Mathematics and the Real World (38 page)

The mathematical tools used today to analyze macroeconomic activity are not essentially different from those developed for understanding how nature functions. These tools include equations of different types, for example, differential or other equations that deal with economic quantities such as consumption, savings, and interest rates. The equations are meant to describe how an economy works and not what type of economy we might be interested in. An analysis of the model sometimes may help us understand what steps the fiscal or monetary policy makers ought to take to achieve the desired objective. The model itself, however, describes the economy as it is. As in a social world, we cannot perform proper controlled
experiments, so economists use data provided by bureaus of statistics. The technique used to analyze these parameters is known as
econometrics
, which is a development of the statistics that we described in the
previous chapter
. The methods developed for purposes of economic analysis are very advanced, but the approach is not essentially different than that which developed over the years for use in the natural sciences and technology. Success in describing macroeconomic conduct is lagging behind the level of the success of mathematics in describing physics and its uses in technology. Is this merely a question of time, and will the gap be closed with the improvement of the existing models, or is there possibly a need for a new mathematics to describe human behavior? There is no unequivocal answer.

We will not describe the macroeconomic models in detail. We will just present two examples of considerations specific to social sciences. Both are related to the recipients of the Nobel Prize in Economics in 2011. The official name of the prize is the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (the Sveriges Riksbank is the central bank of Sweden). Alfred Nobel did not specify the social sciences as one of the fields in which a prize would be awarded. We cite these examples because of the special way in which mathematics is used to describe the complexity of human conduct, and they do not reflect the entire range of uses of mathematics in macroeconomics.

An inherent element of human decision making is the assessment of what is likely to happen in the future, when in many cases the individual considers that his influence on the future is negligible. For years there was an understanding that macroeconomic developments are affected by individuals’ expectation regarding the future, but that understanding was not translated into a component in equations. Robert Lucas of the University of Chicago, who was awarded the Nobel Prize in Economics in 1995, and colleagues who included Thomas Sargent of New York University, winner of the Nobel Prize in Economics in 2011, developed the theory of
rational expectations
. They found a mathematical way to incorporate market expectations in an equation determining the process of development of the economic parameters. These expectations are part of the variables in the model; they influence the other variables and are influenced by them.
Market expectations that affect the development of the market are clearly specific to social sciences. The formulation that describes in mathematical terms the role of such expectations in future developments was adopted by economists and constitutes an integral element in many macroeconomic models.

The second example is merely anecdotal and should not be considered as representing econometric practice. We chose this example because it teaches us something about the link between mathematics and uses. The joint winner of the Nobel Prize in Economics in 2011 with Thomas Sargent was Christopher Sims of Princeton University. The prize citation refers to Sims's contribution to the analysis of time series, that is, statistical series that change with time. The analysis of statistical series in general, and those that change with time in particular, have long interested scientists, and the basis for the mathematical methods of analyzing the errors in these series dates back as early as in the time of Carl Friedrich Gauss. Among other advances, systems were developed for finding whether two series of data were correlated, and that gave quantitative indices for the degree of correlation. In their use in the natural sciences, however, the question did not arise as to which of the two series was dominant, that is, which caused changes in the other series. For example, there is a causal relationship between the Earth's revolving on its own axis and high tide and low tide. Yet no one looked in the two data series for the answer to the question of what is the cause and what is the effect, does the tidal flow cause the Earth to revolve or vice versa. The answer is derived from the laws of nature themselves. In general, in natural sciences we do not try to derive the cause and effect from the two data series themselves, but we try to derive them from their underlying model. Unfortunately, the mathematical models of social and economic occurrences are not reliable enough to enable a similar analysis to be performed. It is natural, therefore, that attempts will be made to derive the cause and determine which of the two series is dominant from the series themselves. Among Sims's contribution to understanding time series was his enhancement of a method proposed by the British economist Clive Granger (1934–2009), a 2003 Nobel laureate in economics. The method was supposed to determine which of two
data series was the cause of the other. The test is known as the Granger-Sims causality test and serves to test and examine causality in many areas of social science and economics.

In 1982 two economists, Richard Sheehan and Robin Grieves, published results of the use of the Granger-Sims causality test to examine possible causality between the appearance of sunspots and the business cycle in the US economy, related to both gross national product (GNP) and the price index. The article was published in the
Southern Economic Journal
(volume 48, pages 775–78). The results were statistically significant and showed that the business cycles in the American economy is the cause of the sunspots. It is clear that this result is inconceivable. Yet to make it quite clear, it does not disqualify the statistical test. What should be learned from it is that we should not rely on a statistical test that is not supported by an independent model. As there is no model that incorporates the effect of GNP on sunspots, the statistical analysis that examines this is not applicable. Such use of a statistical test is limited. The right approach is first to propose a model representing a causal relation, and then to leave it to statistics to confirm or refute the model. A statistical test alone without a possible model of the effect itself can lead to fundamental errors.

45. STABLE MARRIAGES

We will now present an example of a mathematical analysis of the possible actions of a group of people. We use the term
possible
actions and not
desired
actions or
recommended
actions. We will explain the reason for that later on, and here we will just note that following the successful experience of using mathematics in the natural sciences and technology, many hoped that in the social sciences too, mathematical analysis would show how a society should act. The current mathematics of human behavior is very far from being able to fulfill such hopes.

The example relates to an issue that arises in various situations in our lives. Graduates of medical school look for hospitals in which they would like to work in their chosen specialty, and the hospitals are looking for
new interns. The graduates have their individual preferences regarding the hospitals, while the hospitals have their preferences relating to whom they would like as interns. How can and ought the aspirations of the two sides be matched? In a similar vein, how should candidates for a teaching position in a university be selected, how should footballers be taken on to join the team squad, and even how should brides and grooms be matched by a matchmaker?

The natural question that arises in this context is what is the best way of filling available positions. To answer this we need to define the criteria for “best,” and then to find a way to arrive at the solution. Two famous mathematicians, David Gale (1921–2008), of the University of California, Berkeley, and Lloyd Shapley, of the University of California, Los Angeles, tackled this problem using the following mathematical approach. It earned Shapley the Nobel Prize in Economics in 2012.

As this falls within the sphere of mathematics, we should first clearly define the framework of the discussion. Here we will focus on a particular instance, but it is not difficult to extend the solution, when we reach it, to other cases that are closer to reality. Assume that we have a group of
N
women and a group of
N
men, and our task is to find a match for each member of the one group with a member of the other. Each man has his priorities regarding the qualities he would like his potential partner to have, and likewise with the women. There is not necessarily any correlation between the preferences of the members of the groups, and if we were to match them randomly, many would be unhappy. In effect, in any system of matchmaking between them, there are likely to be men and women who would not end up with their ideal match and who may even be matched with someone very far from their initial preferences. In such a situation, what is the optimal match? In our context, “match” means the overall matching of all the men and all the women.

The first contribution made by Gale and Shapley was to change the question! Instead of trying to find a criterion for optimality, they formulated a condition, which in the days of the ancient Greeks was called an axiom, that the match had to satisfy. They called this the stability condition. It can be described simply, as follows. The match will be considered
not stable if a man and a women can be found who prefer each other to the partner that the system proposed for them. The condition that Gale and Shapley set was that the match should be stable, that is, it should not be unstable. The reason for that condition is apparent: a match that is not stable will not last. The man and woman who can improve their situation by getting together will do so, and it will disturb the order that the matchmaking had determined. The next stage is to define a match as optimal if every man and woman is matched with the best partner among the stable matches. In other words, first of all the match must be stable, and among the stable matches the match has to be the best for every man and every woman. In a paper published in 1962, titled “College Admissions and the Stability of Marriage,” Gale and Shapley presented an algorithm that led to a stable match. The algorithm can be stated without formulae or equations, as can the proof that it results in a stable match. We will show the algorithm in a quasi-visual form, although clearly, in the computer age it can be produced in an instant, even if it relates to very large populations.

In the first stage every man places himself next to the woman who heads his list of priorities. Some women may have more than one man standing next to them. Each woman selects the man she prefers from those standing next to her, and she sends the others back to their places. In the next stage every man rejected by his first choice, his top priority, stands by the woman who is next in his list of priorities. Again there may be some women with more than one man near them, and each one of these selects the one she prefers from among those men. It may be that the one she chooses was the one she chose in the first stage, but it could also be that one of the newcomers around her now tops her list. The others she sends back to their places. This continues until there is only one man next to each
woman. The algorithm ends, and the result is a stable match. The proof that this is so is that as long as there are at least two men by any woman, the one who is rejected will at the next stage go to the woman who is lower in his order of priorities. The number of such “declines” in the priorities of every man is finite, so that the process is completed when there can be no further declines, that is, there is only one man by each woman. The stability derives from the fact that at each stage the women either stay with the man who chose them previously, or they select a man higher in their own scale of priorities. If a man prefers a woman with whom he is not matched at the end of the process, the fact is that he had already offered himself to her at an earlier stage, but she had preferred to remain with a man who was higher in her own priorities. That woman will therefore not prefer him to the man whom the procedure matched with her, and hence the stability.

What about the optimality conditional on the stability condition, as defined above? Gale and Shapley showed that there are instances in which it is impossible to find an optimal match. They also showed that the algorithm we have described has the characteristic of partial optimality, meaning that every man gets the woman who is highest in his order of priorities from among those he could be matched within a stable match. Herein too lies the difficulty with this particular proposal. If at the outset, instead of every man approaching the woman he preferred we had decided that every woman would approach the man who headed her list, and so on, that procedure would also result in a stable match, which might be different from the final outcome in the process we described. It will be partially optimal in as much as every woman gets the man highest on her priority list from among those with whom she could have been matched as part of a stable match. Which of the two possibilities is preferable? Mathematics does not answer that question; it just offers alternatives and describes their features.