Your mathematics test is tomorrow and will cover the following topics: game theo
ID: 3256244 • Letter: Y
Question
Your mathematics test is tomorrow and will cover the following topics: game theory, linear programming, and matrix algebra. You have decided to do an "all-nighter" and must determine how to allocate your eight hours of study the among the three topics. If you were to spend the entire eight hours on any one of these topics (thus using a pure strategy), you feel confident that you would earn a 90% score on that portion of the test but would not do so well on the other topics. You have come up with the following table, where the entries are your expected scores. (The fact that linear programming and matrix algebra arc used in game theory is reflected in these numbers.) You have been told that the test will be weighted as follows: game theory: 25%; linear programming: 50%; matrix algebra: 25%. (a) If you spend 25% of the night on game theory, 50% on linear programming, and 25% on matrix algebra, what score do you expect to get on the test? (Round your answer to the nearest percent.) Is it possible to improve on this by altering your study schedule? Yes, you can improve on this strategy by studying only game theory. Yes, you can improve on this strategy by studying only linear programming. Yes, you can improve on this strategy by studying only matrix algebra. Yes, you can improve on this strategy by using a different mixed strategy. No, it is not possible to improve on this strategy. If it is possible, what is the highest score you can expect on the test? (If not possible, enter IMPOSSIBLE.) (c) If your study schedule is according to part (a) and your teacher decides to forget her promises about how the test will be weighted and instead base it all on a single topic, which topic would be worst for you, and what score could you expect on the test? The worst topic would be and you could expect to earn on the test.Explanation / Answer
Solution:
a) If you spend 25% of the night on the game theory, then your score will be
=25/100 * (90+40+70)
=25/100 * 200
=200/4
=50
If you spend 50% of the night on linear programming, then your score will be
=50/100 * (60+90+50)
=50/100 * 200
=200/2
=100
If you spend 25% of the night on matrix algebra, then your score will be
=25/100 * (80+30+90)
=25/100 * 200
=200/4
=50
Expected average score will be
=50+100+50 / 3
=200 / 3
=66.667
Rounding off to two significant digits to get 67.
Thus your expected score on the test is 67%
b) From the above payoff matrix, we can observe that studying game theory for the whole night you can improve your scores.
The highest score would be
=(25/100)*90+(50/100)*60+(25/100)*80
=2250/100 + 3000/100 + 2000/100
=22.50+30+20
=72.50
Thus the expected highest score is 72.50%
c) If the study schedule is according to part (a) and your teacher made the test based on the single topic, then score would be as follows
If the teacher made the test based on game theory, then the score will be
=90 + 40 +70
=200
If the teacher made the test based on linear programming, then the score will be
=60 + 90 +50
=200
If the teacher made the test based on matrix algebra, then the score will be
=80 + 30 + 90
=200
All the scores are same in three topics so we can say that there is no worst topic.
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