Math 480                         Final Examination (150 points)                           Fall, 2000
        Due: No later than 5pm, Thursday December, 21, 2000
        **UNAUTHORIZED LATE WORK MAY BE PENALIZED.
        GROUND RULES: 1. You may consult a)your classmates, b)your notes, c)textbooks, and d) myself.
        2. You may not consult any other persons (student or faculty) than those allowed in rule 1.
        3. All collaborations and consultations should be acknowledged.
        4. Submitted work should reflect your own understanding.

        1. (30  points) A land property consists of three contiguous pieces: A square with corners (0,0), (1,0), (1,1), and (0,1); a triangle with vertices (0,1), (1,1) and (0,2) and a triangle with vertices (1,0), (1,1), and (2,1). Here's how the land will be divided between Rose and Colin.
        Rose will choose a vertical line V with an x coordinate in the interval [0,1]. At the same time Colin will choose an horizontal line H with y coordinate in the interval [0,1].
        Rose will be given all the land that is both above H and to the left of V as well as all the land that is both below H and to the right of V. Colin gets whatever is not given to Rose.
        As usual assume the Rose and Colin both wish to be given as much land as possible. Find the value of this game  and the optimal strategies for Rose and Colin.

        2. ( 30 points) Consider the two person matrix game with matrix:
         

        (1,2) (8,3)
        (4,4) (2,1)

         a. Consider the zero sum games consisting of the payoffs to Rose and Colin to find the security values of this game for Rose and Colin.
         b. Draw the payoff polygon for this game. and find the negotiation set for this game played cooperatively without transferable utility.
         c.  Find the Nash arbitration solution to this game based on using the security values for the status quo..

        3. ( 30 points) Consider the three person game with tranferable utility in which v(1)= 4, v(2)=v(3)=0; v(12)=5, v(23)=6, v(13)=7 and v(123)=10.
         a. Find the Shapley value of this game. [Show relevant work.]
         b. Find the core for this game  or explain why the core is empty.
         c. Show (prove) that {(4,6-x,x), where x is in [0,6]} is a stable set.

        4. ( 60 points) Read Section 31 in Straffin. Do exercise 1 showing all work and explaining your conclusions with enough detail for a reader of the section to follow your computations.