= Consider a game played on a network and a finite set of players N =...

 

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= Consider a game played on a network and a finite set of players N = {1,2,...,n}. Each node in the network represents a player and edges capture their relationships. We use G = (gij) 1≤ij≤n to represent the adjacency matrix of a undirected graph/network, i.e., gij = gji € {0, 1}. We assume gii = 0. Thus, G is a zero-diagonal, squared and symmetric matrix. Each player, indexed by i, chooses an action x € R. Let x = (x₁, x2,...,xn)', Xi > 0, Vi (the transpose of a vector x is denoted by x') be the corresponding vector. Each player i obtains the following payoff T₁(x) = αixi 127x2² centrality vector can be defined as: +8 Σ G i j X i X j z jEN where a > 0. The parameter 8> 0 captures the strength of the direct links between different players. For simplicity, we assume 0 < d < /1(n-1). A Nash Equilibrium is a profile x* = (x,x) such that, for any i = 1,...,.., ₂ T₂(x₁,xn) ≥ Ti (x₁,...,x_1, X₁, Xi+1; In other words, at a Nash equilibrium, there is no profitable deviation for any player i choosing x. Let w = (W₁, W2,, wn)', wi > 0, Vi, and In the n x n identity matrix. Define the weighted Katz-Bonacich centrality vector as: 2 ,x), for any x₁ € R. b(G, w) = [In - SG]-¹w. Let M = (mij)<i<n = [I-8G]-¹ denote the inverse Leontief matrix associated with network G, while mi denote its ij entry, which is equal to the discounted number of walks from i to j with decay factor d. Let 1n = (1, 1,..., 1)' be a vector of 1s. Then, the unweighted Katz-Bonacich (1) b(G, 1) = [I-SG]-¹1n. (a) Show that this network game has a unique Nash Equilibrium x*(G). Can you link this equi- librium to the Katz-Bonacich centrality vector defined above? [4 pt.] = Consider a game played on a network and a finite set of players N = {1,2,...,n}. Each node in the network represents a player and edges capture their relationships. We use G = (gij) 1≤ij≤n to represent the adjacency matrix of a undirected graph/network, i.e., gij = gji € {0, 1}. We assume gii = 0. Thus, G is a zero-diagonal, squared and symmetric matrix. Each player, indexed by i, chooses an action x € R. Let x = (x₁, x2,...,xn)', Xi > 0, Vi (the transpose of a vector x is denoted by x') be the corresponding vector. Each player i obtains the following payoff T₁(x) = αixi 127x2² centrality vector can be defined as: +8 Σ G i j X i X j z jEN where a > 0. The parameter 8> 0 captures the strength of the direct links between different players. For simplicity, we assume 0 < d < /1(n-1). A Nash Equilibrium is a profile x* = (x,x) such that, for any i = 1,...,.., ₂ T₂(x₁,xn) ≥ Ti (x₁,...,x_1, X₁, Xi+1; In other words, at a Nash equilibrium, there is no profitable deviation for any player i choosing x. Let w = (W₁, W2,, wn)', wi > 0, Vi, and In the n x n identity matrix. Define the weighted Katz-Bonacich centrality vector as: 2 ,x), for any x₁ € R. b(G, w) = [In - SG]-¹w. Let M = (mij)<i<n = [I-8G]-¹ denote the inverse Leontief matrix associated with network G, while mi denote its ij entry, which is equal to the discounted number of walks from i to j with decay factor d. Let 1n = (1, 1,..., 1)' be a vector of 1s. Then, the unweighted Katz-Bonacich (1) b(G, 1) = [I-SG]-¹1n. (a) Show that this network game has a unique Nash Equilibrium x*(G). Can you link this equi- librium to the Katz-Bonacich centrality vector defined above? [4 pt.]

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First we need to prove that the payoffs of each player in the network game are concave functions To do this we can take the second derivative of the p... View the full answer