Research projects

Project CEMAPRE internal

TitleGeneral method to identify numbers in CGT
ParticipantsAlda Carvalho (Principal Investigator), Carlos Santos
SummaryIt 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.

[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.