800 023 01/04 Mathematics for Decision Making [Campbell] Homework Assignments

No Homework will be collected, but you should look at the following problems

[We shall only consider cakes which are 1/2 V and 1/2 C, so you may omit the 2nd and 5th problems (2/3 V and 1/3 C; ask me whether you are responsible for problems of the ilk of the 3rd problem.]

• A cake is 1/2 white and 1/2 chocolate. Adam likes white cake thrice as much as chocolate; Betsy likes chocolate cake quintice (may not be a word, but I like it) as much as white.
  1) Is cutting the cake down the middle (1/4 chocolate, 1/4 white each) Pareto optimal?
  2) Describe all Pareto optimal solutions.
  3) How much could Adam get if he knew Betsy's preferences and cut the cake (Betsy picks)?
  4) How much could Betsy get if she knew Adam's preferences and cut the cake (Adam picks)?
  5) what is the range of possible negotiated Pareto optimal divisions if they start with an equal division (1/4 chocolate, 1/4 white each)?
• A cake is 2/3 white and 1/3 chocolate. Adam likes white cake twice as much as chocolate; Betsy likes chocolate cake four times as much as white.
  1) Is cutting the cake down the middle (1/6 chocolate, 1/3 white each) Pareto optimal?
  2) Describe all Pareto optimal solutions.
  3) How much could Adam get if he knew Betsy's preferences and cut the cake (Betsy picks)?
  4) How much could Betsy get if she knew Adam's preferences and cut the cake (Adam picks)?
  5) what is the range of possible negotiated Pareto optimal divisions if they start with an equal division (1/6 chocolate, 1/3 white each)?
• Alfreds preference order for wooden blocks in an estate is ABCDEFGH, while Bonnie's preference order is DBFCHAEG.
  1) What is the resulting division if they alternate picks and Alfred picks first? If Bonnie picks first?
  2) If Alfred knows Bonnie's preferences, what is the best he can get if he picks first? If Bonnie picks first?
  3) If Bonnie knows Alfred's preferences, what is the best she can get if Alfred picks first? If She picks first?
• 1) What division of the 1/2 white 1/2 chocolate cake would result from the adjusted winner procedure reflectiong Adam's 3:1 preferance for white and Betsy's 5:1 preference for chocolate?
  2) How much could Adam get if he modified his "preference" based on knowing Betsy's preference?
  3) How much could Betsy get if she modified her "preference" based on knowing Adam's preference?
• 1) What division of the 2/3 white 1/3 chocolate cake would result from the adjusted winner procedure reflectiong Adam's 2:1 preferance for white and Betsy's 4:1 preference for chocolate?
  2) How much could Adam get if he modified his "preference" based on knowing Betsy's preference?
  3) How much could Betsy get if she modified her "preference" based on knowing Adam's preference?
• A fruit tray contains Applesauce, Blueberries, Cherries, Dates, and Figs. Sally's relative preferences are given by the following breakdown of 100 points: A-30, B-13, C-17, D-22, F-18. Theodore's relative preferences are given by the following breakdown of 100 points: A-30, B-40, C-7, D-13, E-10.
  1) What is the resultant adjusted winner partition?
  2) If Sally knew Theodore's preferences and modified her point allotment, how much could she get?
  3) If Theodore knew Sally's preferences and modified his point allotment, how much could he get?
  5) If none of the five servings of fruit could be divided, what is the most equal partition?
• An estate has chalets in France, Germany, Hungary, and Italy. The heirs Alfred, Beatrice, and Colin value the Chalets as A: F-250, G-150, H-150, I-200; B: F-200, G-200, H-180, I-180; C: F-225, G-150, H-100, I-150 (all values are in Euros).
  1) What is the result of the Knaster inheritance procedure for these values.
  2) How much more money could Colin get if he adjusted his bids based on knowing the other heirs' bids?
  3) How much more money could Alfred get if he adjusted his bids based on knowing the other heirs' bids? (This may be manifested as a smaller payment rather than larger income.)
  
• Imagine a track meet between Argentina, Brazil, Chile, Denmark, England and France, which has eight events. The orders of finish in those events are (each row is an event):
  ABDACBEFED
  BAEFBFECCD
  EAFDFBACCD
  DABCEFCFBA
  FFDEBACCBF
  CAAEBFDFAB
  DABFCCEEBF
  EABCDFFBCE
  1) What is the result of this meet using 6-4-3-2-1 scoring?
  2) Who would win a dual meet between England and France extracted from this meet? (5-3-1 scoring)
  3) who would win a dual meet between Argentina and Brazil extracted from this meet? (use 5-3-1 scoring)
• Assume the preferential rankings for 5 candidates by 12 voters are:
  ABCDE
  ACBED
  ACEBD
  AECDB
  EBACD
  BACDE
  BCEAD
  CEBAD
  DECBA
  EDCBA
  BCEDA
  CEDAB
  1) Who would win a plurality election?
  2) Who would win a run-off between the top two?
  3) Who would win by the Hare method (sequential run-off)
  4) Who would win employing the Borda count?
  5) Is there a Condorcet winner? (who?)
  6) How could the two voters whose first choice is E vote in a plurality election to obtain a more favorable outcome?
  7) How could the two voters who favor C vote in an election determined by a runoff between the top two vote getters to obtain a more favorable outcome?
  
• The populations (in thousands) and number of assigned representatives for NY, NJ, PA, MD and DE were NY:332,6; NJ:180,4; PA:433,8; DE:56,1; and MD:279,6.
  1) Which state is most overrepresented as measured by ai - qi?
  2) Which state is most overrepresented as measured by ai/qi?
  3) Is there a fairer allocation of the seats as measured by ai-qi?
  4) Is there a fairer allocation of the seats as measured by ai/qi?

July 09
campbell@math.uni.edu

University of Northern Iowa is an equal opportunity educator and employer witha comprehensive plan for affirmative action.