M
ASSACHUSETTS
I
NSTITUTE OF
T
ECHNOLOGY
15.053 – Optimization Methods in Management Science (Spring 2007)
Problem Set 6
Due April 4
th
, 2007 at 4:30 pm.
You will need 160 points out of 188 to receive a grade of 5.
Problem 1: RockPaperScissors Play (15 Points, 5 Each)
As indicated in lecture, rockpaperscissors is a classic problem studied in game theory.
Part A:
Play 10 to 25 rounds of the game with your partner and record your outcome. If you do
not have a partner, play with a friend or if you get really desperate you can play with an
online opponent at
http://www.playrps.com/
.
Part B:
The winner of the game should answer the following questions: Was your win luck or
strategy/skill? If you used a strategy, what was it and why do you think it was
successful?
Fun Trivia:
In Japan there are two variations of RPS:
Snake, frog , and slug
Warrior, tiger, and warrior’s mother
There is also an Indonesian variation as well:
Elephant, human, and ant.
Part C:
Write down the payoff matrix for the game.
Next week in Recitation we will have a rock, paper, scissors
tournament and find out who the ultimate champion is. Practice
with all the “friends” you meet over spring break.
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Problem 2: Game Theory, LPs and Excel (45 Points)
Consider the following payoff matrix. The numbers represent payoffs to the row player
R.
2
2
1
5
2
1
0
2
1
0
1
1
Part A: (5 Points)
Formulate a linear program that will find an optimal mixed strategy for the row player.
Call this R’s problem.
Part B: (5 Points)
Formulate a linear program that will find an optimal mixed strategy for the column
player. Call this C’s problem
Part C: (10 Points)
Solve both R’s problem and C’s problem in Excel. What are the optimal solutions?
Format and turn in your spreadsheet and show your work.
Part D: (10 Points)
How do the two answers relate to each other? Is this always the case? Explain your
answer using duality theory.
Part E: (5 Points)
What is the payoff to the row player if the row player chooses the optimal strategy and
the column player decides to choose column 4?
Part F: (10 Points)
Is it possible to change one entry such that either the column player or row player has
multiple optimal strategies?
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 Spring '07
 JamesOrli
 Management, Graph Theory, payoff matrix, optimal mixed strategy, S. Turkey Tim

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