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The University of British Columbia Final Examinations - April 2007
Mathematics 340–
Closed book examination Time: 2.5 hours
Name Signature
Student Number Instructor’s Name
Section Number
Special Instructions:
Calculators, notes, or other aids may not be used. Answer questions on the exam. Note that the exam is two-sided! A note sheet is provided with the exam. You have two extra pages at the back for additional space.
Rules governing examinations
- Each candidate should be prepared to produce his library/AMS card upon request.
- Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expi- ration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those au- thorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
- Smoking is not permitted during examinations.
Total 92
Marks
[8] 1. Consider a zero sum game with M being player A’s payout matrix.
(a) Show that for any matrix M and probability vectors α, ~ ~β (of appropriate dimen- sion) we have ScreamA(~α) ≤ ~αTM β~ ≤ ScreamB (β~) (1).
(b) Assume equation (1) always holds. Show that if
ScreamA(~α) = ScreamB (β~) and ScreamA(α~′) = ScreamB (β~′)
then ScreamA(~α) = ScreamA(~α′)
[20] 3. Consider the problem maximize x 1 + 2x 2 subject to x 1 + x 2 ≤ 1 and x 1 , x 2 ≥ 0.
(a) Write the dual LP.
(b) Consider a proposed optimal solution x 1 = 1 and x 2 = 0. Use complementary slackness to find the dual solution to this proposed solution. Does the dual solution verify that the propsed solution is optimal?
(c) Same questions as in part (b) for a proposed optimal solution x 1 = 0 and x 2 = 1.
(d) Consider the problem maximize x 1 + 2x 2 subject to x 1 + x 2 ≤ 1 and x 1 + x 2 ≤ 1 and x 1 , x 2 ≥ 0 (yes, this is essentially the same LP, except that the constraint x 1 + x 2 ≤ 1 appears twice). Write the dual LP. Will complementary slackness always yield a unique dual solution? Explain.
(e) Consider the LP maximize x 1 + x 2 subject to 2x 1 ≤ 1 and x 1 + x 2 ≤ 1 and 2 x 2 ≤ 1 and x 1 , x 2 ≥ 0. Will complementary slackness always yield a unique dual solution? Explain. (You are not required to write down the dual LP if you can argue convincingly without it.)
[12] 5. Consider the LP maximize x 1 + ax 2 subject to x 1 + x 2 ≤ 1 and x 1 + 2x 2 ≤ b and x 1 , x 2 ≥ 0, where a and b are real parameters. Introducing slack variables, x 3 , x 4 (x 3 for the first inequality), here is one possible dictionary:
x 1 = 2 − b − 2 x 3 +x 4 x 2 = b − 1 +x 3 −x 4 z = 2 − a − b + ab +(a − 2)x 3 +(1 − a)x 4
(a) For what range of a, b is the above dictionary optimal? Explain.
(b) Perform one regular simplex pivot to obtain a dictionary optimal when b = 3/ 2 and a = 3. Explain how you are choosing entering and leaving variables. For what range of a, b is this dictionary optimal?
(c) Perform one dual pivot to obtain a dictionary optimal when b = 3 and a = 3/2. Explain how you are choosing entering and leaving variables. For what range of a, b is this dictionary optimal?
[28] 6. Consider the matrix game associated to the matrix
M =
[
3 c
]
where c is a given real number. (a) Assuming that all pure strategies are involved in a unique equilibrium, what is player A’s equilibrium strategy, α~ = (α 1 , α 2 )?
(b) For what values of c is it not the case that all pure strategies are involved in an equilibrium? Use domination to explain what are the equilibria for those values of c.
(f) Consider an LP with an optimal solution in which every slack variable is zero and every decision variable is positive. Can there be more decision variables than slack variables? [Hint: in a dictionary, how many variables are basic?]
(g) Explain how part (e) and (f) can reduce any to LP to linear algebra provided that you can guess which primal and dual decision variables can be 0 in certain optimal (primal and dual) solutions. Explain what this has to do with a simi- lar observation made in class (and the notes) about matrix games that motives part (a) of this problem.
