A network has N potential members. By joining the newtork, you expect the networ

Question

A network has N potential members. By joining the newtork, you expect the network will have n members (of the N possible).  Everytime you request information from the network, 3 people (of the N) could provide a useful answer. If any of the three are members of the network, your utility from the network is u, u > 0. If none of the people of the three people who could provide a useful answer are in the network, then your utility for that request from the network is $0. Let lambda be the expected size of the network measured as a fraction of the entire population. That is, lambda = n/N.

a. What is the expected utlity per request you recieve from the network? (dont use lambda instead write the expected utlity per request in terms of n, N, and u)

b. What is a nice approximation of the expected utility per request you recieve from the network? (Hint: can be written in terms of lambda)

c. Show that the expected value per request of the network is increasing but at a decreasing rate as lambda increases (up to 1).

Explanation / Answer

utility = u when any of three are members of network

utility = 0 when none of three are members of network

probability ( none of three are members of network) = (N-n)/N * (N-n-1)/(N-1) * (N-n-2)/(N-2)

so expected utility per request =  (N-n)/N * (N-n-1)/(N-1) * (N-n-2)/(N-2)* 0

+ (1 - ((N-n)/N * (N-n-1)/(N-1) * (N-n-2)/(N-2))) * u = (1 - ((N-n)/N * (N-n-1)/(N-1) * (N-n-2)/(N-2))) * u

b) to approximate it

n/N = lambda then approximating n-1/(N-1) as lambda and n-2/(N-2) as lambda

then expected utiltiy = u * (1- (1-lambda)^3)

c) as lambda increases expected utlityincreases

since differentiating w.r.t lambda we have

u * (3(1-lambda)^2) which is anywhere > 0

again differentiating w.r.t lambda we have

u * (-6)(1-lambda) which is < 0 when lambda <1

So rate of growth is decreasing.

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