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Expected Utility

in this lecture we're going to explore how some formal methods borrowed from decision theory might enhance our understanding of the discoveries about action
we're also going to explore, conversely, how those discoveries might bear on our use of these formal models.
Although this is going to seem quite dry at the beginning, when i taught this course in the past this was one of the areas that most interested people. So do bear with me.
Decision Theory is about how individuals decide which of several available actions to perform (Jeffrey, 1983, p. textbook:).
so I say ‘decision theory’ but sometimes people talk about ‘rational choice theory’. Same thing.

decision theory

How do rational agents decide which of several available actions to perform?

simple question: How do rational agents decide which of several available actions to perform ?
what i want to do in this section is just to introduce you to the kinds of representation which are used But it's important first of all to talk a little bit about the motivation the people who study decision theory ...

‘we wish to find the mathematically complete principles which define “rational behavior” for the participants in a social economy, and to derive from them the general characteristics of that behavior’

(Neumann, Morgenstern, Rubinstein, & Kuhn, 1953, p. 31).

von Neumann & Morgenstern, 1953 p. 31

so here we have a super lofty aim and of course one of our questions will be to what extent that is possible and what assumptions they're relying on
Maybe not everyone who uses their mathematical principles shares the founders’ ambitions. Wombles abound.
so as i said we're going to start with trying to understand how actions are represented from the point of view of somebody who wants to go on and construct decision theory. what we're doing now is just a simple background what does the representation look like. let's be sure we understand it

decision theory

How do rational agents decide which of several available actions to perform?

game theory

When two or more agents interact,
so that which outcome one agent’s choice brings about depends on how another chooses,
how do their preferences guide their choices?

Expected utilities can be derived from desirabilities and probabilities. One procedure for choosing an action is to compute expected utilities (in the way illustrated) and then perform the action with the highest expected utility.

I don’t understand expected utility

I can illustrate it using gambling

A True Story

about How to Profit from Bias
First illustration is a simple bet.
Which gamble would you prefer? If you are like me, you are interested in the expected gain ...
Second illustration is about which of two bets you should select if you can only choose one.
OK so that was an illustration with money.
I used money to simplify the idea of expected utility.
But what we are really interested in is preferences.

expected utility
is a bit like
the value of a bet

This is because your preferences do not necessarily vary linearly with monetary amounts. Once you have the first £100, the next £100 might not make nearly as much difference to you and so it might be rational to care a lot less about whether you have it.

Terminology

actions have outcomes

(Jeffrey: outcome = ‘consequence’)

which outcome an action causes depends on **conditions**

subjective probabilities attach to conditions.

preferences rank outcomes

expected utilities attach to actions.

(Jeffrey: expected utility = ‘estimated desirabilities’)

Jeffrey (1983)

Make sure you understand the terminology and can relate it to the scenario.
Be sure to use the terminology consistently, and with precision, in your writing.
It is important to understand that we haven’t done anything more than introduce a way of representing things.

so far: the representation

next step: the theory

We have not yet considered a theory. But understanding the representation is an essential first step towards von N & Mo’s objective.

‘we wish to find the mathematically complete principles which define “rational behavior” for the participants in a social economy, and to derive from them the general characteristics of that behavior’

(Neumann et al., 1953, p. 31).

von Neumann & Morgenstern, 1953 p. 31

Why is what we have done so far not sufficient?
(Because we have not anchored preference or subjective probability to subjects.)