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Please show all of your work for full credit. MATH 105: Final Exam Name: A.(50 points total) Voting Methods Consider the preference schedule given below for the questions in this section. Number of voters 14 1st A B 2nd 3rd C D 4th 10 C B D A 8 D C B A 4 B D C A 1 C D B A 1.(10points) Find the full ranking of candidates using plurality. 2.(20points) Find the full ranking of candidates using the Borda count method. 3.(15points) Find the full ranking of candidates using plurality with elimination. 4.(5 points) Identify whether or not there is a Condorcet candidate. If there is, which candidate is it? B.(50 points total) Power Use the following weighted voting system for the questions in this section. [10 : 5, 4, 2, 2] 5. (20 points) Find the power distribution using Banzhaf power. 6. (20 points) Find the power distribution using Shapley Shubik power. 7. (10 points) Is there any player who has veto power? Why or why not? Page 2 C.(50 points total) Sharing 8. (20 points) A sandwich has two flavors, meatball and vegetarian. Adam and Bill wish to split the sandwich using the divider chooser method with Adam as the divider (making a single cut). Adam dislikes vegetarian and Bill has no preference. What shares do Adam and Bill receive? 9. (20 points) Suppose a third person wishes to share the sandwich. What method(s) could be used to split the sandwich between the three players. Breifly describe how the distribution of the sandwich would play out using one of your methods. 10. (10 points) At an estate sale there are three main items. A cabinet, a table, and a bookshelf. Three people are putting bids on the items as shown in the table below. Describe the distribution of assets using the method of sealed bids. Player cabinet table bookshelf A 100 200 50 B 100 100 100 C 200 100 50 Page 3 D.(30 points total) Apportionment Consider a health service that provides nurses across four different districts (A,B,C, and D). There are 50 nurses to go around to these districts and the populations are given below District Population A 40 30 B C 60 D 20 11.(10 points) Find the apportionment of nurses using Hamilton’s method. 12.(20 points) Find the apportionment of nurses using Adams’ method.(Hint: Change your divisor by .1) E.(20 points total) Game Theory −2 5 13.(20 points) For the payoff matrix give the optimal strategies and the value of the 0 −1 game. (15 point bonus) What is the most useful thing that you learned in this course? How will you apply it in your daily life? Page 4