Nexus of Conflicts: A Variational Inequality Approach

50 Pages Posted: 31 Aug 2017 Last revised: 30 Jun 2018

See all articles by Jin Xu

Jin Xu

Shandong University

Junjie Zhou

National University of Singapore (NUS), Department of Economics

Date Written: January 28, 2018


We study a contest game among players fighting in a nexus of conflicts. Each player confronts different competitors in heterogeneous battlefields, and decides how much effort to exert in order to maximize the expected value of winning prizes net the cost of efforts. We show that the set of pure strategy Nash equilibria is nonempty and convex, and provide equivalent characterizations using techniques from Variational Inequality (VI). We illustrate that strong monotonicity of cost functions always implies uniqueness of equilibrium regardless of the structure of conflict. We provide examples in which the set of Nash equilbira is a continuum when strong monotonicity is violated. In equilibrium, players may strategically choose to abandon certain battlefields, and closed form solutions are in general not available. Nevertheless, using VI approach, we conduct extensive comparative statics analysis with respect to the parameters of the model on equilibrium efforts and payoffs, and discuss several applications in terms of prize allocation and contest design . Our model incorporates many existing models of single or multi-battle contests as special cases when either or both the conflict network and cost function take particular forms.

Keywords: network games, multi-battle contests, Variational Inequality, monotone operator, uniqueness

JEL Classification: C72, D74, D85

Suggested Citation

Xu, Jin and Zhou, Junjie, Nexus of Conflicts: A Variational Inequality Approach (January 28, 2018). Available at SSRN: or

Jin Xu (Contact Author)

Shandong University ( email )

27 Shanda Nanlu
South Rd.
Jinan, Shandong 250100

Junjie Zhou

National University of Singapore (NUS), Department of Economics ( email )



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