Lecture notes 3

Lecture notes 3 - Economics 112 Lecture Notes Lecture 3...

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Copyright David Scoones 2010 Page 1 Economics 112 Lecture Notes Lecture 3. Probability and Expected Utility Something of an aside to consider how the role of luck, chance, and deliberate randomization are modeled in game theory. This allows us to consider both natural uncertainty and mixed strategies . Key terms: uncertainty, probability, expected value, “nature”, risk aversion, expected utility 1. Probability Probability is surprisingly hard to define a general way, if we are seeking the “meaning” of the concept. It seems clear that the probability of an event says something about how certain we should be to see it occur (but notice that saying it is the “likelihood” or “chance” of the event just substitutes new words into the same puzzle). In some circumstances things are fairly clear. The easiest case is for events that are the outcome of a trial that can be repeated, in principle at least, forever. For example, consider the toss of a coin. The probability of a head, then, can be sensibly defined as the limiting frequency of heads, i.e. the proportion of heads in a very long series of repeated trials. If the coin is “fair” the share of heads should approach ½ as the number of repetitions grows. But other times probability is used to refer to beliefs about intrinsically unrepeatable events, or at least for which repeated trials are hard to imagine. In still other uses, it seems to be a logical claim about the strength of evidence. Whatever meaning we ascribe to it, it is easier to agree that in its operational definition probability must adhere to certain rules. Operationally, probability refers to attaching numbers, or “weights” to events to describe their likelihood of occurring. First, it makes no

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Copyright David Scoones 2010 Page 2 sense to talk of there being “less than zero” chance that some particular event will occur, so the weights must all be positive. Second, if the set of events is well defined, something must happen,
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Lecture notes 3 - Economics 112 Lecture Notes Lecture 3...

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