by Andreas Hamel
Suppose you want to go to a restaurant for dinner, and all what counts for you is the quality of the meals. Suppose further that there are two restaurants A and B in town, and by looking at their menus on your smartphone you realize that for each meal on the menu of A there is one on the menu of B which you like better or at least as much as the one from A. Would you reserve a dinner table in restaurant A?
Under the given circumstances, you certainly wouldn’t. Of course, there could be other criteria; for example, you are on a budget and prices in A are considerably lower than the ones in B. However, the assumption here is that money’s no object for your decision (for once) – only the quality of the meals counts.
There are plenty of examples for similar decisions: in which shop to buy shoes or furniture, or which bank or (component) supplier to choose, or in which region to look for a vacation spot. In such cases, the decision has a two-stage character. First, one must choose a provider for a product or service among several (a restaurant, a shoe or furniture shop, a bank or contractor, a region), and only in a second step one chooses a particular product or service (a meal in a restaurant, a pair of shoes or a bed, a special (financial) service, a hotel).
The special feature of the first stage decision is that one has to compare sets instead of single items: a set is a whole bunch or a collection of things like the set of meals on the menu of a restaurant or the set of hotels in a region. It may seem difficult to compare sets; however, with the restaurant example above in mind the following comparison method suggests itself. If for each item in the set A there is one in the set B which is a least as good, then one should prefer the set B over the set A.
This comparison method for sets has been known to computer scientists, economists and decision makers for quite some time; in fact, the restaurant example above is from an article written by Stanford professor for economics David Kreps
which was published already 1979 in Econometrica, a very high-level scientific journal.
Surprisingly, mathematics contributed very little to theory and applications of such set comparison methods although there is a huge part of mathematics, called order theory, which exclusively deals with comparison methods (order structures, in math terms). In particular, the question how to choose the “best” set among several candidates has not been investigated from a mathematical point of view until recently. Why so? What are the difficulties when it comes to comparing sets?
The alert reader may have realized that the comparison method for restaurant menus does not provide an answer to the question which restaurant is ”the best;” it only admits to exclude some restaurants (such as A above). It could happen that for some dish on the menu of B there is no dish on the menu of another restaurant C which is equally good or better, and the same vice versa. In this case, the set comparison method does not produce a favorite; the two restaurants B and C cannot be compared with respect to this method (only B is preferred over A, and therefore A is excluded). This might happen, for example, if the two restaurants B and C are both pizza-and-pasta places, but B has (much) better pasta whereas C offers the (far) superior pizza.
This shows that the dilemma already appears on the level of the single items: it might not make much sense to force a decision if an excellent pizza ”Quattro Stagioni” or delicious “Spaghetti Carbonara” dish is preferable for someone who likes pizza as much as pasta – but on different occasions. So, very often there are already non-comparable single items: high heels and sneakers are not comparable, they just serve different purposes.
Human beings usually become very stressed when faced with decisions among alternatives including some which are not really comparable. At a university, one might think of hiring procedures for new professors or the admission procedure for new students. An easy way out is to introduce another, in many cases quite artificial, ranking which makes everything comparable with everything else. Often, we just compare prices or “points” instead of items with several features or persons, respectively. In economics, this has been a standard assumption for a long time: the preference of a decision maker or an economic agent is assumed to be total, i.e., there are no non-comparable alternatives. Economists, like most human beings, do not like to be stressed. David Kreps also made this assumption for comparisons of single items in his 1979 Econometrika article, and it was inherited to the set comparison method.
In the past 15 years, a new part of mathematical optimization theory emerged which deals with two questions simultaneously. First, how to compare sets and select “best” ones. And secondly, how to deal with such questions in the presence of non-comparable alternatives (on the single-item level as well as on the set level). This new area is called “Set Optimization” and is the main research topic of the author. It already led to many new concepts as well as to new applications in finance, economics, game theory and even statistics.
This new theory can help to reduce the stress in such two-stage decisions by designing a model of the situation and then apply some mathematical results to the model; incidentally, it often turns out that many decision-making procedures which are usually done on the basis of a total ranking are just set comparison methods in disguise or could be done better using the latter.
So, why not design a smartphone app for restaurant or netflix movie selection based on set optimization methods? Any ideas and contributors are welcome. At Free University of Bozen, Faculty of Economics and Management, a “Center for Set Optimization and Applied Sciences” is one of the very few places in the world which bundles expertise and focuses research in this new and exciting area.
A conference series has been established whose fourth edition took place at Friedrich Schiller University Jena in February 2019 (the second one was held 2014 in Bruneck), see
and a state-of-the-art volume (if you like to see some serious math) is here