Villages A, B, C and D are located in a remote area and need to be connected to the main power line to be able to receive electricity for their respective populations (see Figure 1).

The cost of installing the respective connecting lines (in £ thousands) are given in Figure 1.

where a = 1

b = 2

c = 3

d =4

Use the principles of co-operative game theory to determine a fair allocation of cost for each village in order for all the villages to receive electricity. Please show all working when answering the following questions:

Find all the characteristic functions

Is the core empty, have a unique point or have a set of infinite points? Justify your answer.

Calculate the Shapley values in order to allocate costs to the four villages.

Is there a simplier way to determine the allocation of costs? Justify your answer.

Is the coalition of the 4 villages stable? Justify your answer.

From your calculations, summarise your recommendations to the four villages.

Power Line (500 + 2a) k A (200 + a + b + c) k (300 + b) k k = 1000 B (400 + 2a + 3d) k C D Figure 1: Map showing the power line and respectice cabling needed to connect the villages A, B, C and D