Contests In Networks:A Variational Inequality Approach

Contests are widely adopted to model R&D,rent seeking,political campaigns,patent races,advertisement,conflicts,etc.We study a contest game among players fighting in a nexus of conflicts.The conflict game can be represented by the bipartite graph,where there are both the set of players and the set of battles.In this paper we assume that for a given battle,the player who allocates the higher measure of resources is more likely to win that battle,known as Contest Success Function(CSF).Cost function is twice continuously differentiable,and it is weakly monotone and weakly convex.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.Given the generality of our conflict model,there are a few challenging points in showing the existence of an equilibrium:(1)The winning probability function has a discrete jump when contestants exert zero effort in that battle.Therefore the payoffs in the conflict game CF are not continuous.Furthermore,the dimensionality of discontinuity can be very high as each player participates in multiple battles simultaneously.Also,the strategy space can be unbounded.(2)The conflict game,due to multiple battles and multi-dimensional efforts,is not an aggregate game.(3)The conflict game is not a supermodular game.The first challenging point implies that standard existence theorems based on Kakutani’s fixed-point theorem cannot be applied.We approximate the original game by truncating the strategy space:the game CF∈ is defined by requiring that the effort of every player on each battle has a minimal lower bound ∈>0;the other components such as cost,values and conflict topology remain the same.The major part of the proof is to show that:(1)the truncated game CF∈ has an equilibrium for every ∈;(2)the limiting strategy x*must avoid points of discontinuity in the original game,i.e.,for every battle there exists at least one participant exerting strictly positive effort;and(3)there is no profit deviation under the limiting strategy x*in the original conflict game CF.Truncation of efforts removes the discontinuity in the winning probabilities.The equivalent characterization of a Nash equilibrium of a generic smooth concave game using Variational Inequality is known in the literature.Our proposition extends the VI characterization of equilibrium accommodating non-closeness of domain and disconti-nuity in payoffs.Under the assumptions of the model,the operator F is monotone.The set of equilibria is non-empty and convex.Moreover,assuming strong monotonicity of cost functions,the equilibrium of a conflict game is unique.Suppose conflict structure is com-plete,and each player’s cost is of the pure budget case.Any equilibrium must be of type S2.Moreover,the equilibrium is unique.On the other hand,multiple equilibria arise due to the existence of a powerful player.Given any conflict structure,contest technology and value,there exists a cost function for each player,that satisfies Assumption 2.1,such that the resulting conflict game has a continuum of equilibria.Since an equilibrium may not be interior,the standard tool to derive comparative statics using the Implicit Function Theorem does not directly apply here.VI is obtained from the complementarity slackness conditions for each player’s payoff maximization in equilibrium.If equilibrium is non-degenerate,then there exists an open neighborhood such that,the unique equilibrium is continuously differentiable.Conduct Implicit Function Theorem partially on the set of slack variables,because the non-degeneracy condition guarantees that the binding variables indeed remain binding..An interior equilibrium is clearly non-degenerate,so we can employ classical Implicit Function Theorem approach to conduct comparative statics.In addition,both equilibrium payoffs and aggregate efforts are continuously differentiable in the neighbour of equilibrium.Given these sensitivity analyses,we are able to analyze several problems on contest design,and subsidy policy from the perspective of the planner.If the optimal battle subsidy policy is based on aggregate efforts,the planner should target the battle with larger size to incentive greater aggregate effort.If the optimal battle subsidy policy is based on aggregate payoffs,the planner should target the battle with smaller size to induce higher aggregate payoff.If an accuracy level of the contest is greater than 1,the payoff function may not be concave and the solution is not necessarily a Nash equilibrium.In addition,we briefly inves-tigate the issue of stability for the Nash equilibrium in the conflict game.If the adjustment process starts at any point inside the neighbourhood,the solution to the system converges to the unique equilibrium(locally asymptotically stable).Finally,we can adopt VI techniques to analyze the equilibrium under a general feasibility set.The rest of the article is organized as follows.The model with its assumptions is presented in Section 2.In Section 3 we show that there exist Nash equilibrium in the game and by Variational Inequality,a unique one is obtained under some moderate conditions.Section 4 conducts comparative statics.Section 5 concludes.Our paper makes five contributions to the literature:First,we contribute to the contest literature by providing a more general conflict game among multiple players competing in multiple battles.The structure of conflict,which describes who participates in which battles,can be arbitrary.Consequently,many existing models of single-or multi-battle contests are just special cases when the conflict network takes on particular forms.Moreover,the contest production function is an increasing function.Second,although payoff is discontinuous,we prove the existence of NE by truncating the strategy space.Third,under strong monotonic-ity we can characterize the equilibrium set and show the uniqueness using techniques from VI.Fourth,component of the game is changed due to an exogenous shock,and we show how to use the VI approach to conduct comparative statics of an equilibrium which may be in the corner.Fifth,we provide an example to show the importance of sufficient condition of NE.