. 6. Dstance Iteraton Convergence behavor startng wth dfferent ntal prces. TABLE IV PRACTICAL CONVERGENCE BEHAVIOR OF ALGORITHM 1 WITH VARYING INITIAL ESTIMATES FOR STARTING PRICES. Model parameters held constant D base = f11; 15; 1; 95; 9; 85; 8; 75; 75; 85g f = f1:2; 1:15; 1:1; 1:5; 1;:95;:9;:85;:8; :75g ff = f fc1;c2g = f1; 5g Model parameters vared Startng estmate of prces p t 8 2 I and t 2 T =; 5; 1; 15

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