Winning in Retail Market Games: Relative Profit and Logit Demand

Transcription

1 205 IEEE Symposum Seres on Computatonal Intellgence Wnnng n Retal Market Games: Relatve Proft and Logt Demand Jasper Hoogland CWI Amsterdam and TU Delft Emal: hoogland@cw.nl Mathjs M. de Weerdt TU Delft Emal: M.M.deWeerdt@tudelft.nl Han La Poutré CWI Amsterdam and TU Delft Emal: han.la.poutre@cw.nl Abstract We examne retalers that maxmze ther relatve proft, whch s the (absolute) proft relatve to the average proft of the other retalers. Customer behavor s modelled by a multnomal logt (MNL) demand model. Although retalers wth low retal prces attract more customers than retalers wth hgh retal prces, the retaler wth the lowest retal prce, accordng to ths model, does not attract all the customers. We provde frst and second order dervatves, and show that the relatve proft, as a functon of the retaler s own prce, has a unque local maxmum. Our experments show that relatve proft maxmzers beat absolute proft maxmzers,.e. they outperform absolute proft maxmzers f the goal s to make a hgher proft. These results provde nsght nto market smulaton compettons, such as the Power TAC. I. INTRODUCTION In some stuatons, retalers (also referred to as frms) have the ncentve to wn, by makng a hgher proft than other retalers, rather than to maxmze ther own proft. In the lterature, several reasons for ths are suggested. Frst, managers of frms may be rvalrous by nature []. Second, frms wth an ncentve to maxmze ther proft n the long term may sacrfce ther proft n the short term, n order to gan a larger market share [2]. Ths way, other frms are pushed out of the market and new frms are prevented from enterng, eventually to the beneft of the frm s proft. Another possblty s that managers are pad accordng to ther performance relatve to other managers [], [2]. Fnally, n tournaments where the partcpants are ranked by ther proft (e.g. [3], [4]), the ncentve s to make more proft than others, rather than to maxmze one s own proft. If the goal s to wn, frms are known to perform better f they maxmze ther relatve proft nstead of ther (absolute) proft [5]. Relatve proft s the absolute proft mnus the average absolute profts of the other frms (possbly weghted by some factor). Though relatve proft maxmzaton s not necessarly the same as wnnng, t s ntutve that relatve proft maxmzers, by takng nto account the other frms profts, outperform absolute proft maxmzers f the goal s to wn. In ths paper, we study relatve proft maxmzers n retal markets. To model customer behavor, we use the multnomal logt (MNL) demand model [6], whch s among the most wdely used stochastc demand models (see Secton II). It has the desred property that customers do not necessarly buy a product at the lowest possble prce, though the probablty of buyng a product s hgher f ts prce s lower. There are several reasons why a customer may buy a more expensve product, despte the avalablty of a cheaper alternatve. They may do ths ether ntentonally, e.g. f they beleve the more expensve product s better, or unntentonally, e.g. f they are unaware of the cheaper alternatve. In addton, the MNL model has the mathematcally elegant property that the rato between the probabltes of choosng among two products s ndependent of the probabltes of buyng other products. Due to ts desred propertes, MNL demand s a useful model to study n combnaton wth relatve proft. In partcular, an nterestng applcaton of MNL demand and relatve proft maxmzaton s the Power TAC competton [4]. The partcpants of ths tournament create retaler agents that trade electrcty n a smulaton of a smart grd electrcty market. Customers n ths smulaton behave accordng to the MNL model. The wnner of the competton s the retaler wth the hghest proft, so partcpants have the ncentve to create relatve proft maxmzers. To our knowledge, relatve proft maxmzers have not yet been studed n combnaton wth the MNL demand model, despte ts desred propertes. The goal of ths paper s to provde ths connecton. The next secton descrbes prevous work on relatve proft and MNL demand. In Secton III, we formally descrbe the MNL demand model, and we ntroduce the concept of relatve proft. In Secton IV, we gve the frst and second order dervatves of the relatve proft wth respect to the own prce, and we show that relatve proft has a unque local maxmum. We also show that, unlke absolute proft maxmzers, a relatve proft maxmzer may, under certan condtons, have a retal prce lower than the cost prce. Furthermore, we show ths retal prce may be part of a Nash equlbrum. We gve necessary and suffcent condtons under whch a relatve proft maxmzer s more compettve than an absolute proft maxmzer n the same stuaton. As opposed to what prevously has been assumed, condtons exst under whch a relatve proft maxmzer s less compettve. Secton V descrbes and dscusses the experments we performed n order to examne the performance of relatve proft maxmzers f the goal s makng more proft than other retalers. The results suggest that relatve proft maxmzers outperform absolute proft maxmzers n all settngs. Fnally, n Secton VI, we summarze and conclude. II. RELATED WORK Though relatve proft maxmzaton has not been studed as extensvely as absolute proft maxmzaton, a consderable number of papers have been publshed over the years. Bshop /5 $ IEEE DOI 0.09/SSCI

2 [7] suggests that relatve proft maxmzaton as a warfare strategy s a plausble outcome of duopoly market games where the frms compete on quantty. Shubk and Levtan [8] descrbe beat-the average games. Lundgren [] suggests that colluson can be prevented by nstallng ncentves to maxmze relatve proft. Donaldson and Neary [9] show that, f relatve proft ncentves are mposed, effcency s acheved n a socalst ndustry. Matsumura et al. [0] examne relatve proft maxmzers n the context of R&D expendture. Several papers have shown that, f the goal s to have a hgher proft than other frms, relatve proft maxmzers outperform absolute proft maxmzers. Jones [3] descrbes a classroom game where frms compete on quantty, and the frm wth the hghest proft wns. In ths game, successful players am for relatve proft maxmzaton. In an evolutonary settng, Schaffer [5] and Vega-Redondo [] show that, n Cournot-models, a frm outperforms another frm by unlaterally devatng from the Cournot equlbrum. By dong so, the frm decreases ts own proft, but decreases the other frm s proft even more. It s shown that games wth only relatve proft maxmzers result n the Bertrand equlbrum, n whch frms sell ther products at the cost prce. Rechmann [2] analyzes a Cournot model n whch the frms are a mx of absolute and relatve proft maxmzers, and shows that the equlbrum les between the Cournot and Bertrand equlbra. All the work mentoned so far uses Cournot models, n whch the market prce s determned by the total output of all frms. In such models, all frms wth a non-zero output sell ther products at ths prce. However, n many stuatons ths s unrealstc, because frms can have dfferent retal prces and stll have a non-zero output. A few exceptons, though, exst n the lterature. Frst, Appendx C n the paper by Lundgren [] analyzes a case wth heterogeneous products wth dfferent prces. It s shown that relatve proft maxmzers are more compettve than absolute proft maxmzers, provded that both frms have a postve prce-cost margn. Besdes ths concluson, however, no extensve analyss s provded. Second, Nakamura and Sato [2] use a duopoly model allowng dfferent prces, but ther paper manly focusses on capacty choce, whch s not drectly related to our problem. Fnally, Satoh and Tanaka [3] analyze relatve proft maxmzaton for an olgopoly market that allows dfferent prces. They show the exstence of pure strategy equlbra, but they do not address the ssue of outperformng other frms. To our knowledge, our paper studes relatve proft for MNL demand for the frst tme. Ths model, as opposed to Cournot models, has the desred property that frms can have dfferent prces and stll have a non-zero output. For ths model, we derve some propertes that have not been reported for other models, such as the possblty of prces lower than the cost prce and stuatons n whch relatve proft maxmzers are less compettve than absolute proft maxmzers. Furthermore, we study the performance of relatve proft maxmzers f the goal s to outperform each other. Despte ther absence n the relatve proft lterature, MNL demand models [6] are common models for customer behavor n the economcs lterature. In most of ths research, frms are assumed to be absolute proft maxmzers [4] [20]. Some research has been conducted, though, on frms wth dfferent objectve functons (e.g. rsk averse frms [2]). III. RETAIL MARKET MODEL Here we descrbe the MNL demand model [6], whch expresses the demand of customers as a functon of the retal prces of the retalers n the market. We also ntroduce the concept of relatve proft. Though more research has been conducted on absolute proft, n some stuatons, relatve proft maxmzers outperform absolute proft maxmzers. In partcular, f retalers have the goal to have a hgher proft than other retalers, then relatve proft maxmzers perform better than absolute proft maxmzers. A. MNL Demand The MNL demand model s a stochastc choce model, whch means the choce of a customer s sampled from a probablty dstrbuton. A customer chooses from a fnte set of products or chooses not to buy any of them. The latter s called the ext opton, and t means that the customer ether buys elsewhere or does not buy at all. The MNL model assgns a probablty to every opton. The model can also be used to model a group of customers wth the same choce model. In that case, the MNL demand s nterpreted as the proporton of customers that buy the product. In our model, every retaler sells a sngle type of product. These products are homogeneous,.e. all products have the same value to the customer, and preferences are only determned by the retal prces. Although cheaper products are bought more often than more expensve products, customers do not only buy the cheapest product. Furthermore, the rato between the demand of two products s ndependent of the demand of other products. We now gve an expresson for the MNL demand. Let n be the number of retalers and let p = p,...,p n be the vector of retal prces of the retalers products. The probablty q ( p) that a customer buys from retaler s e λp q ( p) = e λp0 + n, () e λpj j= where λ>0 and p 0 > 0 are parameters of the model. Note that 0 <q ( p) <. The parameter λ determnes the amount of stochastcty. In the extreme case of λ =0, the customer s choce has a unform dstrbuton, regardless of the retal prces. The other extreme, λ =, s the determnstc case, where the customer buys the cheapest product wth probablty. The parameter p 0 specfes the prce of the ext opton. Dependng on the applcaton, customers that choose the ext opton buy the product elsewhere, or they do not buy t at all. B. Absolute and Relatve Proft Here, we gve a formal defnton of relatve proft. We assume each retaler has a constant margnal cost c. The absolute proft π of retaler s π ( p) =(p c )q ( p). (2) The relatve proft ρ α ( p) of retaler s defned as ρ α ( p) =π ( p) α π k ( p). (3) The parameter α specfes the degree of relatvty. It can have any real value (even less than 0 for colludng retalers). In 795

3 ths paper we assume α 0. Three specal cases are to be dstngushed. Frst, f α =0, then ρ α ( p) concdes wth the absolute proft. Second, f α = then ρα ( p) s the retaler s own proft mnus the average proft of the other retalers. If the goal s to maxmze relatve proft wth α =, then the game s zero-sum, because n = ρ ( p) =0. Fnally, f α =, then an ncrease of another retaler s proft has the same mpact on ρ α ( p) as a decrease by the same amount of the retaler s own proft. In some papers, the defnton of relatve proft s restrcted to α = (e.g. [], [3]), whle others use the same defnton as ours (e.g. [0], [2]). For a more general verson of relatve proft, we refer to Koçkesen [22]. IV. ANALYSIS p player 2 α =0 player 2 α = player α =0 player α = Here we gve some useful propertes of the MNL demand model, the absolute proft, and the relatve proft. The dervatve of the demand q ( p) wth respect to p s q q = λq ( p)( q ( p)). (4) Note that < 0, so the hgher the retal prce, the lower the probablty that the customer buys from. Furthermore, for k, the dervatve of q k ( p) wth respect to p s q k = λq k ( p)q ( p). Ths mples q k > 0, so the hgher the retal prce of one retaler, the hgher the probablty that the customer buys from another retaler. The dervatve of the absolute proft wth respect to p s [ ] π = q ( p) λ(p c )+λπ ( p). The dervatve of the proft of retaler k wth respect to the prce p of another retaler k s π k = λq ( p)π k ( p). Hence, the frst dervatve of the relatve proft s ρ α [ ] = q ( p) λ(p c )+λρ α ( p). (5) The second dervatve of the relatve proft of retaler s 2 ρ α [ ] p 2 = λ (2q ( p) ) ρα q ( p). (6) The followng proposton shows that the relatve proft has exactly one maxmum. Proposton : Let 0 α. Then, the relatve proft ρ α ( p) has exactly one maxmum for p, vz. for whch ρα = 0. Proof: Frst, we show a statonary pont exsts, then we show ths pont s a unque local maxmum. By substtuton of Equatons (2) and (3) nto Equaton (5), any p satsfyng λ(p c )( q ( p)) = λα π k ( p) Fg.. Ths dagram shows the best responses of both players n 2 player games. The parameters of the games are p 0 =3and λ =. The cost prce s c =for both players. For each player, the best response s shown as a functon of the other player s prce for absolute (α =0) and relatve (α =) proft. s a statonary pont of the relatve proft. The left hand sde of ths equaton approaches 0 f p, and approaches f p. The rght hand sde approaches f p, and approaches some real number f p. Snce both sdes of the equaton are contnuous, there s at least one p that satsfes the equaton, and hence s a statonary pont of the relatve proft. By Equaton (6), ρα p =0mples 2 ρ α p 2 = λq ( p) < 0, so any statonary pont s a local maxmum. Moreover, snce all statonary ponts are local maxma, there s at most one local maxmum. Fgure shows the best responses for two retalers n duopoly games. Nash equlbra exst where the graphs of retalers and 2 ntersect. For Cournot games, t has been reported that relatve proft maxmzers wth α = result n the Bertrand equlbrum [5], [],.e. they have retal prces equal to the cost prce. Ths means equlbrum prces are lower for relatve proft maxmzers than for absolute proft maxmzers. The next proposton shows a smlar property for MNL demand duopoly markets for retalers wth the same cost prces. Proposton 2: For duopoly markets (n =2) wth p 0 = n whch the two retalers are relatve proft maxmzers wth the same α = α and the same cost prce c = c for {, 2}, a symmetrc Nash equlbrum exsts such that, for both retalers {, 2}, 2 p c = λ( + α). (7) 796

4 Proof: Snce p = p 2, t holds that q ( p) =q 2 ( p) = 2. Therefore, by Equaton 5, t holds that ρ α =0 (p c) π ( p)+απ k ( p) = λ 2 p c = λ( + α) for, k {, 2}, k. Hence, by Proposton, Equaton (7) for {, 2} s a Nash equlbrum. From Proposton 2, under the condtons assumed by ths proposton (e.g. the same cost prce for both retalers), t follows that equlbrum prces are lower for relatve proft maxmzers (α > 0) than for absolute proft maxmzers (α = 0), because p decreases wth respect to α. Unlke Cournot games, however, the retal prces are stll hgher than the cost prce. The retal prces only approach the cost prce for hgh values of λ. Under dfferent condtons, though, relatve proft maxmzers have retal prces that are even lower than the cost prce. For example, f a relatve proft maxmzer has a cost prce much hgher than the cost prce of another retaler, then the relatve proft maxmzer may accept a negatve proft, n order to prevent the other retaler from havng a hgh market share and, consequently, a hgh proft. Moreover, there are games for whch a Nash equlbrum exsts such that the retal prce of one of the retalers s lower than the cost prce. Ths s shown by the followng proposton. Proposton 3: There are games wth relatve proft maxmzers, n whch a Nash equlbrum exsts such that one of the retalers has a retal prce lower than the cost prce. Proof: We gve an example of such a game wth two relatve proft maxmzers wth α =. Let the parameters of the MNL demand model be λ =and p 0 =, and let the cost prces be c =5and c 2 =. Then, p = p 2 =4s a Nash equlbrum. By Equaton (5), f p 2 =4, then p =4s a statonary pont of the relatve proft ρ α ( p) of retaler, and by Proposton, t s the optmal value. Smlarly f p =4, then p 2 =4maxmzes the relatve proft ρ α 2 ( p) of retaler 2. Thus, p = p 2 =4s a Nash equlbrum. In ths equlbrum, the retal prce p =4of retaler s lower than the cost prce c =5. A retaler s more compettve than another retaler f t has a lower retal prce than the other retaler n the same stuaton. Retaler n Proposton 3 s more compettve than an absolute proft maxmzer, because absolute proft maxmzers never have retal prces lower than the cost prce. In addton, for n = 2 retalers and α =, Lundgren has shown that a relatve proft maxmzer s more compettve than an absolute proft maxmzer, provded that both retalers have postve prce-cost margns [] (Appendx C). For MNL demand and arbtrary n 2 and α > 0, t can be shown that a relatve proft maxmzer s more compettve than an absolute proft maxmzer f the sum of the profts of the other retalers s postve. However, f ths sum s negatve, then the opposte holds. In that case a relatve proft maxmzer s less compettve than an absolute proft maxmzer. Both cases are covered by Proposton 4. Frst, we prove the followng property of MNL demand. Property : The sgn of k π k( p) does not depend on p. Proof: By Equatons () and (2), the sum of the profts of the other retalers k s π k ( p) = (p k c k )q k ( p) = n j=0 e λpj (p k c k )e λp k. Though k π k( p) depends on p, whether or not t s postve s ndependent of p. Ths holds, because n j=0 e λp j s always postve, so the sgn of k π k( p) only depends on k (p k c k )e λp k, whch s ndependent of p. Due to Property, we can speak of k π k( p) > 0 regardless of p. Proposton 4: Gven the prces p,...,p,p +,...,p n of the other retalers, the retal prce of retaler s lower for a relatve proft maxmzer (wth α 0) than for an absolute proft maxmzer f and only f the sum of the absolute profts of the other retalers s postve. Proof: Let p a be the retal prce of an absolute proft maxmzer (α =0), let p a = p,...,p,p a,p +,...,p n be the retal prce vector wth p = p a, and let qa = q ( p a ) be the demand gven ths prce. Smlarly, let p r be the retal prce of a relatve proft maxmzer for some α>0, let p r = p,...,p,p r,p +,...,p n be the retal prce vector wth p = p r, and let qr = q ( p r ) be the demand gven ths prce. Thus, our goal s to prove that p r <p a π k ( p) > 0. (9) (8) By Proposton, ρα =0for p = p r, and ρ0 =0for p = p a. Therefore, by substtuton of Equatons (2) and (3) nto Equaton (5), t holds that λ(p r c )( q r )= λα π k ( p r ), (0) wth p r = p,...,p,p r,p +,...,p n, and, by choosng α =0, λ(p a c )( q a )=. () Note also that p a >c holds by Equaton () and 0 < q a <. Frst, we assume p r c. Then, p r <p a and, by Equaton (0), k π k( p) > 0, so Equaton (9) follows from p r c. We now assume p r > c. Then, for p n the nterval between p r and p a, both (p c ) and ( q ( p)) are postve and ncrease monotoncally wth respect to p (By Equaton (4)), so ther product (p c )( q ( p)) also ncreases monotoncally wth respect to p n ths nterval. Therefore, p r <p a (p r c )( q r ) < (p a c )( q a ). Furthermore, by equatons () and (0), t holds that π k ( p) > 0 (p r c )( q r ) < (p a c )( q a ). 797

5 Therefore, Equaton (9) follows from p r >c. Thus, regardless of p r > c, Equaton (9) holds, so Proposton 4 has been proven. V. EXPERIMENTS We performed experments n order to examne the performance of relatve proft maxmzers f ther goal s to wn,.e. to outperform each other. We evaluate retalers by computng ther relatve performance S, whch we defne as the retaler s proft dvded by the hghest proft of the other π retalers: S = max k π k. In addton to expressng whether a retaler has the hghest proft (S >), the relatve performance expresses, n case of a vctory, how much better a retaler s than the second best retaler, or, n case of a loss, how close the retaler s to the wnner. A. Settngs We analyzed retal market games wth 3 players and wth 4 players. We tested best response strateges for relatve proft wth α-values n the nterval [0, ]. For the 3 player games, we used step sze 0.0 for α, resultng n 0 dfferent relatve proft maxmzers. For the 4 player games, we used step sze 0.02 for α, resultng n 5 dfferent relatve proft maxmzers. We ran games for all combnatons of retaler strateges. Snce we examne relatve proft maxmzers, a strategy corresponds to a choce for α. Every game conssted of at most 30 teratons. In each teraton, the retalers choose the best-response retal prce gven the retal prces of the prevous teraton. The game s termnated f the retal prces have converged before the last teraton. As parameter values, we chose λ {, 2}, p 0 {2, 4}, and the same unt cost prce c =for all retalers. The best response relatve proft maxmzers were mplemented n Java usng gradent ascent. As ntal prce for the gradent ascent algorthm, we chose p nt =2. For each game, we recorded the number of teratons and, for each player, the fnal retal prce and proft. B. Results In all games, the retal prces converged before the last teraton. Ths provdes emprcal evdence for the exstence of a Nash equlbrum. We tred dfferent values for the parameters λ and p 0, but ths had no major mpact on the results. Here, we present the results for the games wth λ =and p 0 =4. For each game n the experments, we computed the π relatve performance max k π k. The results of the experments are summarzed n Fgure 2. For each α, the best case and worst case relatve performances over all games are shown. In games wth 3 players, the best case and worst case relatve performances have a peak at α =0.5. Around ths pont, the worst case relatve performance s approxmately, so, at worst, these retalers are never far from havng the hghest proft. The hghest relatve performances were acheved n games aganst opponents wth α-values close to ether 0 or. In these games the proft was up to 5.95% hgher than the opponents profts. Fgure 3 shows the relatve performance of a retaler wth α = 0.5 aganst all combnatons of opponent strateges. In games wth 4 players, the best case and worst case relatve performances have peaks around α = 0.32 and α =0.34 respectvely. For the retaler wth α =0.34, the worst case relatve performance s approxmately, so, at worst, ths retaler s never far from havng the hghest proft. The hghest relatve performances were acheved n games aganst opponent retalers wth α-values close to, n whch the relatve proft maxmzer wth α = 0.32 had a proft up to 8.72% hgher than the opponents profts. Ths s not a surprse, as these α-values are the furthest from the best performng retalers. Hgh performance s also acheved n games aganst absolute proft maxmzers, n whch the relatve proft maxmzer wth α =0.32 had a proft up to 3.50% hgher than the opponents profts. The good performance of the retaler wth α = 0.5 n 3-player games and the retalers wth α =0.32, 0.34 n 4- player games can be explaned as follows. If the goal s to outperform other retalers, then t s a zero-sum game, and an ncrease of the retaler s absolute proft s equally benefcal as a decrease of all opponents absolute profts by the same amount. Ths s acheved by optmzng the retaler s own proft mnus the average proft of the other retalers, whch s equvalent to maxmzng relatve proft wth α =. Though extrapolaton to games wth n>4 players seems lkely, more experments are requred to prove ths. The worst case performances of the best strateges (α = 2 for 3 players and α =0.32, 0.34 for 4 players) were slghtly lower than.0, so they dd not wn n all stuatons. At fve dgts after the comma (e.g. S = for α = 2 3 player games, and for α = 0.34 n 4 player games), we found that the relatve performance was slghtly less than n some stuatons. In games n whch ths occurs, retaler has α =, retaler 2 has an α-value slghtly hgher than, and other retalers have α-values close to 0 or (e.g. n the 3 player game wth α 2 =0.52 and α 3 =0). These results can be explaned as follows. In these games, only the strateges of retalers and 2 are good enough to compete for the hghest proft, and the other retalers can be gnored. Therefore, an α- value slghtly hgher than performs better than α = n these cases. VI. CONCLUSIONS We analyze retalers that maxmze ther relatve proft for the MNL demand model. We derve frst and second order equatons of the relatve proft, and show that relatve proft has a unque local maxmum wth respect to the retaler s own prce. We also show that, under certan condtons, the optmal retal prce of a relatve proft maxmzer s lower than the cost prce. In addton, we gve a necessary and suffcent condton n whch a relatve proft maxmzer s more compettve than an absolute proft maxmzer. Ths holds f and only f the sum of the profts of other retalers s postve. We performed experments n order to examne the relaton between relatve proft maxmzaton and wnnng. We smulated games wth 3 and 4 players wth dfferent combnatons of α-values n the range of [0, ]. The results show that retalers wth α = 2 have a hgh performance n games wth 3 players, and retalers wth α = 3 have a hgh performance n games wth 4 players. Furthermore, by extrapolatng to games wth n>4 retalers, these experments suggest that retalers outperform others f 798

6 Relatve performance worst case best case α α Relatve performance α (a) 3 players worst case best case α (b) 4 players Fg. 2. Relatve performance of retaler for dfferent values of α. Relatve performance s defned as the retaler s proft dvded by the maxmum of the profts of the other retalers n the game. For each α, the graphs show the best and the worst relatve performance of the retaler over all games. they maxmze the dfference between ther absolute proft and the average proft of the other retalers (α = ). An applcaton of relatve proft maxmzaton to MNL demand s the Power TAC competton. The Power TAC s a smart grd electrcty market smulaton n whch customers behave accordng to the MNL model, and partcpants have the ncentve to outperform others rather than to maxmze ther own proft. In order to deal wth the complexty of the smulaton, partcpants agents have used computatonal ntellgence technques, such as renforcement learnng [23], [24], partcle swarm optmzaton [25], and other adaptve strateges [26] [28]. However, none have so far addressed the ssue of relatve proft maxmzaton. The nsghts provded by our analyss can be used to enhance computatonal ntellgence-based Power TAC agents. π Fg. 3. Ths dagram shows the relatve performance of retaler max k 2,3 π k wth α = 2 n games wth 3 players. The axes show the α j-values of the other retalers j =2, 3. The dagram shows that retaler has the hghest relatve performance n games n whch the other retalers have α j -values close to ether 0 or. REFERENCES [] C. Lundgren, Usng relatve proft ncentves to prevent colluson, Revew of Industral Organzaton, vol., no. 4, pp , 996. [2] T. Rechmann, Mxed motves n a cournot game, Economcs Bulletn, vol. 4, no. 29, pp. 8, [3] M. Jones, Note on olgopoly: Rval behavor and effcency, The Bell Journal of Economcs, pp , 980. [4] W. Ketter, J. Collns, P. P. Reddy, and M. de Weerdt, The 205 power tradng agent competton, ERIM Report Seres Reference No. ERS LIS, February 205. [5] M. E. Schaffer, Are proft-maxmsers the best survvors?: A darwnan model of economc natural selecton, Journal of Economc Behavor & Organzaton, vol. 2, no., pp , 989. [6] D. McFadden, The revealed preferences of a government bureaucracy: Theory, The Bell Journal of Economcs, pp , 975. [7] R. L. Bshop, Duopoly: Colluson or warfare? The Amercan Economc Revew, pp , 960. [8] M. Shubk and R. Levtan, Market structure and behavor. Harvard Unversty Press Cambrdge, MA, 980. [9] D. Donaldson and H. Neary, Decentralzed control of a socalst ndustry, Canadan Journal of Economcs, pp. 99 0, 984. [0] T. Matsumura, N. Matsushma, and S. Cato, Compettveness and r&d competton revsted, Economc Modellng, vol. 3, pp , 203. [] F. Vega-Redondo, The evoluton of walrasan behavor, Econometrca: Journal of the Econometrc Socety, pp , 997. [2] Y. Nakamura and M. Sato, Capacty choce n a mxed duopoly: The relatve performance approach, Theoretcal Economcs Letters, vol. 3, no. 02, p. 24, 203. [3] A. Satoh and Y. Tanaka, Choce of strategc varables under relatve proft maxmzaton n asymmetrc olgopoly, Economcs and Busness Letters, vol. 3, no. 2, pp. 5 26, 204. [4] S. P. Anderson and A. De Palma, The logt as a model of product dfferentaton, Oxford Economc Papers, pp. 5 67, 992. [5] F. Bernsten and A. Federgruen, A general equlbrum model for ndustres wth prce and servce competton, Operatons research, vol. 52, no. 6, pp , [6] D. Besanko, S. Gupta, and D. Jan, Logt demand estmaton under compettve prcng behavor: An equlbrum framework, Management Scence, vol. 44, no. -part-, pp , 998. [7] A. Capln and B. Nalebuff, Aggregaton and mperfect competton: 799

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