Social Network Games Krzysztof R. Apt CWI & and University of Amsterdam Joint work with Sunil Simon
A Caveat A story should have a beginning, a middle and an end, but not necessarily in that order. Jean-Luc Godard Krzysztof R. Apt Social Network Games
Social Networks Facebook, Hyves, LinkedIn, Nasza Klasa, . . . Krzysztof R. Apt Social Network Games
But also . . . An area with links to sociology (spread of patterns of social behaviour) economics (effects of advertising, emergence of ‘bubbles’ in financial markets, . . . ), epidemiology (epidemics), computer science (complexity analysis), mathematics (graph theory). Krzysztof R. Apt Social Network Games
Example (From D. Easley and J. Kleinberg, 2010). Collaboration of mathematicians centered on Paul Erd˝ os. Drawing by Ron Graham. Krzysztof R. Apt Social Network Games
Social networks Essential components of our model Finite set of agents. Influence of “friends”. Finite product set for each agent. Resistance level in (threshold for) adopting a product. Krzysztof R. Apt Social Network Games
Social networks Essential components of our model Finite set of agents. Influence of “friends”. Finite product set for each agent. Resistance level in (threshold for) adopting a product. 4 0.4 1 0.6 0.5 3 2 0.3 Krzysztof R. Apt Social Network Games
Social networks Essential components of our model Finite set of agents. Influence of “friends”. Finite product set for each agent. Resistance level in (threshold for) adopting a product. {•} 4 0.4 {• , •} 1 0.6 0.5 3 2 0.3 {• , •} {• , •} Krzysztof R. Apt Social Network Games
Social networks Essential components of our model Finite set of agents. Influence of “friends”. Finite product set for each agent. Resistance level in (threshold for) adopting a product. {•} 4 0 . 5 0.4 {• , •} 1 0 . 3 0.6 0.5 0 . 2 3 2 0 . 4 0.3 {• , •} {• , •} Krzysztof R. Apt Social Network Games
The model Social network [Apt, Markakis 2011] Weighted directed graph: G = ( V , → , w ), where V : a finite set of agents, w ij ∈ (0 , 1]: weight of the edge i → j . Products: A finite set of products P . Product assignment: P : V → 2 P \ {∅} ; assigns to each agent a non-empty set of products. Threshold function: θ ( i , t ) ∈ (0 , 1], for each agent i and product t ∈ P ( i ). Neighbours of node i : { j ∈ V | j → i } . Source nodes: Agents with no neighbours. Krzysztof R. Apt Social Network Games
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , Krzysztof R. Apt Social Network Games
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , � 0 if s i = t 0 ◮ if i ∈ source ( S ), p i ( s ) = if s i ∈ P ( i ) c Krzysztof R. Apt Social Network Games
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , � 0 if s i = t 0 ◮ if i ∈ source ( S ), p i ( s ) = if s i ∈ P ( i ) c ◮ if i �∈ source ( S ), p i ( s ) = 0 if s i = t 0 � w ji − θ ( i ) if s i = t , for some t ∈ P ( i ) j ∈N t i ( s ) N t i ( s ): the set of neighbours of i who adopted in s the product t . Krzysztof R. Apt Social Network Games
Example {•} 4 0.4 {• , •} 1 0.5 0.5 {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Network Games
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Network Games
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 p 1 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Network Games
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 p 1 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 0.4 0.4 p 2 ( s ) = 0 . 5 − 0 . 3 = 0 . 2 6 5 p 3 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Network Games
Social network games Properties Graphical game: The payoff for each player depends only on the choices made by his neighbours. Join the crowd property: The payoff of each player weakly increases if more players choose the same strategy. Krzysztof R. Apt Social Network Games
Does Nash equilibrium always exist? {•} 4 0.4 {• , •} 1 0.5 0.5 {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Network Games
Does Nash equilibrium always exist? {•} 4 0.4 {• , •} 1 Observation: No player has the 0.5 0.5 incentive to choose t 0 . {• , •} {• , •} 3 2 Source nodes can ensure a 0.5 payoff of c > 0. 0.4 0.4 6 5 Each player on the cycle can ensure a payoff of at least 0 . 1. {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Network Games
Does Nash equilibrium always exist? {•} ( • , • , • ) 4 0.4 {• , •} 1 Observation: No player has the 0.5 0.5 incentive to choose t 0 . {• , •} {• , •} 3 2 Source nodes can ensure a 0.5 payoff of c > 0. 0.4 0.4 6 5 Each player on the cycle can ensure a payoff of at least 0 . 1. {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Network Games
Does Nash equilibrium always exist? Best response dynamics {•} ( • , • , • ) ( • , • , • ) ( • , • , • ) 4 0.4 ( • , • , • ) ( • , • , • ) ( • , • , • ) {• , •} 1 Observation: No player has the 0.5 0.5 incentive to choose t 0 . {• , •} {• , •} 3 2 Source nodes can ensure a 0.5 payoff of c > 0. 0.4 0.4 6 5 Each player on the cycle can ensure a payoff of at least 0 . 1. {•} {•} Reason: Players keep switching Threshold is 0 . 3 for all the players. between the products. Krzysztof R. Apt Social Network Games
Nash equilibrium Question: Given a social network S , what is the complexity of deciding whether G ( S ) has a Nash equilibrium? Krzysztof R. Apt Social Network Games
Nash equilibrium Question: Given a social network S , what is the complexity of deciding whether G ( S ) has a Nash equilibrium? Answer: NP-complete. Krzysztof R. Apt Social Network Games
Nash equilibrium Question: Given a social network S , what is the complexity of deciding whether G ( S ) has a Nash equilibrium? Answer: NP-complete. The PARTITION problem Input: n positive rational numbers ( a 1 , . . . , a n ) such that � i a i = 1. Question: Is there a set S ⊆ { 1 , 2 , . . . , n } such that a i = 1 � � a i = 2 . i ∈ S i �∈ S Krzysztof R. Apt Social Network Games
Hardness Reduction: Given an instance of the PARTITION problem P = ( a 1 , . . . , a n ) , construct a network S ( P ) such that there is a solution to P iff there is a Nash equilibrium in S ( P ). Krzysztof R. Apt Social Network Games
Hardness Reduction: Given an instance of the PARTITION problem P = ( a 1 , . . . , a n ) , construct a network S ( P ) such that there is a solution to P iff there is a Nash equilibrium in S ( P ). {• ′ } {•} 4 ′ 4 0 . 4 0 . 4 {• ′ , • ′ } {• , •} 1 ′ 1 0 . 5 0 . 5 0 . 5 0 . 5 {• ′ , • ′ } {• , •} {• ′ , • ′ } {• , •} 3 2 3 ′ 2 ′ 0 . 5 0 . 5 0 . 4 0 . 4 0 . 4 0 . 4 6 5 6 ′ 5 ′ {• ′ } {• ′ } {•} {•} Krzysztof R. Apt Social Network Games
Hardness Reduction: Given an instance of the PARTITION problem P = ( a 1 , . . . , a n ) , construct a network S ( P ) such that there is a solution to P iff there is a Nash equilibrium in S ( P ). {• , • ′ } {• , • ′ } {• , • ′ } · · · i 1 i 2 i n {• ′ } {•} 4 ′ 4 0 . 4 0 . 4 {• ′ , • ′ } {• , •} 1 ′ 1 0 . 5 0 . 5 0 . 5 0 . 5 {• ′ , • ′ } {• , •} {• ′ , • ′ } {• , •} 3 2 3 ′ 2 ′ 0 . 5 0 . 5 0 . 4 0 . 4 0 . 4 0 . 4 6 5 6 ′ 5 ′ {• ′ } {• ′ } {•} {•} Krzysztof R. Apt Social Network Games
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