Description
Some classical 2-player combinatorial games (alternating play, last move
wins etc) display interesting number theoretical properties in perfect
play. This holds in particular for the game of Wythoff Nim (Wythoff
1907), with its many variations, and connections with the Golden ratio,
the Fibonacci numbers, Beatty sequences, and more. We review recent
development, in particular with respect to the question: when does a
given sequence of numbers represent the solution to a game? For the
second part of the talk, we discuss a class of `economic combinatorial'
games: by moving, the players gather pebbles from a common finite heap
in the pursuit of maximizing their final utilities. We show that, if the
subtraction set is finite, then the sequence of optimal play utilities
is eventually periodic, which confirms a conjecture by Stewart 2011. A
game converges at heap size $x$ if `the maximal optimal action' is the
same for all heap sizes of size $x$ or larger. We finish off with a
conjecture of duality between convergence of individual games and
numbers of games with certain convergence numbers. The second half of
the talk is a collaboration with Gal Cohensius, Reshef Meir and David
Wahlstedt. The first part is a blend of many projects including
collaborations with the following authors: P. Hegarty, A. S. Fraenkel,
J. Wästlund, M. Cook, A. Landsberg, E. Friedman, R. J. Nowakowski, M.
Fisher, M. Weimerskirch, D. Iannucci, and E. Duchene.