Text: Solution
An example illustrating the VCG algorithm. Consider the network shown in Figure 2.11, where the four links are owned by four different players. Suppose that the network wants to establish a communication path from node 1 to node 3. If a link is selected for transmission, it incurs a cost of p, where I {a, b, c, d}. The overall system goal, also known as the social welfare problem, is to find the minimum-cost path. Define x to be a variable such that x = 0 if link I is selected and x = 1 otherwise. Therefore, the utility function associated with player I is:
-pI, if x = 0, i.e., link I is selected,
Ux = 10, if x = 1, i.e., link I is not selected.
Figure 2.11: A simple network
a
b
c
d
The utility function revealed by player I is:
-pI, if x = 0,
Ux = 10, if x = 1,
where p is the cost claimed by player I. The objective of the network is to maximize U(x).
subject to:
xa + xc = 1,
xb + xa = 1,
xa, xb, xc, xd ∈ {0, 1}.
The equalities (2.63) and (2.64) guarantee that there is a feasible path from node 1 to node 3. Assume pa > pc and pb > po. Let w denote the price charged to link I. Write the value of w under the VCG pricing mechanism. Note: You will find that the prices are non-positive. In fact, -w can be interpreted as the payment link I receives when it is selected by the network.
