Volume 5 • Issue 3 | November 2021

Social choice using moral metrics

Kenneth Halpern


This paper aims to develop a geometry of moral systems. Existing social choice mechanisms predominantly employ simple structures, such as rankings. A mathematical metric among moral systems allows us to represent complex sets of views in a multidimensional geometry. Such a metric can serve to diagnose structural issues, test existing mechanisms of social choice or engender new mechanisms. It also may be used to replace active social choice mechanisms with information-based passive ones, shifting the operational burden.

Under reasonable assumptions, moral systems correspond to computational black boxes, which can be represented by conditional probability distributions of responses to situations. In the presence of a probability distribution over situations and a metric among responses, codifying our intuition, we can derive a sensible metric among moral systems.

Within the developed framework, the author offers a set of well-behaved candidate metrics that may be employed in real applications. The author also proposes a variety of practical applications to social choice, both diagnostic and generative.

The proffered framework, derived metrics and proposed applications to social choice represent a new paradigm and offer potential improvements and alternatives to existing social choice mechanisms. They also can serve as the staging point for research in a number of directions.


  1. Arrow, K.J. (1950), “A difficulty in the concept of social welfare”, Journal of Political Economy, Vol. 58 No. 4, pp. 328-346.
  2. Crippen, G. (1978), “Rapid calculation of coordinates from distance matrices”, Journal of Computational Physics, Vol. 26, pp. 449-452.
  3. Eckart, C. and Young, G. (1936), “The approximation of one matrix by another of lower rank”, Psychometrika, Vol. 1 No. 3, pp. 211-218.
  4. Hamming, R.W. (1950), “Error detecting and error correcting codes”, The Bell System Technical Journal, Vol. 29 No. 2, pp. 147-160.
  5. Levenshtein, V.I. (1966), “Binary codes capable of correcting deletions, insertions and reversals”, Soviet Physics–Doklady, Vol. 10 No. 8, pp. 707-710.
  6. Matousek, J. (2002), Lectures on Discrete Geometry, Springer-Verlag, New York, NY, available at: https://link.springer.com/book/10.1007/978-1-4613-0039-7#about.
  7. Matousek, J. (2013), “Lecture notes on metric embeddings”, available at: https://kam.mff.cuni.cz/∼matousek/ba-a4.pdf.
  8. Mitchell, T.M. (1997), Machine Learning, McGraw-Hill, New York, NY, available at: https://www.worldcat.org/title/machine-learning/oclc/36417892.
  9. Mohri, M. (2018), Foundations of Machine Learning, 2nd ed., MIT Press, Cambridge, MA, available at: https://dl.acm.org/doi/10.5555/2371238.
  10. Rao, C.R. (1945), “Information and accuracy attainable in the estimation of statistical parameters”, Bulletin of the Calcutta Mathematical Society, Vol. 37 No. 3, pp. 81-91.
  11. Vapnik, V.N. (1999), The Nature of Statistical Learning Theory, 2nd ed., Springer-Verlag, New York, NY, available at: https://link.springer.com/book/10.1007/978-1-4757-3264-1#about.
  12. Young, G. and Householder, A. (1938), “Discussion of a set of points in terms of their mutual distances”, Psychometrika, Vol. 3, pp. 19-22.