Influencing Social Evolutionary Dynamics

Transcription

1 Influencing Social Evolutionary Dynamics Jeff S Shamma Georgia Institute of Technology MURI Kickoff February 13, 2013

2 Influence in social networks Motivating scenarios: Competing for customers Influencing mindsets/beliefs Shaping behaviors...over networks Many models of evolution: Evolution of convention Contagion Social learning Belief diffusion etc...shift discussion to influence 1

3 Influence in social networks, cont Setup: Network of customers Competing firms Firms spend resources on customers Customer propensities evolve according to: Propensity of neighbors Received resources Intrinsic compliance Goal: Effective resource allocation policies Emphasis: Network effects Dynamic allocation Model uncertainty Competition 2

4 Selected related work Dubey, Garg, De Meyer (2006): Competing for customers in a social network Setup: Steady state of diffusion with time-invariant resource allocation Analysis: Resulting Nash equilibrium of resource allocation Goyal & Kearns (2011): Competitive contagion in networks Setup: Steady state of contagion with initial one-time resource allocation Analysis: Efficiency of infection at Nash equilibrium (Price-of-Anarchy/Budgets) Fazeli & Jadbabaie (2012): Game theoretic analysis of a strategic model of competitive contagion and product adoption in social networks Setup: Reversible contagion model Analysis: Game over bounds on rates of adoption 3

5 Model State variable: x l i (t) Node index: i {1, 2,..., n} Stage: t = 0, 1, 2,... Firm: l {a, b} Associate with propensities, allegiance,... Larger increased allegiance 4

6 Model, cont Update rule: n x l i(t+1) = θ i w ij x l j(t)+(1 θ i )ϕ i (u l i(t), u l (t)) j=1 Non-negative weights: j w ij = 1 Node compliance: θ i [0, 1) Resource allocation: u l i (t) 0 Budget: j ul j (t) M Influence function: ϕ i : R + R + (, α i ] Node capacity: α i > 0 5

7 Influence function ϕ Relative saturation: ϕ i (u l i, u l i ) = sat(u l i u l i ; α i ) 6

8 Model features x l i(t + 1) = θ i n w ij x l j(t) + (1 θ i )ϕ i (u l i(t), u l (t)) j=1 States subject to network effects States naturally diminish over time Marginal influence diminish/remain constant with increasing resources Influence diminishes with effort of rivals Missing: Influence diminishes with propensity 7

9 Optimal allocation under monopoly State update: x l i(t + 1) = θ i n w ij x l j(t) + (1 θ i )ϕ i (u l i(t), u l (t)) j=1 or x l (t + 1) = Ax l (t) + Bϕ(u l (t), u l (t)) Monopoly: u l (t) = 0 Stage reward: g(x l, u l ) = v T x l c T u l Intrinsic value: v i > 0 Costliness: c i > 0 8

10 Optimal allocation under monopoly, cont Dynamic objective: over policies µ l ( ) β t g(x l (t), µ l (x l (t))) t=0 Assumption: Immediacy βv T (I Θ) c T > 0 Ranking vector: h T = βv T (I βa) 1 (I Θ) Theorem: Optimal allocation policy is to invest budget up to saturation in the order of h T c T 9

11 Discussion h c T = βv T (I βa) 1 (I Θ) c T Ranking vector h c T combines Network value Intrinsic value Costliness Compare to α-centrality: 1 T (I βa) 1 10

12 Sketch of derivation Optimal reward-to-go J k (x) = γ T k x + δ k Dynamic programming value iterations: γ T k+1 = βγ T k A + v T δ k+1 = βδ k + max u α (βγt k (I Θ) c T )u Analysis also provides optimal (time-varying) finite-horizon policy 11

13 Variations Broadcasting: Resource distribution interrelated, e.g., B str = Optimal policy: arg max B str u α ht Bu c T u Nonlinear allocation cost: Optimal policy: Special case: ρ(u) = 1 2 ut u g(x l, u l ) = v T x l ρ(u l ) arg max u α ht u ρ(u) Both cases: h T plays important role 12

14 Competition Evolving states: Stage rewards: x a (t + 1) = Ax a (t) + Bϕ(u a (t), u b (t)) x b (t + 1) = Ãxb (t) + Bϕ(u b (t), u a (t)) g(x a, u a ) = v T x a c T u a g(x b, u b ) = ṽ T x b c T u b Dynamic objectives: β t g(x a (t), µ a (x a (t))) t=0 β t g(x b (t), µ b (x b (t))) Equilibrium strategies: Best response to fixed allocation policy is fixed allocation policy. Can construct of equilibria in special cases. t=0 13

15 Discussion: Model mispecification & available information Did not use evolving state...are these measurable? Did use detailed network structure...is this known? Did use diffusion dynamics...is this critical? e.g., What if... x i (t + 1) = θ i Avg[x i (t T, t)] + (1 θ i )Avg[u i (t T, t)] 14

16 Path dependencies and emergent behavior Illustration: Reinforcement learning with recency Path dependency consequences: Stabilize disequilibrium behavior (JSS & Arslan, 2005) Affect equilibrium selection (Chasparis & JSS, 2012) 15

17 Natural model misspecification? Setup: Population games Collection of populations {1, 2,..., n} Within population distributions x i (e.g., resource allocations) Population vector payoffs: p i = F i (x i, x i ) (can think of agents & mixed strategies) Stable game (SG) : For convenience deal with single population: p = F (x) Definition: For 1 T z = 0, z T F (x)z 0 Theorem (Hofbauer & Sandholm): Various learning rules of the form converge for stable games. ẋ = V (x, F (x)) 16

18 Illustration: EPT dynamics Excess payoff: ˆF (x) = F (x) ( x T F (x) ) 1 EPT: ẋ = τ( ˆF (x)) ( 1 T τ( ˆF (x)) ) x τ( ) is EPT protocol Theorem (H & S): Define Positive correlation (PC): ẋ 0 ẋ T F (x) > 0 (natural?) Integrability (IN): τ = γ for some γ : R n R EPT + PC + IN + SG = convergence Note: EPT + PC + SG not enough! 17

19 Stable games & EPT revisited Stable game: For p = F (x), along trajectories ṗ = F (x)ẋ ẋ T ṗ = ẋ T F (x)ẋ z T F (x)z 0 Implication: For stable games 0 ṗ T ẋ Payoffs and population flows are negatively correlated along trajectories Think of evolutionary dynamic as mapping from ṗ to ẋ Theorem (Fox & JSS): An evolutionary dynamic (EPT is special case) with 0 ṗ T ẋ coupled with a SG results in convergence. (Proof: Adaptation of passivity theory ) Enabler: Positive correlation along trajectories 18

20 Rewired interconnection DF(x) dx/dt x(0) + + x dp/dt - + x1 T ( ) τ( ) ^ F T x( ) F + + F(x(0)) Game: Maps population flows to payoff flows Evolutionary dynamic: Maps payoff flows to population flows Outcome: Robust conclusions without specifics 19

21 Generalizations Natural dynamics: 0 ẋ T ṗ i.e., flow of population and flow of payoff is positively correlated along trajectories Perspective: Existing analyses were special cases of passivity theory Instant correlation (PC) was not enough: 0 < ẋ T p Opportunity: Broader notions of stables games, e.g., path dependencies, delays, etc. with passivity p(t) = F(x([0, t])) 20

22 Concluding remarks Recap: Formulated social influence as optimal decision problem Monopoly solution follows generalized network centrality Explored competitive scenario Future direction: Robustness, dynamics, & time Restricted measurements Misspecified network Uncertain dynamics Temporal sensitivity 21