Price of Anarchy for the N-player Competitive Cascade Game with Submodular Activation Functions

Xinran He, David Kempe{xinranhe, dkempe}@usc.edu

12/14/2013

Diffusion In Social Network

• The adoption of new products can propagate in the social networkDiffusion in the social network

Competitive Diffusion In Social Network

• Different products compete for acceptance in a social network.

• Competitive Diffusion in the social network

Competitive cascade game• Given a social network • The players are N companies, with their products .• The individuals can be in state and .• The players simultaneously allocate resources to individuals in the

social network in order to seed them as initial adopters of their products.• The adoption of products propagates according to diffusion model.• The goal for each player is to maximize the coverage of his own

product.• In this paper, we study the Price of Anarchy of this game.

Main contribution

The upper bound on the coarse Price of Anarchy is 2 for the N player competitive cascade game under the Goyal/Kearns diffusion model.

Improvement over [Goyal/Kearns 2012]:• Improve PoA upper bound from 4 to 2.• Generalize result from 2 player game to N player game.• Simple and clear proof by resorting to valid utility game and general

threshold model.

Competitive cascade game

• Given a social network .• N players, each player is a company with limit budget .• Strategy vector for players: • is the set of nodes selected by company .• .

• Payoff function :• Expected number of people who adopt product .

• Social utility function : • Expected number of people who adopt a product.

General adoption model

• Seeding stage:• Each company selects a set of individuals .• The initial state of node is inactive if no company selects it.• Otherwise, the node becomes in state uniformly at random.

• Diffusion stage:• Given a fixed update sequence .• Nodes change states with the order in according to local dynamics.

General adoption model: Local Dynamic• Let be current sets of nodes in state .• Adoption function:• = Prob{ adopts product }

• Total activation probability:

• A still inactive node changes into states with probability , and remains inactive with probability .

General adoption model: ExampleDiffusion stage

DCD

FC

END

Seeding stage

𝑺𝑩

𝐒𝐑

A

FC

D

E

BG

Useful properties

Additivity of total activation probability, activation function is monotone.

Competitiveness of adoption function:

Submodularity of activation function:

= Prob{ }? Prob{ }?

Prob{ } Prob{ }

Main resultsTheorem: Assume the following conditions hold:

1. For every node , the total activation probability is additive.2. For every node , the activation function is submodular.3. For every player and node , the adoption function is competitive.

Then, the upper bound on the coarse PoA is 2 in the competitive cascade game.

Improvement over [Goyal/Kearns 2012]:• Improve PoA upper bound from 4 to 2.• Generalize result from 2 player game to N player game.

Proof roadmap

Set GameValid utility

gamePoA

bounds

Submodularity of social utility function

By reduction to general threshold model

By global competitiveness

By definition of social utility function

[Vetta 2002]

[Roughgarden 2009]

Proof roadmap

Set GameValid utility

gamePoA

bounds

Submodularity of social utility function

By reduction to general threshold model

By global competitiveness

By definition of social utility function

[Vetta 2002]

[Roughgarden 2009]

By definition.

Proof roadmap

Set GameValid utility

gamePoA

bounds

Submodularity of social utility function

By reduction to general threshold model

By global competitiveness

By definition of social utility function

[Vetta 2002]

[Roughgarden 2002]

Submodular : General Threshold model• General Threshold (GT) Model [KKT 03]• Each node has a threshold uniform in [0,1]• Each node has an activation function, is the set of activated nodes.• A node becomes active if and only if .• is expected number of activated nodes at the end of the process.

Theorem [Mossel/Roch 2007]: Under the general threshold model with monotone and submodular , σ(S) is monotone and submodular.

Submodular : reduction to GT model

𝒖𝟎

𝒗𝟎

𝒘 𝟎

𝒙𝟎

𝒖𝟏

𝒗𝟏

𝒘𝟏

𝒙𝟏

𝒖𝟐

𝒗𝟐

𝒘𝟐

𝒙𝟐

𝒖ℓ

𝒗 ℓ

𝒘 ℓ

𝒙 ℓ

𝒖

𝒗

𝒘

𝒙

Update sequence: 𝒗 𝒘 … ActiveInactive

Proof roadmap

Set GameValid utility

gamePoA

bounds

Submodularity of social utility function

By reduction to general threshold model

By global competitiveness

By definition of social utility function

[Vetta 2002]

[Roughgarden 2009]

Proof of

• Global competitiveness: • Similar to Lemma 1 in [Goyal/Kearns 2012]• Couple two process with and with .• By induction,

𝑿 𝒕

𝒀 𝒕

Proof: wrap up

Lemma: social utility function is submodular, if is additive and is submodular.

Lemma: and is competitive.

The competitive cascade game is a valid utility game

The pure PoA is bounded by 2 [Vetta 2002]

The coarse PoA is bounded by 2 [Roughgarden 2009]

Conclusion

• Improvement over [Goyal/Kearns 2012]:• Improve PoA upper bound from 4 to 2.• Generalize from 2 players to N players.• Generalize from pure PoA to coarse PoA.• With a much simpler and clear proof.

• Further extensions:• Strategy as multiset: • Budget limit on nodes: • Different node weight ,

Future work

• Open question• What is the PoA upper bound for competitive cascade game without

submodularity of activation function?• Upper bound 4 with additive total activation probability and competitive adoption

function for 2 player games. [Goyal/Kearns 2012]• Lower bound 2 by simple example.

• Results on cascade without submodularity• Influence maximization:

• Single product: submodularity -> . [KKT 2003]• Competitive cascade game

Questions?