Allocating scarce budgetary resources across the strategic objectives of the balanced scorecard (BSC) and the sustainability balanced scorecard (SBSC) is a central problem in sustainability-oriented strategic management. Existing approaches typically operate at the level of aggregated perspectives, rely on subjective weighting schemes, or ignore the causal structure of the strategy map. This article formalizes the problem as a cooperative game in which each strategic objective is a player and the value of every coalition combines the individual performance of its members with the synergies arising from the causal relationships of the strategy map, estimated through the decision-making trial and evaluation laboratory (DEMATEL). The Shapley value is adopted as the allocation rule. The resulting game is shown to be superadditive and convex, the Shapley value belongs to the Core, and the rule satisfies exhaustiveness, fairness, sensitivity to the strategy map, monotonicity, and convergence to the proportional rule. Computational experiments and sensitivity analyses support the robustness of the model. In an illustrative case—a sustainability balanced scorecard with ten objectives organized in five perspectives, whose dedicated sustainability perspective hosts a carbon-emission and an employee-safety objective, and a budget of USD 1M—the framework shifts $2.7\%$ of the total budget relative to a proportional allocation (individual objectives change by up to $12.3\%$), funds employee safety above financial cost reduction for any synergy weight $\gamma \ge 0.15$, keeps the combined share of the sustainability objectives stable within $0.6$ percentage points across the entire synergy range, and grants seed funding to sustainability objectives that enter the map without a measured baseline, providing a transparent and auditable basis for sustainable budget allocation.