5. (25 pts) We consider a slightly modied version of network from Braess's paradox. Total trafc is now in, a natural number, instead of 1. See the figure below for the details of our new network. First assume the agents have negligible size. (Nonatomic selfish routing). (a) (5 pts} Find an equilibrium ow. (b) (5 pts} Find the optimal ow and the POA. (c) (5 pts) Consider a much more complicated network where cost edge functions are ctr} = 3: or C(33) = 11. If there are multiple equilibrium ows for this network, argue that the total travel time of these equilibrium ows should be all equal. Now assume there are n = 3 agents on the same network given above but each agent consumes one unit of trafc each (Atomic selsh routing). (d) (5 pts) Suppose 3: people take the top path (0 A v ) d), 3; people take the zigzag path (0 ) v ) w > d}, and 2 people take the bottom path (0 > a: > d) (i.e. ow f = (3:, y,z)). Write the potential function (equation 13.6 in the book} @(f) = 296E Elf; (43(1') as a function of :r,y, z. (e) (5 pts) Using part (d) find the worst equilibrium flow (i.e. the equilibrium ow with the highest total cost)