In the world of business, the exigency to ensure a fair sailing ship is essential, especially in stormy commercial waters. However, there are instances when the vessel, metaphorically referred to as an enterprise, is found floundering amidst the tempest. This is where the need for a competent captain, commonly known as a business turnaround consultant, becomes indispensable.

A business turnaround consultant is a professional who assists an organization in its recovery from a period of poor performance. They specialize in diagnosing the root causes of a business's problems, formulating strategic plans, and implementing changes in a way that it leads to restoration, growth, and eventual prosperity.

Creating a budget for hiring a business turnaround consultant is a critical, albeit challenging, task. It involves striking a balance between the allocation of financial resources and the expected return from the consultant's intervention.

First, there's a need to understand the gravity of the situation. The financial woes of the business must be quantitatively defined to determine the extent of assistance required. This could be gauged through the analysis of financial ratios, trend lines, or through more advanced statistical methods such as regression analysis or survival analysis.

Next comes the identification and quantification of the cost components associated with hiring a consultant. This includes direct costs such as fees and expenses, and indirect costs like potential opportunity costs.

The consultant’s fees could be structured in various ways, each with its own trade-offs. A fixed fee arrangement assures certainty of cost but may not incentivize the consultant to deliver quick results. A performance-based fee, on the other hand, ties the consultant’s remuneration to the achievement of specific targets, incentivizing better performance but potentially leading to higher costs. Another alternative could be an equity-based arrangement, where the consultant receives a share in the company instead of a fee. While this aligns the interests of the consultant with the long-term success of the business, it might dilute the ownership structure.

Determining the budget should also take into account the potential return on investment. This involves estimating the potential increase in profits or reduction in losses that could accrue as a result of the consultant’s intervention. Applications of game theory, specifically the concept of Nash equilibrium, may aid in predicting the potential outcomes of different strategies employed by the consultant, aiding the decision-making process.

It's crucial, however, to remember that these return estimations are based on assumptions and predictions, and therefore, carry an inherent risk of not materializing.

In the final step, the estimated costs and potential returns are compared and analyzed. This is done using metrics such as net present value (NPV), internal rate of return (IRR), or payback period. Each metric offers a different perspective on the budget and should be used according to the company's specific situation and objectives. For instance, while the NPV provides an estimate of the net increase in the company's value, the IRR provides the rate of return that would make the investment in a consultant economically neutral.

In conclusion, budgeting for a business turnaround consultant requires a deep understanding of the company's current financial state, a clear idea of the cost structure, an estimation of potential returns, and a robust method of comparing these costs and benefits. By taking a systematic, analytical approach, business leaders can help ensure that they make a sound, informed decision.