UtilityAndProbability - Probabilities Utilities and...

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Probabilities, Utilities and Decision Making 1 Probabilities 1.1 Example in this course We have encountered probability theory, and have seen how it takes the idea of elementary events, counts them up,, and produces measures called probability. We have applied this to figuring out whether a thief stole socks from one person or another, based on the color of the socks. And to figuring out which island a drunken bird has land on, from the color of the grass. Most relevant to this course, we have used it to figure out which of several documents is most likely to be relevant for a particular query, based on the words in the query, and the words in the document (Language Models). 1.2 Origins in gambling But probability ahs many other applications, some if which are very important. Probability originated with the efforts of gamblers to decide how to bet. For example, betting on a “7” when two dice are rolled is a much safer bet than betting on a “3” because there are more ways it can occur. 1.3 Utility: Von Neumann and Morganstern About 70 years ago, the mathematician John von Neumann, and the economist, Oskar Morgenstern worked together to put the ideas that gamblers used into a form that would work for many other kinds of problems. To do this they had to introduce a new idea: “utility”. In ordinary language utility means several things (electricity, gas, water) and also “usefulness”. [Utility programs on the computer are so named because they are useful.]. This sense of “usefulness” is closest to what the economists mean. Let’s see why they introduce it. 2 Expected value of uncertain alternatives Consider two possible alternatives: (A) you definitely get one dollar. (B) with probability 50% you get two dollars, and with probability 50% you get nothing. On average, how much money is each of these worth, per play? To figure that out we consider how much money we get in each case, multiply it by the probability, and add them up. So the calculations look like this Case A Outcomes 1 2 3 4 5 Probability 100% Value 1.00 Product 1.00 0.00 0.00 0.00 0.00 1.00 So, not surprisingly, this sure thing is worth $1
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But the second alternative, Case B, is, on the average, also worth one dollar. Case B Outcomes 1 2 3 4 5 Probability 50% 50% Value 0.00 2.00 Product 0.00 1.00 0.00 0.00 0.00 1.00 We call the value computed by multiplying value by probability, and summing them up, the expected value of the alternative. A gambler must take risks, but he wants the expected value to be in his favor. And, according to this principle, he is just as happy with case B, as with case A. Because the expected value is the same 3 Problems with expected monetary value But this principle may not be an accurate or complete description of his behavior. According to expect value, he should prefer another alternative (Case C) to either of these, because it is worth (in terms of expected value) 40 cents more, and “every little bit helps”.
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This note was uploaded on 02/20/2012 for the course 790 373 taught by Professor Boros during the Fall '09 term at Rutgers.

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UtilityAndProbability - Probabilities Utilities and...

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