• Target Audience

• Learning Objectives

• 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?

• Utility

• 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.

• Switching Point

• Probability and Quantifying Uncertainty

• Geometric Distribution

• Expectations

• The Law of Large Numbers

• Maximum Expected Utility

• Summary