Choosing products in social networks
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- Scot Warner
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1 Joint work with Krzysztof Apt CWI, Amsterdam
2 How do you decide as a new company entering the market whether to give your articles a high price or a low one, reckoning with the strategy of the existing competing company? Underlying question: How do agents choose products?
3 Social networks Facebook LinkedIn Google+
4 Social networks Facebook LinkedIn Google+ Tupperware party 1960s (Source: Wikipedia)
5 Social networks Essential components of the model Finite set of agents In uence of friends Product set for each agent Resistance level in adopting a product
6 Social networks Essential components of the model Finite set of agents In uence of friends Product set for each agent Resistance level in adopting a product
7 Social networks Essential components of the model Finite set of agents In uence of friends Product set for each agent Resistance level in adopting a product 4 {, } {, } 03 2 {, }
8 Social networks Essential components of the model Finite set of agents In uence of friends Product set for each agent Resistance level in adopting a product 4 05 {, } {, } 03 2 {, } 04
9 The model Social network [Apt, Markakis 2011] Weighted directed graph: G = (V, ) consisting of a nite set of agents V = {1,, n} and a weight function w ij [0, 1]: weight of the edge i j Products: A nite set of products P Product assignment: A map P V 2 P { } which assigns to each agent a non-empty set of products Threshold function: For each agent i and product t P(i) the threshold value 0 < θ(i) 1 Neighbour of node i: {j V j i} Source nodes: Agents with no neighbours
10 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 }
11 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:
12 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: c if s i P(i) For i source(s), p i (s) = 0 if s i = t 0
13 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: c if s i P(i) For i source(s), p i (s) = 0 if s i = t 0 0 if s i = t 0 For i / source(s), p i (s) = w ji θ(i) if s i = t, for some t P(i) j N t(s) i Notation: Ni t (s) is the set of neighbours of i who adopted in s the product t
14 Example {, } 05 {, } {, } 04 5 Threshold is 03 for all the players P = {,, }
15 Example 05 {, } {, } {, } 04 5 Payoff: p 4 (s) = p 5 (s) = p 6 (s) = c Threshold is 03 for all the players P = {,, }
16 Example 05 {, } {, } {, } 04 5 Payoff: p 4 (s) = p 5 (s) = p 6 (s) = c p 1 (s) = = 01 Threshold is 03 for all the players P = {,, }
17 Example 05 {, } {, } {, } 04 5 Payoff: p 4 (s) = p 5 (s) = p 6 (s) = c p 1 (s) = = 01 p 2 (s) = = 02 Threshold is 03 for all the players P = {,, }
18 Example 05 {, } {, } {, } 04 5 Payoff: p 4 (s) = p 5 (s) = p 6 (s) = c p 1 (s) = = 01 p 2 (s) = = 02 p 3 (s) = = 01 Threshold is 03 for all the players P = {,, }
19 Social network games Properties Graphical game: The payoff for each player depends only on the choice made by his neighbours Join the crowd property: The payoff of each player weakly increases if more players choose the same strategy
20 Solution concept Best response A strategy s i of player i is a best response to a joint strategy s i if for all s i, p i (s i, s i ) p i (s i, s i ) Nash equilibrium Ạ strategy pro le s is a Nash equilibrium if for all players i, s i is the best response to s i
21 Solution concept Best response A strategy s i of player i is a best response to a joint strategy s i if for all s i, p i (s i, s i ) p i (s i, s i ) Nash equilibrium Ạ strategy pro le s is a Nash equilibrium if for all players i, s i is the best response to s i Non-trivial Nash equilibrium Ạ Nash equilibrium s is non-trivial if there is at least one player i such that s i t 0
22 Solution concept Best response A strategy s i of player i is a best response to a joint strategy s i if for all s i, p i (s i, s i ) p i (s i, s i ) Nash equilibrium Ạ strategy pro le s is a Nash equilibrium if for all players i, s i is the best response to s i Non-trivial Nash equilibrium Ạ Nash equilibrium s is non-trivial if there is at least one player i such that s i t 0 Question: Does Nash equilibrium always exists?
23 Solution concept Best response A strategy s i of player i is a best response to a joint strategy s i if for all s i, p i (s i, s i ) p i (s i, s i ) Nash equilibrium Ạ strategy pro le s is a Nash equilibrium if for all players i, s i is the best response to s i Non-trivial Nash equilibrium Ạ Nash equilibrium s is non-trivial if there is at least one player i such that s i t 0 Question: Does Nash equilibrium always exists? Answer: No
24 Nash equilibrium {, } 05 {, } {, } 04 5 Threshold is 03 for all the players
25 Nash equilibrium 05 {, } {, } {, } 04 5 Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Threshold is 03 for all the players
26 Nash equilibrium 05 {, } {, } {, } 04 5 Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Threshold is 03 for all the players
27 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
28 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
29 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
30 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) (,, ) (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
31 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) (,, ) (,, ) (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
32 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
33 Nash equilibrium {, } {, } {, } Threshold is 03 for all the players Best response dynamics (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) Observation: No player has the incentive to choose t 0 Source nodes can ensure a payoff of c > 0 Each player on the cycle can ensure a payoff of at least 01 Intuitive reason: Players keep switching between the products
34 Nash equilibrium Observation: Nash equilibrium may not always exist Question: Given a social network S, what is the complexity of deciding if G(S) has a Nash equilibrium?
35 Nash equilibrium Observation: Nash equilibrium may not always exist Question: Given a social network S, what is the complexity of deciding if G(S) has a Nash equilibrium? Answer: NP-complete
36 Nash equilibrium 4 04 {, } {, } 05 {, } 2 04 Properties of the underlying graph: 6 5
37 Nash equilibrium 4 04 {, } {, } 05 {, } 2 04 Properties of the underlying graph: Contains a cycle 6 5
38 Nash equilibrium 4 04 {, } {, } 05 {, } 2 04 Properties of the underlying graph: Contains a cycle Contains source nodes 6 5
39 Nash equilibrium 4 04 {, } {, } 05 {, } 2 04 Properties of the underlying graph: Contains a cycle Contains source nodes 6 5 Question: Does Nash equilibrium always exist in social networks when the underlying graph is acyclic? is free of source nodes?
40 Directed acyclic graphs Theorem In a DAG, a non-trivial Nash equilibrium always exist Procedure to generate a non-trivial Nash equilibrium Initialise: Assigns a product for each source node
41 Directed acyclic graphs Theorem In a DAG, a non-trivial Nash equilibrium always exist Procedure to generate a non-trivial Nash equilibrium Initialise: Assigns a product for each source node Repeat until all nodes are labelled: Pick a node which is not labelled and for which all neighbours are labelled Assign the product which maximises the payoff
42 Directed acyclic graphs Theorem In a DAG, a non-trivial Nash equilibrium always exist Procedure to generate a non-trivial Nash equilibrium Initialise: Assigns a product for each source node Repeat until all nodes are labelled: Pick a node which is not labelled and for which all neighbours are labelled Assign the product which maximises the payoff Theorem A strategy pro le s is a Nash equilibrium iff there is a run of the labelling procedure such that s is the pro le de ned by the labelling function
43 Graphs with no source nodes {t 1, t 2 } 3 1 {t 1, t 2 } 2 {t 1, t 3 } Circle of friends : everyone has a neighbour {t 3, t 4 } 7 4 {t 1, t 4 } 6 {t 1, t 3 } 5 {t 1, t 4 }
44 Graphs with no source nodes {t 1, t 2 } 3 1 {t 1, t 2 } 2 {t 1, t 3 } Circle of friends : everyone has a neighbour {t 3, t 4 } 7 4 {t 1, t 4 } Observation: t 0 is always a Nash equilibrium 6 {t 1, t 3 } 5 {t 1, t 4 } Question: When does a non-trivial Nash equilibrium exist?
45 Graphs with no source nodes {t 1, t 2 } {t {t 1, t 2 } 1, t 3 } {t 3, t 4 } {t 1, t 4 } {t 1, t 3 } {t 1, t 4 } Self sustaining subgraph A subgraph C t is self sustaining for product t if it is strongly connected and for all i in C t, t P(i) w ji θ(i) j N (i) C t Threshold=03
46 Graphs with no source nodes {t 1, t 2 } {t {t 1, t 2 } 1, t 3 } {t 3, t 4 } {t 1, t 4 } {t 1, t 3 } {t 1, t 4 } Self sustaining subgraph A subgraph C t is self sustaining for product t if it is strongly connected and for all i in C t, t P(i) w ji θ(i) j N (i) C t Threshold=03
47 Graphs with no source nodes {t 1, t 2 } {t {t 1, t 2 } 1, t 3 } {t 3, t 4 } {t 1, t 4 } {t 1, t 3 } {t 1, t 4 } Self sustaining subgraph A subgraph C t is self sustaining for product t if it is strongly connected and for all i in C t, t P(i) w ji θ(i) j N (i) C t Threshold=03 Theorem There is a non-trivial Nash equilibrium iff there exists a product t and a self sustaining subgraph C t for t
48 Graphs with no source nodes An efficient procedure For a product t, X 0 t = {i V t P(i)} X m+1 t = {i V j N (i) X m j w ji θ(i)} X t = m N X m t Theorem There is a non-trivial equilibrium iff there exists a product t such that X t Complexity For a xed product t, the set X t can be computed in O(n 3 ) Running time: O( P n 3 )
49 Network dynamics Consequence of addition of new products Question Starting at a Nash equilibrium, suppose some additional products become available to some players Does a best response path converge to a (new) Nash equilibrium?
50 Network dynamics Consequence of addition of new products Question Starting at a Nash equilibrium, suppose some additional products become available to some players Does a best response path converge to a (new) Nash equilibrium? Answer For directed acyclic graphs - Yes For graphs with no source nodes - No
51 Network dynamics Addition of products Observation Starting at a Nash equilibrium, suppose an additional product become available to a single player i Following the best response path can lead to a new Nash equilibrium where (almost) everyone is worse off including player i Addition of links Ṭhe same observation holds for addition of new links in a network
52 Summary Think twice before adding someone as a friend on Facebook!
53 THANK YOU
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