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A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis

by   Francesca Parise, et al.

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.


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1 Introduction

In many social and economic settings, decisions of individuals are affected by the actions of their friends, colleagues, and peers. Examples include adoption of new products and innovations, opinion formation and social learning, public good provision, financial exchanges and international trade. Network games have emerged as a powerful framework to study these settings with particular focus on how the underlying patterns of interactions, governed by a network, affect the economic outcome.222See for instance Galeotti (2010); Goyal (2012); Nagurney (2013); Jackson and Zenou (2014); Bramoullé and Kranton (2015); Jackson et al. (2016). Durlauf (2004) offers a survey of the theoretical and empirical literature on peer effects. For tractability reasons, many of the works in this area studied games with special structure (e.g., quadratic cost functions, scalar non-negative strategies) or special properties (e.g., games of strategic complements or substitutes for which the best response of each agent is increasing or decreasing in the other agents strategies).333See Bulow et al. (1985); Milgrom and Roberts (1990); Vives (1990). These works relied on disparate techniques for their analysis which then lead to different focal network properties for existence and uniqueness of equilibria. For example, papers on network games of strategic complements typically relate equilibrium properties to the spectral radius of the (weighted) adjacency matrix of the network (see e.g. Ballester et al. (2006); Jackson and Zenou (2014); Acemoglu et al. (2015)) while papers considering network games of strategic substitutes highlight the role of the minimum eigenvalue of the adjacency matrix (see e.g. Bramoullé et al. (2014); Allouch (2015)). Moreover, there has been relatively little work on network games that feature neither strategic complements nor substitutes and possibly involve multidimensional strategy sets (e.g., models where agents decide their effort level on more than one activity, Chen et al. (2018)).

In this paper, we provide a unified framework based on a variational inequality reformulation of the Nash equilibrium to study equilibrium properties of network games including existence and uniqueness, convergence of the best response dynamics and comparative statics. Our framework extends the literature in multiple dimensions. It applies to games of strategic complements, substitutes as well as games with mixed strategic interactions. It provides a systematic understanding of which spectral properties of the network (spectral norm, minimum eigenvalue or infinity norm of the adjacency matrix) are relevant in establishing fundamental properties of the equilibrium. Moreover, it covers network games with multidimensional and constrained strategy sets.

Our work is built on the key observation that the analysis of network games can be performed in two steps. In the first step we focus on the operator of the variational inequality associated with the game, which is typically referred to as the game Jacobian and we derive sufficient conditions on the network and on the cost functions to guarantee that it possesses either one of three fundamental properties: strong monotonicity (which is a stronger version than the strict diagonal concavity condition used in Rosen (1965)), uniform block P-function and uniform P-function.444 While sufficient conditions for the former two properties in terms of the gradient of the game Jacobian have been studied in the literature (positive definiteness and the condition discussed in Scutari et al. (2014), respectively), we here suggest a novel sufficient condition for the uniform P-function property (which we term uniform P-matrix condition). Our result extends the P-matrix condition used in (Facchinei and Pang, 2003, Proposition 3.5.9(a)) and is needed to guarantee existence and uniqueness of the solution of a variational inequality without imposing boundedness assumptions on its closed convex set. Our sufficient conditions are formulated in terms of different network measures, i.e., spectral norm, infinity norm and minimum eigenvalue of the weighted adjacency matrix, as detailed in Assumptions 4.1a), 4.1b) and 4.1c). We highlight the relations between these conditions and we show that for symmetric networks, the condition in terms of the minimum eigenvalue is the least restrictive and hence is satisfied by the largest set of networks, whereas for asymmetric networks conditions in terms of spectral norm and infinity norm cannot be compared and are satisfied by different sets of networks. While the conditions that involve spectral norm and minimum eigenvalue have appeared in the study of special network games, the condition in terms of the infinity norm is new and arises naturally for games played over asymmetric networks where each agent is only influenced by a relatively small number of neighbors.

In the second step, we combine our sufficient conditions (connecting network properties to properties of the game Jacobian) with the theory of variational inequalities to derive equilibrium properties. A summary of our results is presented in Table 1. Specifically, we extend previous literature results on existence and uniqueness of the Nash equilibrium to games with multidimensional constrained strategy sets and mixed strategic effects, we provide sufficient conditions to guarantee convergence of continuous and discrete best response dynamics and finally we study how parameter variations affect the Nash equilibrium. Our first result in this context is to guarantee Lipschitz continuity of the Nash equilibrium under the block P-function property.555This result extends (Dafermos, 1988, Theorem 2.1) which holds under the more restrictive strong monotonicity condition. We then build on a sensitivity analysis result for variational inequalities applied to general games given in Facchinei and Pang (2010); Friesz and Bernstein (2016) to establish conditions under which the Nash equilibrium is differentiable and we derive an explicit formula for its sensitivity.666 Contrary to previous works, our formula for the sensitivity of the Nash equilibrium depends on primal variables only. The only result of such type that we are aware of is (Dafermos, 1988, Theorem 3.1). The formula therein is however obtained following geometric arguments and consequently depends on the orthogonal projection operator on the set of active constraints.

Assumption 4.1a Assumption 4.1b Assumption 4.1c
( Thm. B.1) (Thm. B.2) (Thm. B.3) (Thm. B.4)
Existence and
uniqueness (Prop. 2a) (Prop. 2b) (Prop. 2a) (Prop. 2c)
BR dynamics
(continuous) (Thm. 6.2) (Thm. 6.2) (see assumptions of Thm. 6.1) (see assumptions of Thm. 6.3)
BR dynamics
(discrete, simult.) (Thm. 6.2) (Thm. 6.2) (Ex. E.1A) (Ex. E.1A )
continuity (Thm. 7.1) (Thm. 7.1) (Thm. 7.1) (Thm. 7.1)
Table 1: Summary of the relation between the considered network and cost conditions (i.e. Assumption 4.1a, 4.1b, 4.1c) and properties of the Nash equilibrium of the game.

To illustrate our theory we consider three running examples. First, we consider scalar linear quadratic network games for which the best response is a (truncated) linear function. These games have been extensively studied in the literature as illustrated in the seminal papers Ballester et al. (2006), where a model featuring local complements and global substitutes is considered (mainly motivated by crime applications) and Bramoullé et al. (2014), where games with mixed strategic effects are considered. The results in Bramoullé et al. (2014) are derived by using the approach of potential games, first introduced in Monderer and Shapley (1996). Expanding on Bramoullé et al. (2014), we show that for scalar linear quadratic games the existence of a potential function is related to a condition on the symmetry of the network. When such condition is violated the potential approach suggested in Bramoullé et al. (2014) cannot be applied. The variational inequality approach presented in this work can be seen as an extension of the potential approach for cases when a potential function is not available. In fact, potential games whose potential function is strongly convex are a subclass of strongly monotone games. By leveraging on the theory of variational inequalities (instead of convex optimization) we show how the results in Ballester et al. (2006) and Bramoullé et al. (2014) can be recovered and extended to linear quadratic models that are not potential.

A main feature of linear quadratic games, that significantly simplifies their analysis, is the fact that the best response function is linear. A few works in the literature have extended the analysis of network games with scalar strategies to nonlinear settings by focusing on cases where the best response function is nonlinear but monotone. For example, Belhaj et al. (2014) considers a case where the best response function is increasing leading to a supermodular game. On the other hand, Allouch (2015) focuses on a model of public good games where the best response function is decreasing. Conditions for existence and uniqueness in both these cases have been derived using techniques tailored to the special structure of the problem. As second motivating example, we consider network games with scalar non-negative strategies where the best response is nonlinear and non-monotone, representing e.g. races and tournaments where agents work hardest when network externalities are small (neck and neck race) while they reduce their efforts if they fall behind (discouragement effect).

Finally, all the games mentioned above feature scalar strategies. As third motivating example, we consider a network game where each agent decides on how much effort to exert in more than one activity at the time. This model of multiple activities over networks was first suggested in Chen et al. (2018) and can be used to study agents engagement in activities that are interdependent such as crime and drug use (if activities are complements) or crime and education (if activities are substitutes). Chen et al. (2018) focuses on the particular case when the network effects within the same activity are complements and derives results on existence and uniqueness of the Nash equilibrium by using a similar approach as in the single activity case studied in Ballester et al. (2006). Our framework enables the study of cases where network effects within the same activity can be substitutes and interdependencies are present not only in the cost function but also in the strategy sets (e.g. because of budget constraints).

In addition to the papers cited above, our paper is most related to the seminal works of Harker and Pang (1990); Facchinei and Pang (2003, 2010); Scutari et al. (2014); Friesz and Bernstein (2016) and references therein. These works discuss the use of variational inequalities to analyze equilibria of general games. Our novel contribution is to focus on network games and unfold the effect of network properties on the game Jacobian and consequently on the equilibrium properties.

The only papers that we are aware of that study properties of the Nash equilibrium in network games by using variational inequalities are Ui (2016), Melo (2018) and Naghizadeh and Liu (2017a). All these works consider network games with scalar non-negative strategies. Specifically, Ui (2016)

considers Bayesian network games with linear quadratic cost functions.

Melo (2018) considers existence, uniqueness and comparative statics for scalar network games with symmetric unweighted networks, by focusing on strong monotonicity of the game Jacobian. For games with strategic substitutes, we show via a counterexample that strong monotonicity cannot be guaranteed under the minimum eigenvalue condition, not even for scalar network games with symmetric networks (see Example B.4). In fact our analysis suggests that for scalar games of strategic substitutes, the natural property is the uniform P-matrix condition. In addition to this substantial difference, our paper is distinct from Melo (2018) in the following ways: i) we do not consider only strong monotonicity but also uniform (block) P-functions, ii) we investigate how properties of the game Jacobian relate to network properties considering not only the minimum eigenvalue but also the spectral and infinity norm, iii) we consider networks that might be asymmetric and agents with possibly multidimensional strategy sets, iv) we consider games with mixed strategic effects for which the best response might not be monotone as a function of the neighbor aggregate, v) besides uniqueness and comparative statics, we also study convergence of best response dynamics. The recent paper Naghizadeh and Liu (2017a) focuses on a special case of scalar network games and derives conditions in terms of the absolute value of the elements of the adjacency matrix for uniqueness of Nash equilibrium. This may be overly restrictive since taking the absolute value loses structural properties associated with games of strategic substitutes. Naghizadeh and Liu (2017b) studies public good network games by using linear complementarity problems (which are a subclass of variational inequalities). We remark that variational inequalities have been used to study specific network economic models such as spatial price equilibrium models, traffic networks, migration models and market equilibria, as reviewed for example in Nagurney (2013); Facchinei and Pang (2003). In these settings however the network typically appears as part of the constraints and not in the cost function. Consequently, the network effects studied in this paper do not appear. Finally, Xu and Zhou (2017) uses variational inequality theory to study contest games whose cost function does not possess the aggregative structure considered in this paper (i.e. the cost depends on the strategies of the other agents individually and not on their aggregate).

It is also worth highlighting that network games have similarities with aggregative games. Equilibrium properties in aggregative games have been extensively studied in Novshek (1985); Kukushkin (1994); Jensen (2005); Acemoglu and Jensen (2013); Jensen (2010); Kukushkin (2004); Cornes and Hartley (2012); Dubey et al. (2006). Moreover, motivated by technological applications such as demand-response energy markets or communication networks, several papers have studied distributed dynamics for convergence to the equilibria in aggregative games (see Chen et al. (2014); Koshal et al. (2012, 2016); Grammatico et al. (2016); Paccagnan et al. (2016)). The main difference is that while in aggregative games each agent is affected by the same aggregate of the other agents strategies, in network games this aggregate is agent and network dependent.

Our paper is organized as follows. In Section 2 we introduce the framework of network games and three motivational examples. In Section 3

we recall the connection between variational inequalities and game theory, we summarize properties that guarantee existence and uniqueness of the solution to a variational inequality and we present our technical results on the uniform P-matrix condition. Section

4 presents an overview of our results relating network and cost conditions to properties of the game Jacobian and illustrates these conditions for several networks of interest. Sections 5, 6 and 7 exploit the results of Section 4 to study existence and uniqueness, convergence of best response dynamics and comparative statics, respectively. Section 8 concludes the paper. Some basic matrix properties and lemmas used in the main text are summarized in Appendix A. Appendix B provides the technical statements of the results anticipated in Section 4 and Appendix C their proofs. Appendix D proves our technical result on the uniform P-matrix condition. Appendices E and F expand on Sections 6 and 7, respectively. Definitions, examples and technical statements provided in the appendices are labeled with the corresponding letter.


We denote the gradient of a function by Given , denotes the set of integer numbers in the interval . denotes the

-dimensional identity matrix,

the vector of unit entries and

the th canonical vector. Given , () , denotes the th row of , denotes the spectral radius of and the spectrum. denotes the Kronecker product and . Given matrices , is the block diagonal matrix whose th block is . Given vectors , . Note that is the th block component of . We instead denote by , for , the -th scalar element of . The symbols and denote element-wise ordering relations for vectors. is the boundary of the set . denotes the projection of the vector in the closed and convex set according to the weighted norm ; . For the definition of vector and matrix norms see Appendix A.

2 Network games

Consider a network game with players interacting over a weighted directed graph described by the non-negative adjacency matrix . The element represents the influence of agent ’s strategy on the cost function of agent . We assume for all . We say that is an in-neighbor of if . We denote by the set of in-neighbors of agent . From here on we refer to in-neighbors simply as neighbors since we do not consider out-neighbors and we use the terms network and graph interchangeably. Each player aims at selecting a vector strategy in its feasible set to minimize a cost function


which depends on its own strategy and on the aggregate of the neighbors strategies, , obtained as the weighted linear combination

where is a vector whose -th block component is equal to the strategy of agent . The best response of agent to the neighbor aggregate is defined as

where we recall that does not depend on since . A set of strategies in which no agent has an incentive for unilateral deviations (i.e., each agent is playing a best response to other agents strategies) is a Nash equilibrium, that is, , with for all , is a Nash equilibrium if for all players it holds .

Network games can be used to model a vast range of economic settings. Nonetheless, for tractability of analysis and for generating crisp insights the literature adopted specific structural assumptions on best response functions. The next three examples illustrate such assumptions and show how our framework can unify previous literature results and enable consideration of richer economic interactions.

Example 2.1 (label=ex:lq)

(Scalar linear quadratic games) Consider a network game where each agent chooses a scalar non-negative strategy (representing for instance how much effort he exerts on a specific activity) to minimize the linear quadratic cost function


with . Network games with payoffs of this form have been widely used in the literature to study various economic settings including private provision of public goods (e.g., innovation, R&D, health investments in particular vaccinations; see Bramoullé and Kranton (2007); Bramoullé et al. (2014)) and games with local payoff complementarities but global substitutability (e.g., Ballester et al. (2006) suggests that the cost of engaging in a criminal activity is lower when friends participate in such behavior, yet may be higher when overall crime levels increase). For these games, the best response of agent  is given by a (truncated) linear function of the neighbor aggregate :


From Eq. (3) it is evident that the payoff parameter captures how much the neighbor aggregate affects the equilibrium strategy of agent . When the are the same for all , we say that the game has homogeneous weights. We remark that even in this case, the neighbor aggregate may be heterogeneous across agents. We also note that if for each agent (), this is a game of strategic substitutes (complements), meaning that each agent’s best response decreases (increases) in other agents actions.

Linear quadratic network games have been studied in Bramoullé et al. (2014) using the theory of potential games. This approach entails constructing a proxy maximization problem whose stationary points provide the set of Nash equilibria. By Monderer and Shapley (1996) a necessary condition for the existence of an exact potential function is


For linear quadratic network games, this amounts to the restriction that


Bramoullé et al. (2014) focuses on symmetric networks and (with the exception of the last section)777 Section 6 in Bramoullé et al. (2014) shows that for heterogeneous weights, there exists a linear change of variables such that the game in the new coordinates is potential. More in general, for linear quadratic games this happens if there exists a vector such that , which is a generalization of (4). See Lemma A.5 for more details. on homogeneous weights. In this case condition (5) is met, enabling the use of the potential approach. Our subsequent analysis studies asymmetric networks and heterogeneous weights.888 Instead, we do not explicitly consider cases where the externalities may have different signs for the same agent but different neighbors. This case, corresponds to a matrix with both positive and negative entries. Most of our results hold also in this case, but we find that this additional generality obfuscates the overall presentation.

The linear quadratic model described above has been extensively studied in the literature, partially due to the fact that the linearity of its best response function allows for a simple yet informative analysis. As the second main example, we describe a variation of the previous model, that features a nonlinear best response.

Example 2.2 (label=competition)

(Races and tournaments) Consider a network game where each player has a scalar strategy and a nonlinear best response (as a function of the neighbor aggregate, denoted by for simplicity)


We assume , and . Special cases of this model have been considered in the literature for example in Belhaj et al. (2014) with the additional assumption (so that the best response is increasing in other agents actions, leading to a game of strategic complements) or in Allouch (2015) with (so that the best response is decreasing in other agents actions, leading to a game of strategic substitutes).999 More in detail, therein the author focuses on a public good game for which and where is consumer’ Engel curve and is agent’s income. We do not impose any monotonicity assumption on and focus on cases where the sign of may change (either for different values of or across different agents).

To illustrate the richer class of strategic interactions that can be modeled by using non-monotone best response functions, we consider the special case where , and has the form


with . For simplicity, we additionally assume that the network is such that (so that the neighbor aggregate is given by a convex combination of the neighbor strategies). The corresponding best response function is illustrated in Figure 1 and can be used to model races and tournaments. In the initial phase, when , the players increasing effort motivates a neck and neck race which is then followed by a second phase, when , where agent’s effort level declines capturing a discouragement effect.

Figure 1: The best response function given in (6) for as in (7) plotted for , and different values of .

In Examples LABEL:ex:lq and LABEL:competition agents have a single decision variable. Our last motivating example features agents that engage in multiple activities, as studied in Chen et al. (2018).

Example 2.3 (label=ex:multiple)

(Multiple activities in networks) Consider a network game where each player has a strategy vector with representing his level of engagement in two interdependent activities and , such as crime and education. We assume that each player has lower and upper constraints for its engagement in each activity () and an overall lower and upper budget constraint () so that

This generalizes Chen et al. (2018), where no constraints are considered, and Belhaj and Deroïan (2014) where a strict budget constraint is imposed (i.e., ) so that effectively the strategy of each agent is a scalar (since ). Each agent selects its level of engagement in activities and to maximize the following quantity

where the parameter weights the effect of the neighbor aggregate within each activity, weights the effect of the neighbor aggregate across different activities and captures the interdependence of the two activities for each agent . Chen et al. (2018) assumes so that the effort of each agent and its neighbors are strategic complements within each activity, while can be negative (modeling two complementary activities such as crime and drug use) or positive (modeling two substitutable activities such as crime and education).101010 For the model to be consistent from an economic perspective, the sign of should be consistent with the signs of and in the sense that . To see this, consider the case and . In this case an increase of leads to a decrease in (since ) which leads to an increase in (since ). Overall, and are therefore complements implying that must be positive. In our analysis, we will also consider the case .

3 Connection between game theory and variational inequalities

The main goal of this paper is to provide a unified framework to study existence and uniqueness of the Nash equilibrium, convergence of the best response dynamics and continuity of the Nash equilibrium with respect to parametric variations for general network games. Throughout the paper we use the following assumption.

Assumption 3.1

The set is nonempty, closed and convex for all . The function is continuously differentiable and convex in for all and for all , . Moreover, is twice differentiable in and is Lipschitz in .

Our approach relies on the theory of variational inequalities as defined next.

Definition 3.1 (Variational Inequality (VI))

A vector solves the variational inequality VI with set and operator if and only if


In the following we consider the VI with set


obtained as the cartesian product of the local strategy sets and operator whose -th block component is the gradient of the cost function of agent with respect to its own strategy, i.e.,


This operator is sometimes referred to as the game Jacobian. The relevance of this VI in characterizing Nash equilibria of general games (i.e., not necessarily network games) comes from the following well-known relation, see e.g. (Facchinei and Pang, 2003, Proposition 1.4.2).

Proposition 1 (VI reformulation)

Suppose that Assumption 3.1 holds. A vector of strategies is a Nash equilibrium for the game if and only if it solves the VI with as in (9) and as in (10).

Proposition 1 can be seen as a generalization of potential games as introduced in Monderer and Shapley (1996). In fact, it follows from (Monderer and Shapley, 1996, Lemma 4.4) that a game is potential with potential function if and only if . In other words, a game has an exact potential if and only if the game Jacobian is integrable. In such case, by the minimum principle, the VI condition given in (8) coincides with the necessary optimality conditions for the optimization problem , whose stationary points are the Nash equilibria of the game, see e.g. (Bramoullé et al., 2014, Lemma 1). The VI approach enables the analysis of games that are not potential (i.e. games for which is not integrable). Specifically, Proposition 1 allows one to analyze the Nash equilibria in terms of properties of the game Jacobian and of the set . To this end, we recall the following definitions.

Definition 3.2

An operator is

  • Strongly monotone: if there exists such that for all .

  • A uniform block P-function with respect to the partition : if there exists such that for all .

  • A uniform P-function: if there exists such that for all .

The properties in the previous definition are stated in decreasing order of strength, as illustrated in the first line of Figure 2.111111 In the literature a block P-function is typically referred to as a P-function with respect to the partition ; we added the term “block” to avoid any confusion. We note that for (i.e., for scalar games) the definitions of uniform block P-function and uniform P-function coincide. We note that in the case of potential games strong monotonicity of is equivalent to strong convexity of the potential function . For general games (i.e., not necessarily potential games), such condition is a slightly stronger version of the diagonal strict concavity condition used in the seminal work of Rosen (1965) (see more details in Appendix A). The reason why we focus on strong monotonicity in this work is that it allows us to prove existence of the Nash equilibrium without assuming compactness, as instead assumed in Rosen (1965). Our interest in the properties listed in Definition 3.2 stems from the following classical result for existence and uniqueness of the solution of a VI, see e.g. (Facchinei and Pang, 2003, Theorem 2.3.3(b) and Proposition 3.5.10(b)).121212Statement c) is a consequence of statement b). In fact if is a rectangle then it can be partitioned as with for all and if is a uniform P-function then it is a uniform block P-function with respect to a block partition with blocks of dimension one. We here write this condition separately to avoid any confusion with the block partition induced by the players.

Proposition 2 (Existence and uniqueness for VIs)

Consider the VI where is continuous and is nonempty, closed and convex. The VI admits a unique solution under any of the following statements:

  • is strongly monotone.

  • The set is a cartesian product and is a uniform block P-function with respect to the same partition.

  • The set is a rectangle and is a uniform P-function.

Note that the stronger the condition on is, the weaker the requirement on the set is. Proposition 2 together with Proposition 1 allows one to derive sufficient conditions for existence and uniqueness of the Nash equilibrium in terms of properties of the game Jacobian and of the constraint sets. Our main contribution in Section 4 is to derive sufficient conditions that guarantee that the game Jacobian of a network game has one of the properties listed in Definition 3.2, by imposing conditions on the cost functions of the players and on the spectral properties of the network. To this end, we exploit sufficient conditions for the properties in Definition 3.2 to hold in terms of the gradient as detailed in the next subsection.

3.1 Sufficient conditions in terms of

We start by introducing the definitions of P-matrices, P condition and uniform P-matrix condition.

Definition 3.3 (P-matrix)

(Fiedler and Ptak, 1962, Theorem 3.3) A matrix is a P-matrix if all its principal minors have positive determinant. Equivalently, is a P-matrix if and only if for any , there exists a diagonal matrix such that .

A special class of P-matrices are positive definite matrices. In fact, if a matrix is positive definite then Definition 3.3 holds with . The opposite holds true if the matrix is symmetric.131313 It is shown in (Fiedler and Ptak, 1962, Theorem 3.3) that if is a -matrix then all its real eigenvalues are positive. If is symmetric all its eigenvalues are real. These two properties imply that is positive definite.

The sufficient conditions detailed in the following Proposition 3 amount to ensuring that , which is a matrix valued function, possesses suitable properties “uniformly” in . For example, strong monotonicity of can be guaranteed if the gradient is uniformly positive definite. Similarly, to guarantee that is a uniform P-function we show that it is sufficient to assume that satisfies what we term a “uniform P-matrix condition” which is given in full generality in Definition D.1 in the appendix. We here report the corresponding definition for the affine case . Intuitively, this corresponds to assuming that is a “uniform” P-matrix for all values of .

Definition 3.4 (Uniform P-matrix condition - affine case)

Consider an operator where . Then satisfies the uniform P-matrix condition if and only if there exists such that for any there exists a diagonal matrix such that and .

The condition above is formulated in terms of the full matrix which, in the case of agents with multidimensional strategies, has dimension . The condition introduced next is instead used to summarize the effect that each pair of agents has on one another with a scalar number. In fact the condition in Definition 3.5 relates the properties of a matrix with blocks each of dimension with the properties of a smaller matrix , where the effect of each block is summarized with a scalar number (independent of ). Specifically, let be the block in position . Suppose that for all , and let . Moreover, set for . Then is constructed as follows.

Definition 3.5 ( condition)

Scutari et al. (2014) A matrix valued function satisfies the condition over if , for all and in (11) is a P-matrix.

The following proposition relates the previous conditions to the properties in Definition 3.2.

Proposition 3 (Sufficient conditions for strong monotonicity and (block) P-functions)

If the operator is continuously differentiable and the set is nonempty, closed and convex, the following holds.

  • is strongly monotone if and only if there exists such that .

  • If satisfies the condition (as by Definition 3.5), then is a uniform block P-function.

  • If satisfies the uniform P-matrix condition (as by Definition D.1 in Appendix D), then is a uniform P-function.

Statements a) and b) can be found in (Facchinei and Pang, 2003, Proposition 2.3.2(c)) and (Scutari et al., 2014, Proposition 5(e)). The proof of statement c) is given in Appendix D. The conditions in Proposition 3 are illustrated in the second line of Figure 2.

is strongly monotone is a uniform block P-function is a uniform P-function
s.t. satisfies the satisfies the uniform
condition as by Def. 3.5 P-matrix condition as by Def. D.1
Figure 2: Relation between properties of and sufficient conditions in terms of , see Proposition 3.

It is important to stress that while both strong monotonicity and the condition are sufficient to guarantee that is a uniform block P-function (and hence uniqueness of the Nash equilibrium), there is in general no relation between the two. In fact there are strongly monotone functions whose gradient does not satisfy the condition (see Example B.3) and there are functions whose gradient satisfies the condition but are not strongly monotone (see Example B.1). Besides existence and uniqueness, we show in Section 6 that the condition guarantees convergence of the best response dynamics, while we show in Section 7 that strong monotonicity is useful for sensitivity analysis. Finally, we employ the uniform P-function condition for the analysis of scalar games of strategic substitutes. Consequently, it is important to understand conditions under which each of the properties in Figure 2 is satisfied.

4 Properties of the game Jacobian in network games

Our goal in this section is to find sufficient conditions in terms of the influence of agent on its marginal cost (i.e., ), the influence of the neighbor aggregate on the cost of agent (i.e., ) and the network to guarantee that the game Jacobian posses the properties detailed in Definition 3.2. To this end, we exploit the sufficient conditions on detailed in Proposition 3. In the case of network games, can be rewritten as


where and


Note that by the properties of the Hessian, , hence . Let us define


With this notation it is clear that for network games the quantities in Definition 3.5 can be rewritten as and . The next example provides some intuition.

Example 4.1 (continues=ex:lq)

Consider Example LABEL:ex:lq with for simplicity. According to Proposition 1 a vector is a Nash equilibrium if and only if it solves the VI where and

with . From we obtain and for all . Consequently, the quantities in (14) are for all and , .

To get an intuition on the role of the quantities introduced in (14) in our subsequent analysis, let us consider the strongly monotone case. By Proposition 3a, to prove strong monotonicity of one needs to show that is positive definite for all . By using the gradient structure highlighted in (12) it is immediate to see that for network games


Consequently, the positive definiteness of can be guaranteed by bounding the minimum eigenvalue of the sum of the two matrices on the right hand side of (15). It is clear that such a bound should depend on , as defined in (14) and on the properties of , since . This approach motivates us to study conditions of the form


where is a scalar that captures the network effect. Condition (16) can be understood as a bound on how large can the effect of the neighbor aggregate (measured by ) be on an agent payoff with respect to the effect of its own action (measured by ) where is a weight that scales the magnitude of the two effects. In the following, we consider three different network measures, as detailed next.

Assumption 4.1 (Sufficient conditions in terms of and )

Suppose that at least one of the following conditions holds:

(Assumption 2a)
(Assumption 2b)
(Assumption 2c)

Assumption 4.1a and 4.1c allows us to recover and extend conditions derived in the literature for special instances of network games by using a different type of analysis. Assumption 4.1b is new to our knowledge and provides a natural summary of the network effect since corresponds to the maximum row sum of and is thus a measure of the maximum aggregate influence that the neighbors have on each agent.

Our main technical result in this paper is to show that each of the three conditions in Assumption 4.1 guarantees a different set of properties of the game Jacobian, as summarized in Table 2.

Assumption 4.1a Assumption 4.1b Assumption 4.1c
Strong mon.
(Thm. B.1) (Ex. B.1) (Thm. B.3) (Ex. B.4 )
(Thm. B.1) (Thm. B.2) (Ex. B.3) (Ex. B.4 )
Unif. P-function
(Fig. 2) (Fig. 2) (Fig. 2) (Thm. B.4)
Table 2: Summary of the relation between Assumptions 4.1a, 4.1b, 4.1c and properties of the game Jacobian. The technical statements are provided in Appendix B. In all cases we suppose that Assumption 3.1 holds. For negative cases we provide a counter-example in Appendix B. We list the condition instead of the block P-function property because we show in Section 6 that the former is required for convergence of the discrete best response dynamics. Note that Assumptions 4.1c alone does not guarantee any property, in fact multiple Nash equilibria may arise (see Ex. B.2).

Before providing some intuitions on the results of Table 2, we outline the relation between the conditions in Assumption 4.1 and delineate which classes of networks satisfy each one of them.

4.1 Some comments on Assumption 4.1

Table 2 shows that Assumption 4.1a guarantees both strong monotonicity and the condition. However, depending on the network, this assumption might be restrictive. To see this, note that the larger the value of is, the more restrictive Assumption 4.1a becomes. Figure 3 shows that for some representative networks such as the complete network (Figure 3a) or the asymmetric star (Figure 3d), might actually grow unbounded in the number of players . This means that, for such networks, Assumption 4.1a may not hold as the number of agents grows. We argue in the next subsections that Assumption 4.1b and 4.1c may be used as alternative conditions in these cases by showing that for symmetric networks Assumption 4.1c holds for a broader range of networks, while for asymmetric networks Assumption 4.1a and 4.1b allow addressing different set of networks.

Symmetric networks Asymmetric networks
           a)           b)           c)           d)
(4.1a):      3           2        2.2882 1.7
(4.1b):      3           2           2           1
(4.1c):           1           2           -           -
Figure 3: Spectral properties of some representative networks with . a) Complete network: , ; b) Bipartite network; c) -regular network; d) Asymmetric star: , . We use the convention that corresponds to an arrow from to , since means that agent is affected by the strategy of agent . For example in the network d), all the agents are affected by the strategy of the agent in the top left corner and he is not affected by any other agent. In all the networks above, we use unitary edge weight.

4.1.1 Relation between and

Since it follows from Gershgorin theorem that . If is symmetric, (see Lemma A.3 in the Appendix). Consequently, we get . This shows that whenever is symmetric Assumption 4.1b is more restrictive than Assumption 4.1a. Hence Assumption 4.1b is a viable alternative to Assumption 4.1a only for asymmetric networks. For asymmetric networks, there is no relation between and , i.e., there are networks for which and networks for which . Figures 3c) and 3d) show some examples of the latter case, thus justifying our interest for Assumption 4.1b. Note for example that for the asymmetric star network, we have while independently of , thus Assumption 4.1b can hold for large number of players while Assumption 4.1a does not. More generally, we note that, for the case of regular networks where each agent has the same number of in-neighbors, one gets immediately that . Lemma A.4 in Appendix A shows that . Since for asymmetric matrices we obtain . Hence for asymmetric regular networks Assumption 4.1b is a relaxation of Assumption 4.1a. Figure 3c) shows an example where the inequality is strict.

Figure 4: Summary of the relations between Assumptions 4.1a, 4.1b and 4.1c, for asymmetric and symmetric networks.

4.1.2 Relation between and

If the network is symmetric, then all the eigenvalues of are real and is well defined. In this case, we have already mentioned that . The relation between and is detailed in the next lemma.

Lemma 4.1

Suppose that is a non-zero and non-negative matrix entry-wise with zero diagonal. Then and . Consequently, .

Since is non-negative, it follows from Perron-Frobenius theorem that . Since and , it must be .

In other words, Lemma 4.1 proves that if is symmetric Assumption 4.1c is less restrictive than Assumption 4.1a. Figure 3 shows networks where this is strictly the case. Note for example that for the complete network in Figure 3a) we have while independently of , thus Assumption 4.1c can hold for large number of players while Assumption 4.1a does not. We stress that there are graphs of interest for which . Some examples are the undirected ring with an even number of nodes or any bipartite graph (see Figure 3b). For these cases, Assumption 4.1c and Assumption 4.1a coincide. For general -regular graphs, it is known that if and only if is bipartite. The gap between