Economists opine that the essence of budgeting is to manage the allocation of scarce resources to the most valuable ends. This axiom equally applies to the corporate world, especially in the realm of media asset management software (MAMS). MAMS, for the uninitiated, refers to a specialized software used by organizations to organize, store, retrieve, and share media files such as videos, photos, animations, and music. The fundamental question that any rational entity grappling with the decision to invest in MAMS must answer is: How do we strategically budget for this critical tool?

In answering this question, one must think like a chess grandmaster making a series of calculated moves. Deciding on the when, where, and how of purchasing and implementing MAMS calls for a delicate interplay of historical analysis, law of demand, economic modeling, statistics, and advanced business strategies.

The underlying cornerstone in this budgeting process starts with a thorough understanding of the organization's needs. A careful historical analysis of media asset usage and future projections is crucial. This will call for combing through past data on media asset usage trends, the volumes handled, the cost implications, and how these variables are projected to change in the future. In this stage, concepts drawn from statistical analytics, such as regression analysis, may be employed to give a data-driven perspective on future needs.

Next, comes the examination of potential solutions available in the market. Here, the law of demand and its underlying tenets of price elasticity come into play. Different MAMS vendors will offer varying price points, each with their unique features and capabilities. Some solutions may be cost-effective, but perhaps they lack in advanced features. Others may be feature-rich but come at a hefty price. The key is to strike a balance between price and value, keeping in mind the budget constraints.

Furthermore, one must also understand that the cost of MAMS goes beyond the mere purchasing price. There are additional costs such as implementation, training, maintenance, and possible future upgrades. A comprehensive budgetary plan should, therefore, take into account these ancillary costs. This calls for a holistic understanding of the total cost of ownership (TCO), a concept rooted in financial management and economics.

The decision also calls for a deep understanding of the legal landscape. Different jurisdictions have varying laws about data protection, privacy, and intellectual property rights. It would be a cardinal sin to invest heavily in a MAMS system only to find it's non-compliant with some legal requirements. In such a case, the sunk cost theorem, an economic concept that warns against considering irrecoverable costs in making future decisions, would have to be invoked.

In conclusion, the strategic budgeting for MAMS isn't just about numbers. It's about making smart, informed decisions that are aligned with the organization’s overall strategy. It calls for a deep understanding of a broad array of disciplines, from history and law to economics, mathematics, statistics, and social sciences. Just like a chess grandmaster, the decision-makers need to think several moves ahead, anticipate challenges, and make decisions that ensure a checkmate in the organizational media asset management game.

In the grand scheme of things, MAMS is not just a tool but an investment in the future. Therefore, the budgeting process should be treated with the strategic importance it deserves. It is a journey that calls for intellectual rigor, precision, and sound judgment, traits that Harvard graduates are known for. Remember, a well-executed budgeting strategy for MAMS is a checkmate move in the never-ending game of organizational efficiency.