The Fundamental Axioms of
Utility Theory and the
Expected Utility Theorem
Let
B « A
denote “A is preferred to B” and
B = A
denote “A is equally preferable to B.”
The fundamental axioms of utility theory are:
COMPLETENESS 1. —For any two simple lotteries L
and M, either L « M, L = M, or M « L.
TRANSITIVITY 2. —If L « M and M « N, then L « N.
CONVEXITY/CONTINUITY 3. —If L « M « N then there
is p between 0 and 1 such that the lottery
pL + ( 1 – p)N is equally preferable to M.
INDEPENDENCE 4. —If L = M, then pL + ( 1 – p)N = pM
+ ( 1 – p)N.
These axioms appear to capture the basic properties
of rational decision-making under uncertainty and are
the basis for inferences about rational choices such as
the expected utility theorem:
L « M ↔ u(L) « u(M),
where u(;) represents the expected utility of selecting option ;. In words: an agent will prefer lottery M
to lottery L if and only if the expected utility of M is
greater than the expected utility of L.
to save 100 QALYs. Program B (e.g., cardiac surgery made
available to the uninsured) costs $5,000,000 and is expected
to save 100 QALYs. Program C (e.g., vaccinations and public-hygiene programs in underserved areas of the state) costs
$1,000,000 and is expected to save 500 QALYs, etc. (see Table
4). What initiatives should policymakers choose to fund in
order to maximize the benefits that $10,000,000 in tax dollars can generate?
Once a measure (in this case, QALYs) of benefits is determined, the answer is obvious. Subject to monetary constraints,
policymakers should choose those programs that maximize
well-being. Equivalently, they should choose those programs
with the lowest cost per QALY. That goal can be achieved by
ranking the programs in increasing order of cost per QALY, as
is shown in Table 5, and funding them in this
sequence.
The most effective is Program C, in which
one QALY is gained for an investment of
$2,000; the least effective is Program G, in
which each QALY gained costs $100,000. Accordingly, policymakers should fund programs C, F, H,
K, E, J and A to maximize benefits.
Note that the objective in this example is to
maximize society’s utility—in other words, optimize
improvements in life expectancy and the quality of life
for all Connecticut residents. Maximizing the utility of subgroups clearly leads to a different allocation of resources. So
does changing your definition of utility—to revenue per procedure or per patient, for instance.
The concept of QALYs simultaneously captures health and
duration of life. The metric can be used to quantify health state
utilities (preferences), the value of medical interventions and
benefits in resource allocation studies. Firmly grounded in economic decision theory, QALYs are arguably the best tool for
pricing the priceless. This isn’t to say that QALYs are precise
measures. After all, quantifying subjectivity isn’t, nor can it ever
be, an exact science.
The application of QALYs in the U.S. is currently restricted
to public health studies. But they are poised to become an important tool in policymaking as society demands the optimal
use of scarce resources.
CARLOS SANCHEZ-FUENTES is a fellow of the Society of
Actuaries, a fellow of the Conference of Consulting Actuaries
and a member of the Academy. He holds a master’s in business
administration and is managing partner at Axiom Actuarial
Consulting LLC. He can be reached at carlos-fuentes@axiom-
actuarial.com.
References
Bennett, Torrance, Boyle, and Guscott, “Cost-Utility Analysis in Depression,” Psychiatric Services, Sept 2000, Vol. 51, No. 9, pp. 1171-1176.
Cutler, David. Your Money or Your Life, New York: Oxford University
Press, 2004.
Zeckhauser, Richard, and Shepard, Donald. “Where Now for Saving
Lives?,” Law and Contemporary Problems, 40( 4), Autumn 1976, 5-45.
