Targeting Interventions in Networks

We study games in which a network mediates strategic spillovers and externalities among the players. How does a planner optimally target interventions that change individuals' private returns to investment? We analyze this question by decomposing any intervention into orthogonal principal components, which are determined by the network and are ordered according to their associated eigenvalues. There is a close connection between the nature of spillovers and the representation of various principal components in the optimal intervention. In games of strategic complements (substitutes), interventions place more weight on the top (bottom) principal components, which reflect more global (local) network structure. For large budgets, optimal interventions are simple—they essentially involve only a single principal component.


INTRODUCTION
WE study games among agents embedded in a network. The action of each agent-for example, a level of investment or effort-directly affects a subset of others, called neighbors of that agent. This happens through two channels: spillover effects on others' incentives, as well as non-strategic externalities. A utilitarian planner with limited resources can intervene to change individuals' incentives for taking the action. Our goal is to understand how the planner can best target such interventions in view of the network and other primitives of the environment.
We now lay out the elements of the model in more detail. Individuals play a simultaneous-move game with continuous actions. An agent's action creates standalone returns for that agent independent of anyone else's action, but it also creates spillovers. The intensity of these spillovers is described by a network, with the strength of a link network of strategic interactions. It is then progressively less similar to principal components with smaller eigenvalues. In games of strategic substitutes, the order is reversed: the optimal intervention is most similar to the last (lowest-eigenvalue) principal component. The "higher" principal components capture the more global structure of the network: this is important for taking advantage of the aligned feedback effects arising under strategic complementarities. The "lower" principal components capture the local structure of the network: they help the planner to target the intervention so that it does not cause crowding out between adjacent neighbors; this is an important concern when actions are strategic substitutes.
We then turn to the study of simple optimal interventions, that is, ones where the relative intervention on the incentives of each node is determined by a single network statistic of that node, and invariant to other primitives (such as status quo incentives). Propositions 1 and 2 show that, for large enough budgets, the optimal intervention is simple: in games of strategic complements, the optimal intervention vector is proportional to the first principal component, while in games of strategic substitutes, it is proportional to the last one. 3 Moreover, the network structure determines how large the budget must be for optimal interventions to be simple. In games of strategic complements (substitutes), the important statistic is the gap between the top (bottom) two eigenvalues of the network of strategic interactions. When this gap is large, even at moderate budgets the intervention is simple.
Theorem 1, our characterization of optimal interventions, is derived in a deterministic setting where the planner knows the status quo standalone marginal returns of all individuals. Our methods can also be used to study optimal interventions assuming the planner does not know these returns but knows only their distribution. Propositions 3 and 4 characterize optimal interventions in a stochastic setting. These show that suitable analogues of the main insights extend: the order of the principal components corresponds to how heavily they are represented in the optimal intervention.
We now place the paper in the context of the literature. The intervention problem we study concerns optimal policy in the presence of externalities. Research over the past two decades has deepened our understanding of the empirical structure of networks and the theory of how networks affect strategic behavior. 4 This has led to the study of how policy design should incorporate information about networks. Network interventions are currently an active subject of research not only in economics but also in related disciplines such as computer science, sociology, and public health. 5 The main contribution of this paper is methodological. It lies in (i) using the principal components approach to decompose the effect of an intervention on social welfare and (ii) using the structure afforded by this decomposition to characterize optimal interventions. Of special interest is the close relation between the strategic structure of the game (whether it features strategic complements or substitutes) and the appropriate principal components to target. 6 The rest of the paper is organized as follows. Section 2 presents the optimal intervention problem. Section 3 sets out how we apply a principal component decomposition to our game. Section 4 characterizes optimal interventions. Section 5 studies a setting where the planner has incomplete information about agents' standalone marginal returns. Section 6 concludes. The Appendix contains the proofs of the main results-those in Section 4. The Supplemental Material ) presents the proofs of other results and discusses a number of extensions.

THE MODEL
We consider a simultaneous-move game among individuals N = {1 n}, where n ≥ 2. Individual i chooses an action, a i ∈ R. The vector of actions is denoted by a ∈ R n . The payoff to individual i depends on this vector, a, the network with adjacency matrix G, and other parameters, as described below: (1) The private marginal returns, or benefits, from increasing the action a i depend both on i's own action, a i , and on others' actions. The coefficient b i ∈ R corresponds to the part of i's marginal return that is independent of others' actions, and is thus called i's standalone marginal return. The contribution of others' actions to i's marginal return is given by the term β j∈N g ij a j . Here g ij ≥ 0 is a measure of the strength of the interaction between i and j; we assume that for every i ∈ N, g ii = 0-there are no self-loops in the network G.
The parameter β captures strategic interdependencies. If β > 0, then actions are strategic complements; if β < 0, then actions are strategic substitutes. The function P i (a −i G b) captures pure externalities-that is, spillovers that do not affect best responses. The firstorder condition for individual i's action to be a best response is Any Nash equilibrium action profile a * of the game satisfies We now make two assumptions about the network and the strength of strategic spillovers. Recall that the spectral radius of a matrix is the maximum of its eigenvalues' absolute values.
ASSUMPTION 1: The adjacency matrix G is symmetric. 7 6 Appendix Section OA2.1 of the Supplemental Material ) presents a discussion of the relationship between principal components and other network measures that have been studied in the literature. 7 We extend our analysis to more general G in Supplemental Material Section OA3.2.
ASSUMPTION 2: The spectral radius of βG is less than 1, 8 and all eigenvalues of G are distinct. (The latter condition holds generically.) Assumption 2 ensures that (2) is a necessary and sufficient condition for each individual to be best-responding, and also ensures the uniqueness and stability of the Nash equilibrium. 9 Under these assumptions, the unique Nash equilibrium of the game can be characterized by The utilitarian social welfare at equilibrium is defined as the sum of the equilibrium utilities: The planner aims to maximize the utilitarian social welfare at equilibrium by changing a vector of status quo standalone marginal returnsb to a vector b, subject to a budget constraint on the cost of her intervention. The timing is as follows. The planner moves first and chooses her intervention, and then individuals simultaneously choose actions. The planner's incentive-targeting (IT) problem is given by where C is a given budget. The function K is an adjustment cost of implementing an intervention. The crucial features of the cost function are that it is separable across individuals and increasing in the magnitude of the change to each individual's incentives. We begin our analysis with the simple functional form given above capturing these features, and examine robustness in the Supplemental Material. In Section OA3.3, we further discuss the form of the adjustment costs and give extensions of the analysis to more general planner cost functions. In Section OA3.4, we examine a setting in which a planner provides monetary payments to individuals that induce them to change their actions, and show that the resulting optimal intervention problem has the same mathematical structure as the one we study in our basic model.
We present two economic applications to illustrate the scope of our model. The first example is a classical investment game, and the second is a game of providing a local public good. EXAMPLE 1-The Investment Game: Individual i makes an investment a i at a cost 1 2 a 2 i . The private marginal return on that investment is b i + β j∈N g ij a j , where b i is individual i's standalone marginal return and j∈N g ij a j is the aggregate local effort. The utility of i is The case with β > 0 reflects investment complementarities, as in . Here, an individual's marginal returns are enhanced when his neighbors work harder; this creates both strategic complementarities and positive externalities. The case of β < 0 corresponds to strategic substitutes and negative externalities; this can be microfounded via a model of competition in a market after the investment decisions a i have been made, as in Goyal and Moraga-Gonzalez (2001). A planner who observes the network of strategic interactions-for instance, which agents work together on joint projects-can intervene by changing levels of monitoring or encouragement relative to a status quo level. It can be verified that the equilibrium utilities, U i (a * G), and the utilitarian social welfare at equilibrium, W (b G), are as follows: EXAMPLE 2-Local Public Goods: We next consider a local public goods problem in a framework that follows the work of Bramoullé and Kranton (2007), , and Allouch (2015Allouch ( , 2017. In a local public goods problem, each agent makes a costly contribution, which brings her closer to an ideal level of public goods but also raises the levels enjoyed by her neighbors. Examples include (i) contributions to improve physical neighborhoods, such as residents clearing snow; 10 (ii) knowledge workers acquiring non-rivalrous information (e.g., about job applicants) that can be shared with colleagues. In example (i), the network governing spillovers is given by physical proximity, while in example (ii), it is given by organizational overlap. We now elaborate on the nature of initial incentives and the interventions in the context of example (i). Agents receive some level of municipal services at the status quo. They augment it with their own effort, and benefit (with a discount) from the efforts contributed by neighbors. A planner (say, a city councilor) who observes the network structure of physical proximity among houses can intervene to change the status quo allocation of services, tailoring it to improve incentives.
Formally, suppose that if each i contributes effort a i to the public good, then the amount of public good i experiences is 10 Other examples include workers keeping areas on a factory floor clean and safe, and businesses in a retail area contributing to the maintenance of their surroundings.
We now connect these formulas to the motivating descriptions. The optimal level of public good in the absence of any costs is τ; this can be thought of as the maximum that can be provided. Individual i has access to a base levelb i of the public good. Each agent can expend costly effort, a i , to augment this base level tob i + a i . If i's neighbor j expends effort, a j , then i has access to an additionalβg ij a j units of the public good, whereβ < 1. This is a game of strategic substitutes and positive externalities. Performing the change of variables b i = [τ − b i ]/2 and β = −β/2 (with the status quo equal tob i = [τ −b i ]/2) yields a best-response structure exactly as in condition (2). The aggregate equilibrium utility is W (b G) = −(a * ) T a * .
All the settings discussed in Examples 1 and 2 share a technically convenient property: PROPERTY A: The aggregate equilibrium utility is proportional to the sum of the squares of the equilibrium actions, that is, W (b G) = w · (a * ) T a * for some w ∈ R, where a * is the Nash equilibrium action profile.
Supplemental Material Section OA2.2 discusses a network beauty contest game inspired by  and  which also satisfies this property. While Property A facilitates analysis, it is not essential. Supplemental Material Section OA3.1 extends the analysis to cover important cases where this property does not hold.

PRINCIPAL COMPONENTS
This section introduces a basis for the space of standalone marginal returns and actions in which, under our assumptions on G, strategic effects and the planner's objective both take a simple form.
FACT 1: If G satisfies Assumption 1, then G = UΛU T , where: 1. Λ is an n × n diagonal matrix whose diagonal entries Λ = λ are the eigenvalues of G (which are real numbers), ordered from greatest to least: λ 1 ≥ λ 2 ≥ · · · ≥ λ n . 2. U is an orthogonal matrix. The th column of U, which we call u , is a real eigenvector of G, namely, the eigenvector associated to the eigenvalue λ , which is normalized so that u = 1 (in the Euclidean norm).
For generic G, the decomposition is uniquely determined, except that any column of U is determined only up to multiplication by −1.
An important interpretation of this diagonalization is as a decomposition into principal components. First, consider the symmetric rank-one matrix that best approximates G in the squared-error sense-equivalently, the vector u such that i j∈N (g ij − u i u j ) 2 is minimized. The minimizer turns out to be a scaling of the eigenvector u 1 . Now, if we consider the "residual" matrix G (2) = G − u 1 (u 1 ) T , we can perform the same type of decomposition on G (2) and obtain the second eigenvector u 2 as the best rank-one approximation. Proceeding further in this way gives a sequence of vectors that constitute an FIGURE 1.-(Top) Eigenvectors 2, 4, 6. (Bottom) Eigenvectors 10, 12, 14. Node shading represents the sign of the entry, with the lighter shading (green) indicating a positive entry and the darker shading (red) indicating a negative entry. Node area is proportional to the magnitude of the entry. orthonormal basis. At each step, the next vector generates the rank-one matrix that "best summarizes" the remaining structure in the matrix G. 11 Figure 1 illustrates some eigenvectors/principal components of a circle network with 14 nodes, where links all have equal weight given by 1. For each eigenvector, the shading of a node indicates the sign of the entry corresponding to that node in that eigenvector, while the size of a node indicates the absolute value of that entry. 12 A general feature worth noting is that the entries of the top eigenvectors (with smaller values of ) are similar among neighboring nodes, while the bottom eigenvectors (with larger values of ) tend to be negatively correlated among neighboring nodes. 13

Analysis of the Game Using Principal Components
For any vector z ∈ R n , let z = U T z. We will refer to z as the projection of z onto the th principal component, or the magnitude of z in that component. Substituting the expression G = UΛU T into equation (2), which characterizes equilibrium, we obtain Multiplying both sides of this equation by U T gives us an analogue of (3) characterizing the solution of the game: This system is diagonal, and the th diagonal entry of [I − βΛ] −1 is 1 1−βλ . Hence, for every ∈ {1 2 n}, 11 See Spielman (2007), especially Section 16.5.1, on this interpretation. For book-length treatments of spectral graph theory, see Cvetkovic, Cvetković, Rowlinson, and Simic (1997) and Chung and Graham (1997). 12 The circle network is invariant to rotations (cyclic permutations) of the nodes and so the eigenvectors here are determined only up to a rotation. 13 For formal treatments of this phenomenon, see Davies, Gladwell, Leydold, and Stadler (2001) and Urschel (2018).
The principal components of G constitute a basis in which strategic effects are easily described. The equilibrium action a * in the th principal component of G is the product of an amplification factor (determined by the strategic parameter β and the eigenvalue λ ) and b , which is simply the projection of b onto that principal component. Under Assumption 2, for all we have 1 − βλ > 0. 14 Moreover, when β > 0 (β < 0), the amplification factor is decreasing (increasing) in .
We can also use (4) to give a formula for equilibrium actions in the original coordinates: We close with a definition that will allow us to describe optimal interventions in terms of a standard measure of their similarity to various principal components.
DEFINITION 1: The cosine similarity of two nonzero vectors y and z is ρ(y z) = y·z y z .
This is the cosine of the angle between the two vectors in a plane determined by y and z. When ρ(y z) = 1, the vector z is a positive scaling of y. When ρ(y z) = 0, the vectors y and z are orthogonal. When ρ(y z) = −1, the vector z is a negative scaling of y.

OPTIMAL INTERVENTIONS
This section develops a characterization of optimal interventions in terms of the principal components and studies their properties.
We begin by dispensing with a straightforward case of the planner's problem. Recall that under Property A, the planner's payoff as a function of the equilibrium actions a * is W (b G) = w · (a * ) T a * . If w < 0, the planner wishes to minimize the sum of the squares of the equilibrium actions. In this case, when the budget is large enough-that is, C ≥ b 2the planner can allocate resources to ensure that individuals have a zero target action by setting b i = 0 for all i. It follows from the best-response equations that all individuals choose action 0 in equilibrium, and so the planner achieves the first-best. 15 The next assumption rules out the case in which the planner's bliss point can be achieved, ensuring that there is an interesting optimization problem.
The last part of the assumption is technical; it holds for generic status quo vectorŝ b (or generic G fixing a status quo vector) and faciliates a description of the optimal intervention in terms of similarity to the status quo vector.
Let b * solve the incentive-targeting problem (IT), and let y * = b * −b be the vector of changes in individuals' standalone marginal returns at the optimal intervention. Furthermore, let α = 1 (1 − βλ ) 2 14 Assumption 2 on the spectral radius implies that βΛ has no entries larger than 1. 15 In the local public goods application (recall Example 2), w = −1, and so when C ≥ b , the optimal intervention satisfies b * i = 0. Recalling our change of variables there (b i = [τ −b i ]/2), the optimal intervention in that case is to modify the endowment of each individual so that everyone accesses the optimal level of the local public good without investing. and note that a * = √ α b is the equilibrium action in the th principal component of G (see equation (4)).
THEOREM 1: Suppose Assumptions 1-3 hold and the network game satisfies Property A. At the optimal intervention, the cosine similarity between y * and principal component u (G) satisfies the following proportionality: where μ, the shadow price of the planner's budget, is uniquely determined as the solution to and satisfies μ > wα for all , so that all denominators are positive.
We briefly sketch the main argument here and interpret the quantities in the formula. Define x = (b −b )/b as the change in b , relative tob . By rewriting the principal's objective W (b G) and budget constraints in terms of principal components and plugging in the equilibrium condition (4), we can rewrite the maximization problem (IT) as If the planner allocates a marginal unit of the budget to changing x , the condition for equality of the marginal return and marginal cost (recalling that μ is the multiplier on the budget constraint) is It follows that wα μ−wα is exactly the value of x at which the marginal return and the marginal cost are equalized. 16 Rewriting x in terms of cosine similarity, that equality implies Rearranging this yields the proportionality expression (5) in the theorem. The Lagrange multiplier μ is determined by solving (6). Now, given μ, the similarities ρ(y * u (G)) determine the direction of the optimal intervention y * . The magnitude of the intervention is found by exhausting the budget. Thus, Theorem 1 entails a full characterization of the optimal intervention. 16 It can be verified that, for every ∈ {1 n − 1}, the ratio x /x +1 is increasing (decreasing) in β for the case of strategic complements (substitutes): thus the intensity of the strategic interaction shapes the relative importance of different principal components.
Next, we discuss the formula for the similarities given in expression (5). The similarity between y * and u (G) measures the extent to which principal component u (G) is represented in the optimal intervention y * . Equation (5) tells us that this is proportional to two factors. The first factor, ρ(b u (G)), measures the similarity between the th principal component and the status quo vectorb. This factor summarizes a status quo effect: how much the initial condition influences the optimal intervention for a given budget. The intuition here is that if a given principal component is strongly represented in the status quo vector of standalone incentives, then-because of the convexity of welfare in the principal component basis-changes in that dimension have a particularly large effect.
The second factor, wα μ−wα , is determined by two quantities: the eigenvalue corresponding to u (G) (via α = 1 (1−βλ ) 2 ), and the budget C (via the shadow price μ). To focus on this second factor, wα μ−wα , we define the similarity ratio Theorem 1 shows that, as we vary , the similarity ratio r * is proportional to wα μ−wα . It follows that the similarity ratio is greater, in absolute value, for the principal components with greater α . Intuitively, those are the components where the optimal intervention makes the largest change relative to the status quo profile of incentives. The ordering of the r * corresponds to the eigenvalues in a way that depends on the nature of strategic spillovers: COROLLARY 1: Suppose Assumptions 1-3 hold and the network game satisfies Property A. If the game is one of strategic complements (β > 0), then |r * | is decreasing in ; if the game is one of strategic substitutes (β < 0), then |r * | is increasing in .
In some problems, there may be a nonnegativity constraint on actions, in addition to the constraints in problem (IT). As long as the status quo actionsb are positive, this constraint will be respected for all C less than someĈ, and so our approach will give information about the relative effects on various components for interventions that are not too large.

Small and Large Budgets
The optimal intervention takes especially simple forms in the cases of small and large budgets. From equation (6), we can deduce that the shadow price μ is decreasing in C. For w > 0, it follows that an increase in C raises wα μ−wα . Moreover, the sequence of similarity ratios becomes "steeper" as we increase C, in the following sense: if w > 0, for all , such that α > α , we have that r * /r * is increasing in C. 17 PROPOSITION 1: Suppose Assumptions 1-3 hold and the network game satisfies Property A. Then the following hold: 1. As C → 0, in the optimal intervention, r * r * → α α . 2. As C → ∞, in the optimal intervention: 2a. If β > 0 (the game features strategic complements), then the similarity of y * and the first principal component of the network tends to 1: ρ(y * u 1 (G)) → 1. 2b. If β < 0 (the game features strategic substitutes), then the similarity of y * and the last principal component of the network tends to 1: ρ(y * u n (G)) → 1.
This result can be understood by recalling equation (5) in Theorem 1. First, consider the case of small C. When the planner's budget becomes small, the shadow price μ tends to ∞. 18 Equation (5) then implies that the similarity ratio r * of the th principal component (recall (7)) becomes proportional to α . Turning now to the case where C grows large, the shadow price converges to wα 1 if β > 0, and to wα n if β < 0 (by equation (6)). Plugging this into equation (5), we find that in the case of strategic complements, the optimal intervention shifts individuals' standalone marginal returns (very nearly) in proportion to the first principal component of G, so that y * → √ Cu 1 (G). In the case of strategic substitutes, on the other hand, the planner changes individuals' standalone marginal returns (very nearly) in proportion to the last principal component, namely, y * → √ Cu n (G). 19 Figure 2 depicts the optimal intervention in an example where the budget is large. We consider an 11-node undirected network with binary links containing two hubs, L 0 and R 0 , that are connected by an intermediate node M; the network is depicted in Figure 2(A). The numbers next to the nodes are the status quo standalone marginal returns; the budget is set to C = 500. 20 Payoffs are as in Example 1. For the case of strategic complements, we set β = 0 1, and for strategic substitutes, we set β = −0 1. Assumptions 1 and 2 are satisfied and Property A holds. The top-left of Figure 2(B) illustrates the first eigenvector, and the top-right depicts the optimal intervention in a game with strategic complements. The bottom-left of Figure 2(B) illustrates the last eigenvector, and the bottom-right depicts the optimal intervention when the game has strategic substitutes. The node size represents the size of the intervention, |b * i −b i |; node shading represents the sign of the FIGURE 2.-An example of optimal interventions with large budgets.
18 As costs are quadratic, a small relaxation in the budget around zero can have a large impact on aggregate welfare. 19 When individuals' initial standalone marginal returns are zero (b = 0), we can dispense with the approximations invoked for a large budget C. Assuming that G is generic, ifb = 0, then, for any C, the entire budget is spent either (i) on changing b 1 (if β > 0) or (ii) on changing b n (if β < 0). 20 About 125 times larger than b 2 . intervention, with the lighter shading (green) indicating a positive intervention and the darker shading (red) indicating a negative intervention.
In line with part 2 of Proposition 1, for large C, the optimal intervention is guided by the "main" component of the network. Under strategic complements, this is the first (largesteigenvalue) eigenvector of the network, whose entries are individuals' eigenvector centralities. 21 Intuitively, by increasing the standalone marginal return of each individual in proportion to his eigenvector centrality, the planner targets the individuals in proportion to their global contributions to strategic feedbacks, and this is welfare-maximizing.
Under strategic substitutes, optimal targeting is determined by the last eigenvector of the network, corresponding to its smallest eigenvalue. This network component contains information about the local structure of the network: it determines a way to partition the set of nodes into two sets so that most of the links are across individuals in different sets. 22 The optimal intervention increases the standalone marginal returns of all individuals in one set and decreases those of individuals in the other set. The planner wishes to target neighboring nodes asymmetrically, as this reduces crowding-out effects that occur due to the strategic substitutes property.

When Are Interventions Simple?
We have just seen examples illustrating how, with large budgets, the intervention is approximately simple in a certain sense: proportional to just one principal component. After formalizing a suitable notion of simplicity in our setting, the final result in this section characterizes how large the budget must be for such an approximation to be accurate.
Cu 1 i when the game has the strategic complements property (β > 0), Cu n i when the game has the strategic substitutes property (β < 0).
Such an intervention is called simple because the intervention on each node is-up to a common scaling-determined by a single number that depends only on the network (via its eigenvectors), and not on any other details such as the status quo incentives. 23 Let W * be the aggregate utility under the optimal intervention, and let W s be the aggregate utility under the simple intervention.
PROPOSITION 2: Suppose w > 0, Assumptions 1 and 2 hold, and the network game satisfies Property A.
1. If the game has the strategic complements property, β > 0, then for any If the game has the strategic substitutes property, β < 0, then for any Supplemental Material Section OA2.1 presents a discussion of eigenvector centrality and how it compares to centrality measures that turn out to be important in related targeting problems. 22 The last eigenvector of a graph is useful in diagnosing the bipartiteness of a graph and its chromatic number. Desai and Rao (1994) characterized the smallest eigenvalue of a graph and related it to a measure of the graph's bipartiteness. Alon and Kahale (1997) related the last eigenvector to a coloring of the underlying graph-a labeling of nodes by a minimal set of integers such that no neighboring nodes share the same label. 23 The definition also specifies the interventions in more detail-that is, that they are proportional to the appropriate u i . We could have instead left this flexible in this definition of simplicity and specified the dependence on the network explicitly in Proposition 2.
Proposition 2 gives a condition on the size of the budget beyond which (a) simple interventions achieve most of the optimal welfare and (b) the optimal intervention is very similar to the simple intervention. This bound depends on the status quo standalone marginal returns and on the structure of the network via α 2 α 1 −α 2 (or a corresponding factor for "bottom" α's).
We first discuss the dependence of the bound on the status quo marginal returns. Observe that the first term on the right-hand side of the inequality for C is proportional to the squared norm ofb. This inequality is therefore easier to satisfy when this vector has a smaller norm. The inequality is harder to satisfy when these marginal returns are large and/or heterogeneous. 24 Next, consider the role of the network. Recall that α = (1 − βλ ) −2 ; thus if β > 0, the term α 2 /(α 1 − α 2 ) of the inequality is large when λ 1 − λ 2 , the "spectral gap" of the graph, is small. If β < 0, then the term α n−1 /(α n−1 − α n ) is large when the difference λ n−1 − λ n , which we call the "bottom gap," is small.
We now examine what network features affect these gaps, and illustrate with examples, depicted in Figure 3. The obstacle to the existence of simple optimal interventions is a strong dependence on the status quo standalone marginal returns. This dependence will be strong when two different principal components in the network offer the potential for similar amplification of an intervention. Which of these principal components receives the planner's focus will depend strongly on the status quo. In such networks, interventions will not be simple unless budgets are very large relative to status quo incentives. The implication of Proposition 2 is that this sensitivity occurs when the appropriate gap in eigenvalues (spectral gap or bottom gap) is small. Figure 3 illustrates the role of the network structure in shaping how the optimal intervention converges to the simple one (as C increases). Under strategic complements, a large spectral gap ensures fast convergence. Under strategic substitutes, a large bottom gap ensures fast convergence.
We now describe which more directly visible properties of network topology correspond to small and large spectral gaps. First, consider the case of strategic complements. A standard fact is that the two largest eigenvalues can be expressed as follows: Moreover, the eigenvector u 1 is a maximizer of the first problem, while u 2 is a maximizer of the second; these are uniquely determined under Assumption 2. By the Perron-Frobenius theorem, the first eigenvector, u 1 , assigns the same sign-say, positive-to all nodes in the network. Then the eigenvector u 2 must clearly assign negative values to some of the nodes (as it is orthogonal to u 1 ). In the network on the left side of Figure 3(A), any such assignment will result in many adjacent nodes having opposite-sign entries of u 2 ; as a result, many terms in the expression for λ 2 will be negative, and λ 2 will be considerably smaller than λ 1 , leading to a large spectral gap. In the network on the right side of Figure 3(A), u 2 turns out to have positive-sign entries for nodes in one community and negative-sign entries for nodes in the other community. Because there are few edges between the communities, λ 2 turns out to be almost as large as λ 1 . This yields a small spectral gap. These observations illustrate that the spectral gap is large when the network is "cohesive," and small when the network is, in contrast, divisible into nearly-disconnected communities. 25 In light of this interpretation, our results imply that highly cohesive networks admit near-optimal interventions that are simple. Turning next to strategic substitutes, recall that the smallest two eigenvalues, λ n and λ n−1 , can be written as follows: Moreover, the eigenvector u n is a maximizer of the first problem, while u n−1 is a maximizer of the second; these are uniquely determined under Assumption 2. This tells us that λ n is low 26 when the eigenvector u n = arg min u: u =1 i j∈N g ij u i u j (corresponding to λ n ) assigns opposite signs to most pairs of adjacent nodes. In other words, the last eigenvalue is small when nodes can be partitioned into two sets and most of the connections are across sets. Thus, λ n is minimized in a bipartite graph. The second-smallest eigenvalue of G reflects the extent to which the next-best eigenvector (orthogonal to u n ) is good at solving the same minimization problem. Hence, the bottom gap of G is small when there are two orthogonal ways to partition the network into two sets so that, either way, the "quality" of the bipartition, as measured by i j∈N g ij u i u j , is similar.
We illustrate part 2 of Proposition 2 with a comparison of the two graphs in Figure 3(C). The left-hand graph is bipartite: the last eigenvalue is λ n = −3 and the second-last eigenvalue is λ n−1 = −1 64. In contrast, the graph on the right of Figure 3(C) has a bottom eigenvalue λ n = −2 62, and a second-lowest eigenvalue of λ n−1 = −2 30. This yields a much smaller bottom gap. 27 This difference in bottom gaps is reflected in the nature of optimal interventions shown in Figure 3(D). In the graph with large bottom gap, the optimal intervention puts most of its weight on the eigenvector u n even for relatively small budgets. To achieve a similar convergence to simplicity requires a much larger budget when the bottom gap is small: the second-smallest eigenvector u n−1 receives substantial weight even for fairly large values of the budget C.
We conclude by noting the influence of the status quo standalone marginal returns in shaping optimal interventions for small budgets. For a small budget C, the cosine similarity of the optimal intervention for non-main network components can be higher than the one for the main component. This is true when the status quob is similar to some of the non-main network components; see

INCOMPLETE INFORMATION
In the basic model, we assumed that the planner knows the standalone marginal returns of every individual. This section extends the analysis to settings where the planner does not know these parameters. As before, we focus on network games that satisfy Property A.
Formally, fix a probability space (Ω F P). The planner's probability distribution over states is given by P. The planner has control over the random vector (r.v.) B-that is, a function B : Ω → R n . The cost of the intervention depends on the choice of B. There is a function K that gives the cost K(B) of implementing the random variable B. 28 A realization of the random vector is denoted by b. This realization is common knowledge among individuals when they choose their actions. Thus, the game individuals play is one of complete information. 29 We solve the following incomplete-information intervention problem: Note that the intervention problem (IT) under complete information is the special case of a degenerate r.v. B: one in which the planner knows the vector of standalone marginal returns exactly and implements a deterministic adjustment relative to it. To guide our modeling of the cost of intervention, we now examine the features of the distribution of B that matter for aggregate welfare. For network games that satisfy 27 Intuitively, because u n does not correspond to a perfect bipartition, it is easier for a vector orthogonal to u n to achieve a similarly low value of i j∈N g ij u i u j . 28 The domain of this function is the set of all random vectors taking values in R n defined on our probability space. 29 It is possible to go further and allow for incomplete information among the individuals about each other's b i . We do not pursue this substantial generalization here; see Golub and Morris (2020) and Lambert, Martini, and Ostrovsky (2018) for analyses in this direction. Property A, we can write Note the change from the ordinary to the principal component basis in the second step.
In words, welfare is determined by the mean and variance of the realized components b ; these in turn are determined by the first and second moments of the chosen random variable B. In view of this, we will consider intervention problems where the planner can modify the mean and the covariance matrix of B, and the cost of intervention depends only on these modifications. 30

Mean Shifts
We first consider an intervention where there is an arbitrarily distributed vector of status quo standalone marginal returns and the planner's intervention shifts it in a deterministic way. Formally, fix a random variableB, called the status quo, with typical realizationb. The planner's policy is given by b =b + y, where y ∈ R n is a deterministic vector. We denote the corresponding random variable by B y . In terms of interpretation, note that implementing this policy does not require knowingb as long as the planner has an instrument that shifts incentives.
In contrast to the analysis of Theorem 1, the vectorb is a random variable. But we take the analogue of the cost function used there, noting that in the deterministic setting (see (IT)), this formula held with y = b −b.
PROPOSITION 3: Consider problem (IT-G), with the cost of intervention satisfying Assumption 4. Suppose Assumptions 1 and 2 hold and the network game satisfies Property A. The optimal intervention policy B * is equal to B y * , where y * is the optimal intervention in the deterministic problem with b = E[b] taken as the status quo vector of standalone marginal returns.

Intervention on Variances
We next consider the case where the planner faces a vector of means, fixed atb, and, subject to that, can choose any random variable B. It can be seen from (9) that, in this class of mean-neutral interventions, the expected welfare of an intervention B depends only on the variance-covariance matrix of B. Thus, the planner effectively faces the problem of intervening on variances, which we analyze for all cost functions satisfying certain symmetries.
ASSUMPTION 5: The cost function satisfies two properties: (Analogously to our other notation, we useb for realizations ofB.) Part (a) is a restriction on feasible interventions, namely, a restriction to interventions that are mean-neutral. Part (b) means that rotations of coordinates around the mean do not affect the cost of implementing a given distribution. This assumption gives the cost a directional neutrality, which ensures that our results are driven by the benefits side rather than by asymmetries operating through the costs. For an example where the assumption is satisfied, let Σ B be the variance-covariance matrix of the random variable B. In particular, σ B ii is the variance of b i . Suppose that the cost of implementing B with E[b] =b is a function of the sum of the variances of the b i : The cost function (10)  Let the optimal intervention be B * , and let b * be a typical realization. We have the following: 1. Suppose the planner likes variance (i.e., in (9), w > 0). If the game has strategic complements (β > 0), then Var(u (G) · b * ) is weakly decreasing in ; if the game has strategic substitutes (β < 0), then Var(u (G) · b * ) is weakly increasing in . 2. Suppose the planner dislikes variance (i.e., w < 0). If the game has strategic complements (β > 0), then Var(u (G) · b * ) is weakly increasing in ; if the game has strategic substitutes (β < 0), then Var(u (G) · b * ) is weakly decreasing in .
We now provide the intuition for Proposition 4. Shocks to individuals' standalone marginal returns create variability in the players' equilibrium actions. The assumption that the intervention is mean-neutral (part (a) of Assumption 5) leaves the planner to control only the variances and covariances of these marginal returns with her intervention. Hence, the solution to the intervention problem describes what the planner should do to induce second moments of the action distribution that maximize ex ante expected welfare.
Suppose first that investments are strategic complements. Then a perfectly correlated (random) shock in individual standalone marginal returns is amplified by strategic interactions. In fact, the type of shock that is most amplifying (at a given size) is the one that is perfectly correlated across individuals: a common deviation from the mean is scaled by the vector u 1 -the individuals' eigenvector centralities. Such shocks are exactly what b * 1 = u 1 (G) · b * captures. Hence, this is the dimension of volatility that the planner most wants to increase if she likes variance in actions (w > 0) and most wants to decrease if she dislikes variance in actions (w < 0).
If investments are strategic substitutes, then a perfectly correlated shock does not create a lot of variance in actions: the first-order response of all individuals to an increase in their standalone marginal returns is to increase investment, but that in turn makes all individuals decrease their investment somewhat because of the strategic substitutability with their neighbors. Hence, highly positively correlated shocks do not translate into high volatility. The shock profiles (of a fixed norm) that create the most variability in equilibrium actions are actually the ones in which neighbors have negatively correlated shocks. A planner that likes variance in actions will then prioritize such shocks. Because the last eigenvector of the system has entries that are as different as possible across neighbors, this is exactly the type of volatility that will be most amplified, and this is what the planner will focus on most.
EXAMPLE 3-Illustration in the Case of the Circle: Figure 1 depicts six of the eigenvectors/principal components of a circle network with 14 nodes. The first principal component is a positive vector and so B projected on u 1 (G) captures shocks that are positively correlated across all players. The second principal component (top left panel of Figure 1) splits the graph into two sides, one with positive entries and the other with negative entries. Hence, B projected on u 2 (G) captures shocks that are highly positively correlated on each side of the circle network, with the two opposite sides of the circle being anticorrelated. As we move along the sequence of eigenvectors u , we can see that B projected on the th eigenvector represents patterns of shocks that "vary more" across the network. At the extreme, B projected on u 14 (G) (bottom-right panel of Figure 1) captures the component of shocks that is perfectly anti-correlated across neighbors. 32

CONCLUDING REMARKS
We have studied the problem of a planner who seeks to optimally target incentive changes in a network game. Our framework allows for a broad class of strategic and nonstrategic spillovers across neighbors. The main contribution of the paper is methodological: we show that principal components of the network of interaction provide a useful basis for analyzing the effects of an intervention. This decomposition leads to our main result: there is a close connection between the strategic properties of the game (whether actions are strategic complements or substitutes) and the weight that different principal components receive in the optimal intervention. To develop these ideas in the simplest way, we have focused on a model in which the matrix of interaction is symmetric, the costs of intervention are quadratic, and the intervention itself takes the form of altering the standalone marginal returns of actions. In the Supplemental Material, we relax these restrictions and develop extensions of our approach to non-symmetric matrices of interaction and to more general costs of intervention, including a model where interventions occur via monetary incentives for activity. We also relax Property A, a technical condition which facilitated our basic analysis, and cover a more general class of externalities.
We briefly mention two further applications. In some circumstances, the planner seeks a budget-balanced tax/subsidy scheme in order to improve the economic outcome. In an oligopoly market, for example, a planner could tax some suppliers, thereby increasing their marginal costs, and then use that tax revenue to subsidize other suppliers. The planner will solve a problem similar to the one we have studied here, with the important difference that she will face a different constraint-namely, a budget-balance constraint. In ongoing work, Galeotti, Golub, Goyal, Talamàs, and Tamuz (2020) show that the principal component approach that we employed in this paper is useful in deriving the optimal taxation scheme and, in turn, in determining the welfare gains that can be achieved via tax/subsidy interventions in supply chains. 33 We have focused on interventions that alter the standalone marginal returns of individuals. Another interesting problem is the study of interventions that alter the matrix of interaction. We hope this paper stimulates further work along these lines.

APPENDIX: PROOFS
PROOF OF THEOREM 1: We wish to solve We transform the maximization problem into the basis given by the principal components of G. To this end, we first rewrite the cost and the objective in the principal components basis, using the fact that norms do not change under the orthogonal transformation U T . (The norm symbol · always refers to the Euclidean norm.) Letting y = b −b, By recalling that, in equilibrium, a * = [I − βΛ] −1 b, and using the definition α =

1
(1−βλ (G)) 2 , the intervention problem (IT) can be rewritten as y 2 ≤ C 33 In a recent paper, Gaitonde, Kleinberg and Tardos (2020) use spectral methods to study interventions that polarize opinions in a social network.
We now transform the problem so that the control variable is x where x = y /b . We obtain x 2 ≤ C Note that, for all , α are well-defined (by Assumption 1) and strictly positive (by genericity of G). This has two implications. 34 First, at the optimal solution x * , the resource constraint problem must bind. To see this, note that Assumption 3 says that either w > 0, or w < 0 and n =1b 2 > C. Suppose that at the optimal solution, the constraint does not bind. Then, without violating the constraint, we can slightly increase or decrease any x . If w > 0 (resp. w < 0), the increase or the decrease is guaranteed to increase (resp. decrease) the corresponding (x + 1) 2 (since the α are all strictly positive).
Second, we show that the optimal solution x * satisfies x * ≥ 0 for every if w > 0, and x * ∈ [−1 0] for every if w < 0. Suppose w > 0 and, for some , x * < 0. Then [−x * + 1] 2 > [x * + 1] 2 . Since w > 0 and every α is positive, we can raise the aggregate utility without changing the cost by flipping the sign of x * . Analogously, suppose w < 0. It is clear that if x * < −1, then by setting x = −1, the objective improves and the constraint is relaxed; hence, at the optimum, x * ≥ −1. Suppose next that x > 0 for some . Then [−x * + 1] 2 < [x * + 1] 2 . Since w < 0 and every α is positive, we can improve the value of the objective function without changing the cost by flipping the sign of x * .
We now complete the proof. Observe that the Lagrangian corresponding to the maximization problem is Taking our observation above that the constraint is binding at x = x * , together with the standard results on the Karush-Kuhn-Tucker conditions, the first-order conditions must hold exactly at the optimum with a positive μ: We take a genericb such thatb = 0 for all . If, for some , we had μ = wα , then the right-hand side of the second equality in (11) would be 2b 2 wα , which, by the generic assumption we just made and the positivity of α , would contradict (11). Thus, the following holds with a nonzero denominator: x * = wα μ − wα 34 Note that if Assumption 3 does not hold (i.e., w < 0 and n =1b 2 ≤ C), then the optimal solution is x * = −1 for all . This is what we ruled out with Assumption 3, before Theorem 1. and the Lagrange multiplier μ is therefore pinned down by PROOF OF PROPOSITION 1: Part 1. From expression (6) of Theorem 1, it follows that if C → 0, then μ → ∞. The result follows by noticing that Part 2. Suppose that β > 0. Using the derivation of the last part of the proof of Theorem 1, we write (6) of Theorem 1, it follows that if C → ∞, then μ → wα 1 . This implies that x * → α α 1 −α for all = 1. As a result, if C → ∞, then ρ(y * u (G)) → 0 for all = 1. Furthermore, we can rewrite expression (6) of Theorem 1 as where the first equality follows because x * → α α 1 −α for all = 1. The proof for the case of β < 0 follows the same steps, with the only exception that if C → ∞, then μ → wα n . Q.E.D.

PROOF OF PROPOSITION 2:
We first prove the result on welfare and then turn to the result on cosine similarity.
Welfare. Consider the case of strategic complementarities, β > 0. Define byx the simple intervention, and note thatx 1 = √ C/b 1 and thatx = 0 for all > 1. The aggregate utility obtained under the simple intervention is The aggregate utility at the optimal intervention is The fact b 2 1x 2 1 = C, used above, follows because the simple policy allocates the entire budget to changing b 1 . The inequality after that statement follows because Hence, the inequality C > 2 b 2 α 2 α 1 − α 2 2 is sufficient to establish that W * W s < 1 + . The proof for the case of strategic substitutes follows the same steps; the only difference is that we use α n instead of α 1 and α n−1 instead of α 2 .
Cosine similarity. We now turn to the cosine similarity result. We focus on the case of strategic complements. The proof for the case of strategic substitutes is analogous. We start by writing a useful explicit expression for ρ b * √ Cu 1 : where the last equality follows because, at the optimum, b * −b 2 = C. At the optimal intervention, by Theorem 1, Hence, using this in equation (12), we can deduce that We now claim that the inequality in the above display after the "if and only if" follows from our hypothesis that This claim is established by the following lemma.
LEMMA 1: Assume PROOF OF LEMMA 1: Note that But then we have the following chain of statements, explained immediately after the display: The first inequality follows from substituting the upper bound on C(1 − ), statement (15) above, which we derived from our initial condition on C. The equality after that follows by substituting the condition on the binding budget constraint at the optimum, which we derived in Theorem 1. The next equality follows by isolating the term for the first component in the summation and by noticing that that cancels with the first term. The next equality follows by noticing that b 2 = b 2 . The final inequality follows because, from the facts that μ > wα 1 and that α 1 > α 2 > · · · > α n , we can deduce that for each This concludes the proof of Proposition 2.

Q.E.D.
We consider the case of w > 0 and β > 0; the proof of the other cases is analogous and therefore omitted. The expected welfare is a weighted sum of the variances of the principal components, Var(b * ) = Var(u (G) · b * ), and the weight α on the variance of principal component of G is an increasing function of its eigenvalue λ , because β > 0.
Suppose that the claim in the proposition is violated, that is, there exist an such that < and Var(b * ) < Var(b * ). We construct an alternative intervention that has the same cost and does strictly better. Take the permutation matrix (and therefore an orthogonal matrix) P such that P kk = 1 for all k / ∈ { } and P = P = 1. Define B * * = OB * with O = UP U T . Clearly, O is orthogonal, as U and P are both orthogonal. Hence, by Assumption 5, K(B * ) = K(B * * ). Furthermore, the matrix . Since α > α , intervention B * * does strictly better than B * , a contradiction to our initial hypothesis that B * was optimal. Q.E.D.

OA2. DISCUSSION
We discuss the relation of principal components of the matrix of interactions with other related networks statistics (Section OA2.1). We then provide a different economic example, which complements those in our main text, inspired by beauty contest games (Section OA2.2).

First Principal Component and Eigenvector Centrality
For ease of exposition, let the network be connected, that is, let G be irreducible. By the Perron-Frobenius theorem, u 1 (G) is entry-wise positive; indeed, this vector is the Perron vector of the matrix, also known as the vector of individuals' eigenvector centralities. Thus, our results of Section 4 imply that, under strategic complementarities, interventions that aim to maximize the aggregate utility should change individuals' incentives in proportion to their eigenvector centralities.
It is worth comparing this result with results that highlight the importance of Bonacich centrality. Under strategic complements, equilibrium actions are proportional to the individuals' Bonacich centralities in the network . 1 Within the  framework, it can easily be verified that if the objective of the planner is linear in the sum of actions, then under a quadratic cost function the planner will target individuals in proportion to their Bonacich centralities (see also ). Bonacich centrality converges to eigenvector centrality as the spectral radius of βG tends to 1; otherwise, the two vectors can be quite different (see, e.g., Calvó-Armengol, Martí, and Prat (2015) or Golub and Lever (2010)).
The substantive point is that the objective of our planner when solving the intervention problem (IT) is to maximize the aggregate equilibrium utility, not the sum of actions, and that explains the difference in the targeting strategy. Indeed, our planner's objective (under Property A) can be written as follows (introducing a different constant factor for convenience): where σ 2 a is the variance of the action profile and a is the mean action. Thus, our planner cares about the sum of actions and also their diversity, simply as a mathematical consequence of her objective. This explains the reason why her policies differ from those that would be in effect if just the mean action were the focus. To reiterate this point, we finally note that if we consider problem (IT) but we assume that the cost of intervention is linear, that is, K(b b ) = i |b i −b i |, then the optimal intervention will target only one individual (see the discussion in Section OA3.3 of this supplement); note that the targeted individual is not necessarily the individual with the highest Bonacich centrality.

Last Principal Component
We have shown that in games with strategic substitutes, for large budgets interventions that aim to maximize the aggregate utility target individuals in proportion to the eigenvector of G associated to the lowest eigenvalue of G, the last principal component.
There is a connection between this result and the work of .  studied the set of equilibria of a network game with linear best replies and strategic substitutes. They observed that such a game is a potential game, and they derived the potential function explicitly. From this, they deduced that the lowest eigenvalue of G is crucial for whether the equilibrium is unique, and it is also useful for analyzing the stability of a particular equilibrium. 2 The basic intuition is that the magnitude of the lowest eigenvalue determines how small changes in individuals' actions propagate, via strategic substitutes, in the network. When these amplifications are strong, multiple equilibria can emerge. Relatedly, when these amplifications are strong around an equilibrium, that equilibrium will be unstable.
Our study of the strategic substitutes case is driven by different questions, and delivers different sorts of characterizations. We assume that there is a stable equilibrium which is unique at least locally, and then we characterize optimal interventions in terms of the eigenvectors of G. In general, all the eigenvectors-not just the one associated to the lowest eigenvalue-can matter. Interventions will focus more on the eigenvectors with smaller eigenvalues. When the budget is sufficiently large, the intervention will (in the setting of Section 4) focus on only the lowest-eigenvalue eigenvector. As discussed in Section 4, the network determinants of whether targeting is simple can be quite subtle. To the best of our knowledge, these considerations are all new in the study of network games. Nevertheless, at an intuitive level, there are important points of contact between our intuitions and those of . In our context, as discussed earlier, our planner likes to move the incentives of adjacent individuals in opposite directions. The eigenvector associated to the lowest eigenvalue emerges as the one identifying the best way to do this at a given cost, and the eigenvalue itself measures how intensely the strategic effects amplify. This "amplification" property involves forces similar to those that make the lowest eigenvalue important to stability and uniqueness in .

Spectral Approaches to Variance Control
Acemoglu, Ozdaglar, and Tahbaz-Salehi (2016) gave a general analysis of which network statistics matter for volatility of network equilibria. Baqaee and Farhi (2019) developed a rich macroeconomic analysis relating network measures to aggregate volatility. Though both papers note the importance of eigenvector centrality in (their analogues of) the case of strategic complements, their main focus is on how the curvature of best responses changes the volatility of an aggregate outcome, and which "second-order" (curvature-related) network statistics are important. We use the principal components of the network to understand which first-order shocks are most amplified, and how this depends on the nature of strategic interactions.

OA2.2. Beauty Contest With Local Interactions
This example is inspired by  and . Individuals trade off the returns from effort against the costs, as in the first example, but also care about coordinating with others. These considerations are captured in the following payoff: where we assume thatβ > 0 and γ > 0 and that j g ij = 1 for all i, so the total interaction is the same for each individual. This formulation also relates to the theory of teams and organizational economics (see, e.g., Dessein, Galeotti, and Santos (2016), Marschak and Radner (1972), and Calvó-Armengol, Martí, and Prat (2015)). We may interpret individuals as managers in different divisions within an organization. Each manager selects the action that maximizes the output of the division, given by the first term, but the manager also cares about coordinating with other divisions' actions. 3 This is a game of strategic complements; moreover, an increase in j's action has a positive effect on individual i's utility if and only if a j < a i . It can be verified that the first-order condition for individual i is given by By defining β =β +γ 1+γ and b = 1 1+γb , we obtain a best-response structure exactly as in condition (2). Moreover, the aggregate equilibrium utility is W (b g) = 1 2 (a * ) T a * . Hence, this game satisfies Property A.

OA3. EXTENSIONS
We now extend our basic model to study settings where (a) Property A is not satisfied (Section OA3.1), (b) the matrix G is non-symmetric (Section OA3.2), (c) the exact quadratic cost specification does not hold (Section OA3.3), and (d) the interventions occur via monetary incentives for activity (Section OA3.4).

OA3.1. General Non-Strategic Externalities
Section 4 characterizes optimal interventions for network games that satisfy Property A. We now relax this assumption. Recall that player i's utility for action profile a is whereÛ i (a G) = a i (b i + j g ij a j ) − 1 2 a 2 i . At an equilibrium a * , it can be checked that iÛ i (a * G) ∝ (a * ) T a * . Therefore, a sufficient condition for Property A to be satisfied is that i P i (a * −i G b) is also proportional to (a * ) T a * . Examples 1 and 2, as well as the example presented in Section OA2.2, satisfy this property. However, as the next example shows, there are natural environments in which it is violated. EXAMPLE OA1-Social Interaction and Peer Effects: Individual decisions on smoking and alcohol consumption are susceptible to peer effects (see  for references to the extensive literature on this subject). For example, an increase in smoking among an adolescent's friends increases her incentives to smoke and, at the same time, has negative effects on her welfare. These considerations are reflected in the following payoff function: where β > 0 and γ is positive and sufficienctly large. It can be checked that the aggregate equilibrium welfare is with a * given by expression (3). 4 To extend the analysis beyond Property A, we allow the non-strategic externality term P i (a −i G b) to take a form that allows for flexible externalities within the linearquadratic family: 5 We also make the following assumption on the matrix G: ASSUMPTION OA1: The total interaction is constant across individuals, that is, j g ij = 1 for all i ∈ N .
Using equation (3) and Assumption OA1, we can rewrite the expression for the aggregate equilibrium utility as follows: where w 1 = 1 + m 2 + m 5 + (n − 1)m 4 , w 2 = nm 5 (n − 2), and w 3 = √ n[m 1 + (n − 1)m 3 ]. Observe that Property A clearly holds when w 2 = w 3 = 0. On the other hand, if (say) w 1 = w 2 = 0, then the planner's objective is to maximize the sum of the equilibrium actions, which is a fairly different type of objective. A characterization of the optimal intervention when the planner's objective is to maximize the sum of the equilibrium actions can be found in Corollary OA1 below. Under Assumption OA1, the sum of the equilibrium actions is proportional to the sum of the standalone marginal returns. Because u 1 is proportional to the all-ones vector 1, this sum in turn is equal to b 1 .
Together, these facts allow us to extend our earlier analysis to the case of general w 2 and w 3 . First, we can still express the objective function simply in terms of the singular value decomposition; the only difference is that now b 1 will enter both in a quadratic term and in a linear term. In view of this, we first solve the problem (exactly analogously to the previous solution) for a given value of b 1 , and then we optimize over b 1 .
We maintain Assumption 1 and Assumption 2. Recall that player i's utility for action profile a is whereÛ i (a G) = a i (b i + j g ij a j ) − 1 2 a 2 i and P i (a −i G b) is a non-strategic externality term that takes the following form: Here we have taken local and global externality terms given by second-order polynomials in actions. (We could also accommodate externalities that depend directly on the b i in the same sort of way, as will become clear in the proof, but we omit this for brevity.) The implication of Assumption OA1 for our analysis is summarized next.
LEMMA OA1: Assumption OA1 implies that: 1. for any a ∈ R n , i j g ij a j = i a i and i j g ij a 2 j = i a 2 i , 2. λ 1 (G) = 1 and u 1 The proof of Lemma OA1 is immediate. Using part 1 of Lemma OA1, and that individuals play an equilibrium (actions satisfy expression (3)), we obtain The last equality follows because α 1 = 1/(1 − βλ 1 ) 2 , and Assumption OA1 implies that λ 1 = 1.
with w 1 = 1 + m 2 + m 5 + (n − 1)m 4 w 2 = nm 5 (n − 2) w 3 = √ n m 1 + (n − 1)m 3 Using the decomposition G = UΛU T , together with part 2 and part 3 of Lemma OA1, we obtain The intervention problem reads Using the expression for equilibrium actions, we obtain for every , we finally rewrite the problem as Theorem OA1 characterizes optimal interventions for two cases: (i) w 1 ≥ 0 and (ii) w 1 < 0 and =2b 2 > C. The extension of the analysis for the remaining case w 1 < 0 and =2b 2 < C is explained in Remark OA1, which is presented after the proof of Theorem OA1. Taken together, Theorem OA1, and Remark OA1 following it, constitute our extension of Theorem 1 to games that do not satisfy Property A. THEOREM OA1: Suppose Assumptions 1, 2, and OA1 hold. Suppose that either (i) w 1 ≥ 0 or that (ii) w 1 < 0 and =2b 2 > C. The optimal intervention is characterized as follows: 1.
and, for all ≥ 2, Before the proof, we briefly explain the sense in which this extends Theorem 1 and associated results in the basic model. The formula for x * in part 1 is a direct generalization of equation (5), with the shadow price characterized by an equation analogous to (6). The monotonicity relations on x * in part 2 correspond to Corollary 1. The small-C analysis of part 3 corresponds to Proposition 1. The large-C analysis in parts 4 and 5 corresponds to the limits studied in Section 4.2.
PROOF OF THEOREM OA1: Part 1. For a given x ∈ R n , define The maximization problem can be rewritten as We solve this problem in two steps.

First
Step. We fix x 1 so that C(x 1 ) ≥ 0; that is, In the case in which w 1 = 0, we skip this first step. If w 1 = 0, then we argue in a way exactly analogous to the proof of Theorem 1 that, for all = 1, where, for all = 1, μ ≥ w 1 α and it solves Note that, for all ≥ 2, x * > 0 if w 1 > 0 and x * < 0 if w 1 < 0.
Note also that if w 1 < 0, the constraint binds: the bliss point (x * = −1 for all = 1) cannot be achieved because C < n =2b 2 .

Second
Step. Substituting into the objective function the expression for x * , for all ≥ 2, we obtain max The following lemma is instrumental to the solution of this problem. It characterizes μ, which is implicitly a function of x 1 .
LEMMA OA2: From the budget constraint in the above problem, it follows that PROOF OF LEMMA OA2: The proof of part 1 of Lemma OA2 follows directly by inspection of the budget constraint. Expression 2 in part 2 of Lemma OA2 is derived by implicit differentiation of the budget constraint. Part 3 and part 4 of Lemma OA2 follow by inspection of the expression in part 2, and the fact that μ > w 1 α . This concludes the proof of Lemma OA2. Q.E.D.
Lemma OA2 implies that μ as a function of x 1 ∈ [−C/b 1 Cb 1 ] is U-shaped; the slope is −∞ at x 1 = −C/b 1 and +∞ at x 1 = C/b 1 ; and it reaches a minimum at x 1 = 0.
For w 1 = 0, taking the derivative of the objective function W in expression (OA-2) with respect to x 1 , we obtain Plugging in expression for dμ dx 1 in part 2 of Lemma OA2, we obtain that dW dx 1 = −2μb 2 1 x 1 + 2(w 1 + w 2 )α 1b Hence, the optimal x 1 must be interior, which implies that dW dx 1 = 0 or, equivalently, we obtain that the Lagrange multiplier μ must solve The conclusion for w 1 = 0 is obtained by taking the limits as w 1 → 0 of the expression x * 1 and the expression determining μ. This concludes the proof of part 1 of Theorem OA1. Part 2. We have already proved that, for all ≥ 2, x * > 0 if and only if w 1 > 0. We now claim that x * 1 > 0 if and only if w 1 + w 2 + w 3 2b 1 √ α 1 > 0. Suppose, toward a contradiction, that x * 1 < 0. Suppose, toward a contradiction, that x * 1 < 0. By inspection of the maximization problem note that if w 1 +w 2 + w 3 2b 1 √ α 1 > 0 and x * 1 < 0, then, by flipping the sign of x * 1 , K(x 1 ) increases and the constraint is unaltered; this is a contradiction to our initial assumption that x * 1 was optimal.
We have just established that x * 1 > 0. Now, by (OA3.1) above, x * 1 > 0 if and only if it follows that μ > α 1 (w 1 + w 2 ). Finally, if the game has strategic complements, then α 2 > · · · > α n and so |x * 2 | > |x * 3 | > · · · > |x * n |, and if the game has strategic substitutes, then α 2 < · · · < α n and so |x * 2 | < |x * 3 | < · · · < |x * n |. Part 3. This follows by using the characterization in part 1 and by noticing that if C → 0, then μ → ∞. Part 4 and Part 5. Both parts follow by using the characterization together with the following fact, which we will now establish: lim C→∞ μ = max w 1 max{α 2 α n } (w 1 + w 2 )α 1 To show this, recall from above that we have the following equation for the Lagrange multiplier: If C tends to ∞, it must be that either the first denominator (μ − w 1 α ) or the second denominator (μ − (w 1 + w 2 )α 1 ) tends to zero. Concerning the first one, this is true if either w 1 α 2 or w 1 α n (depending on which one is positive) approaches μ. The second denominator tends to 0 if (w 1 + w 2 )α 1 tends to μ. Both denominators are positive by definition of the Lagrange multiplier, so it will be the greater of w 1 max{α 2 α n } and (w 1 + w 2 )α 1 which tends to μ. This concludes the proof of Theorem OA1.

Q.E.D.
A special case of Theorem OA1 is one where the planner wants to maximize the sum of equilibrium actions. This occurs when w 1 = w 2 = 0. In this case, we obtain the following.
COROLLARY OA1: Suppose Assumptions 1, 2, and OA1 hold. Suppose that w 1 = w 2 = 0 and w 3 > 0, that is, the planner wants to maximize the sum of equilibrium actions. Then the optimal intervention is b * =b + u 1 √ C.
REMARK OA1: Suppose w 1 < 0 and =2 b 2 < C, in contrast to what was assumed in the theorem. If x 1 is sufficiently small, the solution in Step 1 in the proof of Theorem OA1 entails x = −1 for all ≥ 2. That is, fixing x 1 , the bliss point can be achieved with the remaining budget after the cost of implementing x 1 , namely, C(x 1 ), is paid. Thus, when we move to Step 2 and optimize over x 1 , we need to take into account that, for small values of x 1 , Step 1 yields a corner solution. Hence, the analysis of how the network multiplier changes when x 1 changes will need to be adapted accordingly.
COROLLARY OA2: The optimal intervention in Example OA1 is characterized by and, for all ≥ 2, COROLLARY OA3: Consider the optimal intervention in Example OA1. It has the following properties.

OA3.2. Beyond Symmetric and Non-Negative G
In this subsection, we relax the assumption that G is symmetric. Recall that equilibrium actions are determined by When G is not symmetric, we employ the singular value decomposition (SVD) of the matrix M = I − βG. This allows us to obtain an orthogonal decomposition of an intervention useful for examining welfare, analogous to the diagonalization. An SVD of M is defined to be a tuple (U S V ) satisfying where: (1) U is an orthogonal n × n matrix whose columns are eigenvectors of MM T ; (2) V is an orthogonal n × n matrix whose columns are eigenvectors of M T M ; (3) S is an n × n matrix with all off-diagonal entries equal to zero and non-negative diagonal entries S ll = s l , which are called singular values of M . As a convention, we order the singular values so that s > s +1 . It is a standard fact that an SVD exists. 7 For expositions of the SVD, see Golub and Van Loan (1996) and Horn and Johnson (2012). The th left singular vector of M corresponds to the th principal component of M . When G is symmetric, the SVD of M = I − βG can be taken to have U = V , and the SVD basis is one in which G is diagonal.
Let a = V T a and b = U T b; then the equilibrium condition implies that a * = 1 s b 2 and therefore the objective function is It is now apparent that the analysis of the optimal intervention can be carried out in the same way as in Section 4. Theorem 1 applies, with the only difference that now α = 1/s 2 . We can also extend Proposition 1 and Proposition 2. As the budget tends to 0, r * /r * tends to α /α ; on the other hand, when C is very large, the optimal intervention is proportional to the first principal component of M , and a simple intervention that focuses on the first principal component performs (nearly) as well as the optimal intervention. When G is symmetric, the nature of strategic interactions (determined by β) pins down the principal component that most amplifies an intervention. If G is non-symmetric, the singular values s l of M are not equal to 1 − βλ l , where λ l are the eigenvalues of G; the singular vectors of M are not the eigenvectors of G; and the left and right singular vectors need not be the same.

OA3.3. More General Costs of Intervention
In Section 4, we solved the optimal intervention problem under a specific cost function. This section discusses some natural properties on a cost function. We then show that our analysis of the optimal intervention extends to the general class of cost functions defined by these properties, as long as the budget is small.
We begin by developing properties that a reasonable cost function (b b ) → K(b;b) must satisfy.
Translational invariance says that there is no dependence on the starting point. Symmetry across players implies that names do not matter for costs. Non-negativity implies that the planner cannot extract money from the system: κ(0) = 0 is the definition of the status quob: it does not cost anything to enactb. Local separability across individuals requires that there are no spillovers in the costs of interventions. This is reasonable, as it ensures that the complementarities we study come from the benefits side and not from the costs of interventions. Finally, the twice-differentiability of the function is a technical assumption to facilitate the analysis, while the positive value of the second derivative at 0 rules out cost functions such as κ(y) = i y 4 i in which the increase in marginal costs at 0 is too slow.
Consider an example of a cost function that satisfies Assumption OA2: κ(y) = iκ (y i ), whereκ(y) = y 2 + c|y| 3 e y + c y 4 , with c and c being arbitrary constants. Our main result is that the structure of interventions identified in Section 3.1 carries over to such cost functions as long as the budget is small. PROPOSITION OA1: Consider the intervention problem (IT) with the modification that the cost function satisfies Assumption OA2. Suppose Assumptions 1 and 2 hold and the network game satisfies Property A. At the optimal intervention, as C → 0, we have r * r * → α α .
PROOF OF PROPOSITION OA1: First, we state and prove a lemma.
LEMMA OA3: Under the conditions of Assumption OA2, on any compact set the function C −1 κ(C 1/2 z) converges uniformly to k z 2 , as C ↓ 0, where k > 0 is some constant. We call the limit G.
PROOF: Consider the Taylor expansion of κ around 0 (κ is defined by part (1) of the assumption). We will now study its properties under parts (2) to (5) of Assumption OA2. Part (5) ensures that the Taylor expansion exists. Local separability (4) says that there are no terms of the form y i y j . Non-negativity (3) (κ is non-negative and κ(0) = 0) implies that all first-order terms are zero. Also, (5) says that terms of the form y 2 i must have positive coefficients, and symmetry (2) says that their coefficients must all be the same. Q.E.D.
Write y := b −b. Let (y) denote the change in welfare from the status quo. Fix all parameters of the problem, and recall the main optimization problem: We maintain, but do not explicitly write, that welfare is evaluated at a * (y), where a * = [I − βG] −1 (b + y).
Let y(C) be the solution of problem (IT(C)), which is unique for small enough C. Then we claim that, as C ↓ 0, we have where the similarity ratios are defined at the optimum y(C).
We will prove the result by studying an equivalent problem using Berge's theorem of the maximum. Lety = C −1/2 y. We will now define a re-scaled version of the problem, (Ǐ T(C)): This is clearly equivalent to the original problem. Lety * (C) be the (possibly set-valued) solution for C.
The problem (Ǐ T(C)) is not yet defined at C = 0, but we now define it there. Let the objective at C = 0 be the limit of C −1 (C 1/2y ) as C ↓ 0, which we call F . Let the constraint be G(y) ≤ 1, where G is from Lemma OA3.
Let us restrictǏ T(C) to a compact set K such that the constraint set {y : C −1 κ(C 1/2y ) ≤ 1} is contained in K for all small enough C. Now we claim that the conditions of Berge's theorem of the maximum are satisfied: The constraint correspondence is continuous at C = 0 because C −1 κ(C 1/2y ) converges uniformly to G, while the objective function is jointly continuous in its two arguments.
The theorem of the maximum therefore implies that the maximized value is continuous at C = 0. Because the convergence of the objective is actually uniform on K by the lemma, this is possible if and only ify approaches the solution of the problem By the same argument, the same point is the limit of the solutions to By Proposition 1, in that limit this satisfies We next impose an additional restriction on the structure of the costs of intervention and we show that this new restriction together with Assumption OA2 fully characterizes the cost functions that we used in our main analysis.

PROPOSITION OA2: Consider a cost function that satisfies Assumptions OA2 and OA3.
There is a function f : R + → R + such that κ(y) = f y Proposition OA2 implies that the cost of intervention y is the same as the cost of an intervention obtained as an orthogonal transformation of y; that is, κ(y) = κ(Oy) with O being an orthogonal matrix. This allows to rewrite the intervention problem using the orthogonal decomposition of welfare and costs that we employ in Section 4, and all the results developed there extend to this more general environment.
We conclude by taking up the implication of linear costs of intervention. The main result is that with a linear cost function, that is, K(b b ) = i |b i −b i |, the optimal intervention will target a single individual. For ease of exposition, we will restrict attention to Example 1. The analysis can be easily extended to general network games.
We consider the following intervention problem: The solution to problem (IT-Linear Cost) has the property that there exists i * such that b * i =b i * and b * i =b i for al i = i * .
PROOF OF PROPOSITION OA3: Define W (b) = a(b) T a(b). Let F be the set of feasible b, those satisfying the budget constraint K(b;b) ≤ C. Suppose the conclusion does not hold and let b * be the optimum, with W * = W (b * ). Then, because by hypothesis the optimum is not at an extreme point, F contains a line segment L such that b * is in the interior of L. 8 Now restrict attention to a plane P containing this L and the origin. Note that L is contained in a convex set The point b * is contained in the interior of L; thus, b * is in the interior of E. On the other hand, b * must be on the (elliptical) boundary of E because U is strictly increasing in each component (by irreducibility of the network) and continuous. This is a contradiction. Q.E.D.
We now characterize the optimal target for the case of strategic complements, that is, β > 0. Remark OA2 explains how to extend the analysis for the case of strategic substitutes.
In the case of strategic complements, it is clear that the planner uses all the budget C to increase the standalone marginal benefit of i * , that is, b * i =b i + C; reducing someone's effort can never help. Thus, the planner changes the status quob into b =b + C1 i * where 1 i * is a vector of 0 except for entry i * that takes value 1. Let a(1 i ) be the Nash equilibrium when all individuals have b j = 0 and b i = 1, that is, a(1 i ) = [I − βG] −1 1 i . It is easy to verify that the solution to problem (IT-Linear Cost) is This is equivalent to where recall that ρ(a(1 i ) a(b)) is the cosine similarity between vectors a(1 i ) and a(b).
There are two characteristics of a player that determine whether the player is a good target. The first characteristic is a(1 i ) . This is the square root of the aggregate equilibrium utility in the game with b = 1 i , that is, the squared root of a(1 i ) T a(1 i ). So, a player with a high a(1 i ) is a player who induces a large welfare in the game in which he is the only player with positive standalone marginal benefit. We call this the welfare centrality of an individual. It is convenient to express the welfare centrality of individual i in terms of principal components of G. Note that Recall that under strategic complement α 1 > α 2 > · · · > α n and so an individual with a high welfare centrality is one that is highly represented in the main principal components of the network. The second factor is ρ(a(1 i ) a(b)). This measures the vector similarity between (i) the equilibrium action profile in the game with b = 1 i and (ii) the status quo equilibrium action profile. A player with a large ρ(a(1 i ) a(b)) is a player that, in the game in which he is the only player with positive standalone marginal benefit, leads a distribution of effort similar to the distribution of effort in the status quo.
Small C. Suppose C ≈ 0. Then the optimal target is selected based on the first term of expression (OA-3); that is, For small budgets, the optimal intervention focuses on the player who has a large welfare centrality and that, at the same time, leads to a distribution of effort not too different from the status quo equilibrium effort.
Large C. For C sufficiently large, the last term of expression (OA-3) dominates and therefore the player that is targeted is the player with the highest welfare centrality.
REMARK OA2-Extension to the Case of Strategic Substitutes: In the case of strategic substitutes, we know for the targeted player i * , b * i * =b i ± C, but we cannot say, a priori, which (positive or negative), and indeed it is easy to provide examples that both can happen. Under this qualification, the analysis developed for the case of strategic complements extends

OA3.4. Intervention Through Monetary Incentives
In the basic model presented in Section 2, an intervention alters incentives for individual action through a direct change in marginal benefits/marginal costs. The convexity in the cost of changing these marginal benefits plays a key role in the analysis. In this section, we provide a demonstration of how our approach can be applied beyond this cost setting. We do this by using our methods to solve the problem of offering monetary incentives to individuals for choosing between two actions.
Let us reinterpret a node i as a population; thus, N = {1 2 n} is the set of populations. Within population i, there is a continuum of individuals distributed uniformly in I = [0 τ]. Each individual in population i chooses whether to take action 1 or to take action 0. A strategy of an individual in population i is a function q i : [0 τ] → [0 1] that describes the probability that an individual of type τ i ∈ [0 τ] chooses action 1. Without loss of generality, we focus on equilibria in which all the players within a population have the same strategy.
The payoff to an individual who chooses action 0 is normalized to 0. If individual τ i takes action 1, then he incurs a cost τ i and gets a benefit that depends on his population's standalone marginal benefit of action 1, b i , and the number of other individuals he meets who have also taken action 1. We assume that the interaction between populations takes the form of random matching, with the following specification: An individual τ i in population i meets someone from population j with probability g ij , and, within population j, τ i meets an individual selected uniformly at random. Suppose τ i meets type τ j , and let q j be the strategy of individuals in population j. Then individual τ i 's payoff for the interaction with the random partner τ j is βq j (τ j ) + b i − τ i In this expression, βq j (τ j ) represents the payoffs from interacting with peers that have also taken action 1.
First, we show that the conditions for an equilibrium are isomorphic to those of the games we studied in Section 3.1. It is immediate to see that the best reply of each individual in population i is a cutoff strategy: there exists a cutoff a i ∈ I so that q(τ i ) = 1 for all τ i ≤ a i and q(τ i ) = 0 otherwise. The equilibrium condition for these cutoffs is that, for all i ∈ N , β j g ij P τ j ≤ a * j + b i − a * i = 0 ⇐⇒ a i = b i + β τ j g ij a * j Denoting by β = β/τ, the equilibrium threshold profile a * solves The equilibrium expected payoff to group i is where the second equality follows by using the best response of each population. So aggregate equilibrium utility is Suppose the planner, before the players choose their action, commits to a subsidy scheme. The subsidy scheme depends on realized actions, which are taken after the scheme is announced. More precisely, the planner selects a vector y ∈ R n and offers the following scheme: Subsidizing action 1. If y i > 0, then the planner gives a subsidy of s 1 i (τ i ) = τ i − [a i (y) − y i ] to all population i's types τ i ∈ [a i (y) − y i a i (y)] who take action 1.
Subsidizing action 0. If y i < 0, then the planner gives a subsidy of s 0 i (τ i ) = [a i (y) + |y i |] − τ i to all τ i ∈ [a i (y) a i (y) + |y i |] who do not adopt the new technology (take action 0).
We make three observations. First, under intervention y, the profile of thresholds a(y) is a Nash equilibrium. Furthermore, the planner does not waste resources in the sense that she uses the minimum amount of resources to implement a(y). To see this, note that, by construction, the planner provides monetary payments to take action 1 or to take action 0 only to types who need such transfers to satisfy their incentive-compatibility constraint. The monetary payments make these incentive-compatibility constraints just bind for the marginal types. Finally, let 1 y i >0 be an indicator function that takes value 1 if y i > 0 and 0 otherwise, then note that the cost of intervention y is We then consider a planner who intervenes in the system. The planner has complete information about the type of each individual in each population and can subsidize types to take action 1 or to take action 0, in a perfectly targeted manner. In doing this, the planner effectively shifts the b i of some individuals in some populations. The cheapest individuals to influence are those who are close to being indifferent between the two actions, so that they do not need to be paid very much to change their behavior. Indeed, the payment to an individual is proportional to his distance x from the marginal type in equilibrium: integrating across all the individuals whose actions are changed gives y i 0 x dx, a cost that is quadratic in the magnitude of the change. The intervention problem turns out to be mathematically equivalent to (IT), and so all our results apply.
We can now define the intervention problem of the planner as follows. Starting from the status quob, the planner chooses intervention y to maximize aggregate equilibrium utility under the constraint that individuals play according to equilibrium and that the cost of the intervention cannot exceed C. Intervention problem (IT-P) is equivalent to the intervention problem (IT) defined in Section 2.
Note that the specific payoff functions we have taken here make the problem isomorphic to the setting of Example 1, but by suitably modifying the payoffs, we could capture more general externalities, along the lines of Section OA3.1 of this supplement.
We focus throughout on maximizing aggregate utility, but we note that the results have applications to other kinds of objectives, such as implementing Pareto improvements. In some cases, interventions will make everyone better off without modification, when positive externalities are strong enough to overcome any negative welfare impacts. However, even when this is not the case, the planner may be able to achieve Pareto improvements. For example, consider a planner who is able to make lump sum transfers-for example, award or take away discretionary compensation-in addition to any targeted incentives or contingent payments. In such cases, if an improvement in aggregate utility is possible, then the planner can use such transfers to compensate individuals (for instance, those harmed by negative externalities), and achieve a Pareto improvement. In the setting discussed in this subsection, combining lump sum and action-contingent transfers would then implement a range of Pareto improvements. Even beyond the monetary-incentives setting under consideration here, lump sum transfers may be available to the planner in addition to whatever incentive-targeting scheme is being used, and in such a setting our comments here would apply also.