Game theory and Shapley value

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).

Figure 1: Map showing the power line and respectice cabling needed to connect the villages A, B, C and D

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

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 K=1000 Power Line K=1000