High-Dimensional Connectivity and Cooperation

Transcription

1 GAMES 01. Istanbul 4th World ongress of the Game Theory Society High-imensional onnectivity and ooperation Amparo Urbano, Angel Sánchez & Jose Vila ERI-ES / epartment of Economic Analysis University of Valencia (Spain) GIS/Matemáticas, Universidad arlos III de Madrid (Spain) 1

2 Why group connectivity? Imagine a table with four people having a conversation at a restaurant; this event is understood as a social network with four nodes. If every one can hear everyone else, then in graph theory this network is represented by a complete graph on four vertices.

3 Why group connectivity? Now imagine a situation of four people playing the game phone so that each person may only whisper in another person s ear. This scenario is again modeled by a complete graph on four vertices. But the situations are extremely different! 3

4 Group connectivity: Higher dimensionality One-dimensional graph theory does not capture the distinction between a single 4- person conversation and six -person conversation. 4

5 Group connectivity: efinition A situation in which interactions among individuals may be of a multidimensional character, i.e., the interaction of n individuals is not simply the aggregation of the interaction of the n(n-1)/ pairs. 5

6 Group connectivity In most social and economic local structures, a specific set of agents may have some kind of special relation among them. We call this set of agents a group and we say that there exists group connectivity among them. The existence of groups generates specific patterns of heterogeneity and groups externalities. Examples of these groups are families, business partners, committees, associations and lobbies. 6

7 Multidimensional connectivity We introduce the concept of multidimensional connectivity to analyze group connectivity among agents in a network. Our key idea is to consider that groups of agents can be linked beyond pair-wise interactions. In standard networks, a link could be understood as having dimension one. In our model, for instance, a link of dimension two would represent a connection among three nodes and such connection has deeper implications that those of an aggregation of the three one-to-one- connections between them. 7

8 From network to simplicial complex To study the above situations, one must turn from (1-dimensional) graph to a higher-dimensional model. The two most popular such models are hypergraphs and simplicial sets. A hypergraph is like a graph, except that edges can connect more than two nodes (not closed under taking subsets). Simplicial sets, are visualized as multi-dimensional polygonal shapes made up of nodes, edges, triangles, and higher-dimensional triangles like tetrahedra. 8

9 From network to simplicial complex Any graph is a 1-dimensional simplicial set. Suppose that we want to construct a -simplicial set. Begin with a graph G, and choose a set of three edges (a,b), (b,c) and (a,c) which form a triangle inside G: a a a c b c b Given this triangle, one may attach in a -simplex to G, filling in the triangle abc. The solid shape represents the idea that the relationship (the conversation) is taking place between three entities in a shared space. The relationship is closed under taking subsets 9

10 From network to simplicial complex We model multidimensional connectivity by associating an abstract simplicial complex to a network in such a way that when a subset of n+1 nodes has a multidimensional connection, then the n-dimensional simplex formed by these n+1 nodes belongs to the simplicial complex. 10

11 Framework of analysis Evolutionary approach: allows us to go beyond perfect rationality and introduce bounded rationality. Prisoner s ilemma Game: the most difficult framework to analyze cooperation. 11

12 Some research on Evolutionary P Eshel, Samuelson and Shaked (1998), analyze an imitationbased learning model in a circle. Two kind of agents, Altruistics (a) and Egoistics (E), who play a x game with her immediate neighbors. The result is that imitation can yield peaceful coexistence of the two strategies, unlike best response dynamics. a a a E E a a a a a a E a a a cycle a a E E E a a 1

13 Some related research Question: What happens if agents are placed in a plane or in a higher dimensional structure? Nowak and May (199,1993), analyze a spatial evolutionary Prisoner s ilemma game. Individuals play with their neighbors and with themselves. They observe a rich variety of behavior depending on the value of the defection parameter. 13

14 The spatial network game An spatial evolutionary Prisoner s ilemma (P). Players are located on a two-dimensional square lattice of N N nodes, interacting in a Moore neighborhood (a local interaction game). Each agent plays the Prisoner s ilemma with each of her 8 immediate neighbors and her strategy, or, is the same in all these games. 14

15 A Moore neighborhood Each agent interacts with her 8 nearest neighbors i 15

16 The spatial network game 1 S T 0 = cooperate / = no cooperate T = temptation to defect payoff >1 S = sucker s payoff (S=0) Each agent i s payoff is the sum over all interactions with her 8 8 nearest neighbors: i ij j 1 After every round of the game, each player i observes the strategy and payoffs of one randomly chosen neighbor j and updates her strategy following the replicator dynamic rule: P ij =(Prob. i imitates agent j s strategy)= ij ( ) ( ) j i j i 16

17 The two-dimensional simplicial complex game: The triads Associate to this game a -dimensional simplicial complex whose simplices (triangles) represent the groups of three neighbor nodes with -dimensional connection: the triads. Suppose that player i belongs to a triangle (a group). This connection may have consequences: On information: the probability that player i observes an agent inside the simplex may be higher. On imitation: the probability that i imitates the behavior of an agent inside the simplex is higher. On payoffs: the payoffs of the P s played within the group have a higher weight than those coming from neighbors outside the group 17

18 A measure of link intensity Two agents are closer, the higher the number of groups they belong to. enote the intensity of the link between agent i and j as and define: ij ij where 1 and [0, 4] is the number of triangles i and j pertain to

19 Heterogeneous link intensity The model introduces link heterogeneity with some specific structure. If agents i and j have link intensity (they belong to triangles), it means that there exist two other agents m and n to whom i and j are respectively connected with link intensity 1. This argument is extended to any other link intensity 3 and

20 Modeling group connectivity REPLIATOR YNAMIS Information (Probability to observe agent j) Imitation (Probability to imitate agent j s strategy) Payoffs (Aggregation of payoff of the P s games) ij Standard spatial P 1 [ ( 8 i Spatial P with triads Δ Δ ij ij j 1 8 ij j1 payoff of the P played by i against j i aggregate payoff of player i j i )]( j i ) ij 8 j1 ij 1 [ ( i ij j 8 j 1 )]( ( x) 0 if x 0 and ( x) 1 if x i ij ij ij j ) 0 i 0

21 Replicator dynamics We have 8 different replicator dynamics depending on which effects of group connectivity are actually active. Let us denote them by RP( INFO, IMIT, PF ) where INFO,IMIT, PF = 0 if the effect is not active and =1 if it is active. EXAMPLES: RP(1,0,0) Preferential information seeking with both standard imitation and payoffs RP(0,1,0) Preferential imitation with both standard imitation and payoffs RP(0,0,1) Preferential payoffs with both standard information and imitation RP(1,0,1) Preferential information and payoffs with standard imitation RP(1,1,1) Preferential information, imitation and payoffs 1

22 The minimal structures of cooperation For T>1, cooperators can only survive in clusters. Nowak and May (1993, deterministic imitation): the minimal cooperation atoms are square shaped clusters, with a minimum of 4, provided that T is small enough (T<3/). For T in (,3), a x cluster will disappear but a 3x3 cluster will persist.

23 3 A triad of s A 5x5 torus, with a triad of cooperators in a sea of defectors: M T T M 3 3

24 4 Propagation of s Propagation of a triad of cooperators in a sea of defectors: M m T T m M 7 7

25 5 Local contagion T T m M M m Fluctuation area 6 the lattice of border At the upper T

26 6 Influence area

27 7 Two triads of s Two types of cooperators: A A B B M T T M A B 3 3

28 8 Some contagion A A B B M m and for T T T A B m

29 9 Survival of the s in the common face of two triads A A 1 M 1 3 Δ and 1, 7 For Δ )- T(T T T

30 30 Only 1 and 4 are in a common face of two triangles 1} 1, 3 max{ T T T A x cluster Local contagion : 4 3 T T

31 Some relevant features The minimal structures that ensure the survival of cooperative behavior, whenever there exist group payoff externalities with big enough as compared to T are: a triangle of cooperative behavior with link intensity. at least two players with a link intensity of (a multidimensional hub). A triangle of s is not immune to an invasion by a defector, but two triangles with a common face are immune to such invasion provided that the defector is not in the common face. ooperators survive if they form clusters. Therefore defectors are always in the lattice boundaries. 31

32 What to say for large NxN lattices? omplex behavior: Local and global correlations. egree heterogeneity and network clustering give rise to different dynamical processes. Simulations: 1x1 lattice (000 periods, starting from a 50% of cooperators) from A. Sanchez, A. Urbano and J. Vila (010). 3

33 Impact of information seeking : RP(1,0,0) 33

34 Impact of imitation RP(0,1,0) 34

35 Impact of payoffs: RP(0,0,1) 35

36 Impact of information, imitation and payoff: RP(1,1,1) 36

37 Results of the simulations The impact of group connectivity depends on the type of effect of group connectivity: Preferential information has no significant effect on cooperation. Preferential imitation hinders cooperation Group payoff supports cooperation 37

38 Results of the simulation With heterogeneity in the intensity of the links, some cooperators stop changing strategy at some step and thus remain cooperators, while others, mostly defectors oscillate much more frequently. The fluctuations are caused by the changes of strategy of the highly connected individual, which induce cascades of conversion of their neighborhoods. Multidimensional hubs and surrounding players form strategy homogeneous components in the network. These components find it very difficult to synchronize with others, leading to fluctuations. The imitation of highly connected players has very different consequences depending on the focal player being a cooperator or a defector. 38

39 onclusions Group connectivity may explain the coexistence of different and small ethnic groups in the midst of neighborhoods: lumpiness and spatial concentration. Population viscosity (individuals do not move far from their places of birth), facilitates the evolution of cooperation by increasing the degree of relatedness among interacting individuals. Higher strategy correlations than those only based on local neighborhoods: role of group effects on several behavioral outcomes, peer influences, clustering. 39

40 Future research Application to specific frameworks of group interaction. Empirical analysis with laboratory / field data. Grujic, Fosco, Araujo, uesta and Sanchez (010, PLoS One): a 13x13 Moore neighborhood àlamoore. Novak and May setting. Spatial structure does not increase cooperation. ynamic models and co-evolution. Analysis of multidimensional connectivity in general networks beyond Moore neighborhoods 40

41 GAMES 01. 4th World ongress of the Game Theory Society Istanbul. 41