Category Archives: General Economics

Lloyd Shapley (1923-2016)

In March Lloyd Shapley sadly passed away . His paper with D. Gale, “College Admissions and the Stability of Marriage”, is fascinating. I am unsure if Shapley set out in 1962 to answer questions in economics explicitly. I say this because this paper really does speak like an economics paper.  Supposing you are matching pairs of people (i.e. marriage), stability is defined as a position in which any person from at least two pairs can leave the pairing for another also willing to leave. A stable position leaves no more room for ordering. Similar to Pareto optimality, you can’t make another person better off without making some else worse off.

The stable marriage problem and the solution to gain optimality and stability is surprisingly easy to understand. Shapley and Gale wrote the paper in very clear English. In fact, they comment on their use of plain English and precise definitions in the concluding remarks:

Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with “a head of figures,” or that they “know a lot of formulas.” At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1.

Many more results from this seemly simple problem and their solution to it have been discovered, some in fact very simple but only recognized a few years after the paper. The stable marriages problem algorithm is commonly summarized as:

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Result 1: For any equal number of men and women, the algorithm terminates

Simple Proof:

No single man can be rejected by every single woman. This is because that would mean that all the women are married. But since there are equal numbers of woman and men, this is impossible.

Result 2: When men propose all the men have the best pairing they can get in any stable matching, and women have the worst possible

Simple Proof:

Suppose after the first execution of the algorithm, by contradiction, you create a set of pairs so that any man prefers w’. In the original execution this means he was rejected by w’ in place for m’. So in the arbitrary matching there is no stable position for (m’,w’) as both rather be together. Thus in the executed algorithm, and not the contradictory one, not only does giving each man his best possible stable match give the only stable match for the same set over and over but it is also the best the man can do, assuming they are proposing.

For the women’s choice, supposing again a contradictory set of matchings, M’, but compared to the algorithm’s results the woman prefers the m (from the algorithm) matched to her to m’ matched to her. This means that m, w cause M’ to be unstable unless in the initial algorithm (which we know to be optimal for the man) he prefers the match in M’ to w, which contradicts the fact found in the first part of proof, as the man always prefers the outcome from the algorithm (assuming men propose).

There are various other results, and lots of interesting literature by economists (particularly Alvin Roth) discussing stabling matching algorithms applied to various problems. For instance, matching residences to medical students. Also, there is even more literature in computer science on matching.

Some extra reading on Shapley’s influence can be found here and another piece of mathematics he completed to further economics (with Folkman) can be found here.

For detailed proofs (precise in the way Shapely discusses), and a video worth watching discussing some of the theorems.

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Pseudo Dichotomy in Positive Economics

‘What ought to be’ and ‘What is’

The two statements above are clearly equatable with normative economics and positive economics, respectively. This comes to us by Hume, Samuelson and Friedman. These terms I believe are less appreciated outside the glossary list of day one of economics 101 at various institutions. However, more than this I always felt another dichotomy existed. Specifically within positive economics. This secondary dichotomy is not so much a strict split, nor does it muddle the differences between normative and positive economics. There still remains a very distinct difference between what is and what ought to be.

Consider theoretical work on optimal decision making in any specific decision faced by an agent. Now let’s assume the work is game theoretic in nature and the suggestion by the economist is whether a certain decision is the optimal condition. Or rather claims multiple conditions based on multiple parameters are the most optimal. This still is a positive analysis. What is the best is clearly a ‘what is’ statement with no value added of the implications outside the paper for the claims, still that leap could be made easily in this hypothetical case. Though other positive economics statements are sometimes purely empirical and reached through economic tools and econometrics. Saying that “consumers in the tinplate industry faced gains due to increased international trade in the industry” is an empirical positive claim. The dichotomy, existing mostly between empirical and theoretical positive claims, exists only in partial form and is unnecessary to make note of. There are situations when claims by people recently introduced to behavioural economics or any sort of criticism of economic theory, such as expected utility theory come to believe because a purely or mostly theoretical paper using a rational actor is unrealistic it means that it is useless. To borrow Richard Thaler’s phrase for the rational actor, Econs, people who view a positive paper on Econs assume it is replicating reality rather than finding optimality or analyzing choices for the sake of analyzing choices. While there are sometimes gaps in descriptive analysis that behavioural economics has filled in, the pejorative of Econs doesn’t completely apply as a critique of positive economics.

Description isn’t the only acceptable form of economic science. Finding truth and knowledge is. And that can be found even in the strictest and unreal of assumptions. It may seem incredibly unintuitive for an individual who enters economics without any experience. However, if one ponders a research idea or is intrigued by a problem to solve then the economist becomes an engineer in a sense and those “unrealistic” foundations of some models in positive analysis becomes useful guides and benchmarks to normative science. This is why Milton Friedman’s writing on methodology, no matter how brash at first his statements for some on assumptions may seem, the insight into positive analysis in economics is very unique and not a diametric balance between unrealistic (bad/biased) and realistic (good/unbiased) assumptions. The assumptions economists make can be abused similarly to novel cases of p-hacking. However, science is hard and novel examples of unintended or intentional abuse of unrealistic assumptions shouldn’t discredit a field.

So, is the dichotomy I mentioned in positive economics that creates a secondary ‘what ought to be’ and ‘what is’ level of analysis real? Not fully, only a in a crude matter to non-economists like myself. But discussing this crude dichotomy may help some detractors, whether from unorthodox fringe schools of economics or media, understand the scientific method in economics.