A special balancing game, Victor Grinberg's question

TL;DR

This paper investigates a specific vector balancing game where two players select vectors and signs to maximize or minimize the norm of their sum, analyzing strategic interactions and outcomes in this setting.

Contribution

It introduces a particular variant of the vector balancing game and analyzes the strategic dynamics and potential outcomes of this special case.

Findings

Player strategies influence the norm growth or reduction.

Optimal strategies depend on the sequence of chosen vectors.

The game reveals insights into balancing and anti-balancing tactics.

Abstract

In a vector balancing game, on step $k$ player 1 chooses a unit vector $v_k$ and player 2 chooses a sign $\varepsilon_k\in \{-1,1\}$, and the position after $n$ steps is $z_n=\sum_1^n\varepsilon_k v_k$. Player 1's target is to make $\|z_n\|$ large and player 2's target is to make it small. We consider a special case of this game.