Inefficient Alliance Formation in Coalitional Blotto Games

In adversarial resource allocation settings, forming alliances can enhance performance, but the benefits may diminish if alliance formation becomes costly. In this work, we explore this issue using the framework of the coalitional Blotto game, in which two players compete separately against a common adversary across valued contests. Each player aims to win contests by allocating more of their limited budget than their opponent. Previous work on alliance formation in coalitional Blotto games has shown that if one player transfers a portion of their budget to the other, then both players can perform better; however, it is also known that it is never in either players' interest to simply concede a portion of their budget. In this work, we study the setting in which transfers are costly, meaning that if one player donates a portion of their budget, the recipient only receives a fraction of the transferred amount. We show that mutually beneficial costly transfers exist, and we provide necessary and sufficient conditions for the existence of such a costly transfer. Then, we consider the setting in which players can transfer budgets and contests at a cost, and we show that this alliance strategy is mutually beneficial in almost all game instances.