Decisions shape our lives. Mathematics rationalizes the
sifting of information and the balancing of alternatives
inherent in any decision. Mathematical models underlie computer
programs that support decision making, while bringing order and
understanding to the overwhelming flow of data computers produce.
Mathematics serves to evaluate and improve the quality of
information in the face of uncertainty, to present and clarify
options, to model available alternatives and their consequences,
and even to control the smaller decisions necessary to reach a
larger goal.
Mathematical areas like statistics, optimization, probability,
queuing theory, control, game theory, modeling and operations
research --- a field devoted entirely to the application of
mathematics in decision making --- are essential for making
difficult choices in public policy, health, business,
manufacturing, finance, law and many other human endeavors.
Mathematics is at the heart of a multitude of decisions, including
those that generate electric power economically, make a profit in
financial markets, approve effective new drugs, weigh legal
evidence, fly aircraft safely, manage complex construction
projects, and choose new business strategies.
Models of Complex Systems
The costs of the policy decisions surrounding global warming
are high politically and financially. Policy makers must work
through a chain of issues: Is global warming real? Is it caused by
automotive and industrial emissions? If so, which ones? Which
remedial strategies will be effective? What is their true cost?
Individual manufacturers whose products are among the suspected
pollutants face parallel decisions at the corporate level.
Specialized mathematical models link the effects of selected
atmospheric pollutants to predictions of global temperature change.
They are the basis for the growing scientific consensus that
observed increases in the average temperature of the earth are
unlikely to be the consequence of natural variation alone.
Similar models can also be used to simulate and evaluate remedial
strategies. The mathematical tools of modeling, simulation,
and risk analysis validate the cause and effect relationship
upon which policy decisions are based, and they permit the
evaluation of the effects of alternate courses of action.
In addition, chaos theory is providing new lenses through
which to view the behavior of such complicated systems.
Complex decisions arise in more tangible settings as well,
such as choosing among the interrelated options that govern the
process of building a complex system like an office building or
an aircraft. Which sequence of tasks chosen now will best
advance completion of the project? Which are potential
bottlenecks? Operations Research uses critical path analysis to
identify the vital tasks so that each subunit is in place at the
right time at minimum cost: no battles are lost for want of the
proverbial nail.
Complexity is aggravated by uncertainty. For example,
decisions about dynamic control of traffic in telephone and
computer networks are made more difficult by the uncertain patterns
of demand. In a simpler form, a bank faces a similar dilemma in
deciding how many tellers to hire: how should resources be
allocated to maintain adequate service (shorter lines) when only
the random characteristics of customer arrival times are known?
Queuing theory provides guidance for these kinds of decisions.
Testing and Evaluation
How can physicians be sure they are prescribing drugs that
help, not hurt? Statistical analysis of clinical trial data
guides the Food and Drug Administration's approval of every
prescription drug.
To ensure an impartial assessment of dose and response effects,
drug trials are conducted using protocols dictated by the
statistical methodology known as the design of experiments.
Assertions about the efficacy of a particular course of treatment
are then accompanied by well-defined confidence intervals,
statements of the likelihood of treatments being effective in
specified circumstances. For example, such analyses are the
foundation of recent reports that estrogen therapy reduces female
mortality from heart attack and stroke.
Expert witnesses in the courtroom use the language of
probability to argue the value of DNA evidence purporting to
match blood samples to unique individuals. Calculations made
using the deep body of mathematical thinking known as probability
theory can surprise casual intuition, giving probability a
particularly important role in guiding decisions in the face of
uncertainty. As a simple example, the probability of two
individuals chosen at random having the same DNA is not 1 in 5.7
billion, the population of the earth, but about 1 in 75 billion,
the number of possible DNA configurations.
Control and Optimization
The tools of control theory allow humans to delegate some
forms of decision making, such as those of a tactical character
that require assessment of data and action on a time scale too
rapid for humans. For example, control systems in commercial
aircraft make fine adjustments in aileron settings as the pilot
changes course so that the aircraft remains stable. A key
component of this kind of automated decision making is selecting
a control action that is optimal in a precisely defined
mathematical sense. Mathematics is also the language in which
those control systems are designed, evaluated, and implemented.
The number of low-cost tickets an airline will sell for a
journey on that same aircraft is decided by a mathematical model
of anticipated customer traffic and acceptance of various price
levels. The mathematical tools of operations research can define
and analyze the trade-offs between the revenues lost to empty seats
and the costs of overbooking, a choice that has made the difference
between profit and loss for at least one major airline.
The electricity we use every day comes from generators whose
level is set to meet projected electric demand at minimum cost.
An amalgam of mathematical and computational tools solves and
re-solves this complicated optimization problem throughout the day
as the utility control center adjusts to changing demand patterns.
Future demand for airline tickets, electricity and many other
commodities is often predicted using time series, a statistical
tool that extrapolates into the future from historical data. Those
predictions are accompanied by measures of confidence that help
planners provide appropriate contingencies for deviations of the
realized future from the predicted.
Many of the decisions about the design of equipment of all
sorts are left to sophisticated design algorithms that integrate
mathematical models of the device with optimization algorithms in
state-of-the-art computational environments. For example, one
technique links disparate analysis tools, such as one for the
strength of an airplane wing and another for its aerodynamic drag,
with powerful optimization engines in order to achieve a product
goal, an aircraft with maximum range, that balances competing
requirements like strength, weight, lift and drag.
Financial and Economic Analysis
Mutual funds can include investments in derivatives such as
currency repurchase options, financial instruments whose prices
are tied to prices of other commodities in the market. Both the
value and the hedging structure of many derivatives are decided
by models of economic behavior. Stochastic differential
equations are the language of those models because they express
naturally the market's intrinsic uncertainty. They lead to
valuation formulas that balance risk and expected return.
Game theory, a discipline that was given its modern form by
the mathematician John von Neumann, models markets in which the
actions of competing parties influence one another while each
acts in its own self interest. The 1994 Nobel prize in economics
was shared by John Harsanyi and the mathematicians John Nash and
Reinhard Selten for their introduction of several different
concepts of market equilibria, situations in which each player is
in an optimum position relative to its competitors. These
perspectives provide deeper insights into price structures than
simple supply and demand, thereby guiding investment and capital
expenditure decisions.
Analyzing a different competitive setting, a political
scientist and a mathematician have recently extended the age-old
technique for dividing a piece of cake between two individuals -
-- one cuts, the other chooses --- to fair division among many
parties when economics and other complex forces are at work.
Such disputes might center on dividing cities and natural resources
at the close of a multi-nation war. The theoretical solution of the
underlying mathematical problem, that of fair, envy-free
division among many parties, might lead eventually to tools that
heads of state could apply to deciding disputes like the
division of territory in Bosnia.
Mathematics at the Core
Mathematics shows many faces as it works in these diverse
settings. Statistics measures the quality of information.
Optimization finds the best alternative. Probability quantifies
and manages uncertainty. Control automates decision making.
Modeling and computation build the mathematical abstraction of
reality upon which these and many other powerful mathematical tools
operate. Mathematics is indeed the foundation of modern decision
making.
Paul Davis, Worcester Polytechnic Institute, pwdavis@wpi.edu
