Consider a group of individuals who interact strategically, and an external entity—a planner—who seeks to achieve a goal through an intervention that changes individuals’ incentives. For concreteness, we present three applications.
School pupils choose their level of educational effort. A pupil’s incentives to study are affected by his own attributes (ability, etc.) and the efforts exerted by his friends. A school principal seeks to improve total student welfare and the instruments at her disposal include offering additional tutoring or rewards for achievement to some students.
Firms invest in capacity, optimizing in response to costs and benefits idiosyncratic to their own business, as well as other firms’ investment decisions. A government seeks to reduce the volatility of aggregate investment and can affect the variances and covariances of firms’ idiosyncratic incentives for investment—e.g., by stabilizing prices of their inputs.
Firms in a supply chain set prices; these decisions affect the demand, and hence pricing, of other firms. A regulator seeks to maximize total market surplus. As in (2), it can, at a cost, affect the variability and correlations of the prices of firms’ inputs.
These cases can be studied within a common framework of interventions in a network game. Agents’ payoffs depend on their own attributes, their actions, and the actions of some others, whom we call their contacts or neighbors. Actions may be strategic complements or strategic substitutes. A planner can, by allocating limited intervention resources, change the incentives of some targeted individuals. Through the network, this will also change the incentives of their neighbors. The planner seeks to improve some aggregate outcome such as total welfare, volatility, or market surplus. We study the question: how should interventions be targeted?
To describe some of the basic forces, let us start by considering setting (1)—e.g., pupils studying in school—under strategic complements. Suppose that the planner can target a given individual and encourage him to work harder. His raised effort, in turn, pushes up the efforts of his friends due to the strategic complementarity, which, in turn, increases the efforts of their friends, and so forth. It is natural to suppose that the planner’s marginal costs of raising any one individual’s effort are increasing. Thus, given a budget, the planner may be best off targeting multiple individuals, rather than focusing all effort on one. But how should she allocate the effort?
An important observation is that in the strategic complements case, the planner will want to move neighbors’ incentives together, since increasing someone’s effort makes it easier to increase the efforts of his neighbors. In other words, the planner’s investments to encourage two neighbors to work harder are complementary. The optimal policy will, then, exploit this complementarity to amplify the impact of interventions. The first main result says that, under certain conditions, the best amplification is achieved by targeting agents in proportion to their eigenvector centrality, which measures how well-connected they are in the network as a whole. An agent’s eigenvector centrality is the right measure of the total direct and indirect complementarity between changes to this agent’s incentives and those of everyone else.
Next, consider the case of strategic substitutes. To take a concrete example, consider a decentralized team collaborating to develop a software product. Effort by one individual towards documenting and testing code reduces the incentives of others using the same code (his network neighbors) to do the same. A planner can encourage some team members to invest more in this activity, e.g., by directly providing help or public recognition. If the planner targets two individuals who are neighbors and encourages both to work harder, one individual’s increase in effort reduces his neighbors’ incentives for effort. That is, these two parts of the intervention work against each other, potentially wasting intervention resources. Thus, the planner’s intervention may want to move neighbors’ incentives in opposite directions. In this case, we show that the best policy divides agents into two groups, the positively and negatively targeted; this is done in such a way that there are many links between oppositely targeted agents, and few links between agents whose incentives are altered in the same direction. It turns out that, parallel to the complements case, the welfare-optimal incentive targeting is proportional to an eigenvector of the network—in this case, the one associated to its leasteigenvalue.
A key idea above is that the network of strategic interactions causes some interventions to be internally self-reinforcing, while others are self-attenuating. Our main technique is a general approach for identifying and separating these, and using that to design interventions. We now sketch the mathematical ideas involved in deriving these results.
Agents’ private incentives to put in effort can be represented as a vector of real numbers—marginal benefits that each agent derives from the activity. An intervention is a change to this vector. In setting (1), the planner’s problem is to choose the best profile of marginal benefits subject to the constraint of having a totaladjustment cost not exceeding a resource constraint. In settings (2) and (3), the planner affects the distribution of individuals’ idiosyncratic incentives: for example, she may, at a cost, reduce common shocks.111For some recent studies on how the structure of volatility affects strategic interactions, see, e.g., Angeletos and La’O (2013), Bergemann and Morris (2013), and Bergemann, Heumann, and Morris (2015). In both settings, as we have argued, it will matter which changes to the incentives have positive feedback and are amplified. To find these, we choose particular coordinates for the vector space of possible interventions: an orthonormal basis that is obtained by diagonalizing the matrix of strategic interactions. Each basis vector can be seen as a “basic” intervention. The basis has two key features: (i) the vectors can be ordered in terms of how much strategic feedback or amplification they induce; and (ii) interventions along different basis vectors are “separable” or “independent” in a certain sense, which makes it easy to trace their impact on outcomes. In the case of strategic complements, the basis vector that achieves maximum amplification is an eigenvector corresponding to the largest eigenvalue of the network. In the case of strategic substitutes, the basis vector that achieves maximum amplification is an eigenvector corresponding to the smallest eigenvalue of the network. We use this to derive the results sketched above. The eigenvectors that play a key role in these characterizations are related to network measures identified as being important in the existing network literature, in particular Ballester, Calvó-Armengol, and Zenou (2006), Bramoullé, Kranton, and d’Amours (2014) and Acemoglu, Ozdaglar, and Tahbaz-Salehi (2016). We discuss the relation to these papers and others in Section 8.
The decomposition that plays the key role in the arguments has an important interpretation as the principal component decomposition of the network adjacency matrix. The basis vector that turns out to be most important in the strategic complements case is the one that, in a suitable sense, best summarizes individuals’ levels of connectedness in the whole network. Thus it is called the first principal component. In the case of strategic substitutes, the key vector is one that is, in a sense, the “least representative” of overall social interactions, and the most sensitive to local structure. Thus it is the last principal component. Especially for this latter case, we also give graph-theoretic interpretations, which show that the eigenvector relevant for strategic substitutes encodes information about “how bipartite” the network of interaction is. When the network is actually bipartite, the eigenvector precisely identifies the two sides of the network, i.e., the two disjoint maximal independent sets of the network. To summarize, the principal component interpretation of eigenvectors allows us to relate the results on optimal targeting to network structure.
In general, our approach to finding optimal interventions focuses on the singular value decomposition (SVD) of the adjacency matrix, which is the right generalization, in our problem, of diagonalizing the adjacency matrix. This reason for the SVD’s usefulness is that it offers a convenient way to write quadratic forms in the equilibrium actions. Quadratic forms arise naturally in the economic problems we have been discussing. Generally, it will not be only the first or last principal component that matter for optimal targeting. However, the degree to which different components figure in a planner’s optimal intervention can generally be nicely ordered, in a way that depends on the strategic structure of the problem at hand and the planner’s objective. To illustrate the versatility of our approach, we propose and solve two related intervention problems: one, minimizing volatility of aggregate investment (setting (2) above) and two, maximizing consumer and producer surplus in a supply chain (setting (3) above).
The structure of the optimal interventions can, in general, turn out to be complex: the right targeting scheme may depend not only on network structure, but on other details of the problem, such as the status quo incentives. A natural question is: under what circumstances can we say that the optimal intervention will be simple, with the relative degree of focus on different agents determined only by the network, and independent of other details? Focusing on the basic network game of setting (1), we show that this occurs for large enough budgets. The network structure determines how large the budget must be for interventions to be simple. Thus we describe which networks admit particularly simple interventions.
Research over the past two decades has deepened our understanding of the empirical structure of networks and how networks affect human behavior. This naturally leads to a study of how policy interventions can effectively exploit network structure, thereby economizing on scarce resources. Our paper contributes to a broad and exciting body of research, spread across economics, sociology, public health, marketing, and computer science, among other fields, which studies network interventions. In economics, this work includes Ballester, Calvó-Armengol, and Zenou (2006), Banerjee, Chandrashekhar, Duflo, and Jackson (2016), Belhaj and Deroian (2017) Bloch and Querou (2013), Candogan, Bimpikis, and Ozdaglar (2012), Demange (2017), Dziubinski and Goyal (2017), Fainmesser and Galeotti (2017), Galeotti and Goyal (2009), Galeotti and Rogers (2013), Akbarpour, Malladi, and Saberi (2017), as well as many other papers.222We refer to Zenou (2016) for a recent survey on interventions in network games. Prominent contributions in related disciplines include Rogers (1983), Feick and Price (1987), Borgatti (2006), Kempe, Kleinberg, and Tardos (2003), and Valente (2012).
The novelty of the paper in the context of this literature lies in three contributions. The first is a unified approach to a class of intervention problems. Our planner’s problems are rich in several ways: they allow for a variety of planner’s objectives (e.g., total welfare, aggregate volatility, various surpluses); they involve different applications (educational effort, investment, pricing); and they allow both strategic complements and substitutes. Across these problems, the planner’s concerns are summarized by closely related quadratic forms that the interventions seek to maximize. The second contribution is in relating the optimal solutions to network structure using the singular value decomposition—and the corresponding analysis of principal components determined by the adjacency matrix. As we have said, all principal components may, in general, play a role in an optimal targeting scheme. The main results show that the degree of emphasis on various components is ordered according to their eigenvalues, even when the intervention is complex. The third contribution is that, in the classical setting of a network game, we characterize when a planner with a welfare objective will have a simple targeting policy, focusing on the first or the last principal component. The conditions involved depend on statistics of the network’s eigenvalues that capture subtle aspects of its large-scale structure. In this result, and in others, the network statistics and centrality measures that play a key role in our characterizations are, in some cases, standard ones or familiar from recent work; in other cases they are distinct and highlight new aspects of network structure as being economically significant.
The rest of the paper is organized as follows. Section 2 presents the optimal intervention-targeting problem in a canonical network game. Section 3 illustrates, informally, optimal interventions in two simple networks. Section 4 sets out notation and basic facts about the singular value decomposition and presents its application to the network game. In Section 5 we provide general results for the optimal targeting intervention problem. Section 6 presents and solves intervention problems for a planner seeking to minimize volatility in an investment game, and to maximize consumer and producer surplus in a supply chain where producers play a pricing game. Section 7 discusses how the main results can be generalized when we relax the assumptions relating to the adjacency matrix, the costs of intervention, and linear-quadratic payoffs. Section 8 relates our work to existing literature on networks games. Section 9 contains concluding remarks. Appendix A contains omitted proofs of some of the propositions.
2. Basic model
There is a set of individuals with ; the individuals are typically indexed by . Individual chooses an action , simultaneously with others; the vector of actions across all individuals is denoted . The payoffs to individual given an action profile are:
In this formulation, denotes each individual’s marginal benefit from his own action; the corresponding vector across all individuals is denoted by . The weighted, directed network with adjacency matrix has directed links with weights ; it is a representation of the strategic interactions.
When is a nonnegative matrix, the parameter captures the direction of strategic interdependencies. If , then actions are strategic complements; if , then actions are strategic substitutes.333We do not need the assumption of nonnegative , though the remarks on the interpretation of show it can be a helpful case to think about. As Ballester et al. (2006) observed, any (pure strategy) Nash equilibrium action profile satisfies:
If the matrix is invertible, the unique Nash equilibrium of the game can be characterized by:
We will maintain throughout, unless stated otherwise, a standard assumption:
The spectral radius of is less than .444The spectral radius of a matrix is the maximum of its eigenvalues’ absolute values.
This ensures existence of the inverse in (2), and also the uniqueness and stability of the Nash equilibria (Bramoullé et al., 2014).555The game we have presented is an instance of a linear-quadratic game played on a network; papers that study such games include Ballester et al. (2006), Bramoullé et al. (2014), and Goyal and Moraga-Gonzalez (2001). For a survey of research in games on networks, see Bramoulle and Kranton (2016) and Jackson and Zenou (2015).
2.1. A network intervention problem
A status quo vector of individuals’ marginal benefits is given. The planner wishes to maximize aggregate utility of individuals and can modify, at an adjustment cost, every individual’s marginal benefit by changing the status quo to . The timing of the intervention is as follows: the planner moves first and chooses her intervention, and then individuals simultaneously choose actions. Formally, the incentive-targeting problem is:
where is the given resource constraint or budget. Thus, the planner is maximizing welfare, subject to agents’ playing an equilibrium (given her intervention), and subject to adjustment costs not exceeding the budget.
2.2. Discussion of assumptions
The formulation of the adjustment cost reflects the idea that the planner faces increasing marginal costs as she seeks to make larger changes in individuals’ incentives. We extend the analysis to more general cost functions, and to nonlinear strategic interactions, in Section 7.
Note that the planner may intervene either to encourage or discourage action, i.e., increasing or decreasing relative to the status quo of , and that both types of interventions are costly. It is natural to think of the available interventions as changes to the environment that make the action in question more or less appealing. The model is not suited to cases where the marginal benefit is, for example, a wage set by our planner, so that the planner is providing all the incentive to take the action in the first place; in that case, the costs would have a different form (we elaborate on possible applications in this direction in Section 9). Instead, it is designed to capture cases where incentives that the planner takes as given set status quo , and the planner can use certain controls to modify them. For instance, suppose is the level of effort toward studying invested by pupils while they are at school; there are also some outside option activities such as, for concreteness, extracurricular clubs. Many of the reasons to pursue each activity are not within a planner’s (e.g., school principal’s) control, but rather have to do with students’ own preferences and future outcomes. Those set the status quo . Instruments available to the planner for modifying it include facilitating either activity: for example, she can offer some students supplementary tutoring, or offer others improved coaching. Depending on which kind of intervention is done, these interventions tend to increase or decrease , the marginal benefit (relative to an outside option) of studying. Assuming our planner starts with the least costly instruments to achieve her desired influence on a given individual and works up to more costly ones, the marginal costs of changing the player’s incentives will be increasing in the magnitude of the change.
For some applications, it may be natural to assume that actions—such as studying effort or prices—can take on a positive value only. This can be reflected in the constraints of the problem. For example, we may study a setting where is large and is such that all actions are positive at any solution the planner contemplates.666Of course, one could also impose nonnegativity of as a separate constraint, not via : this would make some arguments more complicated, though the basic forces we will identify would be present in that world as well.
3. An example
We give a concrete illustration of the main insights of our analysis by presenting the solution to the optimal intervention problem in the case of two networks: a small random network777An Erdős-Rényi graph with . and a circle. These are shown in Figure 1; in each case, the matrix is the adjacency matrix of the undirected graph, with if and only if and are linked. For the case of strategic complements we set , and for strategic substitutes we set .888All our graphs satisfy the spectral radius condition. It is easier to explain the optimal intervention policy and the intuitions behind it when the intervention budget is large (in a sense we will make precise later), so we set .
Figures 1 and 2 present optimal interventions for the different settings. Define as the change to marginal benefits made at the optimal intervention. The size of a node corresponds to the magnitude of that change: i.e., . The color reflects the direction of change: if node is green (red) it means that the intervention has increased (decreased) the attribute from the initial value of . Tables 1-4 present data on the initial , the optimal intervention , as well as on the associated change in action and change in utility .
Tables 1-4 also present the “first” and “last” eigenvectors of the network . To explain what these are in terms of the network, we introduce some notation. Since each considered in the example is symmetric, it is diagonalizable, and we can write , where (i) is a diagonal matrix, whose diagonal has the eigenvalues of ordered from greatest to least as real numbers, and (ii)
is an orthogonal matrix. Thecolumn of , which we call , is a normalized eigenvector of associated to the eigenvalue . In the examples of this section, all the eigenvalues are distinct and the eigenvectors are uniquely determined.
|node||initial benefit||first eigenvector||scaled intervention||action change||welfare change|
|node||initial benefit||first eigenvector||scaled intervention||action change||welfare change|
3.1. Strategic complements
Figure 1 illustrates optimal intervention in the case of strategic complements. Optimal intervention entails targeting the nodes in proportion to the corresponding entry in the eigenvector . This vector is necessarily positive entrywise.999Since both networks have a symmetric and non-negative and are connected, by the Perron-Frobenius Theorem, is entrywise positive; indeed, this vector is the Perron vector of the matrix. Its entries are individuals’ eigenvector centralities. The eigenvector centrality of an individual measures his overall or global level of connectedness. It is characterized101010Given connectedness of , which holds in these examples. by the condition that it is entrywise nonnegative and, for all , we have for some constant . That is, a node is highly central in proportion to the connections ( it has), weighted by the centrality of the partners in those connections.
Under strategic complementarities, and a large budget, it is by adjusting agents’ incentives in proportion to their eigenvector centralities that the planner best amplifies her intervention to maximize aggregate welfare. In the random network, agent has the highest eigenvector centrality and the change chosen by the planner is , which is very close to ; by contrast, node has the lowest eigenvector centrality and is only , which is very close to . In the circle network nodes have the same structural positions, and thus the same centralities. Thus, any heterogeneity in targeting is only due to differences in .111111In both the networks we study, we choose a nonuniform initial vector . This illustrates that the conclusions about targeting being in line with certain eigenvectors are not reliant on any particular structure of the . As Figure 0(b) illustrates, these initial differences are less important because of the large budget: the magnitude of the intervention and the consequent change in action is similar across nodes.
3.2. Strategic substitutes
Figure 2 illustrates optimal intervention in the case of strategic substitutes. Now the intensity of intervention varies approximately in proportion to the “last” eigenvector, . This entails raising the for some nodes and lowering the of others. Figure 2b shows that in the circle network the optimal intervention is to raise ’s of nodes from their initial levels, and to lower those of nodes . This leads, in turn, to an increase in the actions of nodes and a fall in the action of nodes . Figure 2a shows that a combination of positive and negative interventions is involved in the random network, and here too the interventions track the “last” eigenvector.
To see why this happens, it is instructive to examine the nature of best replies: an increase in raises and this exerts, due to the strategic substitutes property, a downward pressure on neighbor ’s action, . A smaller in turn pushes up further, and that lowers even more, and so forth, until we reach a new equilibrium configuration. This process is amplified if we simultaneously increase and decrease . On the other hand, if we were to raise and simultaneously, then the pressure toward a higher effort by and would tend to cancel each other; that would be wasteful.
The smallest eigenvalue of and the associated eigenvector are familiar objects in matrix algebra. They contain information about the local structural properties of the network. This information is useful in determining the bipartiteness of a graph and its chromatic number.121212Desai and Rao (1994) characterize the smallest eigenvalue of a graph and relate it to the degree of bipartiteness of a graph. Alon and Kahale (1997) demonstrate that the last eigenvector of a graph corresponds to an approximate a coloring of the underlying graph, i.e., a labeling by a minimal set of integers to nodes such that no neighboring nodes share the same label. Consider the circle network in the example; that is a bipartite graph. The associated eigenvector precisely determines the way to partition the nodes of the graph to get the two maximally independent sets. The random network in the example is not bipartite, but one can see that the individuals that have been targeted in the same direction have very few connections among themselves, whereas most of the links are across individuals who have been targeted in opposite directions.
|node||initial benefit||last eigenvector||scaled intervention||action change||welfare change|
|node||initial benefit||last eigenvector||scaled intervention||action change||welfare change|
The rest of the paper formalizes and generalizes these insights. Some of the network statistics that turn out to be relevant for solving the intervention problem have been studied in the literature of networks. We discuss these relations in Section 8.
4. Analysis of the game via the singular value decomposition
This section recalls basic concepts that we use to analyze the optimal intervention policy for general network games. Section 4.1 introduces our notation for the singular value decomposition (SVD) of an arbitrary matrix and the associated notion of principal components. When applied to the matrix , this decomposition helps us identify the highest-feedback interventions. This method—focusing on the SVD of —is general and works for any and any . To relate the results to the structure of the network in a familiar way, we often focus on the special case of a symmetric matrix of interaction, such as those seen in the examples of Section 3. In that case, the SVD of boils down to the diagonalization of , and we develop the implications for the game in Sections 4.2 and 4.3. Section 7 is devoted to stating the more general forms of the main results.
4.1. Singular values and principal components: Notation and definitions
Consider any matrix with real entries. A singular value decomposition (SVD) of is defined to be a tuple satisfying:
is an orthogonal matrix whose columns are eigenvectors of ;
is an orthogonal matrix whose columns are eigenvectors of ;
is an matrix with all off-diagonal entries equal to zero and non-negative diagonal entries , which are called singular values of .
It is a standard fact that an SVD exists.131313Standard references on the SVD include Golub and Van Loan (1996) and Horn and Johnson (2012). We describe, after stating Fact 1, some conventions for selecting a unique SVD in our setting.
We can view as a map sending column vectors in to column vectors in . Then the matrices and can be seen as bases for the domain and codomain, respectively, under which is represented by a diagonal matrix. The columns of are called left-hand singular vectors of , and the columns of are called right-hand singular vectors. The -ranked singular value of a matrix is defined to be the -largest, and the -ranked singular vector on the left or right is the corresponding column of or , respectively.
For any vector , let denote the vector written in the basis of the SVD, and similarly, for , let . The basis of the SVD is one in which the map corresponding to is particularly nice: it simply dilates some components and contracts others, according to the magnitudes of the singular values:
An important application and interpretation of the SVD is principal component analysis. We can think of the columns of as data points. The first principal component of is defined as the -dimensional vector that minimizes the sum of squared distances to the actual . The first principal component can therefore be thought of as a fictitious column that best summarizes the data set . To characterize the other principal components, we orthogonally project all columns of off this vector and repeat this procedure. A well known result is that the left singular vectors of are, indeed, the principal components of ; a singular value quantifies the variation explained by the respective principal component. When we refer to the principal component of we mean the -ranked left singular vector of . From now on, we will refer to as the projection of it onto the th principal component, or the magnitude of it in that component.
4.2. A special case: Symmetric
Assume that the matrix is symmetric,141414Symmetry of entails that the impact of ’s action on ’s incentives is the same as that of ’s action on ’s incentives. i.e. that . The usefulness of this assumption is brought out in the following statement (see, e.g., Meyer (2000)).
If is symmetric and Assumption 1 holds, then there is a SVD of with . This SVD corresponds to a diagonalization of satisfying the following conditions:
is a diagonal matrix whose diagonal elements are the eigenvalues of (which are nonnegative real numbers), ordered from greatest to least;
the column of is a real eigenvector of associated to the eigenvalue of in the position of ;
in the SVD of in which .
For a generic symmetric , all the diagonal entries of are positive, a fact we will sometimes use.151515Assumption 1 implies that has no entries larger than in absolute value, so the only case to worry about is where some entries are exactly equal to , which is not generic. The eigenvector of , which we denote by , corresponds to the principal component of . The decomposition is uniquely determined up to (i) a permutation that reorders the eigenvalues in and correspondingly reorders the columns of ; (ii) a sign flip of any column of .
The implication of Fact 1 is that, when is symmetric and Assumption 1 holds, the SVD of can be taken to have , and the SVD basis is one in which is diagonal.161616Furthermore, if (which corresponds to strategic substitutes if is nonnegative), then the -ranked principal component of is the -ranked principal component of ; in the opposite case of , the -ranked principal component of is the -ranked principal component of . For concrete examples of the principal components involved, recall our application of the decomposition in Section 3.
4.3. Analysis of the game using the SVD: Special case of symmetric
Substituting the expression into the equilibrium equation (1), we obtain:
Multiplying the LHS and the RHS by gives an analogue of (2) characterizing the solution of the game:
This system is diagonal. Hence, for any ,
The equilibrium action in the principal component of is a scaling of the magnitude of in that principal component.171717Note that is the magnitude of the orthogonal projection of onto column of . Hence, changes in in a given principal component are entirely confined to that component in terms of their effect on actions. In terms of magnitudes, suppose changes in the principal component of corresponding to a high value of . In this case, the change in is large. It follows that, for a nonnegative , when actions are strategic substitutes an -increase of increases the action by less than ; under strategic complements, however, an increase of increases the action by more than .
Rewriting in the original coordinates:
Thus individual ’s action is proportional to how much is represented in various components () as ranges across all indices; how large the attribute vector is in those components (); and the magnification from the corresponding factor, .
The same analysis can actually be extended to all normal matrices (ones that satisfy , and are therefore orthogonally diagonalizable) by grouping eigenvalues that are complex conjugates together in the summation. However, the symmetric case illustrates the key ideas, and we leave the general case, which does not require diagonalizability at all, for Section 7.
5. Targeting incentives to maximize welfare
We are now in a position to state our first main result on optimal targeting of interventions to increase utilitarian welfare in the problem (IT) of Section 2.1. Recall that the planner solves the following optimization problem:
where the resource constraint is , and is a fixed vector of status quo attributes.
5.1. The structure of optimal interventions in terms of principal components
Our first result shows that optimal interventions, in a suitable sense, focus on changing in some principal components more than in others. Recall that for an arbitrary vector , its tuple of SVD coordinates (equivalently, the same vector written the SVD basis) is denoted by . Here is the projection of onto the th principal component of , corresponding to its th-largest eigenvalue. The statement of our result pertains to relative changes in these components. Thus, for an arbitrary vector , we let:
when these are well-defined (i.e. when the denominators are nonzero). The quantity describes the relative increment in a given from its status quo level of ; the increase is expressed as a fraction of the initial level.
Suppose is symmetric and Assumption 1 holds. Let be a solution to the incentive-targeting problem (IT) with graph . For a generic181818Nonzero in each component. , we have that for all , and that:
If then is weakly decreasing in ;
If then is weakly increasing in .
The proposition says that, in the relative sense described above, the planner focuses her budget most on changing the contribution of a particular principal component. This is the one corresponding to , the largest eigenvalue of , or , the smallest eigenvalue of , depending on whether or , respectively.191919Recall that for the purpose of indexing eigenvalues and the corresponding eigenvectors, the eigenvalues of are ordered from greatest or least as real numbers. Moreover, the degree of focus on principal components is monotonic in the corresponding eigenvalues. If the degree of focus on a component with a larger eigenvalue is larger. On the other hand, when , it is just the opposite. When is nonnegative, the former case corresponds to strategic complements and the latter to strategic substitutes.
The idea of the proof is as follows: First, we rewrite the problem (IT) in the coordinates of the SVD:
This transformation uses (i) that orthogonal transformation into the SVD coordinates does not change sums of squares of coordinates, so the constraint inequality remains identical in form; (ii) the magnitude of the equilibrium action in the principal component of is simply a scaling of the magnitude of (recall (4)) by a coefficient we call , which depends on a corresponding eigenvalue of . Then we make one more transformation, writing the objective and the constraint equivalently in terms of the relative changes, . After a few steps of simplification, the optimization problem becomes:
From this it is straightforward to argue, using basic optimization theory, that at the optimal solution , the entries are increasing in the corresponding ; meanwhile, the are shown to be monotone in the eigenvalues (decreasing in when , and increasing when ). The details are presented in Section A.1.1 of the appendix.
5.2. Simple interventions under large budgets
Beyond what we know from Proposition 1, the details of the planner’s intervention may be complex: the details of what exactly is—which combination of the principal components—will depend in subtle ways on details of the network , the strategic parameter , and the budget . In this section we ask when the optimal policy is simple, in a sense we make precise.
To facilitate this, we define a standard notion of similarity between two vectors.
The cosine similarity of two non-zero vectors and is:
This is the cosine of the angle between the two vectors in the plane that and define; when this is , one vector is a positive scaling of the other. When the cosine similarity is , the vectors are orthogonal; when the cosine similarity is , one vector is a negative scaling of the other.
Our main result in this section shows that, when the resources available for intervention are appropriately large, the optimal intervention targets individuals according to a single component of the network, in the sense that the optimal intervention has high similarity to that component and low similarity to all others. Before stating the result, we introduce two network-dependent quantities that play a key role in it.202020Both quantities depend on through the eigenvalues, but we drop this argument when there is no risk of confusion.
Suppose is symmetric and Assumption 1 holds. Suppose also that and are strict bounds on the other eigenvalues.212121This assumption holds for generic symmetric . Let be a solution to the incentive-targeting problem (IT) with graph and a generic . Then for suitable choice of signs222222Recall that the eigenvectors and are determined up to a sign flip. for and , we have:
Suppose . For any , if then , where .
Suppose . For any , if , then , where .
This Proposition yields an immediate corollary, which is the main take-away: the optimal intervention is similar to exactly one principal component.
Take the assumptions of Proposition 2.
If , then as , the intervention becomes similar to only; i.e., tends to and tends to for .
Suppose . Then as , the intervention becomes similar to only; i.e., tends to and tends to for .
Thus, large budgets imply that the optimal intervention strategy is simple to describe: the relative emphasis on different agents will stop changing with and , and depends only on a single principal component of the network . If actions are strategic complements, the optimal intervention alters the individual incentives (very nearly) proportionally to the first principal component of , namely . On the other hand, if actions are strategic substitutes, the planner changes the individual incentives (very nearly) in proportion to the last principal component, .232323When the initial attributes are zero (), we can dispense with the approximations. Assuming is generic in the sense used in Proposition 2, if , then all of is spent either (i) on increasing (if ), or (ii) on increasing (if ). To see this, set in the maximization problem (IT-SVD); note that if the allocation is not monotonic then effort can be reallocated profitably among principal components without changing the cost.
Indeed, a heuristic for computing the optimal intervention is simply to set(in the case of strategic complements) or (in the case of strategic complements); that clearly exhausts the budget (since vectors are normalized) and puts all the weight on the component we have shown is dominant. In the example of Section 3 note that, in each table, the suitably scaled optimal intervention is indeed very close to the relevant eigenvector, which illustrates that this heuristic gets very close to the optimal targeting scheme.
5.3. Which networks allow simple interventions?
All the different principal components of the network will play a role in the optimal policy when the budget is not large (Proposition 1). Proposition 2 gives a condition on the size of the budget beyond which this complexity goes away. Loosely, this condition is harder to satisfy when the entries of are larger or, holding the average of the entries fixed, when they are more variable.242424Recall that is equal to plus the variance of the entries of the vector . In this subsection, we analyze the role that the network plays in this condition, via the coefficient for the case of and the coefficient for the case of . The key idea that the discussion will highlight is that in some networks, there are two different structures embedded in a network, each one offering a similar potential for amplifying the effects of interventions. In such networks, the condition on the budget for simplicity will be very stringent—i.e., interventions will not be simple for reasonable budgets.
In the strategic complements case, the simplest obstruction to simple interventions occurs when a network consists of two communities, strongly segregated from each other in terms of externalities. What share of her resources the planner would devote to each of these nearly decoupled components depends not only on the network but on which community’s initial conditions are more hospitable to interventions—i.e., the details of . In that case, the network need not admit simple—i.e. -independent—characterizations of interventions. On the other hand, networks that are more cohesive, in the sense that such segregation is ruled out, will admit simple interventions. Formally, in the case of strategic complements, and therefore the lower bound, is large when is small. The difference is called spectral gap and it is known th measure the extent of the segregation we have intuitively described: The network spectral gap is small when the nodes can be divided into at least two communities that have few links between them (Hartfiel and Meyer, 1998; Levin et al., 2009).252525See also Golub and Jackson (2012b, c, a) for more on how the difference corresponds to network structure, and in particular segregation and homophily, some other applications of that, and further citations to an extensive literature on this issue in applied mathematics.
The analogous phenomenon in the strategic substitutes case is more subtle, and is not as simple as finding nearly disconnected pieces in the network. We therefore discuss this first within an example of a large random graph with group structure. The random graph is formed as follows:
Let be a positive integer and let consist of four disjoint groups , each of size .
For every pair of distinct nodes we independently draw a vertical and horizontal compatibility, .
The probability thatand are vertically compatible is if and are in groups with the same vertical (N/S) coordinate, and otherwise.
The probability that and are horizontally compatible is if and are in groups with the same horizontal (E/W) coordinate, and otherwise.
If are compatible on both dimensions then, i.i.d., a link between them is formed in with a probability , which controls the density of the graph. For example, the probability that and are linked is .
In Figure 3(a), we illustrate schematically one parameter configuration. We take so that has all eigenvalues positive; as we explain in Appendix A.2, one such choice is to set to be a constant less than times the average degree of the graph.
We describe when —i.e., on both dimensions, there is more association across dimensions than within.262626Other configurations of the parameters can also be analyzed using the techniques that appear in our proofs. In this case, we can give a simple formula for that holds asymptotically.
If the conditions just stated hold and (i.e. the graph is dense enough) then, as ,
This lower bound coefficient grows large when and are very close to each other; see Figure 3(b). In this case, recalling Figure 3(a), there is an East/West and a North/South bipartition of the network. Then either of these bipartitions can serve as the one that best exploits the potential for amplification via strategic substitutes. Therefore, for moderate budgets, the planner’s choice will again depend on details of —that will determine which intervention has higher returns. An analogous analysis for the case of strategic complements is relegated to Section A.2 in the appendix.
We now discuss the key idea that generalizes beyond the example. For a fixed value of , the magnitude of is, in a way we are about to make precise, a measure of “how bipartite” the graph is: the associated eigenvector is the vector of a given length that is best at assigning different values to neighboring nodes. The magnitude of is large when it is possible to do this well, and small otherwise.272727The eigenvalue is necessarily negative. All the eigenvalues of are real, and the trace of , which is the sum of the eigenvalues is : thus, since is positive by the Perron-Frobenius Theorem, is negative. The next-smallest eigenvalue of captures how good the next-best vector orthogonal to is at doing the same thing. Thus and will be similar if there are two distinct (in the sense of orthogonality) ways to divide the graph, each of which cuts a similar number of edges. (This is exactly what occurs when and are similar, which is what is needed for to be large in the formula of Fact 2.)
To formalize this, we can use a variant of the Courant-Fischer characterization of eigenvalues, which for simplicity we present in the case of a -regular , one in which the sum of every row and column of is equal to . This characterization says that:
Moreover, generically, the eigenvectors and are the maximizers achieving and , respectively. Thus, achieves the greatest total squared difference of entries across links, and achieves the greatest among those orthogonal to . If and are close, then these maxima are very similar, meaning there are two splits in the graph that are about equally good from this perspective.
6. Two further applications
This section presents two additional network intervention problems: (i) minimizing aggregate volatility in investment, and (ii) maximizing consumers’ and producers’ surplus when production occurs in a supply chain.
6.1. Aggregate volatility
The network game introduced in Section 2 is related to models of how idiosyncratic shocks contribute to aggregate volatility. A recent strand of research studies how idiosyncratic shocks in production networks affect aggregate volatility of the economy—e.g., Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012). In the production network context, firms are price-taking, is a productivity shock to firm , is the log output of firm and the production of good is obtained via combining other goods; indicates how important product is for the production of good . See Acemoglu et al. (2016) for a survey of this literature and a formal connection to network games. Other work has studied how private information shapes aggregate volatility in games with a focus, for tractability, on games with linear best replies, e.g., Angeletos and Pavan (2007) and Bergemann, Heumann, and Morris (2015); recent papers (de Martí and Zenou, 2015; Bergemann, Heumann, and Morris, 2017) explore this problem from a network perspective. The underlying game that is studied is one of asymmetric information. Following Golub and Morris (2017), we can extend our complete-information analysis to linear best-response games in which is private information.
Following this latter strand, we interpret actions as levels of investment. The status-quo vector of exogenous attributes is a shock and its variance-covariance matrix is . The planner, at the time of her intervention, does not observe the realization of , whereas is common knowledge among the individuals at the time of their investment choice. The planner wishes to minimize the volatility in the level of aggregate investment:
She can do this by controlling the variances of the shocks to the exogenous attributes. This control comes at a cost: is the cost of changing the variance-covariance matrix from to . We make the following assumption on :
is invariant to rotations of coordinates: for any orthogonal matrix ,
For example, suppose that the cost is equal to the reduction in the sum of attribute variances. Then . This specification satisfies Assumption 2.282828To see this, note that the trace is the sum of the eigenvalues, and this does not change under conjugacy transformations of the argument. The function on the outside is inessential; we could set , where is any function such that . However, in those cases where changing variance more is more costly, then it will be natural to make this function increasing in the absolute value of its argument.
Under this assumption, we will study the variance-minimization problem described above. Formally:
The proposition says that the amount of variance reduction in a given principal component of is monotone in the ranking of that principal component. In particular, the first principal component will receive the most focus when investments are strategic complements, and the last principal component will receive the most focus when investments are strategic substitutes.
We now present the intuition behind the result. The variation in basic incentives comes from shocks to each firm’s own idiosyncratic costs and benefits. The planner intervention can can adjust (ex ante) the variances and covariances of these incentives. What should the planner do to reduce the volatility of aggregate investments? Suppose first that decisions among the firms are strategic complements. A perfectly correlated shock that raises all costs is amplified: there are knock-on effects because each firm’s decrease in activity reduces the incentives of others to invest, adding to the direct effects of those firms’ own shocks. In fact, the type of shock that is most amplifying (at a given size) is the one that is perfectly correlated across agents, with the magnitude of a given agent’s shock proportional to the first principal component. This is the dimension of volatility that the planner most wants to reduce. In the case of the circle, this can be visualized with the help of Figure 1(b): in a given shock realization, all nodes are shocked in the same (here, positive) direction.
If decisions are strategic substitutes, then a global shock is not so bad. The first-order response of all firms to an increase in costs, is to decrease investment but that, in turn, makes all firms want to increase their investment somewhat due to the strategic substitutability with their neighbors. Hence, the effect of a global shock is attenuated. The most amplifying shock profiles, in this case, are the ones in which neighbors have negatively correlated shocks. To stabilize aggregate outcomes, a planner will prioritize combating that particular type of volatility. It turns out that the most concerning type of volatility will be the one that is strongly correlated with the last eigenvector of the system, and this is what the planner will focus most on. In the case of the circle, this can be visualized with the help of Figure 2(b): in a given shock realization, nodes are targeted in an alternating way.
6.1.2. Illustration in the case of the circle
Proposition 3 says that the planner will also invest in reducing shocks that are neither the totally correlated volatility represented by Figure 1(b) nor the “most local” volatility represented by Figure 2(b). That is, may be nonzero for various . How do the non-extreme components look, and what is the meaning of the statement that the planner’s focus is monotonic in , the index of the eigenvalue? Some intuition is suggested by Figure 4, which shows the corresponding to , for the circle.292929This basis is not uniquely determined because the circle has a great deal of symmetry; a generic perturbation would uniquely pin down the eigenvectors, so we do not dwell on the details of the multiplicity here. The component (top left panel of Figure 4), represents a type of volatility that is not perfectly correlated (in that opposite sides of the circle are anti-correlated) but quite correlated locally. This component will receive more focus than any but the component in the case of strategic complements. Indeed, in the case of strategic complements, the planner’s efforts will intervene with decreasing intensity as we progress along the sequence depicted in Figure 4. Turning now to the end of the sequence, the or component (bottom right panel of Figure 4) depicts volatility that is locally quite anti-correlated—the shocks of neighbors are usually opposite—but not quite as strongly as in the last component. These will receive the most focus in the case of strategic substitutes, and generally the planner’s interventions will increase in intensity as we go along the sequence depicted in the figure.
Another way to summarize what the figure depicts is that the principal components corresponding to smaller (more negative) eigenvalues divide the nodes into more regions where the shock hits with opposite signs. Thus, in the case of strategic complements, a given principal component of volatility is of greater concern for the planner if it has a “longer spatial wavelength.” In the case of strategic substitutes, the volatility is of greater concern if it has a “shorter spatial wavelength.” These patterns in the eigenvectors, which are illustrated in this simple example, hold more broadly, as studied in Urschel (2018) and references cited there.
The idea of the proof is to consider any solution and to show that if it did not satisfy the conclusion, then it would be possible to find a different variance reduction that does better. The strategy for finding the rearrangement is to study the problem in the eigenvector basis. In this basis, the aggregate volatility is a weighted sum of the variance of each principal component of , and the contribution to aggregate volatility of the variance of each component is monotone in the corresponding eigenvalue. This monotonicity dictates how to rearrange variance reductions to achieve a bigger effect. In particular, we permute them among the eigenvectors. Assumption 2 on the cost function ensures that the rearrangement we need is feasible without changing the cost. A full proof appears in Section A.3.
6.2. Pricing in a supply chain
We consider a pricing game between suppliers embedded in a supply chain. In this application, the intervention alters the variability of marginal costs across suppliers in order to maximize consumer surplus, producer surplus and welfare.
Price formation in networked markets is an active area of research. This research has focused on buyer-seller networks and on networks of intermediaries. To the best of our knowledge, existing work does not address the study of optimal intervention in these markets; for surveys of this literature, see Condorelli and Galeotti (2016), Goyal (2017), and Manea (2016).
We consider a set of final goods . Final goods are made using the set of inputs ; supplier produces input . Following Vives (2001) and Singh and Vives (1984), a representative consumer with quadratic utilities chooses how much to consume of each final good. Given price vector , the utility of the consumer is:
Here, for simplicity, we assume that final goods are independent; the analysis can easily be generalized to the case where final goods can be substitutes and complements in consumption. The consumer’s optimization leads to a linear demand for final goods: . The utility of the representative consumer is:
We now describe how inputs are transformed into final goods. Let be a -by- matrix with typical element . In order to produce one unit of final good , firm requires units of each input ; without loss of generality, we set, for each , .303030This is a normalization: we choose the relevant units of each input such that the Euclidean length of each vector is equal to . We assume that the final goods markets are competitive and so the price of final good equals the marginal cost of production of good .313131Constant markups can be added without significantly changing our analysis. We can thus write , or, in matrix notation, . The demand of supplier , which depends on all prices, is:
Thus, the vector of demand for inputs is . For a given price profile, , the profit of supplier with a constant marginal cost is:
Consider the simultaneous-move pricing game among suppliers, each having profit function and each taking action . The Nash equilibrium pricing profile solves the system:
where . This is equivalent to system (1) with , and the endogenous variables being suppliers’ prices, . In other words, the pricing game is a special case of the network games we have studied above.
Two observations follow. First, the matrix of interaction across suppliers, , is symmetric; furthermore, , and so the equilibrium is uniquely defined and stable.323232Indeed, all eigenvalues of are positive: they are one plus the eigenvalues of , which are at most in absolute value by our normalization of . It follows that is invertible, so equilibria are uniquely defined and also stable. Second, the SVD of is related to the SVD of . The SVD of reads , where the columns of and are the right and left singular vectors of , respectively. It follows that the SVD of is given by , where . Hence, the singular values of are the square of the singular values of ; and the principal components of turn out to be the left singular vectors of . These can be interpreted as bundles of inputs. In the sense of principal components, they “summarize” the technology of production.
In this application, we refer to the columns of as fundamental bundles of inputs and we define and . We can then rewrite the equilibrium price system (6) as follows:
Now, suppose that the vector of marginal costs is common knowledge among market participants, but it is random from the perspective of a planner at the time of the intervention. The variance-covariance matrix of marginal costs prior the intervention is and the planner can change it to at a cost , which satisfies Assumption 2.
We study the optimal choice of under the constraint , for three objectives: expected consumer surplus , expected producer surplus , and social welfare . It is worth noting right away that a planner who wishes to maximize say, , has preferences for variance in . This is because variance in generates variance in the quantity consumed of final goods, and the utility of the representative consumer is . With other specifications of preferences, the consumer’s preferences might imply, instead, that it is better to have more stable input prices.
In any case, the change of the three surplus quantities, when we move from to , turns out to be a weighted sum of the respective changes in the variances of the marginal costs of the fundamental bundles of final goods. The weight associated to the -ranked fundamental bundle is a function of the -ranked singular value. Formally:
where we recall from Section 6.1 that under and under .
Assume satisfies Assumption 2.
Suppose maximizes expected producer surplus or expected total welfare. Then the variance change in the marginal cost of the fundamental bundle of final goods is decreasing in .
Suppose maximizes expected consumer surplus. Let the fundamental bundle of final goods be such that and .333333Set if and set if . The variance change in the marginal cost of the fundamental bundle is increasing in if and decreasing otherwise.
Proof of Proposition 4.
Consider expression (9) and note that the weight on the variance change of the marginal cost of the fundamental bundle is increasing in . The proof then follows by replicating the proof of Proposition 3. The same arguments apply to the expected welfare. Next, consider the expression (8) for the expected consumer surplus. Note that the weight on the variance change of the marginal cost of the fundamental bundle is increasing in for and it is decreasing in for . The proof then follows by using the same techniques introduced in the proof of Proposition 3. The three expressions (8)-(10) are derived in Lemma 1 in the Appendix.∎
There are two main effects driving the results of Proposition 4. The first is a pass-through effect across suppliers. The pricing game is a game of strategic substitutes. Therefore, shocks in marginal costs that alter the price of some suppliers are attenuated by the strategic response of other suppliers, changing how much of the price change a consumer ultimately experiences. This effect is summarized in expression (7) that indicates that shocks are attenuated more along the higher-eigenvalue principal components. The second effect is a quantity effect. Any shock to marginal costs is passed through to suppliers’ prices and affects the price of final goods, and so the final consumption of the representative consumers. In particular, the equilibrium prices of final goods are:
where the equivalence follows using the SVD of . Hence, the quantity effects are amplified along the higher-eigenvalue principal components.343434If the prices of the and fundamental bundles both increase by , then the change in equals —larger than the change in , which is equal to .
When the objective of the planner is to maximize consumer surplus, these two effects are countervailing: the pass-through effect pushes the planner to increase variance in less representative fundamental bundles, whereas the quantity effect pushes her in the direction of the more representative fundamental bundles. In this case, the solution is generally to target variance change on intermediate principal components. When the objective is the producer surplus, the planner only cares about the pass-through effect and so the focus of variance increase is on the main principal component.353535This is because the profits of a firm are quadratic in .
7. Generalizations: General , different cost functions, and nonlinear systems
The restrictions to a symmetric and a linear system of best replies facilitate the simplest presentation of the key ideas in our approach. However, these restrictions will often fail to be satisfied in practical applications. To conclude the formal analysis, we show that many of the insights of the previous sections are valid even when those restrictions are relaxed. Our perspective is that the special cases above are most useful for intuition, while the generalizations here are more robust and portable.
We begin by relaxing the assumption of symmetric ; we then study a nonlinear system.
7.1. Incentive targeting: Beyond symmetric
In this subsection we extend the analysis of Section 5 to non-symmetric . We maintain the linearity of the best responses, but also relax the functional form assumptions on the objective function.
Individuals take endogenous actions which, in equilibrium, satisfy:
where is a vector of idiosyncratic characteristics. In the setting of Section 5, let us assume that for an arbitrary matrix such that is invertible.