Thinking and Deciding

RANK-DEPENDENT UTILITY THEORIES 273
Rectangular. Another because “it offers a majority of chances of at least an amount
somewhere between $40 and $159. The odds of winning are best between these
dollar amounts, and the odds indicate an excellent chance of winning an amount
between $80 and $119.” Lopes (1996) notes that subjects tend to think in terms of
the chances of getting a large amount, or a very low amount, or “at least as much
as” some amount. The latter statement refers to an aspiration level, and the idea
of “at least as much” implies that the subject is thinking about the sum of all the
probabilities of that amount or larger.
Lopes (1996) argues that a good approximation of risky decisions can be achieved
by considering how people think about security, opportunity, and aspiration. Security
is reflected in terms of the attention people pay to the worst outcome. Opportunity
is reflected in attention to the best outcome, and aspiration is the attention
they pay to whether a certain desired level is achieved. This is somewhat different
from prospect theory, which roughly divides outcomes into those above and below
the status quo.
Lopes’s theory is a member of a broader class of theories called “rank dependent.”
15 The idea is that people evaluate prospects (possible outcomes of choosing
an option) in terms of their rank, such as best, second best, third best, ...worst.
Rank dependence is usually combined with another idea, that people treat probabilities
as cumulative. That is, when people evaluate prospects with several outcomes,
they behave as if they thought of the probability of doing “at least as well as” some
outcome (or, “no better than”). This contrasts with the view of prospect theory that
people think of the probability of each individual outcome. People still apply a utility
function to the outcomes one by one, as assumed by prospect theory.
The difference between cumulative and non-cumulative models has to do with
the way in which probabilities are transformed. The π function applies to individual
probabilities. In cumulative rank-dependent theories, some other function applies to
the cumulative probability, the probability of doing at least as well as some outcome.
An example of such a function is shown in Figure 11.3 (adapted from Birnbaum and
Chavez, 1997). This figure applies to a gamble with four outcomes. P is the probability
of doing at least as well as that outcome, and w(P ) is a weighing function that
distorts P . W (P ) is like π(P ), but it applies to cumulative probability rather than
to probability itself. The decision weight assigned to each outcome is the difference
between W (P ) for that outcome and and W (P ) for the next higher outcome. (For
the best outcome this difference is just W (Pbest), because the probability of doing
better is zero. In Figure 11.3, the difference is represented with w.) For the function
shown in Figure 11.3, the decision weight is highest for the worst outcome and next
highest for the best outcome. A person with this function would be somewhat risk
averse but would not pay much attention to outcomes other than the worst and the
best.
One property of this function is that the decision weights necessarily add to 1.
This is not true of prospect theory in its original form. Strictly speaking, a person
15 The first published theory of this type seems to be that of Quiggin (1982).