A polynomial algorithm for a two parameter extension of Wythoff NIM based on the Perron–Frobenius theory

Endre Boros (), Vladimir Gurvich () and Vladimir Oudalov ()

International Journal of Game Theory, 2013, vol. 42, issue 4, 915 pages

Abstract: For any positive integer parameters a and b, Gurvich recently introduced a generalization mex b of the standard minimum excludant mex=mex 1 , along with a game NIM(a, b) that extends further Fraenkel’s NIM=NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM=NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a, b) are given by the following recursion: $$x_n={\rm mex}_b(\{x_i, y_i \;|\; 0 \leq i > n\}), \;\; y_n=x_n + an; \;\; n \geq 0,$$ and conjectured that for all a, b the limits ℓ(a, b)=x n (a, b)/n exist and are irrational algebraic numbers. In this paper we prove that showing that $${\ell(a,b)=\frac{a}{r-1}}$$ , where r > 1 is the Perron root of the polynomial $$P(z)=z^{b+1} - z - 1 - \sum_{i=1}^{a-1} z^{\lceil ib/a \rceil},$$ whenever a and b are coprime; furthermore, it is known that ℓ(ka, kb)=kℓ(a, b). In particular, $${\ell(a, 1)=\alpha_a=\frac{1}{2} (2 - a + \sqrt{a^2 + 4})}$$ . In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting $${x_n=\lfloor \alpha_a n \rfloor, \; y_n=x_n + an=\lfloor (\alpha_a + a) n \rfloor}$$ . Here we provide a polynomial time algorithm based on the Perron–Frobenius theory solving game NIM(a, b), although we have no explicit formula for its kernel. Copyright Springer-Verlag 2013

Keywords: Combinatorial and impartial games; NIM; Wythoff’s NIM; Fraenkel’s NIM; Minimum excludant; Algebraic number; Asymptotic; Kernel; C02; C73 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s00182-012-0338-6

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