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1 Ecoomic Theory 19, (2002) Research Article Wier-take-all price competitio Michael R. Baye 1 ad Joh Morga 2 1 Departmet of Busiess Ecoomics ad Public Policy, Kelley School of Busiess, Idiaa Uiversity, 1309 East Teth Street, Bloomigto, IN , USA ( mbaye@idiaa.edu) 2 Woodrow Wilso School for Public ad Iteratioal Affairs ad Departmet of Ecoomics, Priceto Uiversity, Priceto, NJ 08544, USA ( rjmorga@priceto.edu) Received: April 7, 2000; revised versio: September 14, 2000 Summary. We aalyze a oligopoly model of homogeeous product price competitio that allows for discotiuities i demad ad/or costs. Coditios uder which oly zero profit equilibrium outcomes obtai i such settigs are provided. We the illustrate through a series of examples that the coditios provided are tight i the sese that their relaxatio leads to positive profit outcomes. Keywords ad Phrases: Price competitio, Discotiuity, Bertrad, Hotellig. JEL Classificatio Numbers: D43, C72. 1 Itroductio This paper examies the competitiveess of wier-take-all price competitio i homogeeous product oligopoly eviromets where uderlyig buyer demads ad/or firms costs eed ot be cotiuous. Our aalysis is motivated by the observatio that a variety of ecoomic settigs have these features. For example, i 1996 a Ivy League uiversity solicited bids from several vedors for its iitiative to dramatically expad ad stadardize desktop computer use throughout the uiversity. The extet of this stadardizatio effort at the staff level depeded o the uit price of the lowest bid received. I particular, should the price per uit prove too high, the oly faculty ad admiistrators would be icluded i the iitiative. If bids were low eough, the iitiative would be expaded to cover A earlier versio of this paper, etitled Necessary ad Sufficiet Coditios for Bertrad s Paradox, was preseted at the 1997 Ecoometric Society Summer Meetigs i Pasadea. We thak Simo Aderso, Paul Klemperer, Da Koveock, ad Dale Stahl for commets o earlier versios. Correspodece to: M. R. Baye

2 272 M.R. Baye ad J. Morga all staff at the uiversity. Thus, vedors biddig for this cotract faced a jump i demad oce their price fell below this threshold. More geerally, discotiuities i demad may be caused by ocovex prefereces, badwago effects, lumpiess i cosumptio, herd behavior, demad etwork effects, cascades, ad a host of other pheomeo. The uiversity s IT departmet recogized that it was much less expesive to support idetical machies rather tha those of mixed types. Dell ad Gateway computers with idetical specificatios are arguably homogeeous products prior to acquisitio, but i maitaiig existig systems ad esurig that software upgrades work, it is importat to adopt oe stadard platform. As a cosequece, the cotract was awarded etirely o a wier-take-all basis to the firm offerig the lowest price. As oted by Klemperer (2000), may oligopoly eviromets have a similar auctio-like structure. The uiversity s preferece for stadardized machies required the wiig vedor to esure that every machie cotaied idetical compoets, used idetical ports, ad idetically cofigured drivers. This created potetial problems for vedors due to the rapid pace of techological chage as well as temporary shortages of compoets available from subcotractors. Cosequetly, vedors also faced jumps i costs oce the umber of uits sold exceeded some quatity threshold. Ecoomic reasos for cost discotiuities have bee documeted as far back as Brems (1952), ad iclude iflexibility i hirig decisios as a result of collective bargaiig agreemets, impositio of pollutio abatemet taxes for productio beyod a certai scale, lumpiess i productio, ad cogestio effects; Friedma (1972) provides a techical aalysis. Ideed, ocovexities i productio are the essece of Milgrom ad Roberts (1990) study of moder maufacturig. How competitive are eviromets such as these? Harrigto (1989) provides sufficiet coditios for wier-take-all price competitio to result i competitive outcomes whe demad is cotiuous ad firms ejoy costat returs. I cotrast, Dastidar (1995) shows that whe cost fuctios are cotiuous but strictly covex ad demad is divided equally amog competig firms i the evet of a tie, multiple equilibria (some with positive profits) arise. Dasgupta ad Maski (1986) ad Baye, Che, ad Zhou (1993) provide geeral coditios for the existece of equilibrium i games (icludig wier-take-all price competitio) where the uderlyig payoff fuctios are discotiuous. However, it is a ope questio whether wier-take-all price competitio leads to zero profit outcomes whe idetical firms with demad or cost discotiuities compete i a wier-take-all fashio. This paper addresses this questio. Sectio 2 presets our geeral model of wier-take-all price competitio. Theorem 1 provides ecessary ad sufficiet coditios for zero profit equilibria to exist i the geeral model, while Theorem 2 idetifies coditios sufficiet to guaratee the uiqueess of these outcomes. Sectio 3 illustrates, through a series of examples, that the coditios provided are tight i the sese that their relaxatio leads to positive profit outcomes. Whe moopoly payoffs are bouded, oly zero profit equilibrium outcomes obtai with discotiuous (but

3 Wier-take-all price competitio 273 o-icreasig) demad. I cotrast, the presece of discotiuous (but odecreasig) costs ca result i aticompetitive pricig. We coclude i Sectio 4. 2 Geeral model ad results A set N = {1, 2,...} of > 1 idetical, risk-eutral firms compete to supply some homogeeous product to a buyer. Let π(p) deote the operatig profits (that is, profits gross of ay uavoidable suk costs) that a moopolist chargig a price p P [0, ) would ear i this market. At this poit we do ot place ay a priori restrictios o demad or costs. Sice the game is symmetric, it is atural to restrict attetio to the case where all firms have a idetical strategy space, P, which we assume is coected. Thus, P deotes the strategy space of the game. Each firm simultaeously chooses a price, p i P, with the firm chargig the lowest price wiig a cotract from the buyer at that price. I the evet of a tie for low price, the buyer awards the cotract to oe of the firms at radom. 1 This is i cotrast to Dastidar (1995), who studies a class of homogeeous product pricig games where split cotracts are awarded i the evet of a tie. 2 Thus, if (p 1, p 2,..., p ) P are the prices chose by the firms, the profits of firm i are give by: π(p i ) if p i < p j j /= i 1 π i (p 1, p 2,..., p ) = m π(p i ) if i ties m 1 other firms for low price 0 otherwise If we let Π = (π 1,π 2,...π ) deote the vector of these payoff fuctios, the a wier-take-all pricig game is give by Γ N, P,Π. We let Φ be the set of all cumulative distributio fuctios o P, ad Φ deote the set of all tuples of such fuctios. For F Φ, let S F deote the support of the desity associated with F. Thus, ( F1, F 2,..., F ) Φ is a mixed strategy Nash equilibrium of Γ if, give the vector of mixed strategies of oppoets, F i, firm i s expected profits are o less uder Fi tha uder ay other strategy F i Φ; that is, ( Eπ i F i, F i ) ( Eπi F i, F i ) i N. Notice that if S F i is a sigleto for all i N, the ( F1, F 2,..., F ) comprises a pure strategy Nash equilibrium. We ow defie the aalog of Bertrad s paradox for this class of games. 1 We assume that each firm tied for the lowest price has a equal chace of receivig the cotract; however our Theorems 1 ad 2 below hold for all tie-breakig rules where a sigle firm receives the etire cotract. 2 Whe firms do ot ejoy costat returs techologies, these differig assumptios give rise to differet payoff fuctios..

4 274 M.R. Baye ad J. Morga Defiitio 1 Let Γ be a wier-take-all pricig game. ( F1, F 2,...F ) Φ is a zero operatig profit equilibrium if ( a) F 1, F2,...F ) is a Nash equilibrium of Γ, ad b) Eπ i (F1, F 2,...F )=0for all i N. This defiitio geeralizes the idea uderlyig Bertrad s paradox to wiertake-all eviromets where techologies eed ot exhibit costat returs. Notice that the origial two-firm Bertrad paradox is merely the special case where all probability mass is cocetrated at the (costat) margial cost, c, ( i.e. SF 1 = S F 2 = c ). Our first Theorem establishes that a ecessary ad sufficiet coditio for the existece of a zero operatig profit outcome i wier-take-all pricig games is that there exists a lowest price that a moopolist would have to receive i order to cover its operatig costs. We refer to this price as a iitial breakeve price ad formalize the cocept i: Coditio B (Iitial Breakeve Price) The moopoly profit fuctio has a iitial breakeve price; that is, there exists a price c P such that for all p P, p < c implies π(p) π(c) =0. Theorem 1 A zero operatig profit equilibrium of Γ exists if ad oly if Coditio B holds. Proof. ( ) If Coditio B holds, the S F i = {c} for all i N comprises a zero operatig profit equilibrium. ( ) If ( F1, F 2,...F ) is a zero operatig profit equilibrium, a firm chargig p ears operatig profits of at least 1 π(p) whe p is the lowest price charged i equilibrium ad zero otherwise. By defiitio, each firm ears zero expected operatig profits uder ( F1, F 2,...F ). It follows that π(p) = 0 for almost all p S Gmi where S Gmi is the support of the distributio of the lowest price iduced by the equilibrium. Hece, there exists a p S Gmi such that π ( p ) =0. This ad the hypothesis that ( F1, F 2,...F ) comprises a Nash equilibrium implies that for all p < p,π(p) π(p ) = 0; otherwise, a firm could re-allocate probability mass to a p < p where π(p ) > 0 ad ear positive expected operatig profits. Thus, Coditio B is ecessary ad sufficiet for Bertrad s price udercuttig logic to lead to the zero profit outcome as a Nash equilibrium to a wier-take-all pricig game. The ituitio is straightforward: For sufficiecy, otice that if the moopoly operatig profit fuctio has a iitial breakeve price, the there are o gais to udercuttig a rival who prices at that level; hece all firms chargig the iitial breakeve price comprises a Nash equilibrium. For ecessity, simply observe that if for a give zero profit equilibrium price, there did exist a lower price at which a moopolist could ear positive operatig profits, the a firm could profitably deviate by chargig that price. This

5 Wier-take-all price competitio 275 cotradicts the hypothesis that the origial prices comprised a equilibrium i the first place. Thus, a zero profit outcome exists i wier-take-all pricig games i fairly geeral eviromets; either cotiuity (of demad or costs) or costat returs to scale are required. Whe each firm prices at c, where Coditio B holds, o firm ca gai by udercuttig. Fially, ote that Theorem 1 also implies that ay wier-take-all pricig game havig zero profit equilibria must have at least oe equilibrium i pure strategies. I light of Theorem 1, it remais to idetify coditios uder which every Nash equilibrium to a give wier-take-all pricig game is, i fact, a zero profit equilibrium whe there are discotiuities i demad ad/or cost. Before we preset these coditios, we eed the followig defiitio. Defiitio 2 A fuctio π : P R is left lower semicotiuous if, for all p P, lim if p p π(p) π(p ). It is a simple matter to show that either of the followig is sufficiet for π (p) to be left lower semicotiuous: (1) π (p) is lower semi-cotiuous, 3 or (2) π (p) is cotiuous but for dowward jumps. 4 For istace, the fuctio F (x) i Figure 1 is left lower semicotiuous but ot lower semicotiuous. We are ow i a positio to state our mai result. Figure 1. A left lower semicotiuous fuctio Theorem 2 Let Γ be a wier-take-all pricig game i which a) Coditio B holds; b) π (p) is bouded from above; ad c) π (p) is left lower semicotiuous. The every Nash equilibrium of Γ is a zero operatig profit equilibrium. Proof. By Theorem 1, a zero operatig profit equilibrium exists. The followig argumet establishes (by exhaustio) that every equilibrium is a zero operatig profit equilibrium. Case 1: By way of cotradictio, suppose there exists a equilibrium ( ) F1, F 2,..., F such that for some firm i 3 A fuctio π : P Ris lower semicotiuous if, for all p P, lim if p p π(p) π(p ). 4 A fuctio π : P R is cotiuous but for dowward jumps if, for all p P, lim if p p π(p) π(p ) lim sup p p π(p); see Milgrom ad Roberts (1994).

6 276 M.R. Baye ad J. Morga Eπ i ( F i, F i ) = π < 0 The, by Coditio B, there exists a price c such that π(c) =0. By re-allocatig all probability mass to c, firm i ears Eπ i (c, F i )=0. This cotradicts the hypothesis that ( F1, F 2,..., F ) is a Nash equilibrium. Case ( 2: By way of cotradictio, suppose that there exists a equilibrium F 1, F2,..., F ) such that for some firm i Eπ i ( F i, F i ) = π > 0 Defie: p i = if { } p i S F i p i = sup { } p i S F i Subcase A: Suppose that ( F1, F 2,..., F ) etails a positive probability that i ties for the lowest price at p i. The Eπ i (p i, F i )=π, ad by left lower semicotiuity ad Coditio B, there exists p i P where p i < p i such that ( ) Eπ i p i, F i >π. This cotradicts the hypothesis that ( F1, F 2,..., F ) is a Nash equilibrium. Subcase B: Suppose that ( F1, F 2,..., F ) etails a zero probability that i ties for the lowest price at p i. The, lim Eπ i (p, F ( i ) lim 1 F j (p) ) π(p) =π. p p i p pi Lettig π deote the upper boud o moopoly profits, it follows that π ( lim 1 F j (p) ) π. p pi j/=i j/=i Hece, the hypothesis that π > 0, implies that lim p pi Fj (p) < 1 for all j /= i : The implied equilibrium mixed strategies employed by firms other tha i are such that all firms j /= i allocate positive probability to prices which either have o possibility of wiig or, at best, offer some chace of a tie for lowest price at p i. Sice π (p) is left lower semicotiuous, some firm j /= i ca profitably deviate by re-allocatig the probability mass from prices at or above p i to prices just below p i. This cotradicts the hypothesis that ( F1, F 2,..., F ) is a Nash equilibrium. Obviously, Coditio B is ecessary for the existece of a zero operatig profit equilibrium (see Theorem 1). The followig sectio presets examples which highlight the role of the other assumptios i Theorem 2.

7 Wier-take-all price competitio Examples Our first two examples are based o a eviromet where two idetical, riskeutral, price-settig firms compete for the demad fuctio, q = D (p), ad have cost fuctios, C (q). The firm settig the lowest price captures the etire market ad ears the moopoly profits correspodig to that price: π (p) = D (p) p C (D (p)). I the evet of a tie, each firm has a equal probability of beig awarded the etire cotract. Figure 2. Discotiuous demad Example 1 (Discotiuous Demad) Two idetical firms produce at zero costs ad compete for a buyer whose demad fuctio (sketched i Figure 2) is give by { 8 p if 0 p < 2 D (p) = 4 p if 2 p 4. Each firm s strategy cosists of a price p i [0, 4]. I this case, the moopoly profit fuctio is give by π (p) = { p (8 p) if 0 p < 2 p (4 p) if 2 p 4. Notice that, as a result of the discotiuity i demad, π (p) is discotiuous, as show i Figure 3. Noetheless, the coditios of Theorem 2 are satisfied. To see this, ote first that Coditio B holds (lettig c =0,π(c) = 0; hece the iitial breakeve price is zero). Thus, by Theorem 1, we kow that a zero operatig profit equilibrium exists. Furthermore, ote i Figure 3 that π (p) is bouded ad left lower semicotiuous. By Theorem 2, we ca coclude that every Nash equilibrium to the game i Example 1 is, i fact, a zero profit equilibrium. The discotiuous demad i Example 1 does ot udermie the traditioal price udercuttig logic.

8 278 M.R. Baye ad J. Morga Figure 3. Discotiuous moopoly operatig profits due to discotiuous demad Example 2 (Discotiuous Costs) Two idetical firms face a market demad of D (p) =4 p ad have cost fuctios give by { q if 0 q 2 C (q) = q +1.5 if 2 < q 4. Figure 4. Discotiuous costs This cost fuctio is illustrated i Figure 4. Each firm s strategy cosists of a price p i [0, 4], with the firm chargig the lowest price earig { (p 1)(4 p) 1.5 if 0 p < 2 π (p) = (p 1)(4 p) if 2 p 4. Figure 5 illustrates the discotiuity i π (p) caused by the discotiuity i costs. It is clear that the moopoly operatig profit fuctio satisfies Coditio B. Thus, by Theorem 1, we coclude that a zero operatig profit equilibrium exists. 5 However, ote that the coditios of Theorem 2 fail, as π (p) is ot left lower semicotiuous. This leaves ope the possibility that, i additio to the 5 I particular, settig π (c) = 0 ad solvig yields c = Sice π (p) < 0 for all p < c, it follows that p 1 = p 2 = comprises a symmetric zero profit equilibrium.

9 Wier-take-all price competitio 279 Figure 5. Discotiuous moopoly operatig profits due to discotiuous costs zero profit equilibrium, there also exists a positive profit equilibrium. Ideed, this is the case; oe may easily verify that p 1 = p 2 = 2 costitutes a Nash equilibrium i which each firm ears expected profits of π i =1. 6 Here, the jump i costs associated with market expasio beyod 2 uits udermies the usual udercuttig argumet. Examples 1 ad 2 suggest that the impact of discotiuities i π (p) o wier-take-all price competitio critically depeds o whether the discotiuities are caused by discotiuities i demad or costs. These ifereces are, i fact quite geeral, as the followig remarks reveal. Remark 1 Over the rage where demad is o-icreasig ad price exceeds margial cost, demad discotiuities cause π (p) to jump dow whe the price rises above the poit of discotiuity i demad. Thus, discotiuities i demad ted to give rise to a moopoly operatig profit fuctio that is left lower semicotiuous. Provided there are o other discotiuities ad the profit fuctio is otherwise well-behaved, the coditios of Theorem 2 will hold. I short, with discotiuous but dowward slopig demad, wier-take-all price competitio teds to result i zero profit outcomes. Remark 2 If the productio possibilities set is closed ad mootoic, the the correspodig cost fuctio is lower semi-cotiuous ad o-decreasig i q (see Nadiri, 1982). Provided there are o other discotiuities, it follows that cost discotiuities lead to the failure of π (p) to be left lower semicotiuous. The geeral coclusio is that zero profit outcomes are by o meas assured i the presece of discotiuous costs. Thus, with discotiuities i demad, π (p) teds to be left lower semicotiuous. I cotrast, discotiuities i cost ivariably lead to the failure of left lower semicotiuity. Whether such a failure leads to positive profit equilibria depeds o the size of the upward jump i costs. I Example 2, for istace, the jump i costs of 1.5 is sufficietly severe to lead to positive profit equilibria. If the jump i costs had bee smaller (< 1), positive profit equilibria would ot arise. For a 6 For other wier-take-all tie-breakig rules, oe ca costruct similar examples where the failure of left lower semicotiuity results i positive profit equilibria.

10 280 M.R. Baye ad J. Morga larger ( 1) jump i costs, equilibria i which firms ear positive operatig profits emerge. Coclusios regardig the competitiveess of wier-take-all pricig games i settigs with discotiuous costs are, ievitably, model-specific. Our fial example illustrates the importace of bouded moopoly profits. Example 3 (Hotellig Models) Cosider the Hotellig model preseted by Gabszewicz ad Thisse (1992) i the Hadbook of Game Theory. Two firms produce a homogeeous product at zero cost ad are located at distaces a i from the edpoits of a lie of uit legth (a 1 + a 2 1; a i 0). Customers are uiformly distributed over this lie ad have trasportatio costs, T (x) = tx to visit a store that is distace x away. Cosumers have a perfectly ielastic demad for oe uit ad buy from the firm offerig the product at the lowest overall cost (price + trasportatio costs). Whe the two firms set prices p i [0, ), their payoffs are: π i (p i, p j )= ( ) 1 aj +a i ( ) 2 p i + 1 2t pi p j pi 2 if p i p j t (1 a 1 a 2 ) p i if p i < p j t (1 a 1 a 2 ) 0 otherwise (1) Propositio 1 i Gabszewicz ad Thisse asserts that for a 1 + a 2 = 1, the uique price equilibrium is give by p1 = p 2 =0. (p. 286). It is ot readily apparet that our theorems shed light o this issue. Note, however, that whe a 1 + a 2 =1, equatio (1) simplifies to a i p i if p i = p j π i (p i, p j )= p i if p i < p j. (2) 0 otherwise These payoff fuctios are isomorphic to those that arise i a wier-take-all pricig game where C (q) =0, D (p) = 1, ad the moopoly profit fuctio is π (p) = p for all p [0, ). 7 Sice moopoly profits are ubouded, the coditios of our Theorem 2 are ot satisfied. Thus, our theorem leaves ope the possibility that there exist positive profit equilibria i the Hotellig model. I fact, cotrary to the claim by Gabszewicz ad Thisse, positive profit equilibria do exist, as the followig theorem demostrates. Theorem 3 Whe a 1 + a 2 =1, the Hotellig model has a cotiuum of positive profit Nash equilibria. Proof. Fix some k (0, ), ad suppose that both firms price accordig to the cumulative distributio fuctio 7 While the tie-breakig rule we have assumed i our model gives equal weight to each firm beig the wiig firm, Gabszewicz ad Thisse model assumes that firm i has probability a i of beig the wier. Sice firms have zero costs, oe ca show that the results of our theorems exted to this case.

11 Wier-take-all price competitio 281 F(p) = { 0 if p k 1 k p if p > k Notice this is a well defied, atomless probability distributio o [k, ), as F(k) =0,F( ) =1,adF (p) > 0 for all p [k, ). By symmetry, it is sufficiet to show that if firm j adopts F as its strategy, firm i caot gai by choosig a strategy differet from F. Suppose firm j chooses a price accordig to F. Iffirmi charges p i, the with probability F(p i ) firm j s realized price is less tha p i. By equatio (1) firm i ears zero profits i this case. With probability [1 F(p i )] firm j s price exceeds p i, ad i this evet firm i ears profits of p i. Thus, the expected profits that firm i ears by chargig p i whe the rival prices accordig to F is Eπ i (p i )=[1 F(p i )]p i. (Sice F is atomless, we ca igore ties; hece the tie-breakig rule is irrelevat here.) Usig the defiitio of F, Eπ i (p i )=k for p i [k, ). This meas that firm i ears the same expected profits, amely k, by pricig at or above k. Iffirm i sets a price that is strictly less tha k, it is certai to wi the etire market, but the correspodig profits are strictly less tha k. Sice firm i s profits are costat ad equal to k for each p i [k, ), ad strictly less tha k for p i < k, ay p i [k, ) is a best respose by firm i to firm j s strategy, F. Sice F allocates all probability i the iterval [k, ), firm i ca do o better tha to price accordig to F. Thus, F costitutes a symmetric Nash equilibrium i which each firm ears positive expected profits of k. Sice this costructio holds for all k (0, ), there exists a cotiuum of positive profit equilibria. Theorem 3 thus establishes that, i additio to the well-kow zero-profit equilibrium to the Hotellig model where firms have the same locatio, there also exists a cotiuum of positive profit Nash equilibrium payoffs. This game satisfies all of the coditios of our Theorem 2 except for bouded moopoly profits. Abset the assumptio of bouded payoffs, oe caot i geeral rule out positive profit equilibria i wier-take-all pricig games. 8 Oe ca show that the costructio give i Theorem 3 is robust to alterative specificatios of trasportatio costs ad distributios of cosumers. 4 Coclusio How competitive are wier-take-all pricig games? Our results reveal that the aswer is itimately related to the properties of the moopoly profit fuctio. Theorem 1 shows that a zero profit equilibrium exists if ad oly if the moopoly profit fuctio possesses a iitial break-eve price. Theorem 2 shows that two additioal coditios o moopoly profits are required to guaratee that all equilibria etail zero profits. The examples show that the coditios i Theorem 2 are tight, i the sese that if moopoly profits ot bouded (Example 3) or 8 Baye ad Morga (1999) set forth geeral coditios for the existece of a cotiuum of positive profit equilibria i homogeeous product pricig games.

12 282 M.R. Baye ad J. Morga ot left lower semicotiuous (Example 2), positive profit equilibria ca arise. Remarks 1 ad 2 suggest that cost discotiuities are more likely to lead to ocompetitive outcomes tha are demad discotiuities. The usual argumet used to guaratee the zero profit outcome requires that, i the evet of a tie, a firm beefits by udercuttig its rivals. The weak form of cotiuity i Defiitio 2 is eeded to geeralize this reasoig to cover cases where demad or costs are discotiuous. Amog other thigs, it guaratees that udercuttig the rival s price domiates acceptig ay tie outcome where firms ear positive operatig profits. Refereces Baye, M., Tia, G., Zhou, J.: Characterizatios of the existece of equilibria i games with discotiuous ad o-quasicocave payoffs. Review of Ecoomic Studies 60, (1993) Baye, M., Morga, J.: A folk theorem for oe-shot Bertrad games. Ecoomics Letters 65, (1999) Brems, H.: A discotiuous cost fuctio. America Ecoomic Review 42, (1952) Dasgupta, P., Maski, E.: The existece of equilibrium i discotiuous ecoomic games, I: Theory. Review of Ecoomic Studies LIII, 1 26 (1986) Dastidar, K. G.: O the existece of pure strategy Bertrad equilibrium. Ecoomic Theory 5, (1995) Friedma, J. W.: Duality priciples i the theory of cost ad productio revisited. Iteratioal Ecoomic Review 13, (1972) Gabszewicz, J.J., Thisse, J.-F.: Locatio. I: Auma, R.J., Hart, S. (eds.) Hadbook of game theory. Amsterdam: North Hollad 1992 Harrigto, J.: A re-evaluatio of perfect competitio as the solutio to the Bertrad price game. Mathematical Social Scieces 17, (1989) Klemperer, P.: Applyig auctio theory to ecoomics. Mimeo, Oxford Uiversity (2000) Milgrom, P., Roberts, J.: The ecoomics of moder maufacturig: techology, strategy, ad orgaizatio. America Ecoomic Review 80, (1990) Milgrom, P., Roberts, J.: Comparig equilibria. America Ecoomic Review 84, (1994) Nadiri, M.: Producers theory. I: Arrow, K., Itriligator, M. (eds.) Hadbook of mathematical ecoomics Vol. II. Amsterdam: North Hollad 1982