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Modeling Infinite Games on Finite Graphs Using Numerical Infinities

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Numerical Infinities and Infinitesimals in Optimization

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 43))

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Abstract

In his seminal work, Robert McNaughton (see [14] and [10]) developed a model of infinite games played on finite graphs. Here is presented a new model of infinite games played on finite graphs using the Grossone paradigm. The new Grossone model provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner.

This chapter is dedicated in loving memory to my wife Zana, who has and will always be my motivation and my inspiration.

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Notes

  1. 1.

    In [19], Sergeyev formally presents the divisibility axiom as saying for any finite natural number n sets \(\mathbb {N}_{k,n}, \; 1\le k\le n\), being the nth parts of the set \(\mathbb {N}\), have the same number of elements indicated by the numeral \(\frac{\textcircled {1}}{n}\) where

    $$\begin{aligned} \mathbb {N}_{k,n}=\{k,k+n,k+2n,k+3n,...\}, \; 1\le k \le n,\; \bigcup ^n_{k=1}\mathbb {N}_{k,n}=\mathbb {N} \end{aligned}$$

    and illustrates this with examples of the odd and even natural numbers.

  2. 2.

    Here we use the notion of complete taken from [19], that is the sequence contains \(\textcircled {1}\) elements.

  3. 3.

    It is noted here that, as is usual, the \(\subset \) symbol can also imply equality.

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Authors and Affiliations

  1. York College, The City University of New York, Jamaica, Queens, NY, 11451, USA

    Louis D’Alotto

  2. The Graduate Center, The City University of New York, 356 Fifth Avenue, New York, NY, 10016, USA

    Louis D’Alotto

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  1. Louis D’Alotto
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Correspondence to Louis D’Alotto .

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Editors and Affiliations

  1. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. Scuola di Scienze e Tecnologie, UniversitĂ  degli Studi di Camerino, Camerino, Italy

    Renato De Leone

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D’Alotto, L. (2022). Modeling Infinite Games on Finite Graphs Using Numerical Infinities. In: Sergeyev, Y.D., De Leone, R. (eds) Numerical Infinities and Infinitesimals in Optimization. Emergence, Complexity and Computation, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-93642-6_12

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