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K-Fibonacci sequences and minimal winning quota in Parsimonious game

Author

Listed:
  • Flavio Pressacco

    (DIES - DIES - Dept. of Economics and Statistics - Università degli Studi di Udine - University of Udine [Italie])

  • Giacomo Plazzotta

    (Department of Mathematics [Imperial College London] - Imperial College London)

  • Laura Ziani

    (DIES - DIES - Dept. of Economics and Statistics - Università degli Studi di Udine - University of Udine [Italie])

Abstract

Parsimonious games are a subset of constant sum homogeneous weighted majority games unequivocally described by their free type representation vector. We show that the minimal winning quota of parsimonious games satisfies a second order, linear, homogeneous, finite difference equation with nonconstant coefficients except for uniform games. We provide the solution of such an equation which may be thought as the generalized version of the polynomial expansion of a proper k-Fibonacci sequence. In addition we show that the minimal winning quota is a symmetric function of the representation vector; exploiting this property it is straightforward to prove that twin Parsimonious games, i.e. a couple of games whose free type representations are each other symmetric, share the same minimal winning quota.

Suggested Citation

  • Flavio Pressacco & Giacomo Plazzotta & Laura Ziani, 2014. "K-Fibonacci sequences and minimal winning quota in Parsimonious game," Working Papers hal-00950090, HAL.
    • Handle: RePEc:hal:wpaper:hal-00950090
    Note: View the original document on HAL open archive server: https://hal.science/hal-00950090
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    References listed on IDEAS

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    1. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    2. Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.
    3. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
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