Recommending and Targeting Gabrielle Demange Paris School of - - PowerPoint PPT Presentation

Recommending and Targeting

Gabrielle Demange

Paris School of Economics

July 9, 2015

Gabrielle Demange (PSE-EHESS) Recommending and targeting 1 / 48

What ?

a single item, a set of items, an ordered set (PageRank)

Who ?

who recommends ? who is targeted ? who reviews ?

Why ?

attract consumers on a platform, attract attention on a product

How ?

search engines, ads, targeting

Gabrielle Demange (PSE-EHESS) Recommending and targeting 1 / 48

In this talk

I focus on two settings ranking by search engines, journals

information based on the hyperlink structure or citations how to aggregate these citations?

targeting in a social network (exploiting positive externalities)

what are the optimal strategies? what is the value of information on the network?

Gabrielle Demange (PSE-EHESS) Recommending and targeting 2 / 48

Important questions left aside

Are recommendations biased ?

single product

• experimentation Che Horner [2013] Kremer, Mansour, Perry [2014]
• distorted value (Google to YouTube) de Corniere and Taylor [2014]

multi-products (Amazon, Netflix) ? motives for bias between products ? main issue: personal data

What are the consequences of better recommendation on welfare ?

• n prices ? on product design ? Bar-Isaac, Caruana, Cu˜

nat [2012]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 3 / 48

Ranking

Ranking

What does a search engine’s ranking mean? Aggregation of preferences or aggregation of information? Various methods, produce different results. e.g. counting, invariant method (basis for PageRank of Google) How to choose a method? Can we define new interesting methods? Useful approach: Characterization of a method through its properties (axiomatization)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 4 / 48

Ranking

1

Ranking The setting Some methods Illustration: Ranking economic journals Graphs/Web

2

Targeting Various targeting problems Constant returns to exposure Diminishing returns to exposure Increasing returns to exposure

Gabrielle Demange (PSE-EHESS) Recommending and targeting 5 / 48

Ranking The setting

Ranking problems

N = {1, ..., n} be a set of n ’items’ to rank M = {1, ..., m} be a set of m ’experts’ n × m matrix π = (πi,j) column j= j’s statement on the items πi,j = valuation of j on i A ranking method assigns scores to each admissible π r = (ri) ri ≥ 0,

• i

ri = 1

Gabrielle Demange (PSE-EHESS) Recommending and targeting 6 / 48

Ranking The setting

Examples

1 Web : incidence matrix πi,j = 1 if j points to i and 0 otherwise

treated as ’Approval voting’

2 Journal : πi,j= average number of references of an article from j to

articles in i. In both cases M = N ’peers’ method normalized statements [π]i,j = πi,j π+j for each i, j. → Intensity-Invariant method to avoid a form of manipulation

Gabrielle Demange (PSE-EHESS) Recommending and targeting 7 / 48

Ranking Some methods

The counting method

The scores (ri) are proportional to the received valuation totals: ri ∝

• j∈M

πi,j for each i used by the Science Citation Index for ranking journals, by reviewers’ systems treats experts equally whatever their statements

Gabrielle Demange (PSE-EHESS) Recommending and targeting 8 / 48

Ranking Some methods

Experts’ weights

most methods assign not only scores to items, r, but also weights to experts: q = (qj) s.t. ri =

j[πi,j]qj for each i

the ranking is a weighted average of the statements weights may vary with the statements. = counting method: qj = 1/m, whatever π Three methods, invariant, HITS, Handicap-Based Equilibrium relationship between scores r and weights q

Gabrielle Demange (PSE-EHESS) Recommending and targeting 9 / 48

Ranking Some methods

Eigenvector methods

N = M (peer setting), π irreducible Recursive Impact factor or LP (Liebowitz-Palmer) method ri = λ

• j∈N

πi,jrj for each i r principal eigenvector of π, dominant eigenvalue λ Invariant method: LP applied to normalized statement [π] Pinski and Narin [1976] ri =

• j∈N

[πi,j]rj for each i. Expert’s Weight= Expert’s Score axiomatization Altman Tennenholtz [2005] Palacios-Huerta Volij [2004]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 10 / 48

Ranking Some methods

Example 1

π =       10 30 27 9 3 9       [π] =      

10 19 10 13 9 10 3 13 1 10 9 19

      Counting (0.4545, 0.4091, 0.1364) LP=(0.3734, 0.4382, 0.1884) reverse order for 1 and 2; 1 cites 2 a lot and ’conveys’ power; Invariant = (0.3806, 0.3945, 0.2249) ’power’ given to 1 by 3 diminishes, power given to 3 by 2 increases

Gabrielle Demange (PSE-EHESS) Recommending and targeting 11 / 48

Ranking Some methods

The Hyperlink-Induced Topic Search HITS method

assign to each i two indices that distinguish between i’s ability as an item (authority) from that as an expert (hub): The HITS method (Kleinberg [1999]) ri =

• j

πi,jqj each i and qj = λ

• i

πi,jri each j → r principal eigenvector of π π and q principal eigenvector of ππ

Gabrielle Demange (PSE-EHESS) Recommending and targeting 12 / 48

Ranking Some methods

Handicap-based method

N and M can differ. There are unique r and q ri =

• j

πi,jqj each i and 1 qj = λ

• i

πi,j 1 rj each j The Handicap-based method : assigns scores r. Demange [2014-b] axiomatizations:

the counting method and HB satisfy a homogeneity property HB is intensity-invariant, the counting m. is not HB is appropriate to aggregate vectors of proportions

Gabrielle Demange (PSE-EHESS) Recommending and targeting 13 / 48

Ranking Illustration: Ranking economic journals

Ranking economic journals

Rankings of 37 journals, same data as in Palacios-Huerta, Volij (04) r and q (handicap-based) per article Same top 6 handi inv q QJE 10.02 11.44 5.34 Eca 9.65 11.74 3.42 J Econ Lit 9.65 9.26 8.16 JpolE 7.22 7.56 4.15 AER (proper) 7.01 7.52 2.48 RES 5.99 7.42 5.59 average score = 100/37 ≈2.97 =average experts’ weights

Gabrielle Demange (PSE-EHESS) Recommending and targeting 14 / 48

Ranking Illustration: Ranking economic journals

handi inv q

• J. Economic Behavior Organization

0.8 0.59 1.75 Scandinavian J. of Econ 0.77 0.47 3.81 Oxford Bull. of Ecs.Stat. 0.69 0.314 2.84 Economics Letters 0.57 0.355 0.66 Weights are positively correlated with the scores, but moderately rankings of theory journals go down : they receive citations proportionately more from top journals journals that go up : Journal of financial economics, Rand, Oxford Econ statistics, Social of economic and welfare

Gabrielle Demange (PSE-EHESS) Recommending and targeting 15 / 48

Ranking Illustration: Ranking economic journals

Remark on Irrelevance of independent alternatives (IIA)

IIA: ranking between i and j only depends on statements on or by them the counting method satisfies IIA

’almost’ the only relevant one Rubinstein [1980] Brink Gilles [2009]

IIA is NOT met by other methods the score of an item depends on the statements over all items via the values taken by the experts’ weights. IIA is not appealing in many contexts

Gabrielle Demange (PSE-EHESS) Recommending and targeting 16 / 48

Ranking Graphs/Web

Graphs/Web

On a graph, a ranking gives a measure of ’importance’ of a node Counting= in-degree if directed Invariant = random surfer interpretation along directed edges Irreducibility: a directed path joins every two nodes Web graph is not irreducible; Perturbation technic

Gabrielle Demange (PSE-EHESS) Recommending and targeting 17 / 48

Ranking Graphs/Web

PageRank/damping factor Brin and Page [1998]

PageRank perturbs the incidence matrix with a ”damping factor” ri = (1 − α) n + α

• j∈N

[πi,j]rj for each i. r = 1 − α n (I − α[π])−11 1 I= n × n identity matrix, 1 1 = n-vector of ones centrality measure for measuring ’prestige’ in a network (Katz [1953] Bonacich [1987] variation: introduce a personalized perturbation

Gabrielle Demange (PSE-EHESS) Recommending and targeting 18 / 48

Ranking Graphs/Web

Example

6 7 8 9 2 1 4 3 5 Boldi, Santini and Vigna [2007] Perturbed invariant method. 1 − α = 10−3 assigns 0.0049 to 0, 0.4943 to 4 and 0.4939 to 5 HB assigns 0.9595, to 0 with weights 0.2385 for i = 6..., 9 damping factor Google (α = 0.85)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 19 / 48

Ranking Graphs/Web

Could we do better ?

How to interpret the absence of a link/vote ? Can we distinguish between negative votes and the absence of awareness ? How to choose the set of ’experts’ M ? impact of changing M ?

Gabrielle Demange (PSE-EHESS) Recommending and targeting 20 / 48

Ranking Graphs/Web

Other questions

allowing multiple rankings along different dimensions (in Ideas) but also

• n different sets of preferences, on different expert’s sets ? related to

personalized recommendations value of forming close communities to make recommendations (under anonymity) Demange [2010], interaction between current ranking and next statements hence next ranking → dynamics Demange [2012] and [2014-a]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 21 / 48

Targeting

1

Ranking The setting Some methods Illustration: Ranking economic journals Graphs/Web

2

Targeting Various targeting problems Constant returns to exposure Diminishing returns to exposure Increasing returns to exposure

Gabrielle Demange (PSE-EHESS) Recommending and targeting 22 / 48

Targeting

Part II. Targeting

A planner (e.g. firm, government, health authority) aims to enhance agents’ activity Social network under positive externalities Tool: targeting of nodes/agents by allocating a fixed amount of ’resources’ Examples: viral marketing, control of contagion, criminal activity ...

Gabrielle Demange (PSE-EHESS) Recommending and targeting 23 / 48

Targeting Various targeting problems

Examples

linear response

strategic games with quadratic payoffs → a linear ’best’ reply Ballester, Calvo Armengol, Zenou [2006] action= criminal activity, effort ..

• bjective: suppress the ’key player’, i.e. a node

pricing model with discrimination of the nodes (Bloch and Querou [2013], Candogan, Bimpikis, Ozdaglar [2012]) action= probability of purchase or adoption profit objective financial network : Eisenberg and Noe [2001] action= proportion of debt repayments (lower and upper bound) Objective: inject cash into banks to maximize overall repayments Demange [2015]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 24 / 48

Targeting Various targeting problems

Binary variables/Threshold models

adoption/contagion process: 0-1 model threshold models or SIR model Schelling [1969], Morris [2003], Domingos and Richardson [2001] in a marketing context, Dodds and Watts [2004] in biology planner’s strategy: choose a subset to initiate the contagion computational issues in 0-1 threshold models: Kempe, Kleinberg and Tardos [2003],

Gabrielle Demange (PSE-EHESS) Recommending and targeting 25 / 48

Targeting Various targeting problems

A general model with continuous interactions

Individual’s action is a continuous variable Interactions are not necessarily linear Planner’s goal: maximize aggregate action a fixed amount of ’resources’ How is the planner’s amount optimally allocated? Is it concentrated

• n few agents or dispersed among numerous ones?

What is the value of information on the interaction structure?

Gabrielle Demange (PSE-EHESS) Recommending and targeting 26 / 48

Targeting Various targeting problems

Impact and Reaction and Response functions

n agents, take actions, θi for agent i, θ = (θi) Bilateral Impacts : πij ≥ 0 = impact of i on j or j’s attention to i πii = 0 Exposures Given θ = (θi)

i πijθi is the (total) exposure of j.

Reaction to exposures : determined by a response function f : i’s reaction : θi = zi + f (

• j

πjiθj) (1) f continuous from R+ to R+, f (0) = 0: zi i’s action level in isolation

Gabrielle Demange (PSE-EHESS) Recommending and targeting 27 / 48

Targeting Various targeting problems

Equilibrium under strategic complements

Assume f is increasing: actions are strategic complements. An equilibrium: θ = (θi) for which equation (1) is satisfied for each i strategic complementarities: equilibria are ’well behaved’ and easy to find (iterate reactions) Topkis [1979]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 28 / 48

Targeting Various targeting problems

Equilibrium under strategic complements

ρ(π)= dominant eigenvalue of π. Assumption A: f ′(τ)ρ(π) < 1 for τ large enough Assumption A+: true for all τ Under assumption A, an equilibrium exists. There is a greatest equilibrium. Under assumption A+, an equilibrium exists and is unique. I consider decreasing, constant, or increasing returns to exposure : f concave, linear or convex

Gabrielle Demange (PSE-EHESS) Recommending and targeting 29 / 48

Targeting Various targeting problems

The planner’s objective

’Planner’ aims at improving aggregate activity :

i θi

Endowed with amount m ≥ 0 to distribute. a feasible targeting, or injection, strategy n-vector x = (xi),

i xi = m

’constrained’ case: in some problems, each xi must be ≥ 0 xi changes zi into zi + xi, hence equilibrium actions x is optimal if it maximizes equilibrium aggregate activity

i θi over all

feasible strategies.

Gabrielle Demange (PSE-EHESS) Recommending and targeting 30 / 48

Targeting Various targeting problems

Various specifications

On interactions:

π is derived from a network g equal impact total:

j πij equal across i

ex: πij= proportion of time spent to speak with j θi=time equal attention total

i πij equal across j

ex: πij = j’s relative attention to i

On response function: The shape of f and whether the limit return to exposure ω = lim

τ→∞ f ′(τ) is > 0 or null plays an important role.

Gabrielle Demange (PSE-EHESS) Recommending and targeting 31 / 48

Targeting Constant returns to exposure

Linear model: f (τ) = δτ- Positive strategies

Explicit equilibrium: θ = (I − δ π)−1z Let µ = (I − δπ)−11 1 ’multipliers’ µmax = maxjµj and S = {i/µi = µmax}. Optimal targeting strategies: the feasible ones with support included in S. Aggregate activity is linear, increased by µmaxm. Optimal strategy is independent of z or m. Multiplier : centrality index1 in the impact network µ = (I − δπ)−11 1 = 1 1 + δπ1 1 + δ2π21 1 · · · + · · · µi = number of discounted paths from i in the impact network.

1Katz [1953] Bonacich [1987] Gabrielle Demange (PSE-EHESS) Recommending and targeting 32 / 48

Targeting Constant returns to exposure

Linear model: Implications

Actions and multipliers are dual to each other (up to z). Actions= linear in the centrality index in the attention network May widely differ in a non symmetric network Targets not necessarily the individuals with the largest action

Gabrielle Demange (PSE-EHESS) Recommending and targeting 33 / 48

Targeting Constant returns to exposure

Linear model: Value of information. f concave

A uniform (or a random) strategy allocates equal amount to each x = m

n 1

1, For a uniform strategy, the increase in aggregate action is: ( 1

n

• i µi)m.

Value of information on the impact structure π: proportional to µmax − (1 n

• i

µi) Value depends on the heterogeneity in the matrix. Equal impact totals

Multipliers are all equal, any strategy is optimal Null information value Benefit of targeting can be large: µmax =

1 1−δρ(π) (= µi each i)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 34 / 48

Targeting Diminishing returns to exposure

Exploiting the Geometry of equilibria- f concave

Simplify notation: zi = 0. Due to complementarity, given x, the set of actions θ ≥ 0 that satisfy θi ≤ xi + f (

• j

πjiθj) for each i (2) has a greatest element, which is an equilibrium, the greatest one. Value of x, solved by the greatest equilibrium V (x) = max

n

• i=1

θi over θ ≥ 0 that satisfy (2)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 35 / 48

Targeting Diminishing returns to exposure

Planner’s programs

constrained case: positive x The planner’s program writes maxx≥0 V (x) s.t.

n

• i

xi = m. f concave ⇒ V concave f strict concave + π invertible ⇒ V strict concave Work with the multipliers associated to each equilibrium actions in (2) Unconstrained x : (2) +

i xi = m collapse into a single constraint:

• i

θi ≤ m +

• i

f (

• j

πjiθj) (3) → maximize

i θi over θ ≥ 0 that satisfy (3)

then set xi to θi − f (τi)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 36 / 48

Targeting Diminishing returns to exposure

An example: Quadratic -No > 0 constraint on x

f (τ) = δτ − γ

2τ 2 for τ ≤ δ γ and constant thereafter

Define i, j-joint impact by σij =

k πikπjk.

σ = π π. In a network: σij = number of nodes impacted by both i and j Given θ, call

j σijθj i’s weighted joint impact at θ

Under equal impact, optimal targeting strategy is characterized by σ To simplify: m small, π invertible

Gabrielle Demange (PSE-EHESS) Recommending and targeting 37 / 48

Targeting Diminishing returns to exposure

Quadratic response function-cd: Equal impact

There are a unique set I and a positive I-vector θ∗

I s.t.

• j∈I

σijθ∗

j = 1 for each i ∈ I

• j∈I

σijθ∗

j ≥ 1 for each i /

∈ I

On N − I take null action → θ∗ and τ ∗ = πθ∗. Equal impact: A targeting strategy leads to equilibrium actions proportional to θ∗: for some α > 0 xi = αθ∗

i − f (ατ ∗ i ), α adjusted to m

Minimize the weighted joint impact of agents whose actions are > 0 Extract the maximum from others: their actions are null and xi < 0.

Gabrielle Demange (PSE-EHESS) Recommending and targeting 38 / 48

Targeting Diminishing returns to exposure

Quadratic response function-Equal impact cd

Only under equal attention, the optimal strategy is uniform. With unequal attention, positive value of information. Simulate networks according to Erdos Renyi and adjust the rows I is almost never N. The targeting strategy is not necessarily monotone for i in I Root of difficulties in targeting strategies

Gabrielle Demange (PSE-EHESS) Recommending and targeting 39 / 48

Targeting Diminishing returns to exposure

Quadratic response function- Unequal impact

Characterization involves a combination of the degrees and the weighted joint impact (negatively) The equilibrium actions are no longer proportional. The larger m the more the joint impact matrix matters. Given m there are θI ≥ 0 and α such that α ≤ δdi − γ

• j∈I

σijθj each i, with = for i ∈ I +budget

Gabrielle Demange (PSE-EHESS) Recommending and targeting 40 / 48

Targeting Diminishing returns to exposure

Constrained case: The multipliers

dV dxi = the multiplier µi associated to the i-th constraint in (2).

µi = 1

• direct effect

+

• j

πijf ′(τj)µj

• indirect

for each i where τj =

k πkjθk exposure levels at the equilibrium.

FOC of T (m): i targeted, xi > 0 ⇒ µi is a maximal coordinate of µ. .

Gabrielle Demange (PSE-EHESS) Recommending and targeting 41 / 48

Targeting Diminishing returns to exposure

Similar expressions for the multipliers as in a linear model, but depend

• n equilibrium exposure levels

⇒ incentives to equalize the exposure levels Taylor expansion of the FOC shows that the matrix π π matters. Simple properties can be derived

’impact-dominated’ nodes are never targeted characterize the allocation of m between non-interacting groups the allocation to groups is monotone (= with interaction)

Gabrielle Demange (PSE-EHESS) Recommending and targeting 42 / 48

Targeting Diminishing returns to exposure

Undirected Star

a = (1, 1/(n − 1), · · · , 1/(n − 1)). Linear f : target the center (whatever δ) f (τ) ∼ δ√τ: The support is full for m large enough Intuition: marginal return to target 1, µ1, relative to µ2 decreases with m when 1 is the only target not true for a line

Gabrielle Demange (PSE-EHESS) Recommending and targeting 43 / 48

Targeting Diminishing returns to exposure

Summary

The differences with linear response functions: equal impact totals:

the optimal strategy requires the knowledge of the network except if attention totals are also equal

unequal impact totals:

whether the limit return to exposure ω = lim

τ→∞ f ′(τ) is > 0 or null

plays an important role

Gabrielle Demange (PSE-EHESS) Recommending and targeting 44 / 48

Targeting Diminishing returns to exposure

Allocation shares : α(m) = x(m)/m, large m

Let f be isoelastic : δ τ 1−γ

1−γ for τ large enough, 0 < γ < 1. ω = 0

Given shares α, let F be the sum of the externality levels F(α) =

• j

f (

• ℓ

πℓjαℓ) The optimal shares converge to α∗ the maximizer of F over the simplex when m increases.

Gabrielle Demange (PSE-EHESS) Recommending and targeting 45 / 48

Targeting Diminishing returns to exposure

Implications

Isoelastic response function. m large Each node, targeted or not, is exposed to a targeted agent. ⇒ many nodes are targeted when the network is dispersed enough even for a small γ = in the linear model where very few targeted nodes (under unequal impact totals) Aggregate equilibrium actions : θN∗ − m ≈ m1−γF(α∗) Benefit over a uniform strategy θ∗

N − θN ≈ m1−γ[F(α∗) − F(1

n1 1)]

Gabrielle Demange (PSE-EHESS) Recommending and targeting 46 / 48

Targeting Increasing returns to exposure

Increasing returns: f convex

Due to complementarity, equilibrium solves Q(z) : min

θ≥0 n

• i=1

θi s.t. θi ≥ zi + f (

• j

πjiθj) for each i. V (z) =the value of Q(z) (because eq. is unique) ⇒ maximize V f convex ⇒ the value function V is convex in z. f is strictly convex + π is invertible ⇒ V is strictly convex ⇒ the target is a singleton.

Gabrielle Demange (PSE-EHESS) Recommending and targeting 47 / 48

Targeting Increasing returns to exposure

Concluding remarks on targeting

Linear model of interaction leads to quite specific targeting strategies Under diminishing returns to exposure

the differences in attention totals and the ’joint’ impact matter the optimal strategies depend on the limit to marginal exposure ω

The benefit from the knowledge of the network is positive under most circumstances, even for large amount m and null ω

Gabrielle Demange (PSE-EHESS) Recommending and targeting 48 / 48

Targeting Increasing returns to exposure Altman A. and M. Tennenholtz (2005) “Ranking systems: the PageRank axioms” EC ’05 Proceedings of the 6th ACM conference on Electronic commerce. Altman A. and M. Tennenholtz (2008) “Axiomatic foundations of ranking systems”, Journal of Artificial Intelligence Research, 31(1), 473-495. Amir R. (2002) “Impact-adjusted Citations as a measure of Journal quality”, CORE discussion paper 74. Ballester, C., Calvo Armengol, A., & Zenou, Y. (2006). Who’s who in networks. wanted: the key player. Econometrica, 74(5), 1403-1417. Bar-Isaac, H., Caruana, G., & Cu˜ nat, V. (2012). Search, design and market structure. Design and Market Structure American Economic Review102(2): 1140–1160. Belhaj, M., & Deroian, F. (2013). Strategic interaction and aggregate incentives. Journal of Mathematical Economics, 49(3), 183-188. Bloch, F., & Qu´ erou, N. (2013). Pricing in social networks. Games and economic behavior, 80, 243-261. Boldi, Paolo, Massimo Santini, and Sebastiano Vigna (2007) ”A deeper investigation of PageRank as a function of the damping factor.” Web Information Retrieval and Linear Algebra Algorithms http://drops.dagstuhl.de/opus/volltexte/2007/1072 . Bonacich P. (1987) “Power and centrality: a family of measures”, American Journal of Sociology, 92(5), 1170-1182. Brin S., and L. Page (1998) “The anatomy of large-scale hypertextual web search engine”, Comp Networks and ISDN systems, 30(1-7), 107-117. Calvo-Armengol, A., Patacchini, E., & Zenou, Y. (2009). Peer effects and social networks in education. The Review of Economic Studies, 76(4), 1239-1267. Candogan, O., Bimpikis, K., & Ozdaglar, A. (2012). Optimal pricing in networks with externalities. Operations Research, 60(4), 883-905. Gabrielle Demange (PSE-EHESS) Recommending and targeting 48 / 48

Targeting Increasing returns to exposure Yeon-Koo Che and Johannes Horner Optimal, 2013 Design for Social Learning Demange, G. (2010). Sharing information in web communities. Games and Economic Behavior, 68(2), 580-601. Demange, G. (2012) “On the influence of a ranking system”, Social Choice and Welfare, Vol 39, Issue 2, 431-455. Demange, G. (2014) “Collective attention and ranking methods”, forthcoming in Journal of Dynamics and Games. Demange, G. (2014). A ranking method based on handicaps. Theoretical Economics, 9(3), 915-942. Demange G. (2015) Contagion in financial networks: A threat index, Cesifo WP 5307. Dodds, P. S., & Watts, D. J. (2004). Universal behavior in a generalized model of contagion. Physical review letters, 92(21), 218701. van den Brink R. and R. P. Gilles (2009) “The outflow ranking method for weighted directed graphs European Journal of Operational Research 193, 484–491. de Clippel G., H. Moulin, and N. Tideman (2008) “Impartial division of a dollar”, Journal of Economic Theory, 139(1), 176-191. Corni¨ Ere, A., & Taylor, G. (2014). Integration and search engine bias. The RAND Journal of Economics, 45(3), 576-597. Domingos, P., & Richardson, M. (2001). Mining the network value of customers. In Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining. Du, Y., Lehrer E. and Pauzner (2012) “Competitive economy as a ranking device over networks”, mimeo. Katz L. (1953) “A new status index derived from sociometric analysis”, Psychometrika, 18(1), 39-43. Eisenberg L. & T. H. Noe (2001). Systemic Risk in Financial Systems, Management Science, 47(2), 236-249. Gabrielle Demange (PSE-EHESS) Recommending and targeting 48 / 48

Targeting Increasing returns to exposure Kempe, D., Kleinberg, J., & Tardos, E. (2003). Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining . ACM. Kleinberg N. (1999) “Authoritative sources in a hyperlinked environment”, Journal of the ACM, 46(5), 604-632. Kremer, I., Mansour, Y., & Perry, M. (2014). Implementing the “Wisdom of the Crowd”. Journal of Political Economy, 122(5), 988-1012. Laffond, G., Laslier, J. F., & Le Breton, M. (1993). The bipartisan set of a tournament game. Games and Economic Behavior, 5(1), 182-201. Morris, S. (2000). Contagion. The Review of Economic Studies, 67(1), 57-78. Liebowitz, S. J., and J. C. Palmer (1984) “Assessing the Relative Impacts of Economics Journals”, Journal of Economic Literature, 22(1), 77-88. Palacios-Huerta, I., and O. Volij (2004) “The Measurement of Intellectual Influence”, Econometrica, 72(3), 963-977. Pinski, G., and F. Narin (1976) “Citation Influence for Journal Aggregates of Scientific Publications: Theory, with Application to the Literature of Physics”, Information Processing and Management, 12(5), 297-312. Rubinstein, A. (1980) “Ranking the Participants in a Tournament” SIAM Journal on Applied Mathematics, Vol. 38, No. 1, 108-111. Slutzki G. and O. Volij (2006) “Scoring of Web pages and tournaments-axiomatizations”, Social choice and welfare, 26(1), 75-92. Woeginger G.J. (2008) “An axiomatic characterization of the Hirsch-index”, Mathematical Social Sciences, 56(2), 224-232. Richardson, M., & Domingos, P. (2002, July). Mining knowledge-sharing sites for viral marketing. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 61-70). ACM. Gabrielle Demange (PSE-EHESS) Recommending and targeting 48 / 48

Targeting Increasing returns to exposure Schelling, T. C. (1969). Models of segregation. The American Economic Review, 59(2), 488-493. Topkis, D. M. (1979). Equilibrium points in nonzero-sum n-person submodular games. SIAM Journal on Control and Optimization, 17(6), 773-787. Watts, D. J., & Dodds, P. S. (2007). Influentials, networks, and public opinion formation. Journal of consumer research, 34(4), 441-458. Gabrielle Demange (PSE-EHESS) Recommending and targeting 48 / 48