The Robustness of a Nash Equilibrium Simulation Model

Transcription

1 8th World IMACS / MODSIM Congress, Carns, Australa 3-7 July 2009 htt://mssanz.org.au/modsm09 The Robustness of a Nash Equlbrum Smulaton Model Etaro Ayosh, Atsush Mak 2 and Takash Okamoto 3 Faculty of Scence and Technology, Keo Unversty, Kanagawa, Jaan 2 Faculty of Busness and Commerce, Keo Unversty, Tokyo, Jaan 3 Graduate School of Engneerng, Chba Unversty, Chba, Jaan Emal: mak@fbc.keo.ac.j Abstract: Ayosh and Mak (2009) roosed a Nash equlbrum model alyng contnuous tme relcator dynamcs to the analyss of olgooly markets. Ths aer consdered a game roblem under the smultaneous constrants of the allocaton of roduct and market shares. The model assumes that a Nash equlbrum soluton can be aled and derved the gradent system dynamcs that can attan the Nash equlbrum soluton wthout volatng the smle constrants. Models assumed that a mnmum of three frms est wthn a market, and that these frms behave to mamze ther rofts, defned as the dfference between sales and cost functons wth conjectural varatons. Before conductng an emrcal analyss based on observatons of olgooly markets n the real world, we have to assess the robustness of the Nash equlbrum model by changng roft and cost functon arameters, as well as the ntal roducton values and market shares of the frms. Ths s necessary n order to assess how well observatons n the real world match those forecasts by the model. When the model s fragle, no olcy mlcatons could be etracted from the model. The aer consders dfferences of the converged values n the number of frms ncluded n the model, n the numbers of the commodtes ncluded n the model, n the secfcaton of frms roft and cost functons, and n the ntal values for the level of roducton and market share. The aroach facltates understandng of the robustness of attanng equlbrum n an olgooly market. Keywords: Nash equlbrum, relcator dynamcs, olgooly 00

2 . INTRODUCTION Olgooly markets reval n develoed countres n both the ndustral and servce sectors. Dt (988) analyze the U.S. automoble ndustry. Kleer (990) analyze the arlne ndustry. In ths study, we roose a smle Nash equlbrum model and use a smulaton method to derve an otmal soluton for roducton decsons by rval frms n olgooly markets. Ayosh and Mak (2009) roosed a Nash equlbrum model that ales contnuous tme relcator dynamcs to the analyss of olgooly markets. In ths aer, we consdered a game roblem under the smultaneous constrants of allocaton of roduct and market share. Before conductng emrcal analyss usng observaton on olgooly markets n the real world, we have to check the robustness of the Nash equlbrum model by changng the arameters of frms roft and cost functons as well as the ntal values of the amount of roducton and frms market share. Ths rocess s necessary to conduct forecastng and smulaton usng real-world observatons after estmatng the model. We sometmes obtan unrealstc solutons as a result of the fraglty of the model. In these cases, no olcy mlcatons can be drawn. The aer s organzed as follows. Secton 2 rovdes a general elanaton of the double constrant nterference model, whch concentrates on the double allocaton roblem of roducton caacty and market share. Secton 3 ntroduces a normalzed Nash equlbrum soluton for the roft mamzaton of layers functons modeled n the double constrant nterference model. Secton 4 descrbes the alcaton of numercal methods to the Nash equlbrum soluton. Secton 5 rooses a smulaton model and reorts the results. Secton 6 resents the conclusons. 2. NONCOOERATIVE NASH EQUILIBRIUM MODEL AND RESOURCE ALLOCATION Consder a contnuous game roblem wth layers and N strategy varables, governed by dulcate smle constrants. The th layer s strategy varables are = (,, ) N N R and =,, N, =,,. The varable matr that contans all varables s = (,, ) = = N N N () (where, =,, are column vectors; and, =,, N are row vectors). Let the th layer s roft functon be E ( ). An unconstraned game roblem s formulated as ma (,,, ), =,, E, (2) where s the th layer s only known varable and the other layers varables (,,, + )are unknown arameters. Consder the game roblem constraned by the smultaneous allocaton of roducts and market share as ma E (,,, ) (3a) N,, b = = subj. to = a, =, =, =,, N, (3b) 0, =,, N, =,, (3c) where, by defnton, N a = b. = = (4) As an eamle, let frms roduce N tyes of roducts n a market; reresents the number of frms, and reresents a roduct tye. The roblem, reresented by Eq. (3), s called a game roblem wth double allocaton constrants, n whch the allocaton of roducton ablty and market share s consdered smultaneously. The roertes of Nash equlbrum solutons, whch are assumed to be ratonal solutons for non-cooeratve game roblems, dffer deendng on ther equalty constrants. 0

3 3. THE NORMALIZED NASH EQUILIBRIUM SOLUTION FOR THE CONSTRAINT INTERFERENCE ROBLEM The Nash equlbrum soluton for the nterference-tye roblem s denoted by Eq. (3), wth double smle constrants. In ths case, statonary condtons for each layer do not est, unlke n the constrant ndeendent tye roblem. A mamzaton roblem for the th layer, under the condton that other layers strateges,,, +,, are gven, s eressed as ma E (, +,,,, ) (5a) N q,, q = = q subj. to = a, =, = b, =,, N, (5b) 0, =,, N, =,,, (5c) N a = b. = = (6) In Eq. (5), the th layer s strategy that satsfes Eq. (5b) s determned unquely, because the other layers strategy varables are already gven. There s no freedom to mamze the functon E. Therefore, to defne the nontrval Nash equlbrum solutons for constrant nterference-tye roblems, we ntroduce the normalzed Nash equlbrum soluton roosed by Rosen (965), whch has the fleblty of mamzaton and for whch statonary condtons can be derved. The normalzed Nash equlbrum soluton for the constrant nterference tye roblem s defned by relang nterference among layers n the constrant Eq. (5b) and consderng the roblem of mamzng the sum of all layers roft functons: (7a) ma (,,,, + E, ) = N = = subj. to = a, =,,, = b, =,, N, (7b) 0, =,, N, =,, (7c) N a = b. = = (8) Notce that Eq. (7a) s deendent on unknown arameters = (,, ), and the varable N N = (,, ) s mamzed smultaneously. Let the functon F : R R R be defned by ( ; ) (,,,, + = E, ). = F (9) We defne as the local normalzed Nash equlbrum soluton for the constrant nterference tye roblem N n Eq. (3), when the neghborhood B ( ) R of ests such that the followng nequalty holds: F( ; ) F( ; ) B( ) S, (0) where S = { satsfes Eq.(7b)(7c) }. Note that the normalzed Nash equlbrum soluton s not a soluton for the smle mamzaton of the sum of all layers roft functons, F, but defnes the mamum ont, F, wth resect to varable n F, gven the value of n F as a arameter (that s, t s defned as a fed ont of the mamzaton oeraton). 4. SEARCH DYNAMICS OF THE NASH EQUILIBRIUM SOLUTION FOR THE CONSTRAINT INTERFERENCE TYE ROBLEM We nvestgate the dynamc used to search for the normalzed Nash equlbrum soluton for the constrant nterference tye roblem eressed n Eq. (3), n whch the double constrants of allocatng roducton ablty and market share are mosed smultaneously. In order to aly the results of the above constrant nterference tye roblem to the double constrant case drectly, we transform the N matr varable T T T nto the N dmensonal column vector varable as = (,, ), and reformulate the double constrants of Eq. (3b) as the lnear equalty constrant of the vector-matr form A = c wth a ( + N) ( N ) coeffcent matr A, eressed as 02

4 = A= c = a a b b N T T T T (,, ), , (,,,,, ). Here an arbtrary element of equalty A = c must be satsfed, under the balancng condtons of Eq. (8), and then A = N + ranked. Let A be the ( + N ) ( N ) matr n whch an arbtrary row of matr A s deleted. We can roose a dynamc to search for the normalzed Nash equlbrum soluton of a game roblem wth a double nterference constrant allocaton tye roblem as follows: where d () t M = Q ( ( t)) M ( ( t)) F( ( t); ( t)), (2) A dt d () t dt d () t =, dt d () t dt (3a) M T T Q ( ) = I M ( ) A ( AM ( ) A ) A, A (3b) M ( ) = dag(/ ) ( N ) ( N )matr (3c) E (,, ) F( ; ) =. E (,, ) M Here the ( N ) ( N ) varable metrc rojecton matr Q A ( ) cannot be eressed by a smle T formula, because the nverse ( AM ( ) A ) cannot be formulated elctly. 5. SIMULATIONS OF THE NORMALIZED NASH EQUILIBRIUM SOLUTION FOR THE CONSTRAINTS OF DOUBLE RESOURCE ALLOCATION 5. The Three-erson, Three-Strategy Game (Benchmark) Consder a three-erson ( = 3) game wth three roducts (N = 3). Even n the smlest model, there s no loss of generalty from the model descrbed n Secton 2. As a concrete eamle, consder three automoble comanes, each of whch roduces three tyes of automobles: budget, mdlevel, and luury. The decson varables are (, 2, 3) T T =, = (, 2, 3) and = (, 2, 3) T, where the subscrts ndcate the roduct and the suerscrts ndcate the frm. The roft functons of each frm are N N q = q = = = q q E ( ) f ( ), () (3d) (4) where q s the loss arameter suffered by the th roduct when layer roduces and layer q roduces q. In the economcs of frms, gan s the cororate roft and loss reresents the varous knds of conjectural costs. The constrants are roducton caacty and market share as eressed by Eq. (3b), and 2 3 a = a = a = and b = b2 = b3 = for smlcty. The gan for frm from roducts, 2, and 3 s ndcated, resectvely, as 2 f ( ) = 2(.4) + 3.2, f ( ) =.9(.3) + 2.8, f ( ) =.8(.2) (5) 03

5 The dfference n the functon, f s due to the roducton technology dfferences among roducts and frms. 2 3 The roft functon E (,, ) for frm s secfed as E (,, ) = f ( ) + f ( ) + f ( ) ( ), where, 2, and 3 are assumed to be zero. For frm 2, the gan functons for roducts, 2, and 3, resectvely, are f ( ) = 2.(.5) f ( ) =.9(.4) f ( ) =.7(.3) (6) The roft functon, E (,, ), for frm 2 s secfed as E (,, ) = f ( ) + f ( ) + f ( ) ( ), where 22, 222, and 223 are assumed to be zero. For frm 3, the gan functons for roducts, 2, and 3, resectvely, are f ( ) = 2.2(.6) f ( ) =.9(.5) f ( ) =.6(.4) (7) The roft functon, E (,, ), for frm 3 s secfed as E (,, ) = f ( ) + f ( ) + f ( ) where 33, 332 ( ), , and 333 are assumed to be zero. To choose the values for the arameters, jk (, j, k =, 2, 3), 22, 222, 223, 33, 332, and 333, we conducted many eerments before selectng two ecet, 2, 3 sets of arameters. The set ndcated n the frst smulaton shows the nternal solutons for frms and commodtes. The set of arameters ndcated n the second smulaton shows that each frm secalzes n the roducton of at least one commodty. Ths s a case for roduct dfferentaton wthn an olgooly market. In the frst smulaton, we assgned the 8 values of q as 2.0, resultng n the normalzed Nash equlbrum soluton for the decson varable equlbrum value for 2 3 = (,, ). Table ndcates the changes n the normalzed Nash 2 3 = (,, ) from the ntal values to the converged values. Table. Benchmark Values of Intal values of (0) Converged values = ( ) ˆ = Changes n Converged Values Caused by Changes n q In the second smulaton, we change the conjectural varaton of frm aganst frm q from 0.0 to 5.0. Fgure 2 3 ndcate the normalzed Nash equlbrum soluton for = (,, ) by changng the arameters of q. The horzontal as ndcates the values of conjecture from 0 to 5; the vertcal as ndcates the market share for commodtes 3. Fgure (a) ndcates changes n the market share of commodty by the three frms. When the value of q s relatvely small, all frms roduce commodty. When the conjecture eceeds the ont of 4 on the horzontal as, frm 3 (blue lots) roduces only commodty, and frms (red lots) and 2 (green lots) do not roduce commodty (Fg. (a)). The tendency s the same for commodty 2 and 3 (Fgs. (b) and (c)). In the olgooly market, conjectural varaton lays an mortant role n determnng the share 04

6 of the roducts wthn and among frms. When the values of conjectural varaton are small, each frm roduces a full set of commodtes. In contrast, when the values of the conjectural varaton are large, roduct secalzaton takes lace wthn an olgooly market. Fgure. Changes n the Market Share as a Result of Changes n the arameters of q (a) Commodty (b) Commodty Intal Values (c) Commodty 3 In the thrd smulaton, we change the ntal values of matr. Table 2 ndcates the converged values of q q n a dfferent set of arameters of q, namely, = 3.6 and = 4.2. In both cases although the ntal q q values of (0) are dfferent, the converged values are the same for = 3.6 and = 4.2. Ths eerment shows that the convergence method s robust wth resect to changes n the ntal values. However, when the ntal values of (0) are n the neghborhood of the boundary, the converged values are sometmes dfferent from one another. Intal values (0) ( 0) = ( 0) = Table 2. Intal Values of (0) and Converged Values of q q Converged values ( = 3.6) Converged values ( = 4.2) ˆ = ˆ = ˆ = ˆ =

7 5.4 Changes n Equalty Constrants We have two knds of data for quantty roduced: share data and quantty data. Ths eerment eamnes the alcablty of the calculaton method for both shares and quantty. The sum of the share becomes unty, whle the sum of the quanttes need not become unty. We set a = (.2,.0, 0.8) and b = (0.9,.0,.) that s, the total of vector a or vector b s not unty but three. Ths eerment shows that the method s alcable not only the case of shares but also to the case of quantty roduced. 5.5 Three-erson, Four-Strategy Game Ths eerment etends the number of frms from three to four, whch yelds the converged values ndcated n Table 3. ( ) Intal values (0) = Four-erson, Three-Strategy Game Table 3. Values of n a Three-erson, Four-Strategy Game q q Converged values ( = 2.0) Converged values ( = 4.2) ˆ = ˆ = Ths eerment etends the number of commodtes from three to four, whch yelds the converged values ndcated n Table 4. Intal values (0) ( 0) Table 4. Values of n a Four-erson, Three-Strategy Game = q q Converged values ( = 2.0) Converged values ( = 4.2) ˆ = ˆ = CONCLUSIONS Because olgooly frms usually roduce a varety of roducts, t s mortant to smultaneously understand both the determnaton of market shares among frms and the roduct m wthn a frm. As the total amount of roducton for both frms and commodtes s a ror gven n the model as the constrant, managers are able to consder roducton strategy relyng on the roft functons n the model. After the functonal form and arameters are fed, the convergent rocess s managed by the relcator dynamcs algorthm. Usng the Nash equlbrum smulaton model, we can generate varous otmal aths for roducton by changng the conjectures of frms. We confrmed that the resent algorthm s good n assessng olcy smulaton. REFERENCES Ayosh. E., and A. Mak A Nash Equlbrum Soluton n an Olgooly Market: The Search for Nash Equlbrum Solutons wth Relcator Equatons Derved from the Gradent Dynamcs of a Smle Algorthm. Mathematcs and Comuters n Smulaton, forthcomng. Dt, A. K Otmal Trade and Industral olces for the U.S. Automoble Industry. In R. Feenstra ed. Emrcal Methods for Internatonal Trade, Kleer, G Entry nto the Market for Large Transort Arcraft. Euroean Economc Revew 34 (4): Rosen, J. B Estence and Unqueness of Equlbrum onts for Concave N-erson Games. Econometrca 33 (3):