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
network Diffusion 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 𝐺 = 𝑉, 𝐸 .
• The players are N companies, with their products 1, … , 𝑁.
• The individuals can be in state 1, … , 𝑁 and 0.
• 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 𝐺 = (𝑉, 𝐸).
• N players, each player is a company with limit budget 𝐵𝑖 .
• Strategy vector for players: 𝑺 = 𝑆1 , … , 𝑆 𝑁 .
• 𝑆 𝑖 is the set of nodes selected by company 𝑖.
• 𝑆 𝑖 ≤ 𝐵𝑖 .
• Payoff function 𝜎 𝑖 𝑺 :
• Expected number of people who adopt product 𝑖.
• Social utility function 𝛾 𝑺 =
𝑁
𝑖
𝜎
(𝑆):
𝑖=1
• Expected number of people who adopt a product.
General adoption model
• Seeding stage:
• Each company 𝑖 selects a set of individuals 𝑆 𝑖 .
• The initial state of node 𝑣 is inactive if no company selects it.
• Otherwise, the node 𝑣 becomes in state 𝑖 uniformly at random.
• Diffusion stage:
• Given a fixed update sequence 𝑄 = 𝑣1 , … , 𝑣ℓ .
• Nodes change states with the order in 𝑄 according to local dynamics.
General adoption model: Local Dynamic
• Let 𝑺 = (𝑆1 , … , 𝑆 𝑁 ) be current sets of nodes in state 𝑖.
• Adoption function:
• ℎ𝑣𝑖 𝑺 = Prob{𝑣 adopts product 𝑖}
• Total activation probability:
• 𝐻𝑣 𝑺 =
𝑁
𝑖
ℎ
𝑖=1 𝑣
𝑺
• A still inactive node 𝑣 changes into states 𝑖 with probability ℎ𝑣𝑖 (𝑺),
and remains inactive with probability 1 − 𝐻𝑣 (𝑺).
General adoption model: Example
𝑺𝑩
Diffusion stage
G
D
B
F
C
A
Seeding stage
D
C
D
F
C
END
E
𝐒𝐑
Useful properties
Additivity of total activation probability
𝑖 ), activation function 𝑓 𝑆 is monotone.
𝐻𝑣 𝑺 = 𝑓𝑣 (∪𝑁
𝑆
𝑣
𝑖=1
Submodularity of activation function:
𝑓𝑣 𝑆 ∪ 𝑣 −Prob{
𝑓𝑣 𝑆 ≥} 𝑓𝑣 𝑇≤
∪ 𝑣
Prob{
− 𝑓𝑣 𝑇 , ∀ 𝑆
⊆𝑇
}
Competitiveness of adoption function:
𝑖 , 𝑆 −𝑖 ⊆ 𝑆 −𝑖 , 𝑆 −𝑖 =∪
𝑘
ℎ𝑣𝑖 𝑆 ≥ ℎ𝑣𝑖 𝑆 Prob{
, ∀𝑆 𝑖 ⊆
𝑆
𝑆
=
𝑘≠iProb{ ? }
? }
Main results
Theorem: Assume the following conditions hold:
1.
2.
3.
For every node 𝑣, the total activation probability 𝐻𝑣 𝑆 is additive.
For every node 𝑣, the activation function 𝑓𝑣 𝑆 is submodular.
For every player 𝑖 and node 𝑣, the adoption function ℎ𝑣𝑖 𝑆 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
Submodularity of social utility function 𝛾(⋅)
By reduction to general threshold model
Set Game
𝜎𝑖
𝑺𝟎 ≥ 𝛾 𝑺𝟎 −
𝛾(𝑺𝟎−𝒊 , ∅𝑖 )
By global competitiveness
𝑖𝜎
𝑖
𝑆0 ≤ 𝛾(𝑆0 )
By definition of social utility function
Valid utility
game
[Vetta 2002]
[Roughgarden 2009]
PoA
bounds
Proof roadmap
Submodularity of social utility function 𝛾(⋅)
By reduction to general threshold model
Set Game
𝜎𝑖
𝑺𝟎 ≥ 𝛾 𝑺𝟎 −
𝛾(𝑺𝟎−𝒊 , ∅𝑖 )
By global competitiveness
𝑖𝜎
𝑖
𝑆0 ≤ 𝛾(𝑆0 )
By definition of social utility function
By definition.
Valid utility
game
[Vetta 2002]
[Roughgarden 2009]
PoA
bounds
Proof roadmap
Submodularity of social utility function 𝛾(⋅)
By reduction to general threshold model
Set Game
𝜎𝑖
𝑺𝟎 ≥ 𝛾 𝑺𝟎 −
𝛾(𝑺𝟎−𝒊 , ∅𝑖 )
By global competitiveness
𝑖𝜎
𝑖
𝑆0 ≤ 𝛾(𝑆0 )
By definition of social utility function
Valid utility
game
[Vetta 2002]
[Roughgarden 2002]
PoA
bounds
Submodular 𝛾(⋅): General Threshold model
• General Threshold (GT) Model [KKT 03]
•
•
•
•
Each node has a threshold 𝜃𝑣 uniform in [0,1]
Each node has an activation function, 𝑓𝑣 𝑆 , 𝑆 is the set of activated nodes.
A node becomes active if and only if 𝑓𝑣 𝑆 ≥ 𝜃𝑣 .
𝜎(𝑆) 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
𝑓𝑣 (S) , σ(S) is monotone and submodular.
Submodular 𝛾(⋅): reduction to GT model
𝒖𝟐
𝒖𝟏
𝒖𝟎
𝒗𝟏
𝒗𝟎
𝒗𝟐
𝒙𝟏
𝒙𝟎
Update sequence:
𝒘𝟐
𝒘𝟏
𝒘𝟎
𝒗
𝒖ℓ
𝒘ℓ
𝒗ℓ
𝒙ℓ
𝒙𝟐
𝒘
𝒖
…
𝒘
𝒗
𝒙
Active
Inactive
Proof roadmap
Submodularity of social utility function 𝛾(⋅)
By reduction to general threshold model
Set Game
𝜎𝑖
𝑺𝟎 ≥ 𝛾 𝑺𝟎 −
𝛾(𝑺𝟎−𝒊 , ∅𝑖 )
By global competitiveness
𝑖𝜎
𝑖
𝑆0 ≤ 𝛾(𝑆0 )
By definition of social utility function
Valid utility
game
[Vetta 2002]
[Roughgarden 2009]
PoA
bounds
Proof of
𝑖
𝜎
𝑺𝟎 ≥ 𝛾 𝑺𝟎 −
−𝒊 𝑖
𝛾(𝑺𝟎 , ∅ )
𝑘)
• Global competitiveness: 𝜎 𝑖 𝑺𝟎 ≤ 𝜎 𝑖 (𝑺−𝒌
,
∅
𝟎
• Similar to Lemma 1 in [Goyal/Kearns 2012]
𝑘
• Couple two process 𝑌𝑡 with 𝑺𝟎 and 𝑋𝑡 with (𝑺−𝒌
,
∅
).
𝟎
• By induction, 𝑋𝑡𝑖 ⊇ 𝑌𝑡𝑖 , 𝑋𝑡0 ⊆ 𝑌𝑡0 ⇒ 𝑋𝑡−𝑖 ⊆ 𝑌𝑡−𝑖
𝑿𝒕
ℎ𝑣𝑖 (𝑿𝒕 )
ℎ𝑣−𝑖 (𝑿𝒕 )
1 − 𝐻𝑣 (𝑿𝒕 )
ℎ𝑣𝑖 (𝒀𝒕 )
ℎ𝑣−𝑖 (𝒀𝒕 )
1 − 𝐻𝑣 (𝒀𝒕 )
𝒀𝒕
Proof: wrap up
Lemma: social utility function 𝛾(⋅) is
submodular, if 𝐻𝑣 𝑺 is additive and 𝑓𝑣 (𝑆)
is submodular.
𝑖 ,
Lemma: 𝜎 𝑖 𝑺𝟎 ≥ 𝛾 𝑺𝟎 − 𝛾 𝑺−𝒊
,
∅
𝟎
if 𝐻𝑣 𝑺 is additive and ℎ𝑣 (𝐒) 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: 𝛼𝑣𝑖
𝑖 ≤ 𝐾𝑖
• Budget limit on nodes: 𝑁
𝛼
𝑣
𝑖=1 𝑣
• Different node weight 𝜔𝑣 , 𝜎 𝑖 𝑀 = 𝐄[
𝑣∈𝑆ℓ𝑖
𝜔𝑣 ]
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 -> 1 − 1/𝑒. [KKT 2003]
• Competitive cascade game