Copyright David Scoones 2010
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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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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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 Winter '08
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 Game Theory, Utility, probability density function, Copyright David Scoones

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