KöMaL 2019 A757

For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit 1. Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\ge k$. Let $X$ denote the set of integers $\{0,1,\dots,2^n-1\}$, let $Y$ denote the set of integers $\{0,1,\dots,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\le i\le k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$. Player $B$ wins the game, if for every $1\le i\le k$ and $1\le j\le k$ $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy?