**Psychology in Choice Theory**

**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

Psychology in Choice Theory

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
( 1
4 > $10; 3
4 $0). In general, the reduced form of a lottery s is denoted by

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 yields $10 if heads and $0 if tails. Yet both have the same reduced
form.

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:

4 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.

**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

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.

**Psychology in Choice Theory**

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:
s ≺ (1>HY (s))=
Similarly it is said to be risk loving (respectively, risk neutral) if the above
expression holds with Â (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
(1>FH(s)) ∼ s=
25The 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.27
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
27 Indeed, this suggests risk aversion — the agent values the lottery less than its expected
outcome. Assuming that more money is preferred to less, it is not a surprise that s ∼
(1> $10) ≺ (1> $50)> that is, s ≺ (1>HY (s))

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