Mathematics and Decision Making
Mathematics and Decision Making
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, firstname.lastname@example.org