If the costs of budget and contest transfers were asymmetric, the dynamics of alliance formation in the coalitional Blotto game would likely become more complex. The existence of mutually beneficial transfers would depend on the specific cost structures associated with each type of transfer. For instance, if the cost of budget transfers (1 - β) were significantly higher than that of contest transfers (1 - γ), players might be less inclined to engage in budget transfers, even if such transfers could lead to improved payoffs. This could result in a scenario where players prioritize contest transfers over budget transfers, leading to a shift in strategy.
In this asymmetric cost setting, the conditions for mutually beneficial transfers would need to be re-evaluated. Theorems similar to Theorem 1 and Theorem 2 would need to be adapted to account for the different cost structures, potentially leading to new necessary and sufficient conditions for the existence of beneficial transfers. The analysis would also need to consider how the adversary's optimal allocation strategy might change in response to these asymmetric costs, as the players' payoffs would be influenced by the relative effectiveness of budget versus contest transfers.