On a question of Matt Baker regarding the dollar game

TL;DR

This paper establishes a tight lower bound on the minimum number of moves needed for an optimal strategy in the dollar game on a graph, improving understanding of move efficiency beyond the borrowing binge strategy.

Contribution

It provides the first lower bound relating optimal move count to the borrowing binge strategy in the dollar game, advancing theoretical understanding.

Findings

Lower bound: M_min ≥ M_0 / (n-1)

Bound is tight, confirming optimality

Enhances understanding of move strategies in the dollar game

Abstract

In an introductory paper on dollar game played on a graph, Matt Baker wrote the following: ``The total number of borrowing moves required to win the game when playing the 'borrowing binge strategy' is independent of which borrowing moves you do in which order! Note, however, that it is usually possible to win in fewer moves by employing lending moves in combination with borrowing moves. The optimal strategy when one uses both kinds of moves is not yet understood.'' In this article, we give a lower bound on the minimum number $ M_{\text{min}} $ of such moves of an optimal algorithm in terms of the number of moves $ M_0 $ of the borrowing binge strategy. Concretely, we have: $ M_{\text{min}} \geq \frac{M_0}{n-1} $ where $ n $ is the number of vertices of the graph. This bound is tight.