A special balancing game, Victor Grinberg's question

In a vector balancing game, on step $k$ player 1 chooses a unit vector $v_k$ and player 2 chooses a sign ${\epsilon }_k\in \{-1,1\}$, and the position after $n$ steps is $z_n={\sum }_1^n{\epsilon }_k{v}_k$. Player 1’s target is to make $\mathrm{\parallel }{z}_n\mathrm{\parallel }$ large and player 2’s target is to make it small. We consider a special case of this game.