In many practical situations, we need to change the spatial distribution of some goods. In such situations, it is desirable to minimize the overall transportation costs. In the 1-D case, the smallest transportation cost of such a change is proportional to what is known as the Wasserstein metric. The same metric can be used to describe the distance between two probability distributions. In the last decades, it turned out that this metric can be successfully used in many economic applications beyond transportation. These successes are somewhat of a mystery: in principle, there are many different metrics on the set of all probability distributions, and it is not immediately clear why namely Wasserstein metric is so successful. In this paper, we show that the Wasserstein metric naturally appears in decision making situations. This fact explains the usefulness of the Wassterstein’s metric in economic applications.

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References

1. R. Belohlavek, J. W. Dauben and G. J. Klir , Fuzzy Logic and Mathematics: A Historical Perspective (Oxford University Press, New York, 2017). Google Scholar
2. P. C. Fishburn , Utility Theory for Decision Making (John Wiley & Sons Inc., New York, 1969). Google Scholar
3. P. C. Fishburn , Nonlinear Preference and Utility Theory (The John Hopkins Press, Baltimore, Maryland, 1988). Google Scholar
4. A. Galishon , Optimal Transport Methods in Economics (Princeton University Press, Princeton, New Jersey, 2018). Google Scholar
5. G. Klir and B. Yuan , Fuzzy Sets and Fuzzy Logic (Prentice Hall, Upper Saddle River, New Jersey, 1995). Google Scholar
6. V. Kreinovich , “ Decision making under interval uncertainty (and beyond)”, in Human-Centric Decision-Making Models for Social Sciences, P. Guo and W. Pedrycz (eds.) (Springer Verlag, 2014), pp. 163–193. Crossref, Google Scholar
7. R. D. Luce and R. Raiffa , Games and Decisions: Introduction and Critical Survey (Dover, New York, 1989). Google Scholar
8. J. M. Mendel , Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions (Springer, Cham, Switzerland, 2017). Crossref, Google Scholar
9. H. T. Nguyen, O. Kosheleva and V. Kreinovich , “ Decision making beyond Arrow’s ‘impossibility theorem’, with the analysis of effects of collusion and mutual attraction”, International Journal of Intelligent Systems 24(1) (2009) 27–47. Crossref, Web of Science, Google Scholar
10. H. T. Nguyen, V. Kreinovich, B. Wu and G. Xiang , Computing Statistics under Interval and Fuzzy Uncertainty (Springer Verlag, Berlin, Heidelberg, 2012). Crossref, Google Scholar
11. H. T. Nguyen, C. L. Walker and E. A. Walker , A First Course in Fuzzy Logic (Chapman and Hall/CRC, Boca Raton, Florida, 2019). Google Scholar
12. V. Novák, I. Perfilieva and J. Močkoř , Mathematical Principles of Fuzzy Logic (Kluwer, Boston, Dordrecht, 1999). Crossref, Google Scholar
13. H. Raiffa , Decision Analysis (McGraw-Hill, Columbus, Ohio, 1997). Google Scholar
14. L. N. Vasershtein , “ Markov processes over denumerable products of spaces, describing large systems of automata”, Problems of Infromation Transmission 5(3) (1969) 47–52. Google Scholar
15. C. Villani , Optimal Transport, Old and New (Springer, Berlin, Heidelberg, 2016). Google Scholar
16. L. A. Zadeh , “ Fuzzy sets”, Information and Control 8 (1965) 338–353. Crossref, Web of Science, Google Scholar