Research Article New Analyses of Duopoly Game with Output Lower Limiters

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1 Abstract and Appled Analyss Volume 3, Artcle ID 46743, pages Research Artcle New Analyses of Duopoly Game wth Output Lower Lmters Zhaohan Sheng, Janguo Du,,3 Qang Me, and Tngwen Huang 4 School of Management Scence and Engneerng, Nanjng Unversty, Nanjng 93, Chna School of Management, Jangsu Unversty, Zhenjang 3, Chna 3 Computatonal Experment Center for Socal Scence, Jangsu Unversty, Zhenjang 3, Chna 4 Texas A&M Unversty at Qatar, P.O. Box 3874, Doha, Qatar Correspondence should be addressed to Janguo Du; conphotels@gmal.com Receved 3 October ; Accepted 3 December Academc Edtor: Chuandong L Copyrght 3 Zhaohan Sheng et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. In the real busness world, player sometmes would offer a lmter to ther output due to capacty constrants, fnancal constrants, or cautous response to uncertanty n the world. In ths paper, we modfy a duopoly game wth bounded ratonalty by mposng lower lmters on output. Wthn our model, we analyze how lower lmters have an effect on dynamcs of output and gve proof n theory why addng lower lmters can suppress chaos. We also explore the numbers of the equlbrum ponts and the dstrbuton of condtoned equlbrum ponts. Stable regon of the condtoned equlbrum s dscussed. Numercal experments show that the output evoluton system havng lower lmters becomes more robust than wthout them, and chaos dsappears f the lower lmters are bg enough. The local or global stablty of the condtonal equlbrum ponts provdes a theoretcal bass for the lmter control method of chaos n economc systems.. Introducton Snce the French economst Cournot [] ntroduced the frst well-known model whch gves a mathematcal descrpton of competton n a duopolstc market, there are many research works based on t (see [ 9]). The Cournot duopoly model represents an economy consstng of two quantty-settng frms producng the same good, or homogeneous goods, and each frm chooses ts producton n order to maxmze ts profts (see []). Rand [] may be the frst man who suggested that the Cournot adjustment process may also fal to converge to a Nash equlbrum and may also exhbt cyclcal and even chaotc dynamcs. Puu [, ] suggested a case of Cournot duopoly wth two or there players, and wth a untary elastcty demand functon and constant margnal costs, and showed that these systems also lead to complex dynamcs, ncludng perod doublng bfurcatons and chaos. Kopel [3] nvestgated mcroeconomc foundatons of Cournot duopoly games and demonstrated that cost functons ncorporatng an nterfrm externalty lead to a system of coupled logstc equatons. Besdes extendng Puu s work to n compettors (see [6]), Ahmed et al. [7]contrbuted to develop dynamc Cournot game characterzed by players wth complete ratonalty nto one wth bounded ratonalty. After them, scholars studed Cournot game wth bounded ratonalty, consderng nfluence of dfferent demand functons and dfferent cost functons (lnear and nonlnear) (see [, 3, 9]). Recently, some scholars (see [4, 5, 4 ]) ntroduced heterogeneous players nto Cournot game wth bounded ratonalty. Ams of all the modfcatons prevously mentoned or not are to make Cournot game model become economcally more justfed n the world. Intherealbusnessworld,tscommonlyobservedthat compettve frms would lmt ther producton for steadness or economes of scale. Huang [] found that cautous responses to fluctuatng prces by frms (through lmtng the growth rate of ts output) may result n a hgher longrun average proft for a smple cobweb. He and Westerhoff [] showed that mposton of a prce lmters can elmnate homoclnc bfurcatons between bull and bear markets and hencereducemarketprcevolatlty.itsworthytobenotced that work of [, ] s based on one-dmenson economc system. In ncomplete competton, the number of players s

2 Abstract and Appled Analyss not less than, so the problem may become more ntrcate (see [3]). Ths paper ams at the new output duopoly game by mposng lower lmters on output and focuses on the mpact of lmters on dynamcs and unravelng stablzng mechansm of lmter method to reduce fluctuaton. As He and Westerhoff [] say, output lmters as appled model are dentcal to a recently developed chaos control method, the lmter method, whch has been analytcally and numercally explored by [ 6]. However, the exstent documents do not dscuss the mpact of lmter on equlbrum of economc system. Unfortunately, there exsts no theoretcal result to assure the fact that chaos can be suppressed by addng smple lmter. The results n ths paper may partly answer ths puzzle n a specal case. The remander of ths paper s organzed as follows. Secton ntroduces an output duopoly game wth bounded ratonalty and examnes the dynamcs of the model. Secton 3 turns to a dscusson of the duopoly game havng outputlowerlmtersandthempactofoutputlowerlmters on dynamcs. The fnal secton concludes the paper.. The Output Duopoly Game wthout Output Lmter The output game we ntroduce here s based on the assumpton that the two frms (players) do not have a complete knowledge of the market. Then one frm s labeled by = and the other =. In game, players behave adaptvely, followng a bounded ratonalty adjustment process based on a local estmate of the margnal proft Π / q (see [, 3, 6, 8, 9]). For example, f a frm thnks the margnal proft of the tme t s postve, t s decded to ncrease ts producton of the tme t+or to decrease ts producton f the margnal proft s negatve. If the thfrmattmet s q (t), tsoutputattme t+canbemodeledas q (t+) =q (t) +α q (t) Π (q,q ), =,, () q where Π (q,q ) s the after-tax proft of the th frmat tme t and α s postve parameter representng the speed of adjustment. As usual n duopoly models, the prce p of the good at tme t s determned by the total supply Q(t) = q (t) + q (t) through a demand functon (see [3]): p=f(q) =a bq, () where a and b are postve constant, and a s the hghest prce n the market. We assume that the producton cost functon has the nonlnear form: C (q )=c +d q +e q, =,, (3) where the postve parameter c s fxed cost of the th frm. In general, the cost functon C (q ), clmbng wth the ncrease of the product output, s convex, so ts frst dervatve C (q ) and second dervatve C (q ) are postve. We can assume that the parameters d, e are postve. In order to make the duopolygameontherals,themargnalproftoftheth frm must be less than the hghest prce of the good n the market. Therefore, d +e q <a, =,. Hence the after-tax proft Π of the th frm s gven by Π = [q (t)(a bq(t)) (c+d q (t) +e q (t))] ( r), =,, (4) where r s the tax rate of busness ncome tax, and r<; r=represents pretax proft. The margnal proft of the th frmatthetmets Π = (a bq(t) bq q (t) d e q (t)) ( r), (5) =,. The duopoly model wth bounded ratonal players can be wrtten n the form: q (t+) =q (t) +α q (t) [a bq (t) (b+e )q (t) d ] ( r), =,... Equlbrum Ponts and Local Stablty. In order to make the soluton of the output duopoly model have the economcalsgnfcance,westudythenonnegatvestablestatesoluton of the model n ths paper. The equlbrum soluton of the dynamcs system (6) s the followng algebrac nonnegatve soluton: q (a bq (b + e )q )=, (7) q (a bq (b + e )q )=. where From (7), we can get four fxed ponts: E = (, ), E =( a d b + e,), E =(, a d b + e ), E =(q,q ), q = (a d )(b+e ) b(a d ) 3b +4be +4be +4e e, q = (a d )(b+e ) b(a d ) 3b +4be +4be +4e e. Snce E, E,andE areontheboundaryofthedecson set, J={(q,q ) q, q }, theyarecalledboundary equlbrums. E s the unque Nash equlbrum provded that (a d )(b+e ) b(d d )>, () (a d )(b+e ) b(d d )>. (6) (8) (9)

3 Abstract and Appled Analyss 3 The Nash equlbrum E s located at the ntersecton of the two reacton curves whch represent the locus of ponts of vanshng margnal profts n (5). In the followng, we assume that ()ssatsfed,sothenashequlbrume exsts. The study of the local stablty of equlbrum ponts s basedontheegenvaluesofthejacobanmatrxofthe(6): +α J =[ A α b ( r) q ], () α b ( r) q +α A As regards the condtons for the fxed pont to be stable (see [, 3]), we have the followng result. Theorem. The boundary equlbra E, E, and E are unstable equlbrum ponts. Proof. At the boundary fxed pont E,theJacobanmatrxs J (E )=[ +α (a d ) ( r) +α (a d ) ( r) ]. (3) where A =(a (4b+4e )q bq d ) ( r), A =(a bq (4b+4e )q d ) ( r). () The egenvalues of J(E ) are λ =+α (a d )( r) and λ =+α (a d )( r), whch are greater than unty. Thus E s a repellng node wth egendrectons along the coordnate axes q and q. At the boundary fxed pont E,theJacobanmatrx becomes J (E )= [ α (a d ) ( r) [ [ α b(a d ) ( r) b + e + α ((a d )(b+e ) b(d d )) ( r) b + e ], (4) ] whose egenvalues are gven by λ = α (a d )( r) wth egenvector ξ = (, ) along q axe and λ =+(α ((a d ) b(d d )(b+e )( r))/(b+e ) wth egenvector ξ =([( r)(b+e )+α ((a d )(b+e ) b(d d ))]/α b(a d ), ), thus f α </[(a d )( r)], E s saddle pont, wth local stable manfold along q axs and the unstable tangent to ξ. Otherwse, E s an unstable node. The bfurcaton occurrng at α =/[(a d )( r)] s a flp bfurcaton at whch E from attractng becomes repellng along q axs, on whch a cycle of perod appears. From the smlarty between E and E, E s a saddle pont wth local stable manfold along q axs and the unstable one tangent to ξ = (, [( r)(b + e )+α ((a d )(b + e ) b(d d ))]/α b(a d )),fα </[(a d )( r)], E s saddle pont, wth local stable manfold along q axs and the unstable tangent to ξ. Otherwse, t s an unstable node. Hence Theorem s true. In order to study the local stablty of Nash equlbrum E =(q,q ), we estmate the Jacoban matrx at E,whch s J (E ) =[ α (b + e )q ( r) α bq ( r) α bq ( r) α (b + e )q ( r)]. (5) The characterstc equaton s P (λ) =λ Tr λ+det =, (6) where Tr s the trace and Det s the determnant, and Tr = f α f α, Det = f α f α +f 3 α α, (7) where f =(b+e )q ( r) s postve, f =(b+e )q ( r) s postve, and f 3 equals 4f f +b q q ( r),spostve. Snce Tr 4Det =4[f α f α ] +4(4f f f 3 )α α >, (8) the egenvalues of Nash equlbrum are real. The local stablty of Nash equlbrum s gven by Jury s condton (see [, 3, 6, 7]), whch are (a) Tr +Det =f 3 α α >, (b) +Tr +Det >. The frst condton s satsfed and the second condton becomes 4f α +4f α f 3 α α 4<. (9) Ths equaton defnes a regon of stablty n the plane of the speeds of adjustment (α,α ).Thestabltyregons bounded by the porton of hyperbola wth postve α and α, whose equaton s 4f α +4f α f 3 α α 4=. () For the values of (α,α ) nsde the stablty regon, the Nash equlbrum E s stable and loses ts stablty through

4 4 Abstract and Appled Analyss a perod doublng (flp) bfurcaton. The bfurcaton curve determned by () ntersects the axes α and α,respectvely, whosecoordnatesaregvenby S = (,), S f = (, ). () f Therefore, from the prevously mentoned dervaton, we have followng theorem llustratng the local stablty of equlbrum E. Theorem. The stable regon of equlbrum E s enclosed by hyperbola defned by () and the axes α and α. Theorem and (6) ndcate that when the adjustng speed of α and α ofthetwofrms productonsntheareadefned by () andtheaxesα, α,theoutputofthetwofrmswll tend towards the equlbrum pont E.Themaxmumproft wll be obtaned at ths pont, just as Π = [q (a bq ) (c +d q +e q )] ( r), =,, () where Q =q +q. It s notceable that the game s based on the bounded ratonalty. The two frms cannot reach the Nash equlbrum at once. They may reach the equlbrum pont after rounds of games. But once one player or both players adjust the productontoofastandpushα, α beyond the bfurcaton curve (defned by ()), the system becomes unstable. Smlar argument apples f the parameters α and α are fxed parameters and the parameters d, d are vared... Numercal Smulaton. We can show the stablty of the Nash equlbrum pont E through the numercal experments and show the way of the system to the chaos through perod doublng bfurcaton. The parameters have taken the values α =., a =, b =, c =., c =, d =, d =, e =, e =., andr =. Fgure (a) shows that the bfurcaton dagram of the system (6) s convergent to Nash equlbrum for α < 9.Thenfα > 9, Nash equlbrum becomes unstable. Perod doublng bfurcatons appears, and fnally chaotc behavors occur. Also the maxmum Lyapunov exponent (Lyap.) s plotted. Postve values of Lyapunov exponent show that the soluton has chaotc behavor. Takng the correspondng output on the outputtracen(4), the proft trace bfurcaton dagram can be drawn(asshownbyfgure (b)). In Fgure (b), Π represents the proft of the =frstfrm,andπ represents the proft of the =secondfrm.wecanseethechangeofproftsthesame as the output of the player wth the varance of parameter α.fgures(a) and (b) also show that f the frm does the decson makng more carefully, the output and the proft wll be more stable. Fgure (c) shows strange attractor for the values of the parameters a=, b=, d =, d =, e =, e =., r =, α =., andα = 4. Strange attractors of the system (6)exhbt a fractalstructure.fgure (d) shows the regon of stablty of the Nash equlbrum for the values of the parameters a=, b=, d =, d =, e =, e =., and r =. Equaton() and the economcal sgnfcance of parameters (whch s postve here) defne the regons of stablty n the plane of adjustment (α,α ). 3. The Output Duopoly Game Havng Output Lower Lmters Next, we assume that the th frm wll mpose lower lmter on output for economes of scale, and (6)becomes q mn where q mn q (t+) = Max [f (q (t),q (t)),q mn ], >and =,, f (q,q )=q +α q [a bq (b + e )q d ] ( r), (3) =,. (4) Lmtng the output s economcally justfed n the real world. It can be explaned by capacty constrants, fnancal constrants, breakeven, and steadness. 3.. Equlbrum Ponts. In order to study the qualtatve behavor of the solutons of the nonlnear map (3), we defne the equlbrum ponts of the dynamc duopoly game by lettng q (t+) =q (t), =,, (5) where q (t + ) s determned by (3). Comparng f (q (t), q (t)) (n bref, f (t), defned by (4)) wth q mn, =,,therearefourcases. (a) f (t) q mn and f (t) q mn.whenq mn q and q mn q, the soluton of the system (3) gvesone fxed pont: E =(q,q ), (6) where q = ((a d )(b + e ) b(a d ))/(3b + 4be +4be +4e e ), q = ((a d )(b + e ) b(a d ))/(3b +4be +4be +4e e ),ande s the unque Nash equlbrum when () s satsfed. (b) f (t) < q mn and f (t) > q mn.whenq mn >q and q mn q,wecanobtanafxedpont: F =(q mn,q ), (7) where q =(a bqmn d )/(b + e ). (c) f (t) > q mn and f (t) < q mn.whenq mn q and q mn >q, we have one fxed pont: F =(q,qmn ), (8) where q =(a bqmn d )/(b + e ).

5 Abstract and Appled Analyss q q 5 Π.5 q q 4 Π q,q q Π,Π 3 Π Π Lyapunov α (a) α (b) 5 5 q.6. α q α (c) (d) Fgure : Partal numercal smulaton of the system (6). (a) The bfurcaton dagram of the evoluton of output. (b) The bfurcaton dagram of the evoluton of after-tax proft. (c) The strange attractor. (d) The stable regon of Nash equlbrum of duopoly game n the phase plane of speed of adjustment. (d) f (t) < q mn and f (t) < q mn.whena b(q mn + q mn ) (b + e )q mn d, =,, the soluton of the system (3) gves one fxed pont: F 3 =(q mn,q mn ). (9) From the prevously mentoned analyss, we can see that the exstence of equlbrums E, F, F, or F 3 has relaton to the sze of lower lmters q mn and q mn,sowecall them condtonal equlbrum ponts. The relatons between exstence of equlbrum and lower lmters are summarzed n Fgure. InFgure, EF(a bq mn bq mn =)denotes the maxmal capacty of market, where the prce of the good comes up to zero. When (q mn, q mn )slocatednregoni, whch s enclosed by horzontal lne BN (q mn =q ), vertcal lne AN (q mn =q ), and the axes qmn, =,,thesystem (3) hasnashequlbrume.if(q mn, q mn )fallsnregon II, whch s surrounded by lne AN, lne NC (a b(q mn + q mn ) (b + e )q mn d =),andtheaxsq mn,thesystem (3) gves the condtonal equlbrum F.Iftslocatedn regon III, whch s enclosed by lne NC, lne EF, lne DN (a b(q mn +q mn ) (b + e )q mn d =),andtheaxes q mn, =,,equlbrumpontofthesystem(3) sf 3.If t s stuated n regon IV surrounded wth lne BN, lne DN, and the axs q mn, condtonal equlbrum pont of the system (3)becomesF. It s very nterestng that condtonal equlbrum F perchesonlnenc,f stands on lne DN, and F 3 s stuated n regon III. Furthermore, the feasble regon of condtonal equlbrum ponts conssts of regon III, and ts boundary s convex. What s more, Nash equlbrum E (namely, N n Fgure ) s one of the vertces of the regon. So the equlbra of the system (3) are dfferent from those of the system (6) whch only has boundary equlbra and Nash equlbrum.

6 6 Abstract and Appled Analyss F D Usng the method of reducton to absurdty can prove ths concluson. We now assume that there s one strct nequalty at least for q (t) q mn and q (t) q mn, for example q (t) > q mn.ifa b(q (t) + q (t)) (b + e )q (t) d =, =or, then q mn III a b(q mn +q mn ) (b+e )q mn d >, =or, (33) B O IV I N A II q mn Fgure : The dstrbutons of condtonal equlbrum ponts of the system (3), at parameter values (a, b, d,d,e,e ) = (,,,,,.). 3.. Stablty. Usng the method smlar to Secton.,we can draw the concluson that Nash equlbrum E of the system (3) has the same stable regon n the plane of the speeds of adjustment (α,α ) as system (6). That s to say, ts regon of stablty s bounded by () andaxesα, =,.Although the equlbrum of the system (3) maystllbelookedas a result of learnng and evoluton, the adjustment of producton s restraned due to lower lmter. Moreover, sze of lmters nfluences exstence of Nash equlbrum. In the followng, we wll explore the stablty of condtonal equlbrum ponts F, F,orF 3 of the system (3), respectvely. As for condtonal equlbrum F 3 of the system (3), the followng result can be made. Theorem 3. The condtonal equlbrum F 3 of the system (3) s globally stable n the plane of the speeds of adjustment (α,α ). Proof. Because q mn, =, are postve and the lower lmters of output, q (t) q mn, =,, t=,,,... (3) As Secton 3. shows, when condtonal equlbrum F 3 exsts, q mn, =,must satsfy a b(q mn +q mn ) (b+e )q mn d, (3) =,. Note that b, d,d,e,ande are postve; also note that when q (t) q mn,wehave (a) a b(q (t) + q (t)) (b + e )q (t) d, (b) a b(q (t) + q (t)) (b + e )q (t) d. Now, when one of the prevously mentoned two nequaltes (a) and (b) becomes an equaton, we prove that q (t) =q mn, =,. (3) C E whch s mpossble. Hence f there s an equaton n a b(q (t) +q (t)) (b+e )q (t) d, =,. (34) then q (t) = q mn, =,. If a b(q (t) + q (t)) (b + e )q (t) d <, =and, accordng to condton (a), condton (b), and the map (3), wecanobtantwoseresq (t) and q (t) (t =,,,...,) whch descend monotonously and have lower bound q mn and q mn, respectvely. Therefore, they have lmts when t +.We assume that the lmts are q l, =,,thenql qmn, =,. When there s an nequalty at least n q l qmn, =,,we assume that q l >qmn and q l s the mage of ql followng the map (3). Note that a b (q l (t) +ql (t)) (b+e )q l (t) d <, =,. (35) Then q l <ql, whch s a conflct wth the fact that ql s the lmt of seres q (t). Hence, q (t) q mn, =,.Wecanknowthat Theorem 3 s true. The global stablty of F 3 ndcates that when lower lmters ncrease to a certan extent, chaotc state of the output game dsappears. In real busness world, frms may suppress chaos and reduce volatlty of output and proft by mposng lower lmter. Consderng the symmetry of the system (3), and condtonal equlbrum ponts F and F, we only need to dscuss the stablty of F. It s very dffcult to nfer the stable regon of F. Here, we use numercal experment to show the mpact of lower lmters on regons of stablty of F,whchsllustrated n Fgure 3. The parameter settng s n Fgure (d). Fgure 3 dsplays that magntudes of lower lmters and sze of ntal outputallhavegreatnfluenceonstabltyoff.asntal output lessens, the stable regon of F reduces. In Fgures 3(a) and 3(c), scopeofα s drawn partly. In fact, we have the followng theorem llustratng ths. Theorem 4. In the plane of the speeds of adjustment (α,α ), when ntal output of the =secondfrmsatsfesq () (a (b + e )q mn d )/b,thescopeofα n the stable regon of condtonal equlbrum F of the system (3) s α >.

7 Abstract and Appled Analyss 7 α α (a) q mn =.83, q mn =., q () =.86, q () =.8 α (b) q mn =.83, q mn =., q () =.86, q () =.5 α α α (c) q mn =.846, q mn =.36, q () =.86, q () =.8 α 5 5 (d) q mn =.846, q mn =.36, q () =.86, q () =.5 α Fgure 3: The stable regon of condtonal equlbrum F of the system (3). Proof. Because q mn > q = ((a d )(b + e ) b(a d ))/(3b +4be +4be +4e e ),so a bq mn d b + e > a (b+e)q mn b d. (36) Note that the trajectores of output converge to (q mn,(a bq mn d )/(b + e )).Ifq () (a (b + e )q mn d )/b, as long as the speed of adjustment of the =secondfrm,α s small enough, after fnte teratve tme, we can obtan q (t) > a (b+e )q mn d. (37) b That s to say, a (b+e )q mn bq (t) d <. Note that q (t) q mn. Therefore, gven α >at wll, we have a (b + e ) q (t) bq (t) d <. (38) Accordng to the system (3), we can obtan a seres of q (t) (t =,,,..., ) whch descend monotonously and have lower bound q mn except ntal fnte terms. Thus, they have lmt when t +.Weassumethatthelmtsq l ; then q l qmn.ifq l >qmn, we assume that q l s the mage of q l followngthemap(3). Note that a b(q l (t) +q (t)) (b+e )q l (t) d <. (39) Then q l <ql, whch s a conflct wth the fact that ql s the lmt of seres q (t). Hence,q l =qmn.thatstosay,f q () (a (b + e )q mn d )/b, the speed of adjustment of the = frst frm has no effect on the stablty of condtonal equlbrum F. The proof s complete. By the same way, we can obtan the followng. Theorem 5. In the plane of the speeds of adjustment (α,α ), when ntal output of the =frstfrmsatsfesq () (a

8 8 Abstract and Appled Analyss q q q,q Π,Π Π Π.5 Lyapunov α (a) α (b) Fgure 4: Bfurcaton dagrams of the system (3) wth output lower lmters (q mn,q mn ) = (,.6), at parameter values (a, b, c,c,d,d,e,e,r,α, ) = (,,.,,,,,.,,.). q Fgure 5: The strange attractor of the system (3) wth output lower lmters (q mn,q mn ) = (,.6), at parameter values (a, b, d,d,e,e,r,α,α ) = (,,,,,.,,., 4). (b + e )q mn d )/b, thescopeofα n the stable regon of condtonal equlbrum F of the system (3) s α > Numercal Smulaton. Numercal experments are smulated to show the nfluence of lower lmters on the stablty q of Nash equlbrum, whch are based on the same parameter settngas Fgures (a) and (b). Fgure 4(a) reveals that the trajectores of output converges to the Nash equlbrum when α < 9; for α > 9, the Nash equlbrum becomes unstable, perod doublng bfurcatons appear, and chaotc behavor occurs.however,thedynamcsofthemapwthlowerlmters (q mn =, q mn =.6) s dfferent from that of the map wthout lmter. When α >.6, the perod behavors come forth agan. Also the maxmum Lyapunov exponent (lyap.) splotted,wherepostvevaluesndcatethatthesystemhas chaotc behavors. Fgure 4(b) basedonthesameparameter settng shows the mpact of lower lmters on the evoluton of proft. Comparng Fgures 4(a) and 4(b) wth Fgures (a) and (b), respectvely, we can see that mposng lower lmters can reduce fluctuatons of producton and proft. Fgure 5 shows the nfluence of producton lmters on the strange attractor of the duopoly game. Comparng t wth Fgure 4(c), we can fnd the dfference between the attractor of the system (6) and that of the system (3). 4. Conclusons Ths paper s concerned wth complex dynamcs of duopoly game wthout and wth output lower lmters. We dscussed that f the behavor of producer s characterzed by relatvely low speeds of adjustment, the local producton adjustment process wthout lmters converges to the unque Nash

9 Abstract and Appled Analyss 9 equlbrum. Complex behavors such as cycles and chaos occur for hgher values of speeds of adjustment. Furthermore, we nvestgate how output lower lmters, whch functon dentcally to a recently explored chaos control method: phase space compresson, and the lmter method, affect the output dynamcs. The exstence of Nash equlbrum becomes condtonal. The dstrbuton of the condtonal equlbrum ponts s dsplayed n ths paper, and ther relaton wth lower lmter s studed. It s funny that the feasble regon of condtonal equlbrum ponts s convex and Nash equlbrum s one of the vertces of the regon. We fnd that smple output lower lmters may (a) reduce the fluctuaton of producton and proft; (b) make chaos of the orgnal duopoly game dsappear; (c) help the frms to avod the exploson of the economc system. The sze of lower lmters and ntal output also has relaton wth the stable regon of condtonal equlbrum ponts F and F, and the relaton s explored analytcally and numercally. The condtonal equlbrum ponts (q mn,q mn )areglobally stable n the plane of speeds of adjustment. The stablty of the condtonal ponts gves a theoretcal bass for the phase space compresson and the lmter method to control chaos n a specal case. Acknowledgments Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna under Grants 7799, 7737, 78, and 77735, by the Natonal Socal Scence Foundaton of Chna for major nvtaton-for-bd project under Grant &ZD69, and by Chna Postdoctoral Scence Foundaton under Grant Ths work was also sponsored by Qng Lan Project and 333 Project of Jangsu Provnce, Jangsu Unversty Top Talents Tranng Project, and Qatar Natonal Prorty Program NPRP funded by Qatar Natonal Research Fund. References [] A. Cournot, Researches nto the Mathematcal Prncples of the Theory of Wealth, Hachette, Pars, France, 838. [] H. N. Agza, A. S. Hegaz, and A. A. 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11 Advances n Operatons Research Volume 4 Advances n Decson Scences Volume 4 Appled Mathematcs Algebra Volume 4 Probablty and Statstcs Volume 4 The Scentfc World Journal Volume 4 Internatonal Dfferental Equatons Volume 4 Volume 4 Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Volume 4 Complex Analyss Volume 4 Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Volume 4 Volume 4 Volume 4 Volume 4 Dscrete Mathematcs Volume 4 Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Volume 4 Volume 4 Volume 4 Internatonal Stochastc Analyss Optmzaton Volume 4 Volume 4