Social Network Games

Transcription

1 CWI and University of Amsterdam Based on joint orks ith Evangelos Markakis and Sunil Simon

2 The model Social netork ([Apt, Markakis 2011]) Weighted directed graph: G = (V,,), here V: a finite set of agents, ij (0,1]: eight 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 ith no neighbours.

3 The associated strategic game Interaction beteen agents: Each agent i can adopt a product from the set P(i) or choose not to adopt any product (t 0 ). Social netork games Players: Agents in the netork. 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,

4 The associated strategic game Interaction beteen agents: Each agent i can adopt a product from the set P(i) or choose not to adopt any product (t 0 ). Social netork games Players: Agents in the netork. 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), pi (s) = c if s i P(i)

5 The associated strategic game Interaction beteen agents: Each agent i can adopt a product from the set P(i) or choose not to adopt any product (t 0 ). Social netork games Players: Agents in the netork. 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), pi (s) = c if s i P(i) P if i source(s), pi (s) = 0 if s i = t 0 ji θ(i,t) if s i = t, for some t P(i) j Ni t (s) Ni t (s): the set of neighbours of i ho adopted in s the product t.

6 Example Threshold is 0.3 for all the players. P = {,, }

7 Example Payoff: p 4 (s) = p (s) = p 6 (s) = c Threshold is 0.3 for all the players. P = {,, }

8 Example Payoff: p 4 (s) = p (s) = p 6 (s) = c p 1 (s) = = Threshold is 0.3 for all the players. P = {,, }

9 Example Payoff: p 4 (s) = p (s) = p 6 (s) = c p 1 (s) = = p 2 (s) = = p 3 (s) = = Threshold is 0.3 for all the players. P = {,, }

10 Social netork games Properties Graphical game: The payoff for each player depends only on the choices made by his neighbours. Join the crod property: The payoff of each player eakly increases if more players choose the same strategy.

11 Does Nash equilibrium alays exist? Threshold is 0.3 for all the players.

12 Does Nash equilibrium alays exist? 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. Threshold is 0.3 for all the players.

13 Does Nash equilibrium alays exist? (,, ) 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. Threshold is 0.3 for all the players.

14 Does Nash equilibrium alays exist? Threshold is 0.3 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. Reason: Players keep sitching beteen the products.

15 Nash equilibrium Question: Given a social netork S, hat is the complexity of deciding hether G(S) has a Nash equilibrium?

16 Nash equilibrium Question: Given a social netork S, hat is the complexity of deciding hether G(S) has a Nash equilibrium? Anser: NP-complete.

17 Nash equilibrium Question: Given a social netork S, hat is the complexity of deciding hether G(S) has a Nash equilibrium? Anser: 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 = a i = 1 2. i S i S

18 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork S(P) such that there is a solution to P iff there is a Nash equilibrium in S(P).

19 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork S(P) such that there is a solution to P iff there is a Nash equilibrium in S(P) { } {, } {, } {, } { } { }

20 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork 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 { } {, } {, } {, } { } { }

21 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork 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 a a 1 4 { } {, } {, } {, } { } { }

22 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork 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 a 1 a 2 a a 1 4 { } {, } {, } {, } { } { }

23 Hardness Reduction: Given an instance of the PARTITION problem P = (a 1,...,a n ), construct a netork S(P) such that there is a solution to P iff there is a Nash equilibrium in S(P). θ(4) = θ(4 ) = 1 2. {, } i 1 {, } i 2 {, } i n a 1 a n a 2 a a n a 1 4 { } {, } {, } {, } { } { }

24 Nash equilibrium Recall the netork ith no Nash equilibrium: Theorem. If there are at most to products, then a Nash equilibrium alays exists and can be computed in polynomial time.

25 Nash equilibrium Properties of the underlying graph: 6

26 Nash equilibrium Properties of the underlying graph: Contains a cycle. 6

27 Nash equilibrium Properties of the underlying graph: Contains a cycle. Contains source nodes. 6

28 Nash equilibrium Properties of the underlying graph: Contains a cycle. Contains source nodes. 6 Question: Does Nash equilibrium alays exist in social netorks hen the underlying graph is acyclic? has no source nodes?

29 Directed acyclic graphs A Nash equilibrium s is non-trivial if there is at least one player i such that s i t 0. Theorem. In a DAG, a non-trivial Nash equilibrium alays exists.

30 Graphs ith no source nodes Circle of friends : everyone has a neighbour

31 Graphs ith no source nodes Circle of friends : everyone has a neighbour. Observation: t 0 is alays a Nash equilibrium. Question: When does a non-trivial Nash equilibrium exist?

32 Graphs ith no source nodes Threshold=0.3 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) j N(i) C t ji θ(i,t)

33 Graphs ith no source nodes Threshold=0.3 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) j N(i) C t ji θ(i,t)

34 Graphs ith no source nodes Threshold= 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) j N(i) C t ji θ(i,t) Theorem. There is a non-trivial Nash equilibrium iff there exists a product t and a self sustaining subgraph C t for t. Corollary. An algorithm ith a running time O( P n 3 ).

35 Finite Improvement Property Fix a game. Profitable deviation: a pair (s,s ) such that s = (s i,s i) for some s i and p i (s ) > p i (s). Improvement path: a maximal sequence of profitable deviations. A game has the FIP if all improvement paths are finite.

36 Summary of results NE arbitrary DAG simple cycle no source graphs nodes NP-complete alays exists alays exists alays exists Non-trivial NE NP-complete alays exists O( P n) O( P n 3 ) Determined NE NP-complete NP-complete O( P n) NP-complete

37 Summary of results NE arbitrary DAG simple cycle no source graphs nodes NP-complete alays exists alays exists alays exists Non-trivial NE NP-complete alays exists O( P n) O( P n 3 ) Determined NE NP-complete NP-complete O( P n) NP-complete FIP co-np-hard yes? co-np-hard FBRP co-np-hard yes O( P n) co-np-hard Uniform FIP co-np-hard yes yes co-np-hard Weakly acyclic co-np-hard yes yes co-np-hard

38 Summary of results NE arbitrary DAG simple cycle no source graphs nodes NP-complete alays exists alays exists alays exists Non-trivial NE NP-complete alays exists O( P n) O( P n 3 ) Determined NE NP-complete NP-complete O( P n) NP-complete FIP co-np-hard yes? co-np-hard FBRP co-np-hard yes O( P n) co-np-hard Uniform FIP co-np-hard yes yes co-np-hard Weakly acyclic co-np-hard yes yes co-np-hard FBRP: all improvement paths, in hich only best responses are used, are finite. Uniform FIP: all improvement paths that respect a scheduler are finite. Weakly acyclic: from every joint strategy there is a finite improvement path that starts at it.

39 Paradox of Choice (B. Schartz, 200) [Gut Feelings, G. Gigerenzer, 2008] The more options one has, the more possibilities for experiencing conflict arise, and the more difficult it becomes to compare the options. There is a point here more options, products, and choices hurt both seller and consumer.

40 Paradox 1 Adding a product to a social netork can trigger a sequence of changes that ill lead the agents from one Nash equilibrium to a ne one that is orse for everybody.

41 Example Cost θ is constant, 0 < θ <.

42 Example Cost θ is constant, 0 < θ <. This is a Nash equilibrium. The payoff to each player is θ > 0.

43 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

44 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

45 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

46 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

47 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

48 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

49 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

50 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

51 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

52 Example Cost θ is constant, 0 < θ <. This is not a Nash equilibrium.

53 Example Cost θ is constant, 0 < θ <. This is a Nash equilibrium. The payoff to each player is 0.

54 Paradox 2 Removing a product from a social netork can result in a sequence of changes that ill lead the agents from one Nash equilibrium to a ne one that is better for everybody.

55 Example Cost θ is product independent. The eight of each edge is, here > θ. Note Each node has to incoming edges.

56 Example Cost θ is product independent. The eight of each edge is, here > θ. This is a Nash equilibrium. The payoff to each player is θ.

57 Example Cost θ is product independent. The eight of each edge is, here > θ. This is not a legal joint strategy.

58 Example Cost θ is product independent. The eight of each edge is, here > θ. This is not a Nash equilibrium.

59 Example Cost θ is product independent. The eight of each edge is, here > θ. This is not a Nash equilibrium.

60 Example Cost θ is product independent. The eight of each edge is, here > θ. This is not a Nash equilibrium.

61 Example Cost θ is product independent. The eight of each edge is, here > θ. This is a Nash equilibrium. The payoff to each player is 2 θ.

62 Final remarks Needed: Identify other conditions that guarantee that these paradoxes cannot arise. Open problem: Does a social netork exist that exhibits paradox 1 for every triggered sequence of changes? Alternative approach: Obligatory product selection (no t 0 ). In this setup the above problem has an affirmative anser.

63 References K.R. Apt and E. Markakis, Social Netorks ith Competing Products. Fundamenta Informaticae S. Simon and K.R. Apt,. Journal of Logic and Computation. To appear. K.R. Apt, E. Markakis and S. Simon, Paradoxes in Social Netorks ith Multiple Products. Submitted. K.R. Apt and S. Simon, ith Obligatory Product Selection. Proc. 4th International Symposium on Games, Automata, Logics and Formal Verification (Gandalf 2013). EPTCS.

64 Thank you