I call games similar to the one I describe below to be Markov games. I am selecting just that one or rather a 1-parameter series of games. The open challenge is to find out which of the players $\ 0\ $ or $\ 1\ $ has a winning strategy for each of the given parameter $\ W.$

NOTATION $\ n\%2=0\ $ for $\ n\ $ even, and $\ n\%2=1\ $ for $\ n\ $ odd.;

Let $ d(0)=J(0)=0.\ $ For arbitrary positive integer $\ n,\ $ player $\ n\%2\ $ selects a positive integer $\ d(n)\le d(n-1)+1;\ $ then $\ J(n)=J(n-1)+d(n).$

When players compete at game $\ M(W),\ $ where $\ W\ $ is an arbitrarily fixed positive integer, then the player that gets exactly $J(n)=W\ $ wins.

Let $\ \omega(W)=0\ $ if player $0$ has a winning strategy at $M(W);\ $ otherwise let $\ \omega(W)=1\ $ if player $1$ has a winning strategy at $M(W)$.

PROBLEM:   Compute function $\ \omega:\mathbb N\to\{0\ 1\}.$

For instance: $\ \omega(1)=1;\ \omega(2)=\omega(3)=0;\ \omega(4)=1, $ etc.   However, $\ \omega(120)\ $ or $\ \omega(5553)\ $ is a bit harder (and unknown to me).