劉念平

摘 要 在Bala and Goyal (2000)提出的雙向流網(wǎng)絡(luò)形成模型基礎(chǔ)上，研究當(dāng)個(gè)體存在異質(zhì)性時(shí)對(duì)納什網(wǎng)絡(luò)存在性的影響．分別針對(duì)幾種不同的環(huán)境設(shè)定下的個(gè)體異質(zhì)性進(jìn)行研究，發(fā)現(xiàn)個(gè)體的連接成本異質(zhì)性是決定納什網(wǎng)絡(luò)存在性的重要因素；但相較于個(gè)體的連接成本而言，連接價(jià)值的異質(zhì)性對(duì)納什網(wǎng)絡(luò)存在性的影響不大．

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關(guān)鍵詞 網(wǎng)絡(luò)形成；納什網(wǎng)絡(luò)；異質(zhì)性；非合作博弈

Abstract The importance of heterogeneity in the existence of Nash networks was explained by using the twoway flow model of network formation initiated by Bala and Goyal (2000). Furthermore, the heterogeneity of link costs in different settings was examined and it is found find that the heterogeneity of link costs plays a crucial role in the existence of Nash networks. However, the heterogeneity of values is not as important as that of link costs.

Key words Network formation; Nash networks; Heterogeneity; Noncooperative Games

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中圖分類號(hào) F224.1 文獻(xiàn)標(biāo)識(shí)碼 A

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1 Introduction

There is substantial evidence to support the role that social and economic networks play in individual behavior and payoffs, and in aggregate social outcomes

Network effects on job search (Montgomery(1991)[2]), trade (Nishiguchi (1994)[3]), the granting of credit (McMillan and Woodruff (1999)[4]) and mutual insurance (Bloch and Genicot (2008)[5]).. Recent developments in this area have focused increasing attention on the theoretical analysis of network formation. In particular, Bala and Goyal (2000)[1] proposed a noncooperative model of network formation, in which networks are categorized as either oneway or twoway flow, according to the flow of resources, while the addition and deletion of links are unilateral decisions that allow players to access resources. In a twoway flow network, a link between two players allows both players to access the others resources, regardless of who pays for the link.

This model has been extended in various directions. Most studies have focused on the architecture of networks that result in equilibrium. Compared with this issue, the existence of Nash networks has been less systematically explored. This paper complements the existing literature on the existence of Nash networks for the noncooperative model of network formation.

Bala and Goyal (2000)[1] proved that Nash networks exist for both oneway and twoway flow networks when the values and costs of links are both homogeneous. However, individual differences arise quite naturally in many contexts. For instance, some individuals possess skills that are scarce or the opportunity costs of maintaining their connections are lower, which makes them more valuable to others. Indeed, the Greek proverb “success has many friends” indicates that people prefer to connect with highly rewarded individuals. The heterogeneity of players plays an important role in network formation. In a oneway flow model, Billand et al. (2008) endprint

In this paper, we investigate the existence of Nash networks in a twoway flow model with linear payoffs and different types of heterogeneity. We do not allow for decay and only consider pure strategies. We show that Bala and Goyals (2000) [1] results are not entirely robust when the link costs are heterogeneous or partnerheterogeneous. In a number of different settings, we find that the heterogeneity of the link costs plays a major role in the existence of Nash networks. However, the role played by the heterogeneity of values is not as important as the heterogeneity of the link costs. We also show that a Nash network always exists when the link costs are ownerhomogeneous. We provide bounds on the link costs that guarantee the existence of Nash networks when the link costs are partnerheterogeneous.

This paper is organized as follows. In section 2 we set the basic twoway flow model with heterogeneous costs and values of the links. In section 3, we examine the existence of Nash networks for this model. In section 4, we study this problem more precisely under various heterogeneity conditions for the link costs regardless of the heterogeneity of the values. Section 5 concludes.

2 Model and notations

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5 Conclusion

The literature on twoway flow models shows that over some parameter ranges, for instance when the link costs and values are both homogeneous or the link costs are homogeneous and the values vary freely, Nash networks always exist. However, the question of whether Nash networks exist over the entire parameter range has not been resolved.

This paper complements the existing literature on the existence of Nash networks for a twoway flow model with linear payoff functions. We first find that the heterogeneity of the link costs plays a key role in the existence of Nash networks in pure strategies, and link costs are obviously more important. Indeed, if values vary freely but link costs are ownerhomogeneous, then Nash networks always exist; however, they do not always exist when the link costs and values are both heterogeneous. Second, we demonstrate that Nash networks do not always exist when the link costs are partnerheterogeneous unless we provide bounds on the link costs.

References

［1］ V BALA,S GOYAL. A noncooperative model of network formation [J].Econometrica,2000, 68(5): 1181-1229. 

［2］ James D MONTGOMERY. Social networks and labormarket outcomes: toward an economic analysis.[J]The American economic review ,1991,81 (5): 1408-1418. endprint

［3］ T NISHIGUCHI. Strategic industrial sourcing: the Japanese advantage[M].Oxford:Oxford University Press, 1994.

［4］ J MCMILLAN, C WOODRUFF. Interfirm relationships and informal credit in Vietnam[J]. The Quarterly Journal of Economics, 19991,14 (4): 1285-1320. 

［5］ F BLOCH, G GENICOT, D RAY. Informal insurance in social networks[J]. Journal of Economic Theory ,2008,143(1): 36-58. 

［6］ P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks in oneway flow models[J]. Economic Theory, 2008,37(3): 491-507. 

［7］ J DERKS, J KUIPERS, M TENNEKES,et al. Existence of Nash networks in the oneway flow model of network formation[J]. Modeling, Computation and Optimization, World Scientific,2009forthcoming.

［8］ J DERKS, M TENNEKES. A note on the existence of Nash networks in oneway flow models[J].Economic Theory ,2009,41 (3): 515-522. 

［9］ H HALLER, J KAMPHORST, S SARANGI. (Non) existence and scope of Nash networks[J]. Economic Theory,2007, 31(3): 597-604．

［10］P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks and partner heterogeneity[J]. Mathematical Social Sciences ,2012,64(2): 152-158.

［11］A GALEOTTI, S GOYAL, J KAMPHORST. Network formation with heterogeneous players[J].Games and Economic Behavior, 2006,54(2): 353-372.

endprint

［3］ T NISHIGUCHI. Strategic industrial sourcing: the Japanese advantage[M].Oxford:Oxford University Press, 1994.

［4］ J MCMILLAN, C WOODRUFF. Interfirm relationships and informal credit in Vietnam[J]. The Quarterly Journal of Economics, 19991,14 (4): 1285-1320. 

［5］ F BLOCH, G GENICOT, D RAY. Informal insurance in social networks[J]. Journal of Economic Theory ,2008,143(1): 36-58. 

［6］ P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks in oneway flow models[J]. Economic Theory, 2008,37(3): 491-507. 

［7］ J DERKS, J KUIPERS, M TENNEKES,et al. Existence of Nash networks in the oneway flow model of network formation[J]. Modeling, Computation and Optimization, World Scientific,2009forthcoming.

［8］ J DERKS, M TENNEKES. A note on the existence of Nash networks in oneway flow models[J].Economic Theory ,2009,41 (3): 515-522. 

［9］ H HALLER, J KAMPHORST, S SARANGI. (Non) existence and scope of Nash networks[J]. Economic Theory,2007, 31(3): 597-604．

［10］P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks and partner heterogeneity[J]. Mathematical Social Sciences ,2012,64(2): 152-158.

［11］A GALEOTTI, S GOYAL, J KAMPHORST. Network formation with heterogeneous players[J].Games and Economic Behavior, 2006,54(2): 353-372.

endprint

［3］ T NISHIGUCHI. Strategic industrial sourcing: the Japanese advantage[M].Oxford:Oxford University Press, 1994.

［4］ J MCMILLAN, C WOODRUFF. Interfirm relationships and informal credit in Vietnam[J]. The Quarterly Journal of Economics, 19991,14 (4): 1285-1320. 

［5］ F BLOCH, G GENICOT, D RAY. Informal insurance in social networks[J]. Journal of Economic Theory ,2008,143(1): 36-58. 

［6］ P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks in oneway flow models[J]. Economic Theory, 2008,37(3): 491-507. 

［7］ J DERKS, J KUIPERS, M TENNEKES,et al. Existence of Nash networks in the oneway flow model of network formation[J]. Modeling, Computation and Optimization, World Scientific,2009forthcoming.

［8］ J DERKS, M TENNEKES. A note on the existence of Nash networks in oneway flow models[J].Economic Theory ,2009,41 (3): 515-522. 

［9］ H HALLER, J KAMPHORST, S SARANGI. (Non) existence and scope of Nash networks[J]. Economic Theory,2007, 31(3): 597-604．

［10］P BILLAND, C BRAVARD, S SARANGI. Existence of Nash networks and partner heterogeneity[J]. Mathematical Social Sciences ,2012,64(2): 152-158.

［11］A GALEOTTI, S GOYAL, J KAMPHORST. Network formation with heterogeneous players[J].Games and Economic Behavior, 2006,54(2): 353-372.

endprint