CHOICE UNDER OBJECTIVE RISK

CHOICE UNDER OBJECTIVE RISK

7 Modelling Risk

Choice under uncertainty is a topic of fundamental interest to economists, since most economic decisions are made in the face of uncertainty. For instance, firms have to make decisions regarding prices and production, and investors in the stock market have to decide whether to buy or sell stocks, and if so, then how many, etc. Insurance is a huge industry in developed countries, and it exists only because people are averse to the uncertainty that pervades their everyday lives. But in order to rigorously study the economics of uncertainty, one first needs a formal model of how agents behave in the face of uncertainty, which we develop in this topic. When we turn our attention to subjective uncertainty in later chapters, it will become clear that there are in fact different types of uncertainty. We begin by studying the most basic type of uncertainty, which we refer to as risk. This is the kind of uncertainty we face in the casino while playing the slot machine: there is uncertainty about whether we will win a prize or not, but we know enough about this uncertainty that we can compute the exact probability of winning. In later chapters this will be contrasted with the uncertainty one may feel when buying stocks: this uncertainty relies on fine details of the economy that may not even be able to conceptualize, and as a result, it does not lend itself to calculating the probabilities of outcomes, at least not in the same way as in the casino. Before we can write down a formal model of how people choose among uncertain alternatives, we first need to find a way to formally describe uncertain alternatives.

7.1 Choice Domain: Lotteries

We use the terms “gamble” or “lottery” or “uncertain prospects” for any uncertain alternatives for which the probability of each outcome is known. A prospect of getting an outcome { with probability 1 is referred to as a degenerate lottery. In general, an outcome could be anything — it could be money, or a trip to Vegas, etc. When the outcomes of a lottery are money, we call it a monetary lottery.

A lottery can be viewed as a probability tree with a final outcome at

each terminal node. For instance, suppose there is a gamble where a fair coin is flipped twice. It yields $10 if two heads come up, and $0 otherwise. Then there are two possible final outcomes, namely $10 and $0, and the probabilities of obtaining each are 0.25 and 0.75, respectively

7.2 Reduced Form of a Lottery

Notice that three things go into describing the lottery in the preceding example:

(i) the possible outcomes ($10 and $0), 

(ii) their associated probabilities (0.25 and 0.75, respectively), and

(iii) the structure of the lottery (the coin is flipped at most two times, that is, the uncertainty resolves in up to two stages).

The reduced form of a lottery specifies just the first two and leaves out the third. In the current example, we would write the reduced form as

Notice that two different lotteries can have the same reduced form. The above lottery involving two coin flips is different from one that involves a biased coin that comes up heads (resp. tails) with probability 0.25 (resp. 0.75) and y

7.3 Mixtures of Lotteries 

Take any two lotteries s> t, each of which involves only one stage of uncertainty:

Now consider another lottery that involves two stages of uncertainty. Specifically, suppose that in the first stage, the lottery s is realized with probability  and lottery t is realized with probability 1−, and in the second stage the realized lottery is played out. Thus, in the first stage we learn whether we obtain lottery s or t, and in the second stage, the outcome of the obtained lottery is received. This “mixture of lotteries” or “compound lottery” can be denoted:24

In order to specify the probabilities  and (1−) by which s and t are being mixed, we call it an -mixture of s and t. The reduced forms of s> t are, of course

As a trivial exercise that just requires you to apply definitions and use elementary algebra, you are asked to:

Exercise 5 Show that the reduced form of the compound lottery (> s; 1 − > t) is

In general, a compound lottery could be of the form (1> s1; 2> s2; ==; q> sq) where there are many possibly outcomes of the first stage, not just two. But we will not be needing this much generality in what follows. 93

8 Expected Value Theory

The standard theory of choice under risk in economics is Expected Utility theory, or EU theory for short. We first present, however, the earliest version of that theory in order to introduce all the basic ideas before introducing the full details of EU theory.

8.1 Model

Expected Value Theory (EV for short) posits that the agent has a preference % over some set of alternatives D, and that choice maximizes preference. What makes it a theory of choice under uncertainty is that D is not just any set of alternatives, but rather a set of lotteries. Thus, the theory is one of agents who choose between lotteries.

The primitive of the theory is a preferences % over monetary lotteries. The hypothesis about the preference % is that it admits a utility representation HY where the utility of any lottery is the expected value of its reduced form:

The theory says that when faced with a lottery, the agent only cares about its reduced form, and moreover, ranks lotteries according to their expected value.

The theory has some very nice features. First, the fact that only the reduced form matters to the agent can be viewed as “rational” — to the extent that all that matters is where we get to at the end (as opposed to how we get there) it makes sense that she should concerns herself only with the overall probabilities of possible outcomes, as opposed to the structure of the lottery. Second, the model captures an intuitive idea that if a lottery gives better outcomes with higher probabilities, then it will be more attractive. Indeed, this holds in the model because lotteries that give higher outcomes with higher probabilities will also have higher expected value, and thus the agent would prefer them. Finally, the elegance of the model is to be appreciated. It captures in a simple and highly compact manner some of the essential considerations that one might like from a theory of choice under uncertainty.

However, simplicity usually comes at the cost of sacrificing realism. As we will show now, the cost associated with the simplicity of the EV theory is too high.

8.2 Evidence

First a quick review of definitions:

Definition 1 A preference % over lotteries is said to be risk averse toward a lottery of the form s = (> {; (1 − )> |) if:

Similarly it is said to be risk loving (respectively, risk neutral) if the above expression holds with  (respectively, ∼).

To explain these definitions, consider a lottery s; for concreteness let s = ( 1 2 > 100; 1 2 > 0). The expected outcome of this lottery is HY (s) = $50. 25 Although the expected outcome of s and (1>HY (s)) is identical, (1>HY (s)) gives the expected outcome for sure whereas s yields it with risk. Thus, an agent’s preference between s and (1>HY (s)) comes down to how he feels about risk vs certainty. Risk averse agents will prefer a sure $50 over a lottery that gives an expected $50. Similarly for risk loving and risk neutral agents.26 Another definition:

Definition 2 The certainty equivalent for a lottery s is the sure sum of money, denoted FH(s), such that

The expected outcome is defined as the average of the outcomes you’d get if you played the lottery repeatedly. Don’t let this confuse you: the lottery is actually being played only once. 26Note that risk attitude (that is, aversion, affinity, or neutrality toward risk) is really a psychological notion, but we have defined it in terms of behavior. Risk aversion is properly defined in terms of a distaste for risk. We don’t observe distaste directly, and thus this intuitive definition is useless for scientific purposes. However, by identifying the behavioral expression of distaste for risk, we are able to provide an empirical means of determining an agent’s risk attitude.

The certainty equivalent is a measure of how much you like or dislike the lottery. If you say that playing the lottery s = ( 1 2 > 100; 1 2 > 0) is just as good as receiving $10, then your certainty equivalent for the lottery is $10. The low value of the certainty equivalent (relative to the expected outcome of $50) suggests that the agent doesn’t find himself very drawn to playing the lottery

Let us turn now to the case against EV theory. Write down your responses to the following two questions.

(A) What is your preference between the lottery ( 1 2 > 1000; 1 2 > −1000) and the sure (zero) outcome (1> 0)? Put differently, how do you feel about playing this lottery vs not playing it?

(B) What is your certainty equivalent for the following lottery? Suppose that an unbiased coin is tossed again and again until it lands on tails, and then you are paid an amount that depends on how many tosses it took for the coin to land on tails. Specifically, the payment rule is that you receive $2q if the coin lands on tails in the qwk toss. Thus, you get $2 if it lands on tails in the first toss, $4 if it lands on heads the first time and tails the second, $8 if it lands on heads the first two times and tails the third, etc. Yes, you can potentially win billions of dollars if q is large enough. Note that the probability of getting tails in the qwk toss is 1 2q . Thus this lottery can be written as ( 1 2 > 2; 1 4 > 4; 1 8 > 8; ===; 1 2q > 2q; =====).

Most people would rather not play the lottery in (A) for the simple reason that uncertainty makes them uncomfortable as it is, and facing the possibility of losing $1000 makes them even more uncomfortable. Since you can never lose any money with the lottery in (B), and you only stand to gain, the certainty equivalent will be strictly positive for any reasonable agent. Experiments report that typical certainty equivalents are a few dollars. The following propositions establish that such responses contradict the EV theory, and thus that the EV theory is not a good descriptive theory

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