11.1 Lotteries, Expected Value and Expected Utility
Many economic decisions are made under conditions of risk: individuals do not know with certainty the future consequences of their actions, but can only estimate their probabilities. How can an individual make rational choices without certainty about the outcome of their decision? To analyze such contexts, we need a new tool: the theory of decision-making under uncertainty, which seeks to answer questions such as:
- Whether or not to insure a car against theft;
- How much to invest in risky assets and how much in safe ones;
- Whether to accept a job contract in which the wage depends not only on individual effort, but also on external factors;
All these situations have in common the fact that the individual must choose between risky alternatives: options that lead to different possible outcomes, each with some probability.
Lotteries
In microeconomics, we represent such alternatives mathematically using the concept of a lottery, defined as:
- a set of possible outcomes (payoffs), often represented as monetary amounts;
- a probability distribution Probabilities are numbers between 0 and 1 and must sum exactly to 1. over outcomes, reflecting how likely the individual considers each of them.
We now present two examples showing how choices under risk can be precisely represented using lotteries.
Example 1 – Insurance
Consider the owner of a car worth $10000$ euros. Suppose the probability of the car being stolen is 2%, i.e. $0.02$, For example, statistically, out of one hundred similar cars, about two are stolen each year. and the probability that it is not stolen is therefore 98%, or $0.98$. The “no insurance” alternative is represented by the following lottery:
| Payoff (euros) | Probability |
|---|---|
| 0 | 2/100 |
| 10,000 | 98/100 |
The alternative “purchase full insurance coverage by paying a premium of $500$ euros” corresponds to a lottery in which there is a single certain outcome, with a payoff of $9500$ euros:
| Payoff (euros) | Probability |
|---|---|
| 9,500 | 1 |
Other alternatives, such as insuring only part of the car’s value or purchasing a policy with a deductible, can also be modeled as lotteries. For example, partial insurance covering half of the value, purchased at a price of $200$ euros, corresponds to the following lottery:
| Payoff (euros) | Probability |
|---|---|
| 4,800 | 2/100 |
| 9,800 | 98/100 |
Example 2 – Investment
A small entrepreneur has 1000 euros to invest in opening new retail locations. They can choose between two locations, $A$ and $B$, both characterized by high potential returns but also uncertainty. Each store has two possible outcomes: in case of success, the return is three times the investment; in case of failure, the entire investment is lost. The probability of success is 50% for each location, and the outcomes in the two locations are independent.
The entrepreneur is evaluating two strategies. The first is to open a large store in one location only, say $A$, investing the full amount there. This choice corresponds to the following lottery:
| Payoff (euros) | Probability |
|---|---|
| 0 | 1/2 |
| 3000 | 1/2 |
The second strategy is to split the investment between $A$ and $B$. The entrepreneur opens two smaller stores, one in each location, investing $500$ euros in each. The possible payoffs Since the outcomes in the two stores are independent, each of the four possible (and mutually exclusive) combinations of success and failure has a probability of $1/4$. The two combinations “$A$ fails, $B$ succeeds” and “$B$ fails, $A$ succeeds” both yield a payoff of $1500$. The probability that one or the other occurs is therefore $1/4 + 1/4 = 1/2$. depend on the combination of outcomes:
| Event | Payoff (euros) | Probability |
|---|---|---|
| Both fail | 0 | 1/4 |
| One succeeds, the other fails | 1500 | 1/2 |
| Both succeed | 3000 | 1/4 |
Expected Value
Once a risky alternative has been represented as a lottery, we can ask what a summary measure of its “value” might be. A natural first answer is the expected value of the lottery, i.e., the weighted average of its payoffs, where the weights are the probabilities associated with each outcome. We will denote this expected value by the symbol $EP$, and call it the expected payoff of the lottery. If a lottery offers $n$ possible outcomes $W_1,W_2,\dots,W_n$ with corresponding probabilities $p_1,p_2,\dots,p_n$, the expected payoff is:
\(\begin{gathered} EP = p_1W_1+p_2W_2+\cdots+p_nW_n \end{gathered}\)
In Example 1, the lottery corresponding to “no insurance” has an expected value of $(2/100) \times 0 + (98/100) \times 10000 = 9800$. In Example 2, it is easy to verify that both strategies yield the same expected payoff: $1500$ euros.
The expected payoff of a lottery summarizes in a single number the list of possible outcomes and their probabilities, providing an objective measure of its value. However, it often fails to explain individuals’ actual behavior. Returning to the entrepreneur in Example 2, many would prefer the second strategy over the first, even though both have the same expected value. The concentrated strategy is “all or nothing”: either the entrepreneur triples its capital or loses it entirely. The diversified strategy, on the other hand, reduces the probability of extreme outcomes ($0$ or $3000$ euros) and increases the chance of a partial return (1500 euros).
Indeed, many people prefer a lottery with a lower expected value in exchange for greater certainty. In Example 2, for instance, some might prefer not to open any store and instead keep the initial $1000$ euros. Similarly, in Example 1, many would choose a sure outcome of $9500$ (i.e., pay $500$ for full insurance) over a lottery with expected value $9800$ but a 2% chance of losing everything (no insurance).
This type of behavior reflects a preference for certainty — which we will call risk aversion — that the expected value concept alone cannot capture. To describe such preferences, we need to introduce a new concept.
Expected Utility
How can we explain the preferences of an individual who, in order to avoid or reduce uncertainty, chooses an alternative with a lower expected value? Why, in Example 1, might someone prefer to insure, even if the insurance has a lower expected payoff? Why, in Example 2, might an entrepreneur prefer to diversify or even not invest, even when that means giving up a higher expected value?
The answer is that individuals do not evaluate alternatives based solely on monetary payoffs, but based on the subjective satisfaction associated with each payoff. In other words, what matters is not the expected value of money, but the expected value of the utility that money brings.
To model this behavior mathematically, we introduce a utility function over wealth, denoted by $u(W)$, which assigns a numerical value to each level of wealth $W$, representing the satisfaction it generates. Choices among risky alternatives are then evaluated by computing the expected utility of each alternative: the average of the utilities of the outcomes, weighted by their probabilities.
In these notes, we will restrict ourselves to utility functions of the form:
\(\begin{gathered} u(W) = W^\pi \end{gathered}\)
where $\pi > 0$ is a parameter that, as we will see, reflects the individual’s propensity toward risk. If a lottery assigns probabilities $p_1,p_2,\ldots,p_n$ to monetary outcomes $W_1,W_2,\ldots,W_n$, then its expected utility is:
\(\begin{gathered} EU = p_1W^\pi_1+p_2W^\pi_2+\cdots+p_nW^\pi_n \end{gathered}\)
An individual’s attitude toward risk can fall into one of three categories, and this attitude is reflected in the shape of the utility function. An individual is said to be:
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Based on what we see in the video, we can say that the contestant is ...
risk averse if they prefer receiving the expected value of a lottery with certainty rather than facing the lottery itself; that is, if $u(EP) > EU$. This occurs if and only if the utility function is concave, i.e., if $\pi < 1$. - risk neutral if they are indifferent between the lottery and receiving its expected value with certainty; i.e., if $u(EP) = EU$. This happens when the utility function is linear, i.e., when $\pi = 1$.
- risk loving if they prefer facing the lottery to receiving its expected value with certainty; i.e., if $u(EP) < EU$. This occurs if and only if the utility function is convex, i.e., if $\pi > 1$.
The following graph illustrates the concept of expected utility and shows how different shapes of the utility function (i.e., different values of $\pi$) correspond to different attitudes toward risk. In the graph, we assume the individual is facing a lottery with two possible outcomes, $W_1$ and $W_2$, with respective probabilities $p_1$ and $p_2$ ($=1-p_1$).
Certainty Equivalent and Risk Premium
Not all risk-averse individuals are alike, and the same is true for risk-loving individuals. As we will now illustrate, the higher the parameter $\pi$, the more the individual is willing to accept risk. In other words, a larger value of $\pi$ corresponds to less risk aversion (or greater risk propensity). To understand this, let us return to the previous examples.
In Example 1, consider the alternatives “no insurance” and “full coverage at a cost of $500$ euros,” whose respective expected payoffs are $9800$ and $9500$. A risk-averse individual with utility function $u(W)=W^{0.5}$ prefers not to insure, because
\(\begin{gathered} 0.98 \times 10000^{0.5} + 0.02 \times 0^{0.5} > 1 \times 9500^{0.5} \end{gathered}\)
This person is risk averse, but not so much as to give up $300$ euros of expected payoff just to eliminate the risk.
Another individual, still risk averse but with utility function $u(W)=W^{0.3}$, instead prefers to insure, because
\(\begin{gathered} 0.98 \times 10000^{0.3} + 0.02 \times 0^{0.3} < 1 \times 9500^{0.3} \end{gathered}\)
This second individual is more risk averse than the first.
In Example 2, let us compare the alternative “open a single large store” (expected payoff $1500$ euros) with “do not invest” (expected payoff $1000$ euros). An individual with $u(W)=W^{0.5}$ prefers not to invest:
\(\begin{gathered} 0.5 \times 0^{0.5} + 0.5 \times 3000^{0.5} < 1 \times 1000^{0.5} \end{gathered}\)
An individual with $u(W)=W^{0.8}$, on the other hand, prefers to invest:
\(\begin{gathered} 0.5 \times 0^{0.8} + 0.5 \times 3000^{0.8} > 1 \times 1000^{0.8} \end{gathered}\)
In other words, for the first individual ($\pi=0.5$), the higher expected payoff ($500$ euros more) is not enough to justify taking on risk. For the second individual, who is less risk averse, it is.
At this point it should be clear that an individual’s attitude toward risk is reflected in the comparison between the expected payoff of a lottery and the amount they would be willing to accept with certainty instead. This leads us to introduce two important concepts that will be useful later.
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The certainty equivalent of a lottery is the amount of sure money that makes the individual indifferent between receiving it and facing the lottery. We denote this value by $CE$; by definition, it satisfies:
\( u(CE) = EU \)
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The risk premium is the difference between the expected payoff of the lottery and its certainty equivalent:
\( RP = EP - CE \)
It measures how much the individual is willing to give up, in terms of expected payoff, to eliminate the risk entirely.
For a risk-averse individual, the certainty equivalent of a lottery is always less than its expected payoff: $CE < EP$, and thus $RP > 0$. The individual is willing to give up part of the expected value in exchange for certainty. By the definition of certainty equivalent, we have \(CE=(p_1W^\pi_1+\cdots+p_nW^\pi_n)^{1/\pi}\) It can be shown (although we will not prove it) that the derivative of this expression with respect to $\pi$ is positive.
Moreover, the more risk-averse the individual, the lower the certainty equivalent and the greater the risk premium. Intuitively, a more risk-averse person is less tolerant of unfavorable outcomes and therefore assigns a lower certainty equivalent to the lottery (their $CE$ is smaller). Put differently, they require greater compensation to accept the uncertainty (their $RP$ is larger).
For a risk-neutral individual, $CE = EP$ and therefore $RP = 0$. For a risk-loving individual, we have $CE > EP$ and thus $RP < 0$: the individual is willing to accept a lower expected payoff in exchange for facing risk.
The graph below illustrates these two concepts, assuming the individual faces a lottery with possible outcomes $W_1 = 0$ and $W_2 = 10000$, with respective probabilities $p_1$ and $p_2$ ($=1-p_1$).
Payoff variability
We conclude this section with an observation that will be useful later, particularly in Chapter 13. By definition, a risk-averse individual prefers a sure sum to a risky lottery with the same expected payoff. More generally, Returning to Example 2: investing everything in $A$ yields a more variable lottery than investing half in $A$ and half in $B$; this makes the second alternative more attractive for a risk-averse person, even though the expected value is the same. at equal expected payoff, a risk-averse individual prefers lotteries with less variability: between two lotteries with the same expected payoff, they will choose the one whose possible payoffs are closer to their mean.
This property can also be expressed equivalently: at equal expected utility, if a lottery becomes more variable, then the individual will require a higher expected value to accept it. In other words, greater risk must be compensated by a higher expected payoff.
We illustrate this principle in the following graph. An individual with utility function $u(W)=\sqrt{W}$ (i.e., $\pi = 0.5$) faces a lottery that offers two equiprobable payoffs, $W_1$ and $W_2$. Once a value of $W_1$ is selected, the figure automatically computes the value of $W_2$ required to keep either the expected payoff or expected utility constant.