# Lotteries

Generally speaking, we talk about preferences over *certain* quantities of goods, or amounts of money. But the world isn’t a certain place: chance determines a lot of outcomes.

Some chance we bring upon ourselves: we play the lottery, or we bet on the outcome of sports games, or we invest in a stock that could go up or down. Some chance occurrences are called “acts of God” - whether you get into a car accident, or your house burns down due to a freak accident.

A great deal of economic activity is centered around transferring risk from one person to another. Insurance contracts pay you money if something bad happens to you; hedge funds invest in securities that are negatively correlated with one another. Understanding how *preferences over risk* drive these markets is one of the critical tasks of modern economic theory.

To analyze situations like this, let’s define a “lottery” as a set of possible outcomes, each occurring with a certain probability. For example, suppose we bet $€150$ on a coin toss; heads I win, tails you win. There are two possible outcomes: the coin could come up heads, and you would lose $€150$, or it could come up tails, and you could win $€150$. Each occurs with probability $\frac{1}{2}$.

Suppose, as you consider whether to take this bet, you have $€250$ in your pocket. Therefore, from your perspective, this lottery would give you an outcome of $c_1 = €100$ if the coin comes up heads, and $c_2 = €400$ if it comes up tails. Of course, you could reject the bet, and have $c_1 = c_2 = €250$ regardless of whether the coin comes up heads or tails.

We can picture this lottery in “good 1 - good 2 space,” where “good 1” (or “state 1”) is consumption in the state of the world in which the coin comes up heads (written $c_1$), and “good 2” (or “state 2”) is consumption in the state of the world in which the coin comes up tails (written $c_2$).

Should you take the bet? It depends on your preferences. We can draw indifference curves through the point $(100,400)$. If the “don’t bet” point is preferred, then you shouldn’t take the be; on the other hand, if the “bet” point is preferred, you should take the bet:

Let’s think about this another way. Let’s assume that the way you feel about money doesn’t depend on whether you win the bet or not: that is, you have some utility function $u(c)$ which says how much utility you get from having $c$ dollars, and that this function is independent of the state of the world.

If you *do not* take the bet, therefore, your utility is just $u(250)$. If you *win* the bet, your utility would be $u(400)$; if you *lose*, your utility will be $u(100)$.

Given this framework, your utility gain from winning the bet is $\textcolor{#31a354}{u(400) - u(250)}$, and your utility loss from losing the bet would be $\textcolor{#d62728}{u(250) - u(100)}$. Since each of the two outcomes is equally likely, you should therefore take the bet if \(\textcolor{#31a354}{u(400) - u(250)} > \textcolor{#d62728}{u(250) - u(100)}\) Let’s see when this is the case. The following diagram shows a particular kind of utility function where $u(c) = c^r$. The horizontal axis shows consumption, in dollars; the vertical axis shows utility, in “utils.” Initially it shows the case where $r = 0.5$, but you can use the slider to change $r$ to be anything from 0.25 to 2.

As you can see, for this particular case, it’s better to take the bet if $r > 1$, and better not to take the bet if $r < 1$. In order to think about this more generally, though, we need to introduce the notion of *expected utility*.