Strategyproof Mechanisms for Competitive Influence in Social

Networks (WWW13)

JOINT WORK WITH ALLAN BORODIN, MARK BRAVERMAN, BRENDAN LUCIER

JOEL OREN, UNIVERSITY OF TORONTO

Special Offers and Social Networks

Special Offers and Social Networks

Marketing via Word of MouthSocial Network Advertising: consumers share opinions, product preferences, links to information.The advertiser’s hope: start a cascade of adoption in a population.

Online networks: Faster/easier to share information; built-in support for sharing. Availability of network data could allow advertisers to target

potentially influential individuals.

Seed set

Targeting Influential IndividualsQuestion: Which nodes in the network should be targeted? Theoretical models for the spread of influence in networks

(Independent Cascades, Linear Thresholds, Distance-based, …) Recently: extensions to handle multiple competing products.

Example: OR model [Borodin,Filmus,Oren’10]

– “Knowledge” of each product spreads randomly along edges, independently.

– Afterwards, each user chooses a product from among those he was exposed to, according to some distribution.

Fact: Influence (total and for each product) is a submodular non-decreasing function of the seed set(s).

Targeting Influential IndividualsQuestion: Which nodes in the network should be targeted?

If expected total influence is a submodular function of the targeted nodes, then a greedy algorithm can find approx. optimal seed sets by repeatedly targeting the node that maximizes marginal influence.

Requires knowledge of network topology, influence parameters.

This talk: A social networking platform could provide “special offer” targeting as a service to advertisers.

(Informal) Framework1. Firms wish to give away some number of offers or coupons.2. Firms declare number of offers to an advertising platform.3. Platform chooses how to place offers within the network.4. Spread of influence determines (expected) utilities.

Firm 1

Firm 2Platform

• Advantage: the mediating platform has knowledge of the entire graph; can target any individual node.

• Complication: impact of competition between advertisers? (negative externalities).

Motivating QuestionFor a given model of competitive influence spread,

Can we design a strategyproof polynomial-time mechanism for selecting seed sets

such that the social welfare (total influence) is approximately maximized?

Strategyproof: advertisers maximize their own utility (i.e., influence) by truthfully revealing the number of offers they have available.

Related WorkModels of influence spread: Single advertiser: Kempe, Kleinberg, Tardos (2003,

2005) Competing advertisers: Bharathi, Kempe, Salek

(2007), Carnes et al (2007), Borodin, Filmus, Oren (2010)

Kearns and Goyal (2012): Proposed a model of competitive influence spread, analyzed efficiency of equilibria when advertisers choose seed sets directly.

Singer (2011): Single advertiser, procurement auction. Purchase seed nodes subject to a budget constraint.

The Model for Competing Influence SpreadCompeting firms A and B, with private demands nA,nB. (number of offers)Mechanism selects sets of nodes SA, SB with |SA|=nA, |SB|=nB. Utility of a firm: expected number of final adopters of that firm’s product.

Assumption 1: uA(SA,SB) - non-decreasing in SA and non-increasing inAssumption 2: Social welfare SW(SA,SB) =uA(SA,SB)+uB(SA,SB) , is submodular and non-decreasing in SA and SB.

Goal of mechanism: maximize social welfare.A

BMechanis

m

Network G

SB SA

nAnB

Competitive Spread Processes: A Simple Example Two-step process. Each initial node (Layer 1) tries to

infect all its neighbors (Layer 2). Succeeds w.p. (indep.).

Ties: Flip a coin.

Layer 1:

Layer 2:

𝑝 𝑝 𝑝 𝑝✔✔

✔

Attempt 1: Greedy Mechanism with serial allocations (dictatoriship ordering)

Simple algorithm: for each player at a time: repeatedly allocate ni

nodes to maximize marginal improvement to social welfare, subject to demand constraints. Obtains a 2-approximation, but is not strategyproof.

Example:

• If both players demand 1: high utililty for one of them (say A)• If A increases demand to 2: utility goes down!

Results

Notes:- Not obvious that such a mechanism exists even

without the polytime constraint.- Polytime in the output size (nA+nB) and the time

needed to query the utility functions (e.g., network size).

Theorem: there is a polynomial-time strategyproof mechanism that obtains a 2-approximation to the optimal social welfare.

Other results:• More than 2 advertisers under certain

assumptions.• Disjoint allocations.

A Flexible Greedy AlgorithmThe Locally Greedy Algorithm:1. Select an arbitrary permutation of nA ‘A’s and nB ‘B’s.2. For each entry of the permutation, allocate to the corresponding player the node that maximizes marginal increase in social welfare.

Theorem [Nemhauser,Wolsey,Fisher’78]: for any permutation, the resulting allocation obtains at least half of the optimal social welfare.

A

BMechanis

m

3

2

ABBAA

Attempt 2: The Uniform Greedy Mechanism Run the locally greedy algorithm using a

uniformly random permutation of nA ‘A’s and nB ‘B’s.

Player A’s expected payoff decreases when budget profile goes from (3,1) to (4,1) – the probability that B infects u2 is higher than the probability the A infects u1 and u2 .

v

u1

u2

c1

c2

c3

c4

ε 1 1 11

Leveraging the Locally Greedy Algorithm Most “natural” approaches for

choosing permutations fail: round-robin, dictatorship, uniformly random, smallest remaining budget.

Approach: Construct a distribution over permutations so that the resulting allocation rule is strategyproof.

Constructing Permutations: A DP Approach

D00 D01 D02 D03

D10 D11 D12 D13

D20 D21 D22 D23

D30 D31 D32 D33

nB

nA

0 1 2 3

0

1

2

3

…Distribution

over AAB,ABA,BAA

……

(0,0) (0,3) (0,5) (0,6)

(2,0) (4,3) (6,6) (6,7)

(4,0) (5,2) (7,6) (8,8)

(5,0) (5,5) (7,8) (9,9)

Constructing Permutations

Utilities (uA,uB), in expectation over choice of permutation

nB

nA

0 1 2 3

0

1

2

3

Approach: show how to extend a monotone construction for all nA+nB<k to a construction for diagonal (nA+nB=k).

Constructing Permutations

We will maintain an additional invariant (cross-monotonicity): uA(nA,nB) ≤ uA(nA+1,nB-1) uB(nA,nB) ≤ uB(nA-1,nB+1)

To be strategyproof, must be (≥7, ≥6)!

A Concern: If we aren’t careful, a monotone extension might not exist.

The DP algorithm

The main technical lemma:1. Monotonicity constraints:a) wA(a,b) ≥ wA(a-1,b) Player A is

monotoneb) wB(a,b) ≥ wB(a,b-1) Player B is monotone

2. Cross Monotonicity: wA(a,b) ≥ wA(a-1,b+1)

Constructing Distributions over Permutations – by DP

Lemma: if u’A≥uA and uB≥uB, then there exists an α such that choosing option 1 w.p. α and option 2 w.p. (1-α) generates expected utility (u*A, u*B) , where u*A≥uA and u*B≥u’B .

D(uA,uB)D’(uA’, uB’)

Option 1: draw and

append an A.Utility is (≥uA, ≤ uB)

Option 2: draw and

append a B.Utility is (≤ u’A ≥ u’B)

Cross-monotonicity

ABAB

ABBA

Constructing Permutations

Lemma: if diagonal (nA+nB)=k satisfies cross-monotonicity, then we can implement distributions for diagonal (nA+nB=k+1 ) that preserves monotonicity and also satisfies cross-monotonicity.

(0,0) (0,3) (0,5) (0,6)

(2,0) (2,3) (3,4)

(4,0) (5,3)

(6,0)

nB

nA

The DP Algorithm in Depth

Let wA(a,b) (wB(a,b), w(a,b)) be the expected payoff of agent A (B, social welfare) under the distribution of allocations defined in M[a,b].

Let i.e., the expected diff. in sw when B’s budget is increased by 1.

Can rewrite B’s monotonicity requirement:

Cross Monotonicity: Player A’s expected payoff cannot increase when increasing B’s budget at the expense of A. It makes the induction hypothesis stronger.

First step: project everything to A – all monotonicity conditions are expressed using A’s expected payoff

Constructing the Permutations Table M

24

The Monotonicity Constraints

25

feasible region for

The Constraints Interval - Im

a,b

Claim 1 : (non-emptiness)

Proof: From cross-monotonicity and the induction hypothesis:

Having ensures that the monotonicity conditions for both A and B are maintained.

The “Feasible” Interval - Ic

a,b

We construct two permutations as follows: Sample a permutation from M[a-1,b] and append an ‘A’

at the end – set W1A as A’s expected payoff.

Sample a permutation from M[a,b-1] and append an ‘B’ at the end – set W0A as A’s expected payoff.

W1A and W0A can be thought of as the influence of A for distribution we can construct.

27

Claim 2: Proof: utilities are

non-decreasing functions. The second inequality follows also from the adverse competition assumption.

Claim 3: Proof:

Use Claim 2 to show that no interval is entirely above the other.

The Probability Distribution for M[a,b]

Any point on the intersection is valid: we will take the minimum point in the intersection.

The intersection point is a convex combination of W1A and W0A.

Randomize between appending an ‘A’ to a solution drawn from M[a-1,b] and appending ‘B’ from a solution drawn from M[a,b-1] so as to obtain the value corresponding to the minimum point.

Claim 4: The minimum point upholds cross-monotonicity:

Last step: showing that each entry needs to contain a small number of permutations.

Main claim: each table entry can contain only 3 permutations.

We show this again by induction on k=a+b.

The Probability Distribution for M[a,b]

30

Improving the RuntimeIn principle our recursive construction could result in distributions over exponentially many permutations.

Claim: it suffices to have 3 permutations per distribution.Proof:

DD’ uA

uB𝐷(+𝐴)

+A

+B Caratheodory

Extension: Disjoint Allocations Our mechanism sometimes includes

a graph node in multiple advertisers’ seed sets. What if we require disjoint allocations?

The 2-approximation is no longer order-independent.

Counter-example: two players, budgets of 1, two items: 1,2. Utility values: If B goes first – social

welfare=N+1. If A goes first – social welfare=2+ε.

Social welfare decreases by a factor of O(N).

A B1 1 12 1+ε N

Extension: Disjoint AllocationsDefinition: utility functions are symmetric if they are invariant under relabeling: uA(SA,SB)=uB(SB,SA). Theorem: if utilities are symmetric, there exists a strategyproof 3-approximate mechanism that always returns disjoint allocations.Proof: Show that the locally greedy algorithm with disjointness is a 3-approximation, when utilities are symmetric. The DP algorithm guarantees a strategyproofness.

Extension: More FirmsWhat if there are 3 or more advertisers?

Definition: utility functions satisfy mechanism indifference whenever SW(S1,…,Sk) depends only on (including multiplicities)

Definition: utility functions satisfy agent indifference whenever ui(S1,…Sk) depends only on (including multiplicities).

Theorem: for k > 2 players, if utility functions satisfy both agent indifference and mechanism indifference, then there exists a polynomial-time e/(e-1)-approximate strategyproof mechanism – the uniform greedy mechanism.Proof sketch: algorithm reduces to the standard greedy e/(e-1)-approx for submodular maximization under a cardinality constraint. Expected utilities are proportional to the bids.

Note: algorithm does not hold for k=2 !

Conclusions We consider mechanisms for placing seeds

in a model of competitive influence spread on a social network.

General model of influence spread: we require submodularity of social utility and negative externalities.

A 2-approximate strategyproof mechanism for 2 players; extends to k players under some additional assumptions on the influence spread process.

Future Directions

Approximations or inapproximability results for three players? (Currently there is no lower bound for truthful mechanisms.)

Can it be shown that the problem requires randomization (even for two players)? That is, is there a strong lower bound for all deterministic methods using locally greedy?

Can we convert other (non greedy) allocation algorithms (for submodular maximization) into truthful mechanisms?

Multiple knapsack version of problem where nodes have costs and agents have budgets; agents putting more resources on some initial nodes. 36

More future directions The “Demand satisfaction”

requirement: a fairness condition. Consider a level of approximate fairness.

Other solution concepts such as POA (as in Goyal and Kearns 2012), Bayesian truthfulness.

Thank You

Addendum: Proof of LemmaLemma: if and , then there exists an such that choosing option 1 w.p. and option 2 w.p. generates expected utility , where and .

D’D 𝑢𝐵

𝑢𝐴

𝐷′ (+𝐵)

𝐷(+𝐴)

+B

+A(𝑢𝐴

′ ,𝑢𝐵′ )(𝑢𝐴

❑ ,𝑢𝐵❑)