research article a dynamic evolutionary game model of...

Research ArticleA Dynamic Evolutionary Game Model ofModular Production Network

Wei He

School of Business Administration Jiangxi University of Finance and Economics No 169 East Shuanggang RoadChangbei Nanchang Jiangxi 330013 China

Correspondence should be addressed to Wei He 04hrirene163com

Received 22 May 2016 Revised 30 June 2016 Accepted 3 July 2016

Academic Editor Ricardo Lopez-Ruiz

Copyright copy 2016 Wei He This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a new organization mode of production in the 21st century modular production network is deemed extensively to be a sourceof competitiveness for lead firms in manufacturing industries However despite the abundant studies on the modular productionnetwork there are very few studies from a dynamic perspective to discuss the conditions on which a modular production networkdevelops Based on the dynamic evolutionary game theory this paper constructs a model which incorporates several main factorsinfluencing the development of modular production network By calculating the replicator dynamics equations and analyzingthe evolutionary stable strategies this paper discusses the evolution process of cooperation strategies of member enterprises ina modular production network Furthermore by using NetLogo software to simulate the model this paper verifies the effectivenessof the model From the model we can find that the final stable equilibrium strategy is related to such factors as the initial cost theextra payoff the cooperation willingness of both parties the cooperation efforts and the proportion each party can get from theextra payoff To encourage the cooperation of production integrator and modular supplier some suggestions are also provided

1 Introduction

Since the 1980s a new organization mode of production hasbeen gradually emerging in the manufacturing industries inmany developed countries Namely the lead firms in theindustries focus on their core businesses such as designand creation of products or provision of services while out-sourcing the manufacturing activities to the contract manu-facturers Scholars called this particular production patternmodular production network [1] The popularity of mod-ular production network is mainly drawn from the rapiddevelopment of modern information technology and mod-ularity technology In general the structure of the modularproduction network has the following features [2] firstly itsstructure is flat Within a modular production network therelationship between the production integrator and modularsupplier is built on a relatively equal basis The poweris symmetrically distributed And knowledge flows morehorizontally Secondly it is flexible Flexibility here refers tothe ability of amodular production network to respond to thefast changing environment Because the modular productionnetwork is built onmodular production and flexible modular

organization it can give a quick response to the externalenvironmentThirdly it is quite open In amodular structureunder the premise of the same standards the modular sup-pliers with the same function are highly substitutive andthey have to compete with each other The mechanism ofselecting the superior and eliminating the inferior endows themodular production network with high openness Fourthlyit is a self-organization system Coordination arises out ofthe interactions between members of modular productionnetwork and it can be spontaneous which is not necessarilycontrolled by an auxiliary agent outside of the system Assuch the system is typically robust and able to develop

Nowadays the modular production network is widelyrecognized as a source of competitiveness for lead firms inthe industries [3ndash6] According to scholars and practitionersthe modular production network has a lot of advantagesSome scholars emphasized product innovation function ofmodular production network They noticed that in modularproduction environment the product innovation ways arediversified the products are differentiated and the pro-duction cycles are shortened [3 7 8] Furthermore some

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 6425158 9 pageshttpdxdoiorg10115520166425158

2 Discrete Dynamics in Nature and Society

scholars pointed out that the modular production networkcan help enterprises to obtain strategic flexibility [9] facilitateoutsourcing and the development of industries [10] andoptimize resource allocation [11] What is more modularproduction network can contribute to reducing transactionalcost and business risks of member enterprises as well asbringing external scale effect [1 12] Therefore under thebackground of the global value chain it is not difficult toconclude that modular production network of complicatedproducts is a powerfulweapon for lead enterprises to competein the industries [13]

Originally the modular production network is derivedfrom the concept ldquomodularityrdquo In 1962 Simon put forwardthe concept ldquomodularityrdquo Then modularity technology wasfirst introduced in the computer design field Langlois andRobertson (1992) pointed out that modularity is a set of rulesto manage the complicated objects [2] A complex systemis divided into several independent parts and each partyexchanges information through the interface After studyingthe data of the computer industry between 1950 and 1996Baldwin and Clark (2000) proposed that the modularity ofcomputer stimulated the industry to transform from theinitial highly concentrated and integrated production tothe present highly scattered structure [7] What is morethey pointed out that the modularity phenomenon wasvery popular in industries like information and automotiveindustries and it has important significance for the evolutionof industrial organization Majority studies on product mod-ularity are carried out on the technical level and the productmodularity is thought to be leading tomodularity of industryOn the basis of that the abroad scholars mainly focusedon the influence on the evolution of industrial organizationthe organizational strategy and organizational capability bycase studies while the domestic scholars focused more on theinfluence on enterprise value networks innovation networksand industrial clusters In other words at that time scholarswere more enthusiastic about studying the influence of mod-ularity on industrial organizations Afterwards scholarsgradually shifted the study focus to the organization modeof production itself By studying different cases they haveexplored the reasons behind the formation of modular pro-duction network the characteristics of modular productionnetwork the different forms of modular production networkand the influencing factors of modular production networkand its influence on a single production organization

However we find that previous studies mainly focusedon qualitative and static analysis with case studies being usedextensivelyThere are rarely quantitative and dynamic studieson the formation and evolution of a single modular produc-tion network The modular production network consists oftwo types of enterprises the production integrators and themodular suppliers By nature the formation and evolution ofa modular production network are an interactive and gameprocess of the two organizations Usually a classical gametheory deals with a rational player who is involved in a givengame with other players and has to decide between differentstrategies in order to maximize the payoff which depends onthe strategies of the coplayers In contrast evolutionary gametheory deals with entire populations of players By learning

copying and inheriting strategies or even by infectionstrategies with high payoff will spread within the populationThe payoffs depend on the actions of the coplayers and hencethe frequencies of the strategies within the population [14]The most distinctive difference between evolutionary gametheory and classical game theory is that evolutionary gametheory assumes the subjects have bounded rationality whichis much closer to reality Evolutionary game theory deemsthat participants have limited abilities to find the optimalbehaviors By learning and copying the best strategies finallyeach party can reach the equilibrium point Therefore dueto these advantages this paper adopts evolutionary gamedynamics to analyze the interaction between production inte-grators and modular suppliers By creating a dynamic evo-lutionary game model this paper examines the evolution-ary game process and the stable equilibrium of modularoutsourcing In addition by simulation analysis based onmultiagent modeling method and NetLogo platform thispaper further tests the effectiveness of the game model

This paper proceeds as follows Section 2 is literaturereview Section 3 is the construction of an evolutionarygame model and discussion of the equilibrium of the modelSection 4 is the simulation analysis of the game model byusing NetLogo software Section 5 is the conclusions

2 Literature Review

By analyzing transactional activities between the lead firmsand the ldquoturn-keyrdquo firms in the American electronic infor-mation industry Sturgeon (2002) made detailed analysis onmodular production networks and called it ldquothe new formof American industrial organizationrdquo [4] In general themodular production network can be characterized by func-tional partitioning into discrete scalable reusable modulesrigorous use of well-defined modular interfaces and makinguse of industry standards for interfaces Based onmodularitythe modular production network is a topology structureconnected by interface rules and guaranteed by allocationrules [15] The overall orderliness of this structure and theactive nodes in the structure can facilitate the evolutionof the complicated product system As a global productionsystem production integrators and modular suppliers withina modular production network have formed a new type ofcooperative relationship [16]

Currently the major studies on the modular productionnetwork mainly examined the historical evolution processof modular production networks and discussed the natureand causes of modular production networks [17ndash21] Otherscholars combined modularity with different network formssuch as enterprise value networks industrial value chainsindustrial clusters and innovation networks to study theformation mechanism of modular production network [22ndash26] However despite these abundant researches there isa lack of studies which explore the specific formation andevolution process of a single modular production networkand there is no study from member enterprise perspective toexamine the operation of modular production network Therelated researches are quite limited Next wemake a literaturereview on evolution of modular production network As a

Discrete Dynamics in Nature and Society 3

new form of network organization we extend the scope tonetwork organization

According to Zhu andZhou (2007) network organizationcan be divided into two types one with core enterprise andone without core enterprise [27] Furthermore they dis-cussed in detail the different evolutionary patterns of thesetwo modes Taking a cue from the complex system theoryPan and Yao (2011) studied the evolutionary process ofnetwork organizations and indicated that the formation ofnetwork organizations was a self-organization process [16]It is the interaction of the market mechanism and hierarchymechanism that facilitate the formation of network organi-zations And the interaction between the two governancemechanisms decides the development path and direction ofnetwork organizations Learning fromCAS theory Ruan andGao (2009) pointed out that the network organization was acomplex adaptive system [28] Its evolutionary process canbe seen as a nonlinear interactive process between subjects ofthe network By active adaptation of subjects and their inter-actions with the system the evolution of the whole system ispromoted Plowman et al (2007) applied complex adaptivesystem theory to explain the problem of mutual cooperationand decision participation in a network organization [22]They deemed that bymutual cooperation a network organiza-tion could accomplish the synergy effect that is the creationof benefits that are greater than the simple sum of all partsFurthermore they revealed that it was the synergy effectthat stimulated the formation of a network organizationAnderson noted that to understand the networking processof organizations we needed to initiate the studies from theperspectives of their form and the cooperative relationshipseparately Johanson and Mattsson developed Andersonrsquosstudy and for the first time proposed a mutual aid modelwhich well connected the networking relationship betweennodes with the interactive behaviors of them [1]

Wu et al (2009) established an evolutionary game modelfor information sharing in a network organization [29] Theyfound that the excess profits from information sharing initialcost of cooperation differences in service abilities of memberenterprises and discount factor are the key factors that affectthe evolution of the network organization Zhang and He(2010) combined the views of different theory perspectivesand analyzed in detail the characteristics and influencing fac-tors of the four spiral upward stages in the evolution processof network organizations [30] Koka et al (2006) pointed outthat when the environment was uncertain and the resourcesupply capacity changed interfirm networks generally expe-rienced four patterns of network change (network expan-sion network churning network strengthening and net-work shrinking) [14] Furthermore based on environmentalcontext and strategic action they further expounded theevolutionary path of network organizations Hakansson andSnehota (1995) reckoned that the formation of network orga-nizations needed to experience the resource searching coor-dination and integration stage [10] Salancik (1995) pointedout that it was the division of labor that drove the formationof a network [31] Networking can not only produce synergyeffect but also develop diversified contracts to govern thelong-term interdependent relationship between enterprises

Evolutionary game theory is the application of game the-ory to evolving populations of life forms in biology [32] A keydifference between evolutionary game theory and classicalgame theory is that evolutionary game theory focuses moreon the dynamics of strategy change as influenced not solelyby the quality of the various competing strategies but by theeffect of the frequency with which those various competingstrategies are found in the population [33 34] Originallyevolutionary game theoretical framework stems from biol-ogy Maynard Smith and Pricersquos (1973) seminal article inNature ignited further development of evolutionary gametheory [35] Since the 1990s evolutionary game theory hasbeen enriched with phenomena such as pattern formationequilibrium selection and self-organization and has gainedgreat development [36] For instance Tanimoto (2007) dis-cussed the coevolution mechanisms of both networks andstrategy and proved that this mechanism can solve dilemmasin several game classes [37] Wang et al (2015) proposednew universal scaling parameters for the dilemma strength[36] Moreover some textbooks and papers have appeareddescribing the theory of evolutionary games in relation toeconomics or more broadly the social sciences For exampleRitzberger and Weibull (1995) investigated stability prop-erties of evolutionary selection dynamics in normal-formgames [38] Samuelson (1997) described the usefulness ofevolutionary game theory as a tool for equilibrium selection[39] By building an evolutionary model Guttman (2000)showed that preferences for reciprocity could survive in apopulation partly consisting of ldquoopportunistsrdquo [40] Levineand Pesendorfer (2007) discussed evolutionarily stable out-comes for a class of games that admit cooperation and conflictas possible Nash equilibria [41] Recently what needs to benoticed is a shift of the research interest to the discussionson the cooperation problems in network organizations andenterprise alliances [42 43] For example Qin et al (2011)analyzed the influence of absorptive capacity on the behaviorof partners in RampD alliances based on evolutionary gametheory [44]Wang and Jiang (2013) constructed aHawk-Dovegame model on the interorganizational resource sharingof the technological innovation network and analyzed thedynamic evolutionary process of resource sharing gamestrategies of both parties [45] Han and Peng (2015) analyzedthe factors which affect the stability of network organizationby building a single group evolutionary model [46]

Based on the literature review we can find that in termsof the evolution of network studies were mainly drawn fromeconomics organizational ecology theory evolutionary gametheory and complex system theory However in generalthese studies are relatively autonomous from each otherStudies which synthesize these diverse theories are still rareWhat is more network organizations have various forms andmodular production networks are quite different fromothersIt develops with the maturation of modularity technologyand the development of information technology It is notrestricted to regions and is closely related to the decom-posable and integrated features of complicated product orcomplicated technologyTherefore there is a necessity to dis-cuss the evolutionary law of modular production network byfocusing on its unique characteristics Besides in terms of the


evolutionary game theory currently there were affluent stud-ies discussing the cooperation problems in network organiza-tions However until now there is no specific study focusingon the formation of modular production network which is aunique type of network organization and discussing the influ-encing factors Moreover most studies centered on buildingan evolutionary model to discuss the equilibrium conditionsThe validity of the evolutionary model is seldom tested So byfocusing on the particularity of modular production networkand combining the dynamic evolutionary game theory andmultiagent simulationmethod this paper constructs a modelto explore the cooperation between production integratorsandmodular suppliers in a modular production network anddiscuss the evolution of the modular production networkFurthermore it also tests the effectiveness of the model

3 Replicator Dynamic Evolutionary GameModel of Modular Production Network

31 Model Assumptions We suppose that within the mod-ular production network there are two types of individualenterprises the production integrator (enterprise 119860) and themodular supplier (enterprise 119861) They game with each otherin the network The two types of enterprises are enterpriseswith different production scales The set of their coopera-tive strategies is cooperation noncooperation Under thecooperation strategy both sides cooperate with each otherto produce Under the noncooperation strategy each partymay produce independently or one party refuses to continueto cooperate because of the other partyrsquos contract breachbehaviors During the evolutionary game process both sidesselect and adjust their cooperative strategies according to theother partyrsquos strategies In this model we simplify the inter-action and cooperation between production integrators andmodular suppliers and do not consider the social networks ornetwork reciprocity that may exist in the population mainlyfor the following two reasons firstly although with largepopulation there may be spatial structures and networks forproduction integrators and modular suppliers in a modularproduction network it is hard to obtain some key informa-tion such as the cost of opponent from a network Thus theasymmetry of information still remains which may affectcooperation choice Secondly even if the key informationof opponent can be obtained from a network due to thebounded rationality the agentrsquos next action will not totallydepend on the economic calculation of the outcome of itsformer action It is influenced bymany factors such as its cog-nition risk preference and environmental factors In otherwords it is still hard to predict an agentrsquos cooperation actionSo considering the two reasons we decide tomainly focus onthe economic interaction betweenproduction integrators andmodular suppliers and have not taken network reciprocityinto account

The normal payoffs of enterprise 119860 and enterprise 119861 are120587119860and 120587

119861 If the two enterprises cooperate with each other

to form themodular production network they can obtain theextra payoffs 120587 and the proportion each party can get is 119886119887 And 119886 + 119887 = 1 To cooperate with each other each party

Table 1 The general payoff matrix for enterprise 119860 and enterprise119861

Enterprise 119860 Enterprise 119861Cooperation (119910) Noncooperation (1 minus 119910)

Cooperation (119909) 119880119860 119880119861

120587119860minus 119862119860 120587119861

Noncooperation (1 minus 119909) 120587119860 120587119861minus 119862119861

120587119860 120587119861

needs to pay a certain amount of initial cost (119862119860and119862

119861 resp)

and effort cost which is decided by both sidesrsquo cooperationwillingness (120575

119886and 120575119887 resp) and their level of effort (120572 and 120573

resp) So the payoffs both sides can obtain when networkingand cooperating with each other are

119880119860= 120587119860+ 119886120587 minus 119862

119860minus 120575119886120572

119880119861= 120587119861+ 119887120587 minus 119862

119861minus 120575119887120573

(1)

32 Model Construction

321 Game Payoff Matrix For enterprise 119860 the probabilityof enterprise119860 to choose to join themodular production net-work to cooperate and to produce is 119909 Then the probabilityfor enterprise 119860 not to join the modular production networkis 1 minus 119909 Suppose the probability of enterprise 119861 to choose tojoin themodular production network is 119910 and the probabilityfor enterprise 119861 not to join the modular production networkis 1 minus 119910 Based on the above assumptions the strategy setsof both sides are cooperationnoncooperation The twoenterprises game with the proceeding of negotiation Thenegotiation space is an uncertain and bounded rationalityspace The two enterprises continuously select and adjusttheir own strategies with the consideration of their own pay-off and the other partyrsquos strategyTherefore the payoffmatrixof the two enterprises in the modular production network isshown in Table 1

Then the expected payoffof enterprise119860 choosing ldquocoop-erationrdquo strategy is

119864119862119860= 119910119880119860+ (1 minus 119910) (120587

119860minus 119862119860) (2)

The expected payoff of enterprise 119860 choosing ldquononcoop-erationrdquo strategy is

119864119880119860= 119910120587119860+ (1 minus 119910) 120587

119860= 120587119860 (3)

According to the above expected payoff of enterprise 119860we can calculate the average payoff

119864119860= 119909119864119862119860+ (1 minus 119909) 119864119880119860

(4)

Similarly the expected payoff of enterprise 119861 choosingldquocooperationrdquo strategy is

119864119862119861= 119909119880119861+ (1 minus 119909) (120587119861 minus 119862119861) (5)


119864119880119861= 119909120587119861+ (1 minus 119909) 120587119861 = 120587119861 (6)

The average payoff of enterprise 119861 is

119864119861= 119910119864119862119861+ (1 minus 119910) 119864

119880119861 (7)


322 Replicator Dynamics Game Model The replicator dy-namic evolutionary game model describes the progressiveprocess of the transformation of both sidesrsquo favorable strategyBoth sides are not adjusting their strategies at the same timeOne side needs to decide its strategy by considering the othersidersquos strategy and the payoff the strategy brings Hence it is alearning processThe specific calculation process is as follows

Suppose there are two types of individuals who are play-ing a game in competition with one another Both of themadopt pure strategies Let 119876 denote the set of pure strategieslet 119873119905(119902) denote the set of individuals who adopt the pure

strategy 119902 isin 119876 at the 119905 moment The state variable 119899119905(119902)

denotes the proportion of individuals who adopt the purestrategy 119902 at the 119905moment So we have

119899119905(119902) =

119873119905(119902)

sumℎisin119876

119873119905 (ℎ)

(8)

The expected payoff of individuals who adopt the purestrategy 119902 at the 119905moment is

119906119905(119902) = sum

ℎisin119876

119899119905 (ℎ) 119906119905 (119902 ℎ) (9)

where 119906119905(119902) denotes the expected payoff one group gets when

taking the pure strategy 119902 while the other group is taking thepure strategy ℎ

The average expected utility of the group is

119906119905= sum

119902isin119876

119899119905(119902) 119906119905(119902) (10)

According to the above assumptions the individuals whohave fewer payoffs will find the differences sooner or laterand will start to imitate other individuals Therefore the pro-portion of individuals who take the same strategy will changewith time and is the function of timeThe speed changes withtime and depends on the learning speed of individuals

Taylor and Jonker proposed a continuous replicatordynamics model

1198731015840

119905(119902) = 119873

119905(119902) 119906119905(119902) (11)

The derivative of (8) with respect to time is

119889119899119905(119902)

119889119905=1198731015840

119905(119902)sumℎisin119876

119873119905 (ℎ) minus 119873119905 (119902)sumℎisin119876119873

1015840

119905(ℎ)

[sumℎisin119876

119873119905 (ℎ)]2

(12)

Combining with (11) we can simplify (12) into

119889119899119905(119902)

119889119905= 119899119905(119902) [119906

119905(119902) minus 119906

119905] (13)

According to the above deduction we can get the replica-tor dynamics model of the production integrator

119865 (119909) =119889119909

119889119905= 119909 (119864

119862119860minus 119864119860) = 119909 (1 minus 119909) (119864119862119860

minus 119864119880119860)

= 119909 (1 minus 119909) (119910119880119860 minus 119862119860 minus 119910120587119860 + 119910119862119860)

(14)

Similarly the replicator dynamics model of the modularsupplier is

119866 (119910) =119889119910

119889119905= 119910 (119864

119862119861minus 119864119861) = 119910 (1 minus 119910) (119864

119862119861minus 119864119880119861)

= 119910 (1 minus 119910) (119909119880119861minus 119862119861minus 119909120587119861+ 119909119862119861)

(15)

33 Analysis of Game Process In the bounded rationalityrepeated game if the optimal equilibrium strategy can returnas before when there is small amount of interference wecall the equilibrium evolutionary stable strategy (ESS) Letthe replicator dynamics equation be equal to 0 Then wecan get all the replicator dynamic stable states Accordingto the ldquostability theoryrdquo of the differential equations we candiscuss the stability of these states and therefore decide theevolutionary stable strategies of production integrator andmodular supplier in the modular production network

331 Analysis of Game Process of the Production IntegratorFrom the replicator dynamic differential equation of theproduction integrator 119865(119909) we can know when 119910

lowast=

119862119860(119880119860minus120587119860+119862119860) and119865(119909) is always equal to 0This indicates

that within [0 1] 119909 is always in stable stateLet 119865(119909) = 0 we can get 119909 = 0 and 119909 = 1 and they are two

stable points of the dynamic differential equation(1) When 0 lt 119910

0lt 119910lowast (119889119865(119909)119889119909)|

119909=0lt 0 and (119889119865(119909)

119889119909)|119909=1

gt 0 therefore 119909 = 0 is the evolutionary stable strat-egy Namely the production integrator intends to chooseldquononcooperationrdquo strategy

(2) When 119910lowast lt 1199100lt 1 (119889119865(119909)119889119909)|

119909=0gt 0 and (119889119865(119909)

119889119909)|119909=1

lt 0 therefore 119909 = 1 is the evolutionary stable strat-egy Namely the production integrator intends to chooseldquocooperationrdquo strategy

332 Analysis of Game Process of the Modular SupplierSimilarly from the replicator dynamic differential equation ofthemodular supplier119866(119910) we can knowwhen119909lowast = 119862

119861(119880119861minus

120587119861+ 119862119861) and 119866(119910) is always equal to 0 This indicates that

within [0 1] 119910 is always in stable stateMake 119866(119910) = 0 we can get 119910 = 0 and 119910 = 1 and they are

two stable points of the dynamic differential equation(1) When 0 lt 119909

0lt 119909lowast (119889119866(119910)119889119910)|

119910=0lt 0 and (119889119866(119910)

119889119910)|119910=1

gt 0 therefore 119910 = 0 is the evolutionary stable strat-egy Namely the modular supplier intends to choose ldquononco-operationrdquo strategy

(2) When 119909lowast lt 1199090lt 1 (119889119866(119910)119889119910)|

119910=0gt 0 and (119889119866(119910)

119889119910)|119910=1

lt 0 therefore 119910 = 1 is the evolutionary stable strat-egy Namely the modular supplier intends to choose ldquocoop-erationrdquo strategy

333 Discussion From the replicator dynamics models wecan find that there are five local equilibrium points (0 0)(0 1) (1 0) (1 1) and (119862

119861(119880119861minus120587119861+119862119861) 119862119860(119880119860minus120587119860+119862119860))

According to the method proposed by Friedman we cananalyze the stability of the equilibrium points by analyzingthe local stability of the Jacobian matrix of the whole system

From the partial derivatives of (12) and (13) with respectto 119909 and 119910 we can get the Jacobian matrix


Table 2 Analysis of equilibrium points

The equilibrium point The determinant of matrix 119869 Sign The trace of matrix 119869 Sign Result

119909 = 0 119910 = 0 119862119860119862119861

+ minus (119862119860+ 119862119861) minus ESS

119909 = 0 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (119880

119860minus 120587119860) + (119880

119861minus 120587119861) + Not stable

119909 = 1 119910 = 0 119862119860119862119861

+ 119862119860+ 119862119861

+ Not stable

119909 = 1 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (120587

119860minus 119880119860) + (120587

119861minus 119880119861) minus ESS

119909 =119862119861

119880119861minus 120587119861+ 119862119861

=119862119861

119887120587 minus 120575119887120573

119910 =119862119860

119880119860minus 120587119860+ 119862119860

=119862119860

119886120587 minus 120575119886120572

119862119860119862119861(119880119860minus 120587119860) (119880119861minus 120587119861)

(119880119860minus 120587119860+ 119862119860) (119880119861minus 120587119861+ 119862119861)

+ 0 Saddle point

[

[

(1 minus 2119909) (119910119880119860 minus 119862119860 minus 119910120587119860 + 119910119862119860) 119909 (1 minus 119909) (119880119860 minus 120587119860 + 119862119860)

119910 (1 minus 119910) (119880119861minus 120587119861+ 119862119861) (1 minus 2119910) (119909119880

119861minus 119862119861minus 119909120587119861+ 119909119862119861)

]

]

(16)

At each equilibrium point the determinant and trace ofthe above Jacobian matrix are shown in Table 2

There are two stable strategies ldquocooperation coopera-tionrdquo and ldquononcooperation noncooperationrdquo How the pro-duction integrator and themodular supplier game and evolvedepend on the payoff matrix and initial states

Firstly the analysis of the saddle point is importantbecause the saddle point is the threshold point of the systemIf an initial state of the system is closer to an equilibrium statethen it ismore likely to converge to this point From the aboveanalysis we can know that the saddle point of this problem is119909 = 119862

119861(119887120587minus120575

119887120573) and 119910 = 119862

119860(119886120587minus120575

119886120572) For enterprise119860

when 119909 = 119909lowast= 119862119861(119887120587 minus 120575

119887120573) we have 119889119910119889119905 = 0 all the

time This indicates that all are in stable states under thiscondition while 119910 = 0 and 119910 = 1 are the two stable stateswhen 119909 = 119862

119861(119887120587 minus 120575

119887120573) It is the same for enterprise 119861

Secondly when 119909 gt 119862119861(119887120587 minus 120575

119887120573) the expected payoff

of enterprise 119861 to ldquocooperaterdquo minus the expected payoff ofenterprise 119861 to ldquononcooperaterdquo is 119864

119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+

119862119861)minus119862119861gt 0Therefore enterprise119861will choose to cooperate

And the equilibrium strategy is ldquocooperation cooperationrdquoThirdly when 119909 lt 119862

119861(119887120587 minus 120575



119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+

119862119861)minus119862119861lt 0Therefore enterprise 119861will choose not to coop-

erate And the equilibrium strategy is ldquononcooperation non-cooperationrdquo

According to the symmetry of the game this analysiscan also be applied to enterprise 119861 That means when 119910 gt

119862119860(119886120587 minus 120575

119886120572) the equilibrium strategy is ldquocooperation

cooperationrdquo when 119910 lt 119862119860(119886120587 minus 120575

119886120572) the equilibrium

strategy is ldquononcooperation noncooperationrdquo

4 Simulation Analysis

In this section we use NetLogo software to simulate thedynamic evolutionary model In a modular production net-work there are two types of agents the production integratorand the modular supplier The payoff matrix for enterprise119860 and enterprise 119861 shown in Table 3 By setting the valuesof parameters in the payoff matrix and simulating the gameprocess of the two agents we can obtain different evolution-ary equilibrium states In Figures 1 2 3 and 4 the two curvesrespectively denote the cooperation fractions of productionintegrators and modular suppliers We name these curves 119909and 119910 They show the replicator dynamics game process andthe final evolutionary stable strategiesThepopulation of eachagent is 100 and the patches are 100 lowast 100

We suppose in a modular production network the extrapayoff 120587 both parties can get is 1 Because the productionintegrator usually works as lead organization in the modularproduction network it takes bigger pieces of the extra payoffsHence we suppose the proportion the production integratorand the modular supplier can get is 06 and 04 respectivelyAs for the production integrator we suppose the normalpayoff 120587

119860it can get is 4 Its initial cost 119862

119860is 02 Comparing

with the production integrator the modular supplier isusually smaller in size and production scale Therefore thenormal payoffs and initial cost of the modular supplier arerelatively smaller and we suppose 120587

119861is 3 and 119862

119861is 01 In

addition we suppose the cooperation willingness of eachparty is the same and 120575

119886= 120575119887= 05 But the level of effort

they put in the cooperation is different due to the differentroles they play in the network Accordingly we suppose 120572 is06 and 120573 is 04


Table 3 The specific payoff matrix for enterprise 119860 and enterprise119861


Cooperation (119909) 29 23 38 3Noncooperation (1 minus 119909) 4 29 4 3

0

100

Prop

ortio

n

0 126

Time

Member enterprises over time

Figure 1 Simulation result for 1199090= 03 and 119910

0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04

By calculation we can get the payoffmatrix 119909lowast = 05 and119910lowast= 067Based on four groups of initial values of 119909 and 119910

we simulate the game process of the modular productionnetwork The horizontal axis is time and the vertical axis isthe values of 119909 and 119910 The results are as follows

(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast

and 0 lt 1199100lt 119910lowastWe can get Figure 1 and the final game result

is 119909 = 0 119910 = 0 Namely the evolutionary game strategy of thegame subjects is ldquononcooperationrdquo

(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1

and 0 lt 1199100lt 119910lowast We can get Figure 2 and the final game

result is 119909 = 0 and 119910 = 0 Namely the evolutionary gamestrategy of the game subjects is ldquononcooperationrdquo

(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1

and 119910lowast lt 1199100lt 1 We can get Figure 3 and the final game

result is 119909 = 1 and 119910 = 1 Namely the evolutionary gamestrategy of the game subjects is ldquocooperationrdquo

0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast

and119910lowast lt 1199100lt 1We can get Figure 4 and the final game result

is 119909 = 1 and 119910 = 1 Namely the evolutionary game strategyof the game subjects is ldquocooperationrdquo

5 Conclusions

In todayrsquos cooperation society the competition featured asalliances and cooperative network impels major manufactur-ing enterprises to join global production network to acquirecompetitive advantages It is a necessity for these enterprisesto take proper cooperative strategies and safeguard thehealthy development of the modular production networkso that utmost benefits can be obtained Based on theevolutionary game theory by analyzing the game processbetween the bounded rational suppliers and integrators ina modular production network this paper establishes adynamic evolutionary game model By simulating the gamemodel this paper further verifies the validity of the modelFrom the results we can see that through the evolutionarygame between subjects in the modular production networkthe final stable equilibrium strategy is related to the coop-erative strategies of the subjects and the saddle point Fromthe expression of the saddle point we can further knowthat the saddle point is influenced by the initial cost theextra payoff the cooperation willingness of both parties thecooperation efforts and the proportion each party can getfrom the extra payoff For these variables what we can be


sure of is that the bigger the initial cost the higher theprobability that both sides will choose the initial strategy andnot cooperate with each other The bigger the extra payoffthe more the willingness of both sides to cooperate witheach other Moreover we can know that an effective incentiveand punishment mechanism is useful for coordination in amodular production network From the model we can seethat the levels of effort 120572 120573 and the proportions each partycan get from the extra payoffs 119886 119887 can decide the initial stateand the evolutionary path of thewhole systemThemotivatorno matter whether it is a positive motivator or a negativemotivator in the premise of fairness can improve the levelof effort of member enterprises in a modular productionnetwork By the increase of information disclosure level theopportunistic behaviors of member enterprises can greatlybe reduced and a sound cooperative atmosphere can beformed More importantly proper payoff distribution andcost sharing schemes are critical to guarantee the fairness ofdecisions in a modular production network They can notonly well motivate member enterprises to take concertedactions and form a good cooperative relationship but alsoreduce transactional cost and realize the optimization ofoverall benefits of the modular production network

Competing Interests

The author declares no competing interests The author hasno financial and personal relationships with other people ororganizations that can inappropriately influence the work

Acknowledgments

This work was supported by the China Scholarship Councilthe National Natural Science Foundation of China (Grantnos 71462009 and 71361013) and the Natural Science Foun-dation of Jiangxi Province (Grant nos 20142BA217018 and20151BAB207059)

References

[1] J Johanson and L G Mattsson ldquoInterorganizational relationsin industrial systems a network approach compared with thetransaction-cost approachrdquo International Studies of Manage-ment amp Organization vol 17 no 1 pp 34ndash48 2016

[2] RN Langlois andP L Robertson ldquoNetworks and innovation ina modular system lessons from the microcomputer and stereocomponent industriesrdquo Research Policy vol 21 no 4 pp 297ndash313 1992

[3] BH Peng ldquoLiterature reviewonmodular production networkrdquoScience and Technology Management Research no 10 pp 301ndash303 2009

[4] T J Sturgeon ldquoModular production networks a new Americanmodel of industrial organizationrdquo Industrial and CorporateChange vol 11 no 3 pp 451ndash496 2002

[5] W He and S-H Chen ldquoGame analysis of determinants ofstability of semiconductor modular production networksrdquo Sus-tainability vol 6 no 8 pp 4772ndash4794 2014

[6] S-H Chen and W He ldquoStudy on knowledge propagationin complex networks based on preferences taking wechat as

examplerdquo Abstract and Applied Analysis vol 2014 Article ID543734 11 pages 2014

[7] C Y Baldwin and K B Clark Design Rules the Power ofModularity MIT Press Cambridge Mass USA 2000

[8] S-H Chen ldquoThe game analysis of negative externality of envi-ronmental logistics and governmental regulationrdquo InternationalJournal of Environment and Pollution vol 51 no 3-4 pp 143ndash155 2013

[9] S Y Chen ldquoNew forms of contemporary capitalist organizationof production-Modular Production Networksrdquo ContemporaryEconomic Research no 4 pp 32ndash36 2011

[10] H Hakansson and I Snehota Developing Relationships inBusiness Networks Routledge New York NY USA 1995

[11] J Hofbauer and K Sigmund ldquoEvolutionary game dynamicsrdquoBulletin of the AmericanMathematical Society vol 40 no 4 pp479ndash519 2003

[12] MKawakami ldquoExploiting themodularity of value chains inter-firm dynamics of the Taiwanese notebook PC industryrdquo 2008httpsiridegojpdspacebitstream23447403ARRIDE Dis-cussion No146 kawakamipdf

[13] Y Ke and S Y Wang ldquoResearch of a new form of industryorganization modular production networkrdquo China IndustrialEconomy no 8 pp 75ndash82 2007

[14] B R Koka R Madhavan and J E Prescott ldquoThe evolution ofinterfirm networks environmental effects on patterns of net-work changerdquo Academy of Management Review vol 31 no 3pp 721ndash737 2006

[15] R N Langlois ldquoModularity in technology and organizationrdquoJournal of Economic Behavior and Organization vol 49 no 1pp 19ndash37 2002

[16] S T Pan and C X Yao ldquoAnalysis of evolution of enterprise net-work organization-based on complex system theoryrdquo EnterpriseEconomy no 3 pp 34ndash37 2011

[17] Y Ke ldquoStudy on industrial organization effect of automo-bile industry modular production network-enlightenment onoptimization of Guangxi automobile industrial organizationrdquoAcademic Forum no 6 pp 154ndash167 2012

[18] W He ldquoAn inventory controlled supply chain model based onimproved BP neural networkrdquoDiscrete Dynamics in Nature andSociety vol 2013 Article ID 537675 7 pages 2013

[19] S H Chen ldquoThe influencing factors of enterprise sustainableinnovation an empirical studyrdquo Sustainability vol 8 no 5 pp425ndash442 2016

[20] S-H Chen S-H Xu C Lee N N Xiong and W He ldquoThestudy on stage financing model of IT project investmentrdquo TheScientific World Journal vol 2014 Article ID 321710 6 pages2014

[21] H Perks and S Moxey ldquoMarket-facing innovation networkshow lead firms partition tasks share resources and developcapabilitiesrdquo Industrial Marketing Management vol 40 no 8pp 1224ndash1237 2011

[22] D A Plowman S Solansky T E Beck L Baker M Kulkarniand D V Travis ldquoThe role of leadership in emergent self-organizationrdquo Leadership Quarterly vol 18 no 4 pp 341ndash3562007

[23] S-H Chen ldquoA novel culture algorithm and itrsquos application inknowledge integrationrdquo Information vol 15 no 11 pp 4847ndash4854 2012

[24] S-H Chen ldquoEmpirical research on knowledge integrationimproving innovation ability of IT enterprisemdashbased on struc-tural equation modelrdquo Information vol 14 no 3 pp 753ndash7582011


[25] H Wei ldquoChinarsquos technology innovation strategymdashfrom tech-nology transfer perspectiverdquo Information vol 15 no 11 pp4841ndash4846 2012

[26] S Ponte and T Sturgeon ldquoExplaining governance in globalvalue chains a modular theory-building effortrdquo Review of Inter-national Political Economy vol 21 no 1 pp 195ndash223 2014

[27] L L Zhu and D Q Zhou ldquoNetwork organization mode and itsevolution mechanismrdquo Modern Economic Research no 8 pp21ndash25 2007

[28] P N Ruan and J Gao ldquoStudy on evolution of network organ-ization-based on CAS theoryrdquo Enterprise Economy no 10 pp26ndash29 2009

[29] H X Wu H X Dong and Q Zhang ldquoEvolutionary gameanalysis of information-sharing mechanism in network organi-zationrdquo Computer Engineering and Applications vol 45 no 16pp 232ndash234 2009

[30] L Zhang and Q F He ldquoResearch on evolution of enterprisenetwork based on product cooperative innovationrdquo Science ofScience andManagement of Science amp Technology no 7 pp 135ndash139 2010

[31] G R Salancik ldquoWANTED a good network theory of organiza-tionrdquo Administrative Science Quarterly vol 40 no 2 pp 345ndash349 1995

[32] J Tanimoto Mathematical Analysis of Environmental SystemSpringer Tokyo Japan 2014

[33] D Easley and J KleinbergNetworks Crowds andMarkets Rea-soning About a Highly Connected World Cambridge UniversityPress Cambridge Mass USA 2010

[34] J Tanimoto Fundamentals of Evolutionary GameTheory and ItsApplications vol 6 Springer Tokyo Japan 2015

[35] J Maynard Smith andG R Price ldquoThe logic of animal conflictrdquoNature vol 246 no 5427 pp 15ndash18 1973

[36] Z Wang S Kokubo M Jusup and J Tanimoto ldquoUniversalscaling for the dilemma strength in evolutionary gamesrdquo Physicsof Life Reviews vol 14 pp 1ndash30 2015

[37] J Tanimoto ldquoDilemma solving by the coevolution of networksand strategy in a 2 times 2 gamerdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 76 no 2 Article ID021126 2007

[38] K Ritzberger and J WWeibull ldquoEvolutionary selection in nor-mal-form gamesrdquo Econometrica vol 63 no 6 pp 1371ndash13991995

[39] L Samuelson Evolutionary Games and Equilibrium Selectionvol 1 MIT Press Cambridge Mass USA 1997

[40] J M Guttman ldquoOn the evolutionary stability of preferences forreciprocityrdquo European Journal of Political Economy vol 16 no1 pp 31ndash50 2000

[41] D K Levine andW Pesendorfer ldquoThe evolution of cooperationthrough imitationrdquo Games and Economic Behavior vol 58 no2 pp 293ndash315 2007

[42] S H Chen ldquoThe influencing factors of enterprise sustainableinnovation an empirical studyrdquo Sustainability vol 8 no 5 p425 2016

[43] S Chen ldquoConstruction of an early risk warning model oforganizational resilience an empirical study based on samplesof RampD teamsrdquo Discrete Dynamics in Nature and Society vol2016 Article ID 4602870 9 pages 2016

[44] W Qin F Xu and B Song ldquoEvolutionary game analysis ofbehavior in RampD alliances with absorptive capacity perspec-tiverdquo Industrial Engineering and Management vol 16 no 6 pp16ndash20 2011

[45] X T Wang and C Jiang ldquoEvolutionary game analysis oninter-organizational resource sharing decisions in technologicalinnovation networksrdquo Enterprise Economy no 3 pp 27ndash302013

[46] J W Han and Z Y Peng ldquoResearch on the stability and gov-ernance mechanism of enterprise network organization basedon embeddednessrdquo Economic Survey vol 32 no 3 pp 90ndash952015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


scholars pointed out that the modular production networkcan help enterprises to obtain strategic flexibility [9] facilitateoutsourcing and the development of industries [10] andoptimize resource allocation [11] What is more modularproduction network can contribute to reducing transactionalcost and business risks of member enterprises as well asbringing external scale effect [1 12] Therefore under thebackground of the global value chain it is not difficult toconclude that modular production network of complicatedproducts is a powerfulweapon for lead enterprises to competein the industries [13]

Originally the modular production network is derivedfrom the concept ldquomodularityrdquo In 1962 Simon put forwardthe concept ldquomodularityrdquo Then modularity technology wasfirst introduced in the computer design field Langlois andRobertson (1992) pointed out that modularity is a set of rulesto manage the complicated objects [2] A complex systemis divided into several independent parts and each partyexchanges information through the interface After studyingthe data of the computer industry between 1950 and 1996Baldwin and Clark (2000) proposed that the modularity ofcomputer stimulated the industry to transform from theinitial highly concentrated and integrated production tothe present highly scattered structure [7] What is morethey pointed out that the modularity phenomenon wasvery popular in industries like information and automotiveindustries and it has important significance for the evolutionof industrial organization Majority studies on product mod-ularity are carried out on the technical level and the productmodularity is thought to be leading tomodularity of industryOn the basis of that the abroad scholars mainly focusedon the influence on the evolution of industrial organizationthe organizational strategy and organizational capability bycase studies while the domestic scholars focused more on theinfluence on enterprise value networks innovation networksand industrial clusters In other words at that time scholarswere more enthusiastic about studying the influence of mod-ularity on industrial organizations Afterwards scholarsgradually shifted the study focus to the organization modeof production itself By studying different cases they haveexplored the reasons behind the formation of modular pro-duction network the characteristics of modular productionnetwork the different forms of modular production networkand the influencing factors of modular production networkand its influence on a single production organization

However we find that previous studies mainly focusedon qualitative and static analysis with case studies being usedextensivelyThere are rarely quantitative and dynamic studieson the formation and evolution of a single modular produc-tion network The modular production network consists oftwo types of enterprises the production integrators and themodular suppliers By nature the formation and evolution ofa modular production network are an interactive and gameprocess of the two organizations Usually a classical gametheory deals with a rational player who is involved in a givengame with other players and has to decide between differentstrategies in order to maximize the payoff which depends onthe strategies of the coplayers In contrast evolutionary gametheory deals with entire populations of players By learning

copying and inheriting strategies or even by infectionstrategies with high payoff will spread within the populationThe payoffs depend on the actions of the coplayers and hencethe frequencies of the strategies within the population [14]The most distinctive difference between evolutionary gametheory and classical game theory is that evolutionary gametheory assumes the subjects have bounded rationality whichis much closer to reality Evolutionary game theory deemsthat participants have limited abilities to find the optimalbehaviors By learning and copying the best strategies finallyeach party can reach the equilibrium point Therefore dueto these advantages this paper adopts evolutionary gamedynamics to analyze the interaction between production inte-grators and modular suppliers By creating a dynamic evo-lutionary game model this paper examines the evolution-ary game process and the stable equilibrium of modularoutsourcing In addition by simulation analysis based onmultiagent modeling method and NetLogo platform thispaper further tests the effectiveness of the game model

This paper proceeds as follows Section 2 is literaturereview Section 3 is the construction of an evolutionarygame model and discussion of the equilibrium of the modelSection 4 is the simulation analysis of the game model byusing NetLogo software Section 5 is the conclusions

2 Literature Review

By analyzing transactional activities between the lead firmsand the ldquoturn-keyrdquo firms in the American electronic infor-mation industry Sturgeon (2002) made detailed analysis onmodular production networks and called it ldquothe new formof American industrial organizationrdquo [4] In general themodular production network can be characterized by func-tional partitioning into discrete scalable reusable modulesrigorous use of well-defined modular interfaces and makinguse of industry standards for interfaces Based onmodularitythe modular production network is a topology structureconnected by interface rules and guaranteed by allocationrules [15] The overall orderliness of this structure and theactive nodes in the structure can facilitate the evolutionof the complicated product system As a global productionsystem production integrators and modular suppliers withina modular production network have formed a new type ofcooperative relationship [16]

Currently the major studies on the modular productionnetwork mainly examined the historical evolution processof modular production networks and discussed the natureand causes of modular production networks [17ndash21] Otherscholars combined modularity with different network formssuch as enterprise value networks industrial value chainsindustrial clusters and innovation networks to study theformation mechanism of modular production network [22ndash26] However despite these abundant researches there isa lack of studies which explore the specific formation andevolution process of a single modular production networkand there is no study from member enterprise perspective toexamine the operation of modular production network Therelated researches are quite limited Next wemake a literaturereview on evolution of modular production network As a
















Cooperation (119909) 119880119860 119880119861

120587119860minus 119862119860 120587119861


120587119860 120587119861


119861 resp)




119880119860= 120587119860+ 119886120587 minus 119862

119860minus 120575119886120572

119880119861= 120587119861+ 119887120587 minus 119862

119861minus 120575119887120573

(1)




119864119862119860= 119910119880119860+ (1 minus 119910) (120587

119860minus 119862119860) (2)


119864119880119860= 119910120587119860+ (1 minus 119910) 120587

119860= 120587119860 (3)


119864119860= 119909119864119862119860+ (1 minus 119909) 119864119880119860

(4)


119864119862119861= 119909119880119861+ (1 minus 119909) (120587119861 minus 119862119861) (5)


119864119880119861= 119909120587119861+ (1 minus 119909) 120587119861 = 120587119861 (6)


119864119861= 119910119864119862119861+ (1 minus 119910) 119864

119880119861 (7)






119899119905(119902) =

119873119905(119902)

sumℎisin119876

119873119905 (ℎ)

(8)


119906119905(119902) = sum

ℎisin119876

119899119905 (ℎ) 119906119905 (119902 ℎ) (9)




119906119905= sum

119902isin119876

119899119905(119902) 119906119905(119902) (10)



1198731015840

119905(119902) = 119873

119905(119902) 119906119905(119902) (11)


119889119899119905(119902)

119889119905=1198731015840

119905(119902)sumℎisin119876

119873119905 (ℎ) minus 119873119905 (119902)sumℎisin119876119873

1015840

119905(ℎ)

[sumℎisin119876

119873119905 (ℎ)]2

(12)


119889119899119905(119902)

119889119905= 119899119905(119902) [119906

119905(119902) minus 119906

119905] (13)


119865 (119909) =119889119909

119889119905= 119909 (119864

119862119860minus 119864119860) = 119909 (1 minus 119909) (119864119862119860

minus 119864119880119860)


(14)


119866 (119910) =119889119910

119889119905= 119910 (119864

119862119861minus 119864119861) = 119910 (1 minus 119910) (119864

119862119861minus 119864119880119861)


(15)



lowast=




0lt 119910lowast (119889119865(119909)119889119909)|

119909=0lt 0 and (119889119865(119909)

119889119909)|119909=1


(2) When 119910lowast lt 1199100lt 1 (119889119865(119909)119889119909)|

119909=0gt 0 and (119889119865(119909)

119889119909)|119909=1



119861(119880119861minus




0lt 119909lowast (119889119866(119910)119889119910)|

119910=0lt 0 and (119889119866(119910)

119889119910)|119910=1


(2) When 119909lowast lt 1199090lt 1 (119889119866(119910)119889119910)|

119910=0gt 0 and (119889119866(119910)

119889119910)|119910=1



119861(119880119861minus120587119861+119862119861) 119862119860(119880119860minus120587119860+119862119860))






119909 = 0 119910 = 0 119862119860119862119861

+ minus (119862119860+ 119862119861) minus ESS

119909 = 0 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (119880

119860minus 120587119860) + (119880

119861minus 120587119861) + Not stable

119909 = 1 119910 = 0 119862119860119862119861

+ 119862119860+ 119862119861

+ Not stable

119909 = 1 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (120587

119860minus 119880119860) + (120587

119861minus 119880119861) minus ESS

119909 =119862119861

119880119861minus 120587119861+ 119862119861

=119862119861

119887120587 minus 120575119887120573

119910 =119862119860

119880119860minus 120587119860+ 119862119860

=119862119860

119886120587 minus 120575119886120572

119862119860119862119861(119880119860minus 120587119860) (119880119861minus 120587119861)

(119880119860minus 120587119860+ 119862119860) (119880119861minus 120587119861+ 119862119861)

+ 0 Saddle point

[

[



119861minus 119862119861minus 119909120587119861+ 119909119862119861)

]

]

(16)




119861(119887120587minus120575

119887120573) and 119910 = 119862

119860(119886120587minus120575

119886120572) For enterprise119860


119887120573) we have 119889119910119889119905 = 0 all the


119861(119887120587 minus 120575





119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+



119861(119887120587 minus 120575



119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+




119862119860(119886120587 minus 120575











119861is 3 and 119862

119861is 01 In








0

100

Prop

ortio

n

0 126

Time



0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04



(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast



(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1



(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1



0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast



5 Conclusions




Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Cooperation (119909) 119880119860 119880119861

120587119860minus 119862119860 120587119861


120587119860 120587119861


119861 resp)




119880119860= 120587119860+ 119886120587 minus 119862

119860minus 120575119886120572

119880119861= 120587119861+ 119887120587 minus 119862

119861minus 120575119887120573

(1)




119864119862119860= 119910119880119860+ (1 minus 119910) (120587

119860minus 119862119860) (2)


119864119880119860= 119910120587119860+ (1 minus 119910) 120587

119860= 120587119860 (3)


119864119860= 119909119864119862119860+ (1 minus 119909) 119864119880119860

(4)


119864119862119861= 119909119880119861+ (1 minus 119909) (120587119861 minus 119862119861) (5)


119864119880119861= 119909120587119861+ (1 minus 119909) 120587119861 = 120587119861 (6)


119864119861= 119910119864119862119861+ (1 minus 119910) 119864

119880119861 (7)






119899119905(119902) =

119873119905(119902)

sumℎisin119876

119873119905 (ℎ)

(8)


119906119905(119902) = sum

ℎisin119876

119899119905 (ℎ) 119906119905 (119902 ℎ) (9)




119906119905= sum

119902isin119876

119899119905(119902) 119906119905(119902) (10)



1198731015840

119905(119902) = 119873

119905(119902) 119906119905(119902) (11)


119889119899119905(119902)

119889119905=1198731015840

119905(119902)sumℎisin119876

119873119905 (ℎ) minus 119873119905 (119902)sumℎisin119876119873

1015840

119905(ℎ)

[sumℎisin119876

119873119905 (ℎ)]2

(12)


119889119899119905(119902)

119889119905= 119899119905(119902) [119906

119905(119902) minus 119906

119905] (13)


119865 (119909) =119889119909

119889119905= 119909 (119864

119862119860minus 119864119860) = 119909 (1 minus 119909) (119864119862119860

minus 119864119880119860)


(14)


119866 (119910) =119889119910

119889119905= 119910 (119864

119862119861minus 119864119861) = 119910 (1 minus 119910) (119864

119862119861minus 119864119880119861)


(15)



lowast=




0lt 119910lowast (119889119865(119909)119889119909)|

119909=0lt 0 and (119889119865(119909)

119889119909)|119909=1


(2) When 119910lowast lt 1199100lt 1 (119889119865(119909)119889119909)|

119909=0gt 0 and (119889119865(119909)

119889119909)|119909=1



119861(119880119861minus




0lt 119909lowast (119889119866(119910)119889119910)|

119910=0lt 0 and (119889119866(119910)

119889119910)|119910=1


(2) When 119909lowast lt 1199090lt 1 (119889119866(119910)119889119910)|

119910=0gt 0 and (119889119866(119910)

119889119910)|119910=1



119861(119880119861minus120587119861+119862119861) 119862119860(119880119860minus120587119860+119862119860))






119909 = 0 119910 = 0 119862119860119862119861

+ minus (119862119860+ 119862119861) minus ESS

119909 = 0 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (119880

119860minus 120587119860) + (119880

119861minus 120587119861) + Not stable

119909 = 1 119910 = 0 119862119860119862119861

+ 119862119860+ 119862119861

+ Not stable

119909 = 1 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (120587

119860minus 119880119860) + (120587

119861minus 119880119861) minus ESS

119909 =119862119861

119880119861minus 120587119861+ 119862119861

=119862119861

119887120587 minus 120575119887120573

119910 =119862119860

119880119860minus 120587119860+ 119862119860

=119862119860

119886120587 minus 120575119886120572

119862119860119862119861(119880119860minus 120587119860) (119880119861minus 120587119861)

(119880119860minus 120587119860+ 119862119860) (119880119861minus 120587119861+ 119862119861)

+ 0 Saddle point

[

[



119861minus 119862119861minus 119909120587119861+ 119909119862119861)

]

]

(16)




119861(119887120587minus120575

119887120573) and 119910 = 119862

119860(119886120587minus120575

119886120572) For enterprise119860


119887120573) we have 119889119910119889119905 = 0 all the


119861(119887120587 minus 120575





119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+



119861(119887120587 minus 120575



119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+




119862119860(119886120587 minus 120575











119861is 3 and 119862

119861is 01 In








0

100

Prop

ortio

n

0 126

Time



0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04



(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast



(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1



(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1



0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast



5 Conclusions




Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119899119905(119902) =

119873119905(119902)

sumℎisin119876

119873119905 (ℎ)

(8)


119906119905(119902) = sum

ℎisin119876

119899119905 (ℎ) 119906119905 (119902 ℎ) (9)




119906119905= sum

119902isin119876

119899119905(119902) 119906119905(119902) (10)



1198731015840

119905(119902) = 119873

119905(119902) 119906119905(119902) (11)


119889119899119905(119902)

119889119905=1198731015840

119905(119902)sumℎisin119876

119873119905 (ℎ) minus 119873119905 (119902)sumℎisin119876119873

1015840

119905(ℎ)

[sumℎisin119876

119873119905 (ℎ)]2

(12)


119889119899119905(119902)

119889119905= 119899119905(119902) [119906

119905(119902) minus 119906

119905] (13)


119865 (119909) =119889119909

119889119905= 119909 (119864

119862119860minus 119864119860) = 119909 (1 minus 119909) (119864119862119860

minus 119864119880119860)


(14)


119866 (119910) =119889119910

119889119905= 119910 (119864

119862119861minus 119864119861) = 119910 (1 minus 119910) (119864

119862119861minus 119864119880119861)


(15)



lowast=




0lt 119910lowast (119889119865(119909)119889119909)|

119909=0lt 0 and (119889119865(119909)

119889119909)|119909=1


(2) When 119910lowast lt 1199100lt 1 (119889119865(119909)119889119909)|

119909=0gt 0 and (119889119865(119909)

119889119909)|119909=1



119861(119880119861minus




0lt 119909lowast (119889119866(119910)119889119910)|

119910=0lt 0 and (119889119866(119910)

119889119910)|119910=1


(2) When 119909lowast lt 1199090lt 1 (119889119866(119910)119889119910)|

119910=0gt 0 and (119889119866(119910)

119889119910)|119910=1



119861(119880119861minus120587119861+119862119861) 119862119860(119880119860minus120587119860+119862119860))






119909 = 0 119910 = 0 119862119860119862119861

+ minus (119862119860+ 119862119861) minus ESS

119909 = 0 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (119880

119860minus 120587119860) + (119880

119861minus 120587119861) + Not stable

119909 = 1 119910 = 0 119862119860119862119861

+ 119862119860+ 119862119861

+ Not stable

119909 = 1 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (120587

119860minus 119880119860) + (120587

119861minus 119880119861) minus ESS

119909 =119862119861

119880119861minus 120587119861+ 119862119861

=119862119861

119887120587 minus 120575119887120573

119910 =119862119860

119880119860minus 120587119860+ 119862119860

=119862119860

119886120587 minus 120575119886120572

119862119860119862119861(119880119860minus 120587119860) (119880119861minus 120587119861)

(119880119860minus 120587119860+ 119862119860) (119880119861minus 120587119861+ 119862119861)

+ 0 Saddle point

[

[



119861minus 119862119861minus 119909120587119861+ 119909119862119861)

]

]

(16)




119861(119887120587minus120575

119887120573) and 119910 = 119862

119860(119886120587minus120575

119886120572) For enterprise119860


119887120573) we have 119889119910119889119905 = 0 all the


119861(119887120587 minus 120575





119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+



119861(119887120587 minus 120575



119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+




119862119860(119886120587 minus 120575











119861is 3 and 119862

119861is 01 In








0

100

Prop

ortio

n

0 126

Time



0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04



(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast



(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1



(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1



0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast



5 Conclusions




Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909 = 0 119910 = 0 119862119860119862119861

+ minus (119862119860+ 119862119861) minus ESS

119909 = 0 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (119880

119860minus 120587119860) + (119880

119861minus 120587119861) + Not stable

119909 = 1 119910 = 0 119862119860119862119861

+ 119862119860+ 119862119861

+ Not stable

119909 = 1 119910 = 1 (119880119860minus 120587119860) (119880119861minus 120587119861) + (120587

119860minus 119880119860) + (120587

119861minus 119880119861) minus ESS

119909 =119862119861

119880119861minus 120587119861+ 119862119861

=119862119861

119887120587 minus 120575119887120573

119910 =119862119860

119880119860minus 120587119860+ 119862119860

=119862119860

119886120587 minus 120575119886120572

119862119860119862119861(119880119860minus 120587119860) (119880119861minus 120587119861)

(119880119860minus 120587119860+ 119862119860) (119880119861minus 120587119861+ 119862119861)

+ 0 Saddle point

[

[



119861minus 119862119861minus 119909120587119861+ 119909119862119861)

]

]

(16)




119861(119887120587minus120575

119887120573) and 119910 = 119862

119860(119886120587minus120575

119886120572) For enterprise119860


119887120573) we have 119889119910119889119905 = 0 all the


119861(119887120587 minus 120575





119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+



119861(119887120587 minus 120575



119862119861minus 119864119880119861

= 119909(119880119861minus 120587119861+




119862119860(119886120587 minus 120575











119861is 3 and 119862

119861is 01 In








0

100

Prop

ortio

n

0 126

Time



0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04



(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast



(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1



(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1



0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast



5 Conclusions




Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0

100

Prop

ortio

n

0 126

Time



0= 04

0

100

Prop

ortio

n

0 200

Time



0= 04



(1) We give 1199090= 03 and 119910

0= 04 That is 0 lt 119909

0lt 119909lowast



(2) We give 1199090= 07 and 119910

0= 04 That is 119909lowast lt 119909

0lt 1



(3) We give 1199090= 07 and 119910

0= 09 That is 119909lowast lt 119909

0lt 1



0

200

Prop

ortio

n

0 126

Time



0= 09

0

100

Prop

ortio

n

0 288

Time



0= 09

(4) We give 1199090= 03 and 119910

0= 09 That is 0 lt 119909

0lt 119909lowast



5 Conclusions




Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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