Utility
Overview
A utility function is a method for modeling decision situations based on the
axioms of preference. At its simplest formulation,
it does not add anything to the choice thoery given in the preference axioms, however, it does become essential when considering
risky outcomes.
Utility Function
A utility function is a mapping between
options and real numbers. That is, a number is assigned
to each choice, and is called the utility of that choice. The numbers are assigned in such a way that if once choice is preferred
over another, then its utility is a larger number than the less preferred choice.
For a simple utility function, there is no interpretation attached to the specific number assigned to a choice. That is, if one choice is
assigned a utility of 10 while another is assigned a utility of 5, there is no interpretation attached to the fact that 10 is twice 5. All
one can say is that the 10 option is preferred to the 5 option.
It should be clear that a utility function defined this way is not unique. That is, there are several possible functions that each are
equal in ability to model a given agents preferences. In fact, given one such utility function, another valid function can be obtained
by applying a positive monotonic transformation to the utility function.
Modeling Uncertainty
The basic theory of preference an utility outlined above is set in a world with no uncertainty. That is, the available options
from which to choose are all known with certainty. In the real world, many choices involve an element of uncertainty
(or risk). Typically this is the case when the outcome of a choice represents some outcome in the future. For example,
the outcome of purchasing an asset is not known for certain until the future unfolds.
To model risk, the options become bundles of outcomes, each outcome assigned a probability. As a simple example,
consider playing a game where guess the outcome of a coin toss. The outcome can be either heads or tails. If you guess
correctly, you get $1, otherwise nothing.
We can represent the outcomes as a vector, such as
{%
\begin{bmatrix}
Heads \\
Tails \\
\end{bmatrix}
%}
The outcomes listed in the outcome vector are often referred to as states.
Then the outcomes are then represented by two other vectors.
{%
\begin{bmatrix}
1 \\
0 \\
\end{bmatrix}
%}
if heads is chosen
and
{%
\begin{bmatrix}
0 \\
1 \\
\end{bmatrix}
%}
if tails is chosen.
Here, the choices are a list of outcomes, each assigned a probability.
Utility and Uncertainty
When dealing with uncertainty, utility functions are constructed with additional constraints to the simple utility functions above.
Utilities are still assigned to known outcomes. For example, a utility will be assigned to receiving a dollar (as in the case of winning
the bet above) or having 0 dollars (as in losing the bet). But now, the utility function is constructed such that the agent
will choose the choice that represents the largest expected utility, or
{% \mathbb{E}[U(s)] = \sum_{i} p_i \times U(s_i) %}
Here, each state is represented by a variable {% s_i %}
