GAME THEORETICAL INFERENCE OF HUMAN BEHAVIOR IN SOCIAL NETWORKS Symposium on “ Resilience and performance of networked systems ” Zürich, 16.01.2020 [N. Pagan & F. Dörfler, “ Game theoretical inference of human behavior in social networks”, Nature Communications, 2019] NICOLÒ PAGAN FLORIAN DÖRFLER
MOTIVATION How do social networks form? Individual behavior determines the social network structure. The social network structure influences individual behavior. 01
OBSERVATIONS Actors decide with whom they want to interact. directionality : followers followees ≠ 02
OBSERVATIONS Actors decide with whom they want to interact. directionality : followers followees ≠ Ties can have different weights . 02
OBSERVATIONS Actors decide with whom they want to interact. 2 nd degree directionality : followers followees ≠ Ties can have different weights . 1 st degree Limited information is available. 3 rd degree 02
OBSERVATIONS Actors decide with whom they want to interact. directionality : followers followees ≠ Ties can have different weights . Limited information is available. Network positions provide benefits to the actors. 02
SOCIAL NETWORK POSITIONS’ BENEFITS Social Influence The more people we are connected to, the more we can influence them. [Robins, G. Doing social network research, 2015] 03
SOCIAL NETWORK POSITIONS’ BENEFITS Social Influence The more people we are connected to, the more we can influence them. [Robins, G. Doing social network research, 2015] Social Support The more our friends’ friends are our friends, the safer we feel. [Coleman, J. Foundations of Social Theory, 1990] 03
SOCIAL NETWORK POSITIONS’ BENEFITS Social Influence The more people we are connected to, the more we can influence them. [Robins, G. Doing social network research, 2015] Social Support The more our friends’ friends are our friends, the safer we feel. [Coleman, J. Foundations of Social Theory, 1990] Brokerage The more we are on the path between people, the more we can control. 03 [Burt, R. S. Structural Hole (Harvard Business School Press, Cambridge, MA, 1992]
SOCIAL NETWORK POSITIONS’ BENEFITS Social Influence The more people we are connected to, the more we can influence them. Degree Centrality Clustering Coefficient [Robins, G. Doing social network research, 2015] Social Support The more our friends’ friends are our friends, the safer we feel. [Coleman, J. Foundations of Social Theory, 1990] Brokerage The more we are on the path between people, the more we can control. 03 [Burt, R. S. Structural Hole (Harvard Business Betweenness Centrality School Press, Cambridge, MA, 1992]
• Preferential Attachment • Small-World Network Complex Networks • Agent-Based Model SOCIAL NETWORK FORMATION Economics Sociology • Stochastic Actor-Oriented Models • Strategic Network Formation Model • Exponential Random Graph Models
SOCIAL NETWORK FORMATION MODEL Directed weighted network with agents. 𝒪 = {1,…, N } The weight quantifies the importance of the a ij ∈ [0,1] friendship among and from ’s point of view. i j i A typical action of agent : i N − 1 a i = [ a i 1 , …, a i , i − 1 , a i , i +1 , …, a iN ] ∈ = [ 0,1 ] Every agent is endowed with a payoff function i V i and is looking for 04 a ⋆ i ∈ arg max a i ∈ V i ( a i , a − i )
V i ( a i , a − i ) = t i ( a i , a − i ) PAYOFF FUNCTION Social influence on friends t i ( a i , a − i ) = ∑ i ∑ δ i ∑ k ≠ j ∑ l ≠ k ∑ k ≠ j ∑ + δ 2 a ji + a kj a ji a lk a kj a ji j ≠ i j ≠ i j ≠ i k paths of length 2 paths of length 3 k j where [Jackson, M. O. & Wolinsky, A. A strategic model of social δ i ∈ [0,1] & economic networks. J. Econom. Theory 71, 44–74 (1996)] k l j i j j 05 j j
V i ( a i , a − i ) = t i ( a i , a − i ) + u i ( a i , a − i ) PAYOFF FUNCTION Social influence on friends t i ( a i , a − i ) = ∑ i ∑ δ i ∑ k ≠ j ∑ l ≠ k ∑ k ≠ j ∑ + δ 2 a ji + a kj a ji a lk a kj a ji j ≠ i j ≠ i j ≠ i paths of length 2 paths of length 3 where [Jackson, M. O. & Wolinsky, A. A strategic model of social δ i ∈ [0,1] & economic networks. J. Econom. Theory 71, 44–74 (1996)] Clustering coefficient i u i ( a i , a − i ) = ∑ ∑ a ij a ik a kj j ≠ i k ≠ i , j [Burger, M. J. & Buskens, V. Social context and network formation: an experimental study. Social Networks 31, 63–75 (2009)] . 05 j k
V i ( a i , a − i ) = t i ( a i , a − i ) + u i ( a i , a − i ) − c i ( a i ) PAYOFF FUNCTION Social influence on friends t i ( a i , a − i ) = ∑ i ∑ δ i ∑ k ≠ j ∑ l ≠ k ∑ k ≠ j ∑ + δ 2 a ji + a kj a ji a lk a kj a ji j ≠ i j ≠ i j ≠ i paths of length 2 paths of length 3 j where [Jackson, M. O. & Wolinsky, A. A strategic model of social δ i ∈ [0,1] & economic networks. J. Econom. Theory 71, 44–74 (1996)] j Clustering coefficient j i u i ( a i , a − i ) = ∑ ∑ a ij a ik a kj j ≠ i k ≠ i , j [Burger, M. J. & Buskens, V. Social context and network formation: an experimental study. Social Networks 31, 63–75 (2009)] . j 05 j j c i ( a i ) = ∑ Cost a ij j ≠ i
V i ( a i , a − i | θ i ) = α i t i ( a i , a − i ) + β i u i ( a i , a − i ) − γ i c i ( a i ) PAYOFF FUNCTION θ i = { α i , β i , γ i } α i ≥ 0, β i ∈ ℝ , γ i > 0 Social influence on friends t i ( a i , a − i ) = ∑ i ∑ δ i ∑ k ≠ j ∑ l ≠ k ∑ k ≠ j ∑ + δ 2 a ji + a kj a ji a lk a kj a ji j ≠ i j ≠ i j ≠ i paths of length 2 paths of length 3 j where [Jackson, M. O. & Wolinsky, A. A strategic model of social δ i ∈ [0,1] & economic networks. J. Econom. Theory 71, 44–74 (1996)] j Clustering coefficient j i u i ( a i , a − i ) = ∑ ∑ a ij a ik a kj j ≠ i k ≠ i , j [Burger, M. J. & Buskens, V. Social context and network formation: an experimental study. Social Networks 31, 63–75 (2009)] . j 05 j j c i ( a i ) = ∑ Cost a ij j ≠ i
NASH EQUILIBRIUM Definition. The network is a Nash Equilibrium if for all agents : ⋆ i ∀ i , a ⋆ a i ∈ V i ( a i , a ⋆ − i | θ i ) i ∈ arg max ⟹ V i ( a i , a − i ⋆ | θ i ) ≤ V i ( a ⋆ i , a − i ⋆ | θ i ) , ∀ a i ∈ Nodes 𝒪 Individual Payoff Network Game ( 𝒪 , V i ( θ i ), ) ⋆ ( θ i ) Behavior θ i V i ( θ i ) Action Space 06
Question: Given , which is in equilibrium ? ⋆ INDIVIDUAL θ i BEHAVIOR θ i DETERMINE ∀ i , a ⋆ a i ∈ V i ( a i , a ⋆ − i | θ i ) i ∈ arg max STRATEGIC NETWORK FORMATION MODEL STRATEGIC PLAY SOCIAL NETWORK STRUCTURE ⋆ ( θ i )
Question: Given , for which is in equilibrium ? ⋆ ⋆ INDIVIDUAL θ i BEHAVIOR θ i DETERMINE ∀ i , a ⋆ a i ∈ V i ( a i , a ⋆ − i | θ i ) i ∈ arg max STRATEGIC NETWORK FORMATION MODEL STRATEGIC PLAY GAME-THEORETICAL INFERENCE SOCIAL NETWORK STRUCTURE ⋆ ( θ i ) ∀ i , find θ i s.t. V i ( a i , θ i | a ⋆ − i ) ≤ V i ( θ i | a ⋆ − i ) , ∀ a i ∈ i , a ⋆
HOMOGENEOUS V i ( a i , a − i | θ ) = α γ t i ( a i , a − i ) + β γ u i ( a i , a − i ) − c i ( a i ) RATIONAL AGENTS θ = { α , β , γ } α ≥ 0, β ∈ ℝ , γ > 0 Assumptions. (i) Homogeneity : for all agents . θ i = θ , i (ii) Fully rational agents. Derive Necessary and Sufficient conditions for Nash equilibrium stability of 4 stylised network motifs. 07 Empty Complete Complete Balanced Bipartite Star
HOMOGENEOUS V i ( a i , a − i | θ ) = α γ t i ( a i , a − i ) + β γ u i ( a i , a − i ) − c i ( a i ) RATIONAL AGENTS θ = { α , β , γ } α ≥ 0, β ∈ ℝ , γ > 0 Assumptions. (i) Homogeneity : for all agents . θ i = θ , i clustering (ii) Fully rational agents. cost 2 Empty Complete 1 Bipartite Star social influence β / γ 0 cost − 1 07 − 2 0 1 2 3 4 5 6 7 8 α / γ
HOMOGENEOUS V i ( a i , a − i | θ ) = α γ t i ( a i , a − i ) + β γ u i ( a i , a − i ) − c i ( a i ) RATIONAL AGENTS θ = { α , β , γ } α ≥ 0, β ∈ ℝ , γ > 0 Assumptions. (i) Homogeneity : for all agents . θ i = θ , i (ii) Fully rational agents. Definition. The network is a Nash Equilibrium if for all agents : ⋆ i ∀ i , a ⋆ a i ∈ V i ( a i , a ⋆ − i | θ ) . i ∈ arg max ⟹ V i ( a i , a − i ⋆ | θ ) ≤ V i ( a ⋆ i , a − i ⋆ | θ ) , ∀ a i ∈ Using the Variational Inequality approach, it is equivalent to 07 ⟨ ∇ V i ( a i , a ⋆ , a i − a i ⋆ ⟩ ≤ 0, − i | θ ) ∀ a i ∈ . a ⋆ i
EXAMPLE: V i ( a i , a − i | θ ) = α γ t i ( a i , a − i ) + β γ u i ( a i , a − i ) − c i ( a i ) COMPLETE NETWORK θ = { α , β , γ } α ≥ 0, β ∈ ℝ , γ > 0 Theorem. 2 NE Let be a complete network of homogeneous, rational agents. CN N Define: 1 γ NE := { αδ ( 1 + δ (2 N − 3) ) + β ( N − 2), if β > 0 β / γ ¯ 0 αδ ( 1 + δ (2 N − 3) ) + 2 β ( N − 2), if β ≤ 0, − 1 then is a Nash equilibrium if and only if . CN γ ≤ ¯ γ NE − 2 0 1 2 3 4 5 08 α / γ
Recommend
More recommend
