An Introduction to Social and Economic Networks. Lecture 4, Part II

Transcription

1 An Introduction to Social and Economic Networks Lecture 4, Part II Michael D Ko nig University of Zurich Department of Economics June 2017

2 Outline Strategic Network Formation Pairwise Stability Nash Stability Pairwise Nash Stability Strongly Stable Networks Directed Nash Stable Networks Stochastic Strategic Network Formation Noise and Stochastic Stability 2/62

3 Pairwise Stable Networks In the connections model introduced in Jackson and Wolinksy (1996) agents receive information from others to whom they are connected to 1 Through these links they also receive information from those agents that they are indirectly connected to, that is, trough the neighbors of their neighbors, their neighbors, and so on 2 The individual incentives to form or severe links determine the addition or deletion of links Incentives are defined in terms of the utility of the agents which depends on the interactions among agents, ie the network The utility functions assigns a payoff to every agent as a function of the network the agents are embedded in 3/62 1 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp Here only the shortest paths are taken into account

4 The payoff, π i(g), agent i receives from network G with n agents is a function π i : G R with π i(g) = n δ d ij (G) j=1 j N i c, (1) where d ij (G) is the number of edges in the shortest path between agent i and agent j d ij (G) = if there is no path between i and j 0 < δ < 1 is a parameter that takes into account the decrease of the utility as the path between agent i and agent j increases N i is the set of nodes in the neighborhood of agent i 4/62

5 The total payoff of a network is defined by Π(G) = n i=1 πi(g) A network is considered efficient if it maximizes the total utility of the network Π(G) among all possible networks, G with n nodes Definition: Denote the set of networks with n nodes by G n A network G is efficient if Π(G) = n i=1 u i(g) Π(G ) = n i=1 π i(g ) for all G G n 5/62

6 Proposition: The unique efficient network (maximizing aggregate payoffs) in the connections model is (i) the complete graph, K n, if c < δ δ 2, (ii) a star, K 1,n 1, encompassing everyone if δ δ 2 < c < δ + n 2 2 δ2, (iii) the empty graph, K n, (no links) if δ + n 2 2 δ2 < c 6/62

7 Proof of part (i) We assume that δ 2 < δ c Any pair of agents that is not directly connected can increase its utility (the net benefit for creating a link is at least δ c δ 2 > 0) and thus the total utility, by forming a link Since every pair of agents has an incentive to form a link, we will end up in the complete graph K n, where all possible links have been created and no additional links can be created any more 7/62

8 Proof of part (ii) Consider a component of the graph G containing m agents, say G The number of links in the component G is denoted by k, where k m 1, otherwise the component would not be connected Eg a path containing all agents would have m 1 links The total utility of the direct links in the component is given by k(2δ 2c) There are at most m(m 1) k left over links in the 2 component, that are not created yet The payoff of each of these left over links is at most 2δ 2 (it has the highest utility if it is in the second order neighborhood) Therefor the total utility of the component is at most ( ) m(m 1) k2(δ c) + k 2δ 2 (2) 2 8/62

9 Consider a star K 1,m 1 with m agents The star has m 1 agents which are not in the center of the star An example of a star with 4 agents is given below The utility of any direct link is 2δ 2c and of any indirect link (m 2)δ 2, since any agent is 2 links away from any other agent (except the center of the star) Thus the total utility of the star is (m 1)(2δ 2c) + (m 1)(m 2)δ 2 (3) }{{}}{{} direct connections indirect connections Figure: A star encompassing 4 agents 9/62

10 The difference in total utility of the (general) component and the star is just 2(k (m 1))(δ c δ 2 ) This is at most 0, since k m 1 and c > δ δ 2, and less than 0 if k > m 1 Thus, the value of the component can equal the value of the star only if k = m 1 Any graph with k = m 1 edges, which is not a star, must have an indirect connection with a distance longer than 2, and getting a total utility less than 2δ 2 Therefore the total utility from indirect connections of the indirect links will be below (m 1)(m 2)δ 2 (which is the total utility from indirect connections of the star) If c < δ δ 2, then any component of a strongly efficient network must be a star Similarly it can be shown 3 that a single star of m + n agents has a higher total utility than two separate stars with m and n agents Accordingly, if an efficient network is non-empty, it must be a star 10/62 3 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp 44 74

11 Proof of part (iii) A star encompassing every agent has a positive value only if δ + n 2 2 δ2 > c This is an upper bound for the total achievable utility of any component of the network Thus, if δ + n 2 2 δ2 < c the empty graph is the unique strongly efficient network 11/62

12 Pairwise Stable Networks Definition: 4 The graph G is pairwise stable if (i) ij G, π i(g) π i(g ij) and π j(g) π j(g ij) ; (ii) ij / G, if π i (G + ij) > π i (G) then π j (G + ij) < π j (G), and, if π j (G + ij) > π j (G) then π i (G + ij) < π i (G) 12/62 4 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp 44 74

13 Proposition: 5 Consider the connections model in which the utility of each agent is given by Equation (1) (i) A pairwise stable network has at most one (non-empty) component (ii) For c < δ δ 2, the unique pairwise stable network is the complete graph K n (iii) For δ δ 2 < c < δ a star encompassing every agent is pairwise stable, but not necessarily the unique pairwise stable graph (iv) For δ < c, any pairwise stable network that is non-empty is such that each agent has at least two links (and thus is inefficient) 13/62 5 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp 44 74

14 Proof of part (i) Let s assume, for the sake of contradiction, that G is pairwise stable and has more than one non-empty component Let π i(g + ij) denote the marginal utility of agent i from forming a link with agent j, ie π i (G + ij) = π i (G + ij) π i (G) if ij / G and π i (G ij) = π i (G) π i (G ij) if ij G We consider now ij G Then π i (G + ij) 0 Let k, l belong to a different component Since i is already in a component with j, but k is not, it follows that π j (G + jk) > π j (G + ij) 0, because agent j will receive an additional utility of δ 2 from being indirectly connected to agent l (on top of being connected to k) For similar reasons π k (G + jk) > π k (G + kl) 0 Hence, both agents, j and k, in the separate components would have an incentive to form a link This is a contradiction to the assumption of pairwise stability i j k l 14/62

15 Proof of part (ii) The net change in utility from creating a link is at least δ δ 2 c Before creating the link, the geodesic distance between agent i and agent j is at least 2 When they create a link, they gain δ but they lose the previous utility from being indirectly connected by some path whose length is at least 2 So if c < δ δ 2, the net gain from creating a link is always positive Since any link creation is beneficial (increases the agents utility), the only pairwise stable network is the complete graph, K n 15/62

16 Proof of part (iii) We assume that δ δ 2 < c δ and show that the star is pairwise stable The agent in the center of the star has a distance of 1 to all other agents and all other agents are separated by 2 links from each other The center agent of the star cannot create a link, since it has already maximum degree The center has no incentive to delete a link either If it deletes a link, the net gain is c δ, since there is no path leading to the then disconnected agent By assumption, δ δ 2 < c < δ, c δ < 0 and the marginal payoff is negative, and the link will not be removed 16/62

17 We consider now an agent that is not the center of the star This peripheral agent cannot create a link with the center, since they are both already connected The net gain of creating a link to another agent in the periphery is δ δ 2 c, which is strictly negative by assumption So it will not create a link either Thus the star is pairwise stable Now consider the star encompassing all agents Suppose an agent would not be connected to the star If the center of the star would create a link to this isolated agent, the net gain would be δ c > 0 and the benefit of the isolated agent is again δ c > 0 So both will create the link 17/62

18 The star is not the unique pairwise stable network We will show that for 4 agents, the cycle, C 4 is also a pairwise stable network Figure: A cycle of 4 agents (left) and the resulting graph after the deletion of a link from agent 3 to agent 4 (middle) and the resulting graph after the creation of a link from agent 3 to agent 1 (right) If agent 3 removes a link to agent 4, then its marginal payoff is δ 3 δ + c This is negative if δ 3 δ + c < 0, ie c < δ δ 3 18/62

19 For the range of costs of δ δ 2 < c < δ δ 3 < δ, the agent will never do it Similarly, if agent 3 adds a link to agent 1, the marginal payoff is δ δ 2 c < 0 Thus, for n = 4 and δ δ 2 < c < δ δ 3, there are at least two pairwise stable networks: the star and the cycle Figure: A cycle of 4 agents (left) and the star with 4 agents (right) 19/62

20 Proof of part (iv) For δ < c the star is not a pairwise stable network because the agent in the center of the star would gain c δ from deleting a link Moreover, it can be shown 6 that any connected agent has at least 2 links One can see, from the two propositions described above, that a pairwise stable network is not necessarily efficient For high cost c > δ there are non-empty pairwise stable networks but they are not efficient 20/62 6 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp 44 74

21 Moreover, one can show that 7 if the benefit from maintaining an indirect link of length two is greater than the net benefit from maintaining a direct link (δ 2 > δ c > 0) then the probability that the unique efficient network, the star K 1,n 1, is reached vanishes for large n This has important implications Indeed, in the connections model, there can be stable networks that are not necessarily efficient 21/62 7 Alison Watts A Dynamic Model of Network Formation Games and Economic Behavior 342 (2001), pp

22 It can be shown that for δ δ 2 < c < δ a star K 1,n 1 encompassing every agent is not necessarily the unique pairwise stable graph However, in the same range, the only strongly efficient network is the spanning star The existence of inefficient equilibria is of interest because it indicated that the system, let alone to evolve, does not always reach an efficient configuration In this respect, the result is important from the point of view of designing of policies that help the system to reach an efficient configuration Finally, we note that 8 have proposed an extension of the connections model in which stable networks show the properties of a Small-World 22/62 8 Matthew O Jackson and Brian W Rogers The Economics of Small Worlds Journal of the European Economic Association 32/3 (2005), pp

23 Diffusion and Strategic Network Formation Consider the following payoff function π i (G i ) = λ PF (G i ) cd i (4) The returns in Equation (4) are determined by the largest eigenvalue associated with a component G i, λ PF (G i ), and it coincides with the growth rate in the number of walks of length k in a component 9 On the other hand, the payoff in Equation (4) decreases with the degree d i of the agent Therefore, it is best for a agent to reach the other agents through many walks but to have not too many links to pay for 23/62 ( 9 Each component of the power k of the adjacency matrix, A k), gives the number of ij walks of length k from node i to node j Considering the vector u = (1,, 1), we have that n k u T A k u is the number of all walks of length k among all nodes in G When the adjacency matrix is symmetric we have that u = a i w i where w i is the eigenvector of A associated with the eigenvalue λ i It follows that n k = i a i 2 λ k i For large k, we have approximately n k λ k PF, and we get n k n k 1 λ n PF Thus, the largest real eigenvalue λ PF k 1 of the graph measures the growth rate of the number of walks of length k

24 This payoff function has been studied in the context of innovation networks and the diffusion of ideas 10 Motivation: Assume that new knowledge within firm i N is generated in continuous time according to ẋ i(t) = γx i(t) + β n a ijx j(t), γ > 0, β > 0, j=1 where γ and β measure the response of knowledge growth to, respectively, variations in internal and external knowledge Then one can show that ẋ i (t) lim t x i (t) = λ PF(G i ) + γ 24/62 10 Michael D König et al The Efficiency and Stability of R&D Networks Games and Economic Behavior 75 (2 2011), ; Michael D König et al Recombinant knowledge and the evolution of innovation networks Journal of Economic Behavior & Organization 793 (2011), pp

25 We start by defining welfare as the sum of agents payoffs Π(G, c) = n π i(g i) = i=1 n (λ PF(G i) cd i) = i=1 n λ PF(G i) 2mc (5) We are interested in finding the network structures that maximize Equation (5) for a given level of marginal cost c This is because, the level of marginal cost c captures how strong is the trade-off between walks and cost of direct connections in the model More formally, let G(n) denote the set of all graphs with n nodes For a given value of cost c, the efficient graph is defined as G = argmax G G(n) Π(G, c) It can be shown that the efficient graph must be connected, and Equation (5) for total payoffs boils down to i=1 Π(G, c) = nλ PF (G) 2mc (6) 25/62

26 Efficiency of Nested Split Graphs (NSG) Proposition: Let G be the efficient graph for a given number n > 3 of agents and linking cost c 0 [ ] (i) If c 0, ( [0, 05] for large n) then the unique efficient n 2n 1 network G is the complete graph, K n [ ] n (ii) If c, 1 ( [05, 1] for large n) then the efficient network 2n 3 G is a connected nested split graph (NSG) which is not the complete network, K n [ ) (iii) If c 1, then the efficient network G is not empty and n 2 n 1 has one connected component which is a nested split graph (NSG) [ ] n (iv) If c 2, n(n 1) then the efficient network G has at most n 1 2 one non-singleton component which is a nested split graph (NSG) n(n 1) (v) If c > ( n for large n) then the unique efficient network 2 G is the empty graph, K n 26/62

27 Nested Stars A nested star, denoted by F n,d, 11 is the graph obtained from the complete graph K d with d nodes and a subset of n d disconnected nodes, by adding n d links connecting one node in K d to each of the n d disconnected nodes The nested star has a stepwise adjacency matrix, and therefore is a special case of connected nested-split graph The figure shows an example of connected nested split graph (the nested star F 10,7 ) with the associated adjacency matrix The complete graph, K n, and the spanning star, F n,d, are particular cases of connected nested split (and nested star) graphs (NSG) 27/62 11 FK Bell On the Maximal Index of Connected Graphs Linear Algebra and its Applications 144 (1991), pp

28 A = Figure: An example of a connected nested split graph, the nested star F 10,7, (left) and its associated adjacency matrix A (right) 28/62

29 Proposition: Let G be the efficient graph for a given number n of agents, and c 0 the cost of collaboration Denote the relative error of total payoffs between G and the nested star F n,d by ϵ = (Π(G, c) Π(F n,d, c)) /Π(G, c), where d = arg max d Π(F n,d, c) Then ϵ is bounded by ϵ c2 (8n 9) n 2nc n 2 + 2nc c 2 (8n 9), implying that ϵ 0 in the limit of large n 29/62

30 c F 10,10 = K 10 F 10,9 F 10,8 F 10,7 Figure: Examples of graphs F n,d for values of cost c = 05, 065, 075, 085 and n = 10 Note: For these specific values of cost c and industry size n, F n,d is the graph that returns maximal total payoffs 30/62

31 Proposition: Let c denote the linking cost and n the total number of nodes in the network G Then the stability conditions for the different types of graphs identify the following regions in the parameter space (c, n) R + N: (i) For cost c > 1 (and any n) the empty graph K n is the unique stable network (ii) The complete graph K n is stable if and only if 2 c(1 c) n < (7) c If costs are zero, c = 0, then the complete graph K n is the unique stable network (for any n) (iii) The graph consisting of a set of d 2 equally sized, disconnected cliques, Kk, 1 Kk, 2, Kk d (G having n = d k nodes in total), is stable if there exist an integer k < n, with mod(n, k) = 0 such that 1 + c(1 c) 2 c(1 c) k (8) c c 31/62

32 Proposition: Let c denote the marginal cost of a collaboration and n the total number of nodes in the network G Then the stability conditions for the different types of graphs identify the following regions in the parameter space (c, n) R + N: (iv) The star K 1,n 1 is stable if c 2 (6 + c 2 ) n (9) c 4c 2 (v) The graph consisting of d 2 disconnected cliques K 1 k,, K d k with n = kd, and the star K 1,n 1 are both stable for all pairs of integers k, n with k n such that the conditions in Equations (8) and (9) hold (vi) There exists a range of cost c 1(n) < c < c 2(n) < 1 such that the dominant-group architecture D n,k is stable if n = k + 1 but it is not stable if n k + 2 (vii) Any network with two or more isolated nodes is not stable in the cost range c [0, 1) 32/62

33 Figure: Characterization of stable networks for combinations of cost c and network size n In order to make visible the intersections among the curves, both axes are in logarithmic scale The solid, dashed and dot-dashed curve correspond to the bounds in Equations (7) and (9), respectively The dotted curve corresponds to the bound in item (ii) of the proposition 33/62

34 Stability vs Efficiency Proposition: The efficient network G is not stable: (i) if c 05 and n > 2 c(1 c), or if c > 05 and 2 c(1 c) < n c, or c c 2c 1 (ii) if c > 05 and n, or (iii) if 1 < c < n+2 2n 2 (n 1) (with n > 3) 8n 9 34/62

35 Nash Stable Networks Myerson (1991) 12 proposes a normal form game of network formation, where players simultaneously announce all the links they wish to form: Agents in N = {1,, n} individually announce all the links the wish to form For all pairs of agents i, j N let si,j = 1 if i wants to form a link with j, and s i,j = 0 otherwise A strategy of agent i is si = (s i,1,, s i,i 1, s i,i+1,, s i,n ) S i where S i = {0, 1} n 1 is the set of strategies available to i The link ij is created if and only if sij s ji = 1, that is, links are created under mutual consent 35/62 12 Roger B Myerson Game Theory: Analysis of Conflict Harvard University Press, 1991

36 Let S = S 1 S n A strategy profile s = (s 1,, s n ) S induces a network G(s) and a vector of payoffs π(g(s)) A strategy profile s = (s 1,, s n) is a Nash equilibrium if and only if π i(g(s )) π i(g(s i, s i)) for all s i S i and i N, and the network G(s ) is a Nash stable network (NS) The empty network is always a Nash stable network 36/62

37 Let G G n For all i, j N such that ij G: π i(g ij) = π i(g) π i(g ij) is the marginal payoff to i from the link ij G A network G G n is pairwise stable (PS) with respect to the network payoff function π if and only if (i) for all i, j N, if ij G then both π i (G ij) > 0 and π j (G ij) > 0, while (ii) if ij / G then π i (G + ij) > 0 implies π j (G + ij) < 0 We denote by PS(π) the set of pairwise stable networks with respect to the payoff function π 37/62

38 Pairwise Nash Stable Networks In pairwise stable networks agents were only allowed to change one link at a time However, it might be reasonable to assume that agents can delete multiple links at a time A network G G n is a pairwise-nash equilibrium network with respect to the network payoff function π if and only if there exists a Nash equilibrium strategy profile s that supports G, that is, G = G(s ), and, for all i, j N, if ij G, then π i(g + ij) > 0 implies that π i(g + ij) < 0 In other words, G is a pairwise-nash equilibrium network if it is both pairwise stable and a Nash equilibrium outcome We denote by PNE(π) the set of pairwise-nash equilibrium networks with respect to the payoff function π 38/62

39 Pairwise Stable vs Pairwise Nash Stable Networks The set of pairwise-nash (PNS) networks is thus at the intersection of the set of Nash equilibrium (NS) outcomes and the set of pairwise stable (PS) networks PS(π) NS(π) PNS(π) 9/62

40 α-submodularity Let α 0 The network payoff function π is α-submodular in own current links if and only if: π i(g L) α π i(g ij) ij L for all i N and any subset of links L E(G) The condition for α-submodularity states that the joint returns from a group of links L E(G) already in the network G is higher than the sum of the marginal returns of each single link, scaled by α The case α = 1 corresponds to submodularity 40/62

41 PS(π) = PNS(π) α-submodularity says is that if a unilateral deviation (severing one link) does not pay then multilateral deviations (severing several links at the same time) will not pay either Theorem: PS(π) = PNS(π) if and only if π is α-submodular on PS(π), for some α 0 When π is α-submodular on PS(π), then no player is better off by cutting any subset of the existing links simultanously (not only single links) in a pairwise stable network 41/62

42 Strongly Stable Networks Jackson and Van den Nouweland (2005) 13 have introduced the notion of strongly stable networks A strongly stable network is a network which is stable against changes in links by any coalition of individuals A network G is obtainable from G through deviations by S N if (i) ij G and ij / G implies {i, j} S, and (ii) ij G and ij / G implies {i, j} S = 42/62 13 Matthew O Jackson and Anne Van den Nouweland Strongly stable networks Games and Economic Behavior 512 (2005), pp

43 Part (i) implies that any new links only involve agents in S (ie the consent of two agents is needed to form a link) Part (ii) requires that at least one agent of a deleted link must be in S (ie agents can unilaterally sever a link) A network is strongly stable with respect to a profile of utility functions π = (π 1,, π n) if for any S N, G that is obtainable from G through deviations by S, and i S such that π i (G ) > π i (G), there exists a j S such that π j (G ) < π j (G) 43/62

44 Directed Nash Stable Networks Definition: Let a network G be obtainable from network G by agent i if the only changes from G to G involve links that are originating from agent i, ie if a kj a kj then k = i A directed network G is directed Nash stable (NS) if π i (G) π i (G ) for each i and all networks G that are obtainable from G by agent i We can consider two variants of the model: 14 One-Way flow: Costs and benefits are only incurred by the initiator of a link Two-Way flow: Costs are carried by the initiator of a link, while benefits are shared by both parties The payoff function is the distance based utility π i(g) = n f j=1 where f(k) > f(k + 1) > 0 for any k and c 0 ( δ d ij (G) ) d + i c, (10) 44/62 14 Venkatesh Bala and Sanjeev Goyal A Noncooperative Model of Network Formation Econometrica 685 (2000), pp

45 Two Way Flow Model Proposition: The directed Nash stable networks (NS) are given by: (i) networks that have one directed link between every pair of agents if c < f(1) f(2), any directed star encompassing all agents if f(1) f(2) < c < f(1), while there might also exist other stable networks, (ii) a periphery sponsored star (in which no link is formed by the central agent) if f(1) < c < f(1) + n 2 f(2), while there might also exist other 2 stable networks (eg the empty network), (iii) only the empty network if f(1) + n 2 f(2) < c 2 45/62

46 Proposition: The efficient networks are given by (i) networks with one directed link between every pair of agents if c < 2(f(1) f(2)), directed stars encompassing all agents if 2(f(1) f(2)) < c < 2f(1) + (n 2)f(2), and (ii) the empty network if 2f(1) + (n 2)f(2) < c 46/62

47 Figure: Stable and efficient networks in the two-way flow model for n = 4 agents Venkatesh Bala and Sanjeev Goyal A Noncooperative Model of Network Formation Econometrica 685 (2000), pp /62

48 One Way Flow Model Proposition: Assume that δ = 1 in the payoff function in Equation (10) Then the unique efficient network is (i) the wheel encompassing all agents if c < n 1, and (ii) an empty network if c > n 1 48/62

49 Proposition: Assume that δ = 1 in the payoff function in Equation (10) The directed Nash stable (NS) networks are given by (i) If c < 1 then the the wheel encompassing all agents is the only directed Nash stable network, (ii) if 1 < c < n 1 then the n-agent wheel and the empty network are the only directed Nash stable networks, (iii) if c > n 1 then the empty network is the unique Nash stable network 49/62

50 Figure: Stable and efficient networks in the one-way flow model for n = 4 agents Venkatesh Bala and Sanjeev Goyal A Noncooperative Model of Network Formation Econometrica 685 (2000), pp /62

51 Stochastic Strategic Network Formation Assume that n agents meet over time and have the opportunity to form links with each other Time, is divided into periods and is modeled as a countable, infinite set {1, 2,, t, } Let G t represent the network that exists at the end of period t and let each player i receive payoff π i (G t ) at the end of period t In each period, a link ij is randomly identified to be updated with uniform probability If the link ij is already in Gt, then either player i or j can decide to sever the link If ij / Gt, then players i and j can form link ij and simultaneously sever any of their other links if both players agree 51/62

52 Each player is myopic, and so a player decides whether or not to sever a link or form a link (with corresponding severances), based on whether or not severing or forming a link will increase his period t payoff A network G is adjacent to G if it differs by only one link, ie G and G are adjacent if G = G + ij for some ij / G or G = G ij for some ij G A network G defeats an adjacent network G if either (i) G = G ij and π i(g ) > π i(g), or (ii) G = G + ij and π i(g ) π i(g) and π j(g ) π j(g) with at least one inequality being strict A network G is pairwise stable if and only if it is not defeated by any adjacent network 52/62

53 An improving path is a sequence of distinct networks G 0, G 1,, G t,, G T such that each network G t, t < T is adjacent to and defeated by the subsequent network G t+1 A network is pairwise stable (PS) if and only if there is no improving path emanating from it (absorbing state) If no pairwise stable network exists, then there must exist at least one improving cycle, ie a sequence of adjacent networks G 0, G 1,, G t,, G T such that each G t defeats the previous G t 1 and we have that G 0 = G T Proposition: Consider the connections model 17 Assume that 0 < δ c < δ 2 For 3 < n <, there is a positive probability, 0 < P(K n 1 ) < 1, that the network formation process will converge to a star K n 1 However, as n increases, P(K n 1 ) decreases, and as n goes to infinity, lim n P(K n 1 ) = 0 53/62 17 Matthew O Jackson and Asher Wolinsky A Strategic Model of Social and Economic Networks Journal of Economic Theory 711 (1996), pp 44 74

54 Network Potentials 18,19 The network payoff function π admits an exact network potential if there exists a function Φ : G n R such that for every network G G n, every player i N, and every link ij G: π i (G) π i (G ij) = Φ(G) Φ(G ij) The network payoff function π admits an ordinal network potential if there exists a function Φ : G n R such that for every network G G n, every player i N, and every link ij G: Φ(G) > Φ(G ij) π i(g) > π i(g ij) Φ(G) < Φ(G ij) π i (G) < π i (G ij) π i (G) = π i (G ij) Φ(G) = Φ(G ij) Clearly, any exact potential game is an ordinal potential game but not the other way around 54/62 18 D Monderer and LS Shapley Potential Games Games and Economic Behavior 141 (1996), pp S Chakrabarti and RP Gilles Network potentials Review of Economic Design 111 (2007), pp 13 52

55 Example: In the symmetric connections model, if c < δ δ 2, then Φ(G) = n i=1 πi is an ordinal network potential In that case, any link formed increases the payoffs of both players forming the link and does not reduce the payoffs of all the others Example: In the eigenvalue-diffusion model for any connected network G there exists an exact potential function given by Φ(G) = λ PF mc Example: The linear-quadratic payoff function π i(q, G) = η iq i νq 2 i bq i n j i q j + ρ admits an exact potential function given by 20 Φ(q, G) = n (η iq i νqi 2 ) b 2 i=1 n i=1 j i n a ijq iq j ζd i, j=1 n q iq j + ρ 2 n i=1 j=1 n a ijq iq j ζm, for any q Q n and G G n where m denotes the number of links in G 55/62 20 Chih-Sheng Hsie, Michael D König, and Xiaodong Liu Network Formation with Local Complements and Global Substitutes: The Case of R&D Networks Department of Economics, Working Paper No 109 (2012) University of Zurich,

56 An ordinal potential function has the property that G defeats G if and only if Φ(G ) > Φ(G) and G and G are adjacent Proposition: Let Φ : G n R be an ordinal potential function Then (i) there are no improving cycles, and (ii) there exists a pairwise stable network, as any network that maximizes Φ must be undefeated The network payoff function π exhibits no indifference if for any two adjacent networks G and G, either G defeats G or G defeats G Proposition: If payoffs π exhibit no indifference, then there are no improving cycles only if there exists an ordinal potential function 56/62

57 Noise and Stochastic Stability Time is modeled discretely: 1, 2,, t, At time t the state of the process will be given by strategy profile s t specifying the links established by each player At every period t one agent is randomly chosen to revise her strategy When an agent receives this opportunity, she selects a best-response to the strategy profile in the previous period: s i (t) argmax π i (s i, s i (t 1)) s i S i (t) If there are several best-responses, each of them is chosen with equal probability This strategy revision process defines a Markov chain on G n 21 57/62 21 James R Norris Markov chains 2008 Cambridge university press, 1998

58 Proposition: 22 Consider the directed two way flow model 23 under the above Markov chain For general n, from every initial network, the dynamic process converges almost surely to the complete network Kn when c < δ δ 2 and to the empty network Kn when c > δ + (n 2)δ 2 58/62 22 For n = 4 we have convergence to the directed Nash equilibria identified in the earlier proposition 23 Venkatesh Bala and Sanjeev Goyal A Noncooperative Model of Network Formation Econometrica 685 (2000), pp

59 Stochastic Stability Now suppose that, when given the chance to revise her strategy, a player makes a mistake with probability ϵ In this case, the player chooses her strategy at random For each ϵ, the resulting evolutionary process is an aperiodic and irreducible Markov chain, and so has a unique invariant probability distribution µ ϵ 24 We analyze the structure of µ ϵ as the probability of mistakes ϵ converges to zero 24 James R Norris Markov chains 2008 Cambridge university press, /62

60 A network G is called stochastically stable if where ˆµ(G) > 0 where ˆµ = lim ϵ 0 µ ϵ The set of stochastically stable networks is defined as Ĝ = {G G n : ˆµ(G) > 0} Proposition: 25 Let 0 < δ < 1 (i) If c < δ δ 2 then Ĝ is the complete network Kn, (ii) If δ δ 2 < c < δ then the star K 1,n 1 is in Ĝ, and there exists an n such that for all n > n and δ δ 3 < c < δ we have K 1,n 1 = Ĝ, and (iii) if c > δ, there exists an n such that if n > n, then the empty graph is stochastically stable, Ĝ = K n 60/62 25 F Feri Stochastic stability in networks with decay Journal of Economic Theory 1351 (2007), pp issn:

61 s-tree Theorem 27 Consider a Markov chain on a finite state space S, given by the set of networks G n, and a transition matrix Π : S S [0, 1], 26 determined by randomly selecting a link and then following an improving path by adding or deleting a link A set of mutations of Π is a set of transition matrices Π(ϵ), 0 < ϵ < ϵ for some ϵ > 0, such that Π(ϵ) is aperiodic and irreducible for every ϵ, Π(ϵ) converges to Π as ϵ 0, and Π(ϵ)s,s > 0 implies that there exists an r 0 such that 0 < lim ϵ 0 Π(ϵ) s,s ϵ r < for any s, s S 61/62 26 James R Norris Markov chains 2008 Cambridge university press, Mark Freidlin and Alexander D Wentzell Random perturbations of dynamical systems Springer, 1984; WH Sandholm Population games and evolutionary dynamics 2010 MIT Press,

62 The number r is the resistance of the transition from state s to s, and can be thought of as the number of mutations needed to get from s to s for any s, s S Given any state s, an s-tree is a directed graph with a vertex for each state and a unique directed path leading from each state s s to s The resistance of s is the minimum across all s-trees of the summed resistance over directed edges in the s-tree Theorem: 28 Let Π be the transition matrix associated with a Markov chain on a finite state space S with associated set of mutations {Π(ϵ)} and a unique stationary distribution {µ(ϵ)} Then the stationary distribution µ(ϵ) converges to the distribution µ of Π Moreover, a state s S has a positive probability under µ (and is thus stochastically stable) if and only if s has minimum resistance 62/62 28 HP Young Individual strategy and social structure: An evolutionary theory of institutions Princeton University Press, 1998