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Network Economics Assignment 1 2022
发布时间:2022-08-30
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Network Economics
Assignment 1
August 20, 2022
Exercise Lecture 1 [10 points]
Consider a network composed of 3 agents with the following links: 12, 23 31. Compute
(a) the betweenness centrality [2 pt.];
(b) the eigenvector centrality (show all the necessary algebraic steps) [3 pt.];
(c) the Bonacich centrality with 6 = 0.2 (show all the necessary algebraic steps) [3 pt.].
(d) What can you say about individual clustering ? what if a new node, 4, enters in the network and connects with agent 3? [2 pt.]
Exercise Lecture 2 [10 points]
Consider the co-authorship game from class. The total utility of a person i adds up the benefit across each of their links ij in the network g
ui (g) = j:ij∈T ┌ +
+
,
= 1 + j:ij∈T ┌ +
,
In a 5 person version of this game find the pairwise stable network(s) and the efficient net- work(s)? [10 pt.]
Exercise Lecture 3 [10 points]
Consider a game played on a network and a finite set of players Ⅳ = X1, 2, . . . , n}. Each node in the network represents a player and edges capture their relationships. We use G = (gij )1ki_jk京 to represent the adjacency matrix of a undirected graph/network, i.e., gij = gji e X0, 1}. We assume gii = 0. Thus, G is a zero-diagonal, squared and symmetric matrix. Each player, indexed by i, chooses an action xi e R. Let x = (x1 , x2 , . . . , x京 )\ , xi > 0, Ai (the transpose of a vector x is denoted by x\ ) be the corresponding vector. Each player i obtains the following payoff
πi (x) = αixi _ xi(2) + δ gij xixj ,
j∈N
(1)
where αi > 0. The parameter δ > 0 captures the strength of the direct links between different players. For simplicity, we assume 0 < δ < /1(n _ 1).
A Nash Equilibrium is a profile x_ = (x1(_) , . . . , x_京) such that, for any i = 1, . . . , .., n,
πi (x1(_) , . . . , x_京) > πi (x1(_) , . . . , xi(_)一1 , xi , xi(_)+1 , . . . , x_京), for any xi e R. (2)
In other words, at a Nash equilibrium, there is no profitable deviation for any player i choosing xi(_) .
Let w = (w1 , w2 , . . . , w京 )\ , wi > 0, Ai, and I京 the n x n identity matrix. Define the weighted Katz-Bonacich centrality vector as:
b(G, w) = [I京 _ δG]一1w. (3)
Let M = (mij )1ki_jk京 := [I _ δG]一1 denote the inverse Leontief matrix associated with network G, while mij denote its ij entry, which is equal to the discounted number of walks from i to j with decay factor δ . Let 1京 = (1, 1, . . . , 1)\ be a vector of 1s. Then, the unweighted Katz-Bonacich
centrality vector can be defined as:
b(G, 1) = [I _ δG]一1 1京 . (4)
(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.]
(b) Let x_ (G, a) = i(i)1(京) xi(_)(G, a) denote the sum of actions (total activity) at the unique Nash Equilibrium in part 1. Calculate this value. Determine
∂xj(_)(G, a)
∂αi ,
∂x_ (G, a)
∂αi .
Interpret the results. [6 pt.]
Exercise Lecture 4 [10 points]
Consider the following routing game with two routes where l1 ,l2 are the travel times on each route when the fraction of traffic on each route is given by x1 and x2 . The parameters a, b satisfy a > 0 and 0 < b < 1. See Figure 1.
(a) Find the optimal routing [3 pt.]
(b) Find the equilibrium routing [3 pt.]
(c) Suppose everyone values saving an hour of time at $10/hour. What tax and on which route will result in the optimal routing as an equilibrium? [3 pt.]
(d) What is the maximum ratio of inefficiency of the travel time under equilibrium routing relative to the optimal routing ( ) [1 pt.]
Figure 1: Traffic Routing