Decision Problems are Everywhere
Renting vs Buying a Movie
Suppose you want to watch a movie.
You are presented with two options to watch it: rent or buy.
Which do you choose?
Is there a way to determine which choice is best?
In order to determine which choice is best, you must determine how much utility a choice will bring you.
Utility is a means to quantify the benefit you will get from choosing to do something.
In the case of renting versus buying, it might be more natural to start by thinking of cost.
The remainder of this post adopts the following notation:
Let \(i = 1\) represent the choice to rent the movie and let \(i = 2\) be the choice to buy it.
Let \(n\) be the number of times you will watch the film.
Let \(c_i(n)\) be the cost of watching the movie \(n\) times given you have chosen option \(i\).
Let \(u_i(n) = -c_i(n)\) be the utility of watching the movie \(n\) times given you have chosen option \(i\).
You can see from the above notation that utility is the negative of cost, and cost is the negative of utility.
Consider the following concrete example:
Suppose the cost to rent the movie once is \(\$10.99\) and the cost to buy the movie is \(\$19.99\).
In symbols: \(u_1(1) = -10.99\) and \(u_2(1) = -19.99\).
Suppose that after renting and watching the movie you must pay the rental fee again to watch it again.
This relationship can be represented with the following equation: \(u_1(n) = n\times u_1(1)\).
In this specific example you have \(u_1(n) = n\times-10.99\)
In words: the utility of renting and watching the movie \(n\) times is \(n\) times the utility of renting and watching it once.
Alternatively, you own the movie if you buy it.
This means that there are no new costs associated with rewatching the movie.
This relationship can be represented with the following equation: \(u_2(n) = u_2(1)\).
In this specific example you have \(u_2(n) = -19.99\).
The example should have made clear 3 points:
The utility of the choice is the negative of its cost.
The utility of renting a movie is proportional to the number of times you watch the movie.
The utility of watching a movie is constant regardless of the number of times you watch the movie.
Probability and Quantifying Uncertainty
The Law of Large Numbers
Maximum Expected Utility