Thinking and Deciding

CONCLUSION 255
7. On the basis of your answers to questions 4 and 6, what is the relation between the curvature
of the utility function and risk attitude (risk seeking versus risk averse), according to expectedutility
theory?
8. Assume that the utility of yearly income X is X .5 . If you have $50,000 in income to divide
between two people, what division will maximize their total utility? Show your work. (You
can do this graphically, if you do not know calculus.)
Conclusion
Utility theory as a normative model tells us what it means for a decision to be best
for achieving our goals. The theory has been widely applied as the basis of decision
analysis. It plays another important role, however, in justifying the prescriptive
model of actively open-minded thinking. Actively open-minded thinking about decisions
helps us to achieve our goals because it helps us to maximize utility. The
expected utility of a neglected possibility (option) could be higher than that of any
option that was considered, so we do well to search thoroughly for options. Likewise,
neglecting a possible piece of evidence (possible consequence) or goal (attribute), or
weighing the evidence in a biased way, could lead us to choose an option with less
utility than the best option we could choose. Parallel arguments can be made for
the importance of actively open-minded thinking in belief formation. Beliefs formed
after a thorough search for evidence and an unbiased evaluation of that evidence are
the most useful for decision making. They reflect the evidence available most accurately
and correspond to better-calibrated probability judgments, so they help us to
maximize utility in the long run.
Utility theory also justifies my claim that thorough search is a virtue to be practiced
in moderation. The utility of search is negative, and the compensating expected
benefits decline as search continues: There is a point of diminishing returns in the
expected utility of thinking itself.
Utility theory in general can also serve as a prescriptive model for decision making,
as well as a normative model. When we make real decisions, it may usually be
helpful to ask ourselves what produces the best expected outcome. This heuristic —
and it is a heuristic — may lead us to make decisions that produce good outcomes.
In the next chapter we shall look at the descriptive theory of decision making.
Departures from utility theory, we shall discover, provide further reasons not to trust
decisions made without adequate thought or formal analysis.
Answers to exercises
1. The line plotted on the graph should pass through the origin and be curved downward.
2. 0, √ 5, √ 10
3. .5 · √ 10
4. EU($5) is greater, thus showing risk aversion.
5. These are the same. EV ($5) = $5.
6. EU($5) = 25. EU($10,.5) = .5 · 100 = 50 > 25. This shows risk seeking when the utility
function is convex.
7. Concave implies risk aversion; convex implies risk seeking.