We investigate the convexity of cooperative games arising from network flow problems. While it is well-known that flow games are totally balanced, a complete characterization of their convexity has remained an open problem. In this paper, we provide a necessary and sufficient characterization of the networks that induce convex flow games. We show that a flow game is convex if and only if the underlying network is acyclic and admits an arc cover by $s$-$t$ paths that are disjoint at their bottleneck arcs. Specifically, every bottleneck arc must belong to exactly one path, and every non-bottleneck arc must possess sufficient capacity. To derive this characterization, we establish six structural properties of convex flow games. Additionally, we prove that our characterization can be verified efficiently, yielding a polynomial-time algorithm to recognize convex flow games. Since the class of flow games coincides exactly with the class of non-negative totally balanced games, as established by Kalai and Zemel (1982), our structural and algorithmic characterization applies to all such games, provided they are represented in their network form.

翻译：本文研究网络流问题产生的合作博弈的凸性。众所周知流博弈是完全平衡的，但其凸性的完整刻画一直是一个开放问题。本文给出了诱导凸流博弈网络的充分必要条件。我们证明：一个流博弈是凸的当且仅当底层网络是无环的，并且存在由瓶颈弧上不相交的$s$-$t$路径组成的弧覆盖。具体而言，每条瓶颈弧必须恰好属于一条路径，且每条非瓶颈弧必须具有足够容量。为推导该刻画，我们建立了凸流博弈的六项结构性质。此外，我们证明该刻画可被高效验证，从而得到识别凸流博弈的多项式时间算法。由于Kalai与Zemel（1982）已证明流博弈类恰好与非负完全平衡博弈类重合，我们的结构与算法刻画适用于所有以网络形式表示的非负完全平衡博弈。