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You are planning a vacation. You have three options: go to London, go to New Delhi and go to Rome. You have been to London before; you have always enjoyed your time there but it is a very expensive city and you know that it would be an expensive vacation. You have never been to India and would be very excited to visit New Delhi; however, you heard that some friends got a stomach infection while visiting India and are worried that you might get one too. Finally, you have been dreaming about going to Rome ever since you watched the movie “The Gladiator” and would really enjoy visiting Rome, but you worry that it might be too hot and you really suffer in the heat.
The possible outcomes are:
z1 You have an enjoyable but expensive vacation in London
z2 You have an exciting time in New Delhi without health problems
z3 You are in NewDelhi, miserable because of a stomach infection
z4 You have an uncomfortable vacation in Rome because of the heat
z5 You have a great vacation in Rome with pleasant weather
(a) Write your decision problem in terms of states, outcomes and acts. A state should specify what the weather is like in Rome and whether or not you have a good immune response against stomach infections.
(b) Let H be the event that it is hot in Rome (so that its complement H is the event that it is not hot in Rome) and let S be the event that you have a strong immune system (so that S is the event that you don’t have a strong immune system). Write down H, H , S and S . [You don’t get a stomach infection if and only if you have a strong immune system.]
(c) Now assign probabilities to the states making the following assumptions: (1) data over the past 20 years suggests that the probability that it will be hot in Rome at the time you are considering is 1 , (2) your doctor told you that you would have a 25% 3 probability of catching a stomach infection in New Delhi, (3) H is independent of S and is independent of S .
(d) Using the probabilities under (c), represent each choice as a lottery over outcomes.
From now on, assume that the best outcome for you would be to vacation in New Delhi in good health; the second best outcome is a vacation in Rome with good weather; the third best is a vacation in London, the fourth best is an uncomfortable vacation in Rome in hot weather and the worst outcome would be to be in New Delhi with a stomach infection.
(e) Suppose that you satisfy the axioms of expected utility and you are indifferent between going to London and going to New Delhi. Determine, as much as possible, the values of your von Neumann-Morgenstern utility function.
(f) What questions would you have to ask yourself in order to determine all the values of your von Neumann-Morgenstern utility function?
(g) Suppose that the answers to those questions are: 0.9 and 0.6. Where will you spend your vacation?
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