game theory/Graph theory

1. Alice, Bob, and Carly are dividing an estate consisting of a Lamborghini, a luxury yacht, and a chateau using the method of sealed bids. Their bids on each of the items are given in the following table:

	Alice	Bob	Carly
Lamborghini	$183,000	$198,000	$210,000
Luxury yacht	$2,100,000	$2,130,00	$2,070,000
Chateau	$5,100,000	$5,010,000	$5,040,000

a. Find the value of each player's fair share.

b. Describe the first settlement (who gets which item and how much do they owe the estate).

c. Find the surplus after the first settlement is over.

d. Describe the final settlement.

2. Three players (A, B, and C) agree to divide the 13 pieces of candy shown below by lining them up in order and using the method of markers. The players' bids are as indicated.

a) Which items go to A?

b) Which items go to B?

c) Which items go to C?

d) Which items are left over?

3.

a. Give the vertex set V.

b. Give the Edge set E.

c. List the degree of each vertex.

4.

a. Find a path of length 4 from D to E.

b. Find all circuits of length 1.

c. Find all circuits of length 2.