Predicting preferences

Summary: Psychology of choice and predictive models of preferences are exciting areas of mathematics blending social science, economics, and commerce.

Mathematically, preference is an ordering of alternative possibilities. It can refer to conscious choices based on ideas and beliefs, positive emotional responses or liking, or biologically mandated behaviors. Preferences are usually determined statistically: for individuals, based on multiple instances of decisions over time; and for groups, based on aggregated data of members. In 2009, the Netlix Prize contest awarded a team called BellKor’s Pragmatic Chaos $1 million for their preference-predicting algorithm.

Theoretical and Behavioral Economics

Among all sciences that deal with predicting preferences, such as social psychology and education theory, the most developed mathematical apparatus can be found in economics. As any branch of mathematics, theories of economic preferences start with axiomatic assumptions. These abstract axioms do not always apply to all real situations. Economic theories that take into account psychological factors, such as cognitive limitations and emotions, are developed within an interdisciplinary area called “behavioral economics.”

Most abstract theories of preference prediction assume most parts of the so-called total order, which is a group of mathematical axioms and properties from set theory. Let A, B, and C be different choices. Total order assumes that either A≤B or B≤A. In real life, this assumption is a statistical statement at best: today a person can prefer apples, but might prefer bananas tomorrow. The property of transitivity says that if A≤B and B≤C, then A≤C. This property works in some situations; for example, if one prefers $20 over $10, and $100 over $20, it is likely the person will prefer $100 over $10. However, in complex situations with multiple choices, such as elections, transitivity fails to describe real human behavior. Experiments show that, given a choice between one pair of candidates at a time, people may prefer Beth over Alice, Carol over Beth, and Alice over Carol. One axiom of total order, called “antisymmetry,” that almost never makes sense in preference theories is that if D≤E and E≤D, then E=D. For example, when group data shows that people think diesels are worse or the same than electric cars, and electric cars are worse or the same as diesels, it does not mean that diesel cars and electric cars are the same entity. It means that people prefer them about the same. Economic theories call this situation “indifference” and use a separate symbol for it: E-D.

Another assumption frequently made in economic preference predictions comes from topology and is called “continuity.” It is the assumption that if A is preferred over B, then an option that is very similar (close) to A will also be preferred over at option that is very similar to B. Many complex phenomena, including preferences, are discontinuous. They exhibit various “tipping points,” near which minute differences cause radical changes in preferences. These non-continuous phenomena are studied using models from calculus or chaos theory, a branch of differential equations. One frequent example of noncontinuous preference is price near powers of 10: many people choose to buy an object that costs $999 over a similar object that costs $1,001 even though the difference in prices is minuscule compared to the total. Behavioral economics explains this by cognitive limitations: people see 1001 as thousands and 999 as hundreds, which is technically correct but makes less of a difference in this case than intuition leads one to believe.

Paradoxical Preferences

A paradox is a false or contradictory statement that logically follows from a set of true statements. Preference prediction leads to several types of paradoxes. A very frequent type is the situation when an initial model describes the reality well, but its mathematical corollaries do not. Another type, a true logical paradox, occurs when mathematical corollaries contradict one another.

For example, the expected value is the sum of products of probabilities and payoffs. Suppose a fair coin is flipped in a hypothetical game and the player is paid $10 if the coin lands on heads and $20 if it lands on tails. The expected value of winning is $15 because 0.5(10)+0.5(20)=15. When the same game is played many times, it is rational to prefer options with higher expected values. Under this assumption, it is better to play the game where the player is paid nothing for heads and $40 for tails than the first game, because the expected value of winning is higher: 0.5(0)+0.5(40)=20. However, in real life, risk aversion will make many people choose the first game.

To resolve this and other related paradoxes, many preference models account for risk aversion as a separate variable. A utility function is the measure of relative satisfaction of a range of choices. An assumption that people will only want to maximize utility is not realistic, because it does not account for risk aversion. Because marginal choices usually come with higher risks, the utility function that accounts for risk aversion will look like a hump, being concave.

Bounded rationality principle is commonly used to explain paradoxical preferences by taking into account limited information, time, and cognitive abilities of people. Models based on bounded rationality include human limitations, such as computational capacity, and are based on computer science, statistics, and psychology.

Information Theory and Aesthetic Preferences

Information theory is a mathematical science that studies storing, compressing, and processing of data. In the 1990s, its branch called “algorithmic information theory,” which deals with the complexity of algorithms, was applied to explain some aspects of the human sense of beauty and of aesthetic preferences. According to this theory, objects that have shorter algorithmic descriptions in terms of observer’s knowledge will seem more beautiful, compared to objects with longer algorithmic descriptions. For example, it is easier to remember an object with mirror symmetry because only half of the information is original—symmetry provides information compressibility. Therefore, symmetric objects, as well as objects with patterns or fractal self-similarity, are seen as more beautiful.

Algorithmic information theory also models preferences by interest, which are separate from preferences based on beauty. Within these models, interest can be compared to the first derivative of beauty, showing the observer’s perception of change in understanding. People prefer an experience on the basis of interest when it involves better compressibility or predictability of information than before. For example, noticing a new pattern (and therefore better organizing an image) is preferred because it is interesting.

Preferences, Desires, and Motivation

Many preferences and choices are based on needs, wants, and desires, which are explained in theories of motivation. Researching motivation is challenging because of individual differences among people, as well as language ambiguity. There are disagreements among researchers even over relatively straightforward terminology, such as intrinsic and extrinsic motivation. Many motivation theories include taxonomies of needs and desires. For example, in Maslow’s hierarchy, named after Abraham Maslow, unsatisfied physiological needs, such as hunger or thirst, have higher priority than unsatisfied self-esteem needs, such as recognition. Some theories identify long lists of motivators, such as curiosity, tranquility, order, and independence. Other theories only define a few broad classes of needs.

Each category of need can be considered a variable. Graphs of values of these variables versus levels of motivation often demonstrate the characteristic “mirrored C” shape called a “backward bending curve.” For example, as activities provide more order, they first become more motivating (and preferred), but beyond a certain point, more order becomes less motivating. This curve is famously described in the baseball manager Lawrence “Yogi” Berra’s joke about a restaurant: “Nobody goes there anymore. It’s too crowded.” People usually prefer restaurants that are not too empty or too full.

Preferences and Demographics

A number of statistical studies find significant differences in preferences of different demographics within populations, such as males and females, socioeconomic classes, ages, and political affiliations. Because statistical packages make many types of mathematical and statistical analyses of databases very easy, there are many results that demonstrate significant differences in preferences among different demographics. However, determining meanings of these differences is a significantly more difficult research problem. Demographic differences in preferences can also vary from culture to culture. In some cultures, for example, more females than males prefer bright colors in clothes, and in other cultures, it is reversed.

Bibliography

Anthony, Martin, and Norman Biggs. Mathematics for Economics and Finance: Methods and Modeling. New York: Cambridge University Press, 1996.

Berry, M. A. J., and G. Linoff. Data Mining Techniques For Marketing, Sales and Customer Support. Hoboken, NJ: Wiley, 1997.

Netflix Prize. http://www.netflixprize.com.