## Project CEMAPRE internal

Title | General method to identify numbers in CGT |

Participants | Alda Carvalho (Principal Investigator), Carlos Santos |

Summary | It is well known that there are rulesets whose positions only have numbers as game-values and rulesets that may admit values other than numbers [1]. A notable example of the first class is blue-red-hackenbush [1, 2]. However, most rulesets belong to the second class. When analyzing games, an early question is: is it possible that all the positions are numbers? If that is true, then it is easy to determine the outcome of a disjunctive sum of positions, just add up the numbers. It is also easy to find the best move, just play the summand with the largest denominator. The problem is how to recognize when all the positions are numbers. Siegel [3], page 81, states «If every incentive of G is negative then G is a number». This does not provide much insight or intuition. In fact, in most non-all-small-games, there are non-zero positions, some of which are numbers and others not. Let S be a set of positions of a ruleset. It is called a hereditary closed set of positions of a ruleset (HCR) if it is closed under taking options and disjunctive sum. These HCR sets are the natural objects to consider. There are two properties either of which, if satisfied for all followers of a position, tells us that the position is a number. This fact is known, but the proposed general method, based on two fundamental properties, is not described in literature. References: [1] J. Conway. On Numbers and Games. Academic Press, 1976. [2] E. Berlekamp, J. Conway, R. Guy. Winning Ways. Academic Press, London, 1982. [3] A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013. |