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In this issue of Econometrica:
"Targeting Interventions in Networks"
We take an old question: "Whose incentives matter most for aggregate outcomes in a network game?"
and give some new answers and a new approach.
w/ A. Galeotti @LBS and S. Goyal @CambridgeINET
"Targeting Interventions in Networks"
We take an old question: "Whose incentives matter most for aggregate outcomes in a network game?"
and give some new answers and a new approach.
w/ A. Galeotti @LBS and S. Goyal @CambridgeINET
A canonical network game is very simple: the return to more effort (e.g., studying) depends both on a "basic" return (e.g., motivation) & neighbors' effort.
Strategic complements: returns increase when neighbors' effort increases.
Strategic substitutes: returns decrease.
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Strategic complements: returns increase when neighbors' effort increases.
Strategic substitutes: returns decrease.
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Example of complements: students helping each other study. In this case, neighbors = study partners.
Ex. of substitutes: firms invest in R&D for a new product. Neighbors are competitors with overlapping markets.
In either case, we have a network of strategic spillovers.
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Ex. of substitutes: firms invest in R&D for a new product. Neighbors are competitors with overlapping markets.
In either case, we have a network of strategic spillovers.
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Now a planner comes along and gets to adjust the basic incentives: e.g., a school can allocate tutoring or encouragement.
A government can offer incentives or partnerships with some firms.
What should the planner do to maximize the total welfare of the agents?
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A government can offer incentives or partnerships with some firms.
What should the planner do to maximize the total welfare of the agents?
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Since changing YOUR basic incentives changes your action, and therefore spills over to the incentives of your friends, a planner should think about that.
A classic network economics result ( @coballester, Calvo, and @yveszenou1) says: focus on central people, e.g. this one:
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A classic network economics result ( @coballester, Calvo, and @yveszenou1) says: focus on central people, e.g. this one:
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Exactly how incentives should be altered depends on the status quo basic incentives and size of the budget you have to mess with incentives.
But, with complements, targeting will be correlated with one's network centrality. Here node sizes show the amount of intervention:
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But, with complements, targeting will be correlated with one's network centrality. Here node sizes show the amount of intervention:
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With strategic complements, it is clear why this is smart: encouraging a student who is "network-central" to study spills over in a positive way to friends, then friends of friends, etc.
But what about the strategic substitutes case?
That turns out to be very different!
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But what about the strategic substitutes case?
That turns out to be very different!
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If encouraging one agent DIScourages its neighbors, then encouraging a very central agent can reduce total effort/welfare and be counterproductive.
A similar potential for crowd-out can occur with public goods problems and many other kinds of games.
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A similar potential for crowd-out can occur with public goods problems and many other kinds of games.
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Another thing:
if central people are neighbors (as they are in the example above) then, with substitutes, encouraging them both is working against yourself.
These observations suggest that optimal targeting in the substitutes case is very different compared w/ complements.
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if central people are neighbors (as they are in the example above) then, with substitutes, encouraging them both is working against yourself.
These observations suggest that optimal targeting in the substitutes case is very different compared w/ complements.
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Working out the optimal targeting strategy numerically in an example on the same network but with substitutes gives an answer like this:
you target some very central agents negatively (e.g., reduce help or subsidies to them), and target their neighbors positively.
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you target some very central agents negatively (e.g., reduce help or subsidies to them), and target their neighbors positively.
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In the illustration above, node color corresponds to whether an agent's basic incentive is increased (green) or decreased (red), and size corresponds to how much.
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Note an agent's incentives are changed in the opposite direction of neighbors'. That's a general tendency, and can hold exactly when the network is biparite, as this one is.
So we see that the optimal targeting depends greatly on the economic structure (sign of spillovers).
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So we see that the optimal targeting depends greatly on the economic structure (sign of spillovers).
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Our contribution is to organize these two policies - which might at first seem pretty unrelated - under a common umbrella.
How? For any network, we use a certain basis of building blocks: interventions we can add up to make any other.
Here they are for a circle network:
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How? For any network, we use a certain basis of building blocks: interventions we can add up to make any other.
Here they are for a circle network:
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These are the eigenvectors of the underlying network. And we call the "building block" interventions the principal components.
When we think of an intervention as a mix of principal components, the optimal targeting problem looks mathematically nicer.
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When we think of an intervention as a mix of principal components, the optimal targeting problem looks mathematically nicer.
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The ordering of these principal components is defined by their associated eigenvalues. High eigenvalue = 'earlier' principal component.
Here is a statement of the optimal targeting theorem for large budgets.
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Here is a statement of the optimal targeting theorem for large budgets.
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And here are optimal targeting schemes for two networks - a circle network and a more interesting one.
With strategic complements, target according to eigenvector centrality. With strategic substitutes, focus on the LAST component - the flippy one.
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With strategic complements, target according to eigenvector centrality. With strategic substitutes, focus on the LAST component - the flippy one.
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The key features of these examples generalize: top principal components, which (in a precise sense) reflect more global structure, matter more in strategic complements problems.
Bottom components, reflecting local structure, are important in strategic substitutes problems.
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Bottom components, reflecting local structure, are important in strategic substitutes problems.
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More broadly, the paper argues for the value of a certain method (pioneered by many papers that inspired us):
Express your network game in a well-chosen basis, and you often learn something new about how the network shapes the game - and how that depends on the game.
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Express your network game in a well-chosen basis, and you often learn something new about how the network shapes the game - and how that depends on the game.
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Applying that approach to rewrite eq'm welfare, we see that the substitutes and complements problems have tightly analogous solutions - something not obvious a priori.
In general, a whole family of network statistics (all the eigenvectors!) can be important.
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In general, a whole family of network statistics (all the eigenvectors!) can be important.
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But for this problem, we can order them in importance, and in some cases show that one of them is key.
In other economic problems, intermediate eigenvectors can be the ones that "best explain" what a planner cares about.
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In other economic problems, intermediate eigenvectors can be the ones that "best explain" what a planner cares about.
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Paper here: http://bengolub.net/papers/TIN.pdf
Slides here: http://bengolub.net/wp-content/uploads/2020/09/TIN_slides.pdf
New applications of these methods to social learning and coordination games coming soon :)
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Slides here: http://bengolub.net/wp-content/uploads/2020/09/TIN_slides.pdf
New applications of these methods to social learning and coordination games coming soon :)
21/21
