MATH4994 Capstone Projects in Mathematics and Economics

Transcription

1 MATH4994 Capstone Projects in Mathematics and Economics Homework One Course instructor: Prof. Y.K. Kwok 1. This problem is related to the design of the rules of a game among 6 students for allocating 6 coins among this group of Ann, Bob, Carl, Dora, Ed, and Fran. Nature of the game: Ann goes first. She is given a bag that everyone knows contains six gold coins. Ann makes a proposal of how to allocate the six coins among the six contestants, including her. The contestants (including Ann) then vote yes or no on the proposal. If the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Ann has to leave the island empty-handed and she is out of the game. In this case, the bag of six gold coins passes to Bob. He gets to make a proposal of how to allocate the coins among the remaining contestants (i.e., including Bob but excluding Ann) and the remaining contestants (i.e., including Bob but excluding Ann) then vote. As before, if the proposal gets more than half the votes then the coins are allocated according to the proposal and everyone leaves the island. If the proposal gets half or fewer than half the votes then Bob has to leave the island empty-handed and is out of the game. In this case, the bag of six gold coins passes to Carl. And so on, the same voting rules apply, with each failed proposal leading to expulsion of the proposer, and with the role of proposer being passed on alphabetically. Assumptions made: (a) The coins are indivisible, and side contracts to make monetary payments are not allowed. (b) There are no abstentions; each surviving voter must vote yes or no: whenever a voter is indifferent, she or he votes no. (c) The players only care about the gold (and this is common knowledge). For example, leaving empty handed because your proposal fails is the same as leaving emptyhanded because a successful proposal gives you no coins. (d) All the contestants are rational. Question: What proposal should Ann make and why? Hint: When there are only two contestants left, Fran should reject any proposal made by Ed. How about the reaction of Ed and Fran to any proposal made by Dora when there are three contestants left? Using backward induction, find the optimal proposal placed by Ann. 2. (a) Prove that the divide-and-choose procedure does not guarantee an efficient allocation. (b) Prove that Austin s procedure does not guarantee an efficient allocation. 1

2 3. Is the divide-and-choose procedure manipulable? That is, can one player achieve a strictly better outcome by misrepresenting his true valuation of the cake? Prove that it is not, or give an example where one player achieves a strictly better outcome than obtained with an honest application of the divide-and-choose procedure. 4. Give an example to illustrate that the Selfridge-Conway method for three parties fails to be (a) efficient; (b) equitable. 5. Give an example to demonstrate that the Webb moving knife procedure for three parties is not (a) efficient; (b) equitable. 6. This exercise presents yet another valid algorithm described by Kuhn for fairly dividing the cake. Have Tom cut what he considers equal thirds and ask the other two to identify any of the three pieces they find acceptable (worth a third in the player s estimation). Let us organize this information in a matrix where 1 means this piece is acceptable and 0 means this piece is unacceptable. The information might look like the table below, for example. X 1 X 2 X 3 Tom Dick Harry (a) How can the division be accomplished in this case? (b) Why may we assume that all entries in the first row are 1? Can there be a row with no 1? (c) Devise a method of fair division if the table looks like: (Of course, further cuts will be required.) X 1 X 2 X 3 Tom Dick Harry (d) Considering the problem for four players, how could you proceed if the matrix takes the form given? X 1 X 2 X 3 X 4 Tom Dick Harry Amy

3 (e) Assuming that there is always a fair division for three players, consider the problem for four players and show in all cases that a fair division is possible. 7. Suppose the cake is cut in six pieces and you get the first and last choice. (a) How much are you sure to get by your estimation on your first choice? The last choice? (b) Show the two combined choices will always guarantee you at least 1/5 of the cake. Under what conditions will you get only 1/5 of the cake? 8. How many cuts are required in the worst case to divide cake fairly among three persons using the Kuhn Algorithm described in Problem 6? 9. To modify Stromquist s Moving Knife Algorithm so that it applies to the dirty work problem, ask a referee to move a sword from left to right dividing X into two pieces, X 1 to the right of the sword and X 2 to the left. Simultaneously, have the three players adjust parallel knives over X 2 so that each of their knives cuts X 2 in exact halves. Instruct the players that when any one of them says cut, the lawn will be cut in three portions by the referee s sword and the middle knife of the three players, creating partitions X 2 = X 2 X 2 and X = X 2 X 2 X 1. Further instruct them that they should say cut whenever they first think X 1 is smaller than or equal to both X 2 and X 2. Without loss of generality, we can assume that the players knives from left to right belong to P 1, P 2 and P 3, respectively. Consider the three cases generated by which of the three players says cut and show how, in each case, to assign X 2, X 2, and X 1 to the players in an envy-free way. 10. Consider the following fair division procedure for 3-people: Amy, Beth, and Colin. Amy divides the cake into two pieces of equal value in her opinion. Beth takes the larger (in her opinion) of the two pieces, and gives the remaining piece to Amy. Amy and Beth each divide their piece of cake into three pieces they consider to be equally valuable. There are now six pieces of cake. Colin chooses one piece of cake from Amy s three pieces, and one piece of cake from Beth s three pieces. Amy keeps her remaining two pieces and Beth keeps her remaining two pieces. (a) Is this procedure proportional? Why or why not? (b) Is this procedure envy-free? Why or why not? 11. (a) Determine the allocation determined by the adjusted winner procedure for the following example. Ross Item Rachel s Points Points 35 Manhattan Apartment Custody of Daughter Emma Share in ownership of local coffee shop 15 5 Right to spend Thanksgiving with Monica and Chandler Total 100 3

4 (b) Observing that the distributions of points allocated to the items by the two players are quite close, explain why each party gets relatively few points overall in the final distributions. 12. The adjusted winner procedure can be adapted for unequal entitlements. Suppose that Annie and Ben are getting a divorce, but they signed a pre-nuptial agreement that gives Annie 60% of the joint property and Ben 40%. During the equitability adjustment stage of the adjusted winner procedure, Annie s point total should be exactly 1.5 times that of Ben. Determine the allocation dictated by the adjusted winner procedure for the following items. Annie s Points Item Ben s Points 35 Right to retain lease on apartment Entertainment System Pool table Antique Table Washer & Dryer Total Emma and Kate are planning to open a new restaurant, and have several projects to finish before they will be ready to open. They would rather split up the projects between them so that each person has full control of a few specific issues instead of working together on each of the different projects. Each person has devoted 100 points to the projects listed below. Emma s Points Item Kate s Points 20 Menu Design Interior Design Advertising 5 15 Dining Room Layout Bar Layout Hiring Waitstaff Hiring Chefs Total 100 Another method for dividing goods or issues between two people is balanced alternation, wherein the two parties take turns choosing issues and the party that chooses second is compensated by being able to choose two items during his first turn. For example, if persons A and B are dividing six goods between them, then they might choose in the following order: A,B,B,A,B,A. (a) If Emma and Kate use balanced alternation rather than adjusted winner, what is the final allocation of issues? (b) Is Emma better or worse off with balanced alternation than with adjusted winner? What about Kate? 4

5 (c) Describe a particular example of two sets of goods to be divided where adjusted winner is far better than balanced alternation. (d) Is there any situation in which the parties might prefer to use balanced alternation over adjusted winner? 14. Eighteen cookies are to be divided between three good friends (Michael, Mike, and Peter) after a hard night s work in Athens, Georgia. There are six chocolate chip cookies, 6 peanut butter cookies, and 6 sugar cookies with rainbow sprinkles. Michael is thinking of going vegan (he s already a vegetarian), so the chocolate chip cookies are worthless to him (fortunately, the peanut butter and sugar cookies were made without eggs, butter, or milk). He likes the peanut butter and sugar cookies equally. Mike is allergic to peanuts, so he cannot eat the peanut butter cookies. He likes the chocolate chip and sugar cookies equally. Peter likes the chocolate chip and peanut butter cookies equally but does not like the sugar cookies at all the sprinkles fall into his mandolin. Give examples of allocations of cookies (all 18 must be accounted for) that are (a) envy-free but not equitable; (b) equitable but not envy-free. 15. Consider the following marriage problem where all men prefer the same woman as their first choice and all women prefer the same man as their first choice P (m 1 ) = w 1, w 2, w 3 P (w 1 ) = m 1, m 2, m 3 P (m 2 ) = w 1, w 2, w 3 P (w 2 ) = m 1, m 3, m 2 P (m 3 ) = w 1, w 3, w 2 P (w 3 ) = m 1, m 2, m 3 Find the corresponding stable matchings for men-oriented µ M and women-oriented µ W. Hint: Explain why any matching that does not pair m 1 with w 1 is unstable. 16. Man-woman-child matching problem: There are three sets of people: men, women, and children. A matching is a division of the people into groups of three, containing one man, one woman, and one child. Each person has preferences over the sets of pairs he or she might possibly be matched with. A man, woman, and child (m, w, c) block a matching µ if m prefers (w, c) to µ(m); w prefers (m, c) to µ(w), and c prefers (m, w) to µ(c). A matching is stable only if it is not blocked by any such three agents. Consider three men, three women, and three children, with the following preferences: P (m 1 ) = (w 1, c 3 ), (w 2, c 3 ), (w 1, c 1 ),... (arbitrary) P (m 2 ) = (w 2, c 3 ), (w 2, c 2 ), (w 3, c 3 ),... (arbitrary) P (m 3 ) = (w 3, c 3 ),... (arbitrary) P (w 1 ) = (m 1, c 1 ),... (arbitrary) P (w 2 ) = (m 2, c 3 ), (m 1, c 3 ), (m 2, c 2 ),... (arbitrary) P (w 3 ) = (m 2, c 3 ), (m 3, c 3 ),... (arbitrary) P (c 1 ) = (m 1, w 1 ),... (arbitrary) P (c 2 ) = (m 2, w 2 ),... (arbitrary) P (c 3 ) = (m 1, w 3 ), (m 2, w 3 ), (m 1, w 2 ), (m 3, w 3 ),... (arbitrary). 5

6 Show that there is no stable matching in this example. Remark Observe that the preferences in this problem are separable into preferences over men, women, and children; that is, there are no preferences like (m, w, c) is preferred by m to (m, w, c ), but (m, w, c ) is preferred to (m, w, c). 17. Many-to-one matching Consider a set of firms and a set of workers. Each worker can work for at most one firm and has preferences over those firms he is willing to work for. Each firm can hire as many workers as it wishes and has preferences over those subsets of workers it is willing to employ. It is clear what a matching is in this case, and a firm F and a subset of workers C block a matching µ if F prefers C to the set of workers assigned to it at µ, and every worker in C who is not assigned to F prefers F to the firm he is assigned by µ. Consider two firms and three workers with the following preferences: P (F 1 ) = {w 1, w 3 }, {w 1, w 2 }, {w 2, w 3 }, {w 1 }, {w 2 } P (F 2 ) = {w 1, w 3 }, {w 2, w 3 }, {w 1, w 2 }, {w 3 }, {w 1 }, {w 2 } P (w 1 ) = F 2, F 1 P (w 2 ) = F 2, F 1 P (w 3 ) = F 1, F 2. Find the five individually rational matchings without unemployment. Show that each of these matchings can be blocked by some matching pair. Check that any matching that leaves w 1 unmatched is blocked either by (F 1, w 1 ) or by (F 2, w 1 ); any matching that leaves w 2 unmatched is blocked either by (F 1, w 2 ), (F 2, w 2 ), or (F 2, {w 2, w 3 }). Finally, any matching that leaves w 3 unmatched is blocked by (F 2, {w 1, w 3 }). 6