Predicting Preferences
Prediction involves making a statement concerning the likely value of an event or action uncertain or unknown at the time of the statement. Since the theory of probability, (inaugurated by the French mathematicians Blaise Pascal and Pierre Fermat in 1654), was developed to quantify uncertain events in terms of their likelihood of occurrence, formal prediction is now viewed as a mathematical topic involving probabilistic modeling. Indeed, the mathematician Karl Pearson said in 1907 that the fundamental problem in statistics is prediction. Prediction, however, is usually not an end goal itself, but rather means to put probabilistic bounds on the relative frequency or likelihood of occurrence of future uncertain events so that strategies or actions can be taken incorporating these predictions. Risk management needs predictive analysis, as does economic regulation, engineering control, and marketing effectiveness. This latter use of prediction often involves predicting an individual’s choice (or group’s choice) or preference over alterative options.
Preference can be conceptualized as an individual’s (or group’s) attitude concerning a set of objects, and is usually formulated within a choice making context (i.e., X is preferred to Y if one would choose X over Y). In this way choice and preference are linked, and predicting preferences is akin to predicting choices. An auxiliary question is to formulate models that explain “why” or “how” the choices or preferences are formed or exhibited. Applications of preferences modeling are found in consumer behavior, marketing, advertising, economics and finance.
Research on predicting preferences has generally followed two related streams; economic choice modeling (based on rationality axioms) and attribute-based preference modeling (linking product or event attributes to preferences). Forecasting or predicting behavior follows in each stream, but the orientations are slight different.
Economic models for predicting preferences
Early work on choice or preference modeling coincided with the development of probability theory. According to expected value theory developed by mathematician and physicist Christian Huygens in 1657, rational economic agents facing an uncertain event or choice set should enumerate all the possibly outcomes of the event, say x1, x2, …, xn, along with their corresponding probabilities of occurrence p1, p2, …, pn, and then calculate the mathematical expectation or mean value μ = p1*x1+ p2*x1 + …+ pn*xn. This value, μ, was called the fair economic value of a gamble because if one repeated this same gamble over and over again, and paid the amount $μ to enter the gamble each time, then in the long run your winnings would be $0 and your losses would be $0. You would exactly break even according to the relative frequency theory of probability exposited by Pascal and Fermat. If offered this gamble at a cost of less than $μ, you should take it, if offered it at a cost of more than $μ, you should refuse to take it, and you would be indifferent about gambling if offered it at a price of exactly $μ. Using this theory, one could also predict preferences between two uncertain events: calculate the expected value of each and prefer the one with the larger expected value.
A major breakthrough in modeling and predicting preferences occurred in 1738 when mathematician Daniel Bernoulli presented and solved a problem first discussed by his mathematician cousin Nicolas Bernoulli 25 years earlier. With D. Bernoulli’s contribution, the concept of decision maker’s risk aversion came into the prediction of preferences as he recognized people will pay more than the fair or expected value of a hazard for insurance to eliminate the loss possibility.
Bernoulli’s analysis revolves around a theoretical game (called the St. Petersburg paradox game) that leads pay-off sequence with an infinite expected value. However, most individuals only would be willing to pay a small amount to play this game so the predictions made from the expected value rule conflict with the preferences that a real rational person would exhibit. Bernoulli resolved the paradox by postulating that people do not make choices based on the expected rewards alone, but rather based on the value or pleasure that each individual pay-off is likely to yield to them. He said “The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.” He suggested there was a function U(w) (called a utility function) which describes the amount of value or utility that the individual gains by having an amount of wealth w. Rational individuals prefer more money to less money, so U(w) is increasing, but the incremental value of more wealth decreases as one becomes wealthier (decreasing marginal utility or value of wealth) so U(w) is concave. Bernoulli stated that people determined how much to pay for the game or gamble not based on the gamble’s expected value but rather it’s expected utility. Thus, if x1, x2, …, xn are the possible outcomes of an uncertain economic endeavor or gamble, and their probabilities of occurrence are p1, p2, …, pn then a decision maker with an initial wealth of W would make decisions based on their expected utility of their final wealth, U(W+x1)*p1 + U(W+x2)*p2 +…+ U(W+xn)*pn. Using this approach, one can predict preferences between two uncertain events by calculating the expected utility of each with preference going to the one having larger expected utility. Bernoulli suggested a logarithmic function U(x) = ln(x) as an appropriate function exhibiting increasing desire for wealth but with decreasing marginal value for incremental wealth.
Bernoulli’s work on expected utility formed the basis for the economic development of rational decision making under uncertainly, and the prediction of economic choices based on expected utility has become the most widely used paradigm in predicting preferences. The actual determination of the “correct” utility function to use for an individual decision maker and the elicitation of the parameters for the utility function is a still ongoing research area. A major developmental step occurred in 1944 with the publication by mathematician John von Neumann and economist Oskar Morgenstern of the book Theory of Games and Economic Behavior. They developed a set of axioms describing the preferences relations between uncertain events which should hold for a rational decision maker, and proved mathematically that if a decision maker acted rationally, then there existed a utility function U such that the decisions were made as if the decision maker were using the expected utility paradigm with utility function U. Thus, they took the utility function from an assumption about behavior to a conclusion about rational decision making and preferences. If a decision maker is rational (again according to their axioms of rational decision making) then one could predict preferences and choices according to expected utility theory.
Attribute-based preference modeling

Some disciplines, such as psychology and consumer behavior, have developed alternative approaches to predicting (or understanding) preferences, such as multi-attribute modeling. In this cognitive approach attitudes and preferences are based on an object’s (e.g., a brand of detergent) multiple discernable attributes which are examined by the decision maker. The decision maker weights these attributes in order to make a choice in various ways. One technique of modeling and predicting preferences or choices using multiattribute products is the choice modeling technique elucidated by the Nobel Laureate Daniel McFadden. Under this model it is assumed that individuals select among alternatives according to a rational underlying decision process possessing a functional form dependent on the behavioral context. Commonly used functional forms include the multinomial logit model which is often used as an approximation to utility maximization described previously. Individual product or event related attributed effect the choice probability by having a linear functional relationship with the log odds ratio of selecting the various alternatives compared to a base alternative. The parameters of the linear model are determined by fitting to behavioral observations. Once the model has been estimated, the prediction of preferences can be made, and the strength of preferences can be judged by the relative probability of the different choice options. Statistical tests and goodness of fit statistics can be applied to a hold out sample of data to determine the degree of confidence in the preference prediction.
Another methodology (or rather a collection of methodologies) incorporating multiple covariate attributes assumes that the functional form relating attributes to the choice or action behavior is one of the unknowns to be determined in the modeling process. Predictive analytics (also called predictive modeling) uses data on prior decision processes to determine both the functional form and the prediction of preferences. This approach encompasses techniques in data mining and artificial intelligence (neural networks modeling, for example) to make predictions about uncertain events and preferences.
Predictive models in this class of techniques exploit patterns in historical and transactional data to identify revealed preferences. Flexible models are designed to capture the interrelationships between the attributes and the observed choice behavior to guide in forecasting. These models involve complex mathematical functional forms sufficiently flexible that require rigorous data analysis to fit parameters and allow prediction of behavior. They have been used to improve marketing effectiveness by analyzing past consumer behavior to determine the likelihood a customer will exhibit a specific purchase or preference behavior in the future.
SEE ALSO: Advertising, Data Analysis and Probability in Society, Data Mining, Forecasting, Probability, Rankings

FURTHER READING:
Sarma, Kattamuri R., 2007, Predictive Modeling with SAS Enterprise Miner: Practical Solutions for Business Applications, SAS Publishing
Hauser, J.R., 1978. Consumer preference axioms: Behavioral postulates for describing and predicting stochastic choice. Management Science 24, pp. 1331–1341
Manrai, A.K., 1995. Mathematical models of brand choice behavior. European Journal of Operational Research 82 (1),1-17.
McFadden, D., 1980. Econometric models for probabilistic choice among products. Journal of Business 53 (3), 513-530.