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Orders on Multisets and Discrete Cones

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We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston Conder & Arkadii Slinko

  2. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Simon Marshall

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  1. Marston Conder
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  2. Simon Marshall
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  3. Arkadii Slinko
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Correspondence to Arkadii Slinko.

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Conder, M., Marshall, S. & Slinko, A. Orders on Multisets and Discrete Cones. Order 24, 277–296 (2007). https://doi.org/10.1007/s11083-007-9073-1

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