The intricate process of product innovation is characterized by the creation and subsequent marketing of new products. This becomes even more challenging when considering how to formulate an accurate budget for such an endeavor. The endeavor entails a number of complex variables, ranging from research and development (R&D) costs, to market studies, prototyping, testing, and eventual product launch.

In this context, product innovation consultants become critical players. These are seasoned professionals equipped with a unique blend of skills in product development, project management, market research, and financial analyses, helping businesses to navigate the labyrinth of product innovation. While the value they bring is undeniable, attention must be paid to ensure an effective budgeting strategy to reap the full benefits of their services.

To begin with, understanding the type and extent of services provided by product innovation consultants is crucial. Some engagements may revolve around refining and enhancing existing products, which could be less resource-intensive than developing a new product from scratch. Other services could involve comprehensive product innovation, from the ideation phase, through to product testing, launch, and post-launch analysis.

The nature of the product also carries considerable weight in the budgeting process. Products with a technological slant, for instance, may require significantly more investment in R&D, prototype development, and compliance with strict industry standards and regulations.

The application of game theory can guide budgeting decisions in these cases. Game theory, in the context of economics, involves understanding the strategic interactions between different 'players' in a scenario, where the 'players' in this case are the business and the innovation consultant. By understanding the best response of the business to the consultant’s actions, and vice versa, optimal outcomes can be achieved.

The services' cost structure, whether it is fixed, hourly, or contingent on success, impacts the budgeting process. Fixed costs can be planned for with a degree of certainty, while variable costs require a probabilistic approach to budgeting. Various statistical methods and forecasting models could be employed to calculate and cater for these costs.

A scenario analysis offers a robust method of dealing with uncertainties. This involves creating different 'what-if' scenarios, each with its own set of assumptions about costs, timelines, and outcomes. This strategy provides a range of possible budgets, each corresponding to a specific scenario.

The presence of economies of scale or scope in the consultant's service delivery model must also be accounted for. The principles of economies of scale and scope, drawn from microeconomics, suggest that the cost per unit of output decreases with an increase in the scale or variety of output. If these exist, it may be more cost-effective to bundle multiple projects or tasks together.

However, the budgeting process does not end at the planning phase. A time-series analysis, a statistical technique that deals with time-based data, can be used to track the efficiency and effectiveness of the consultant's services over time, allowing for real-time adjustments to the budget.

In conclusion, budgeting for product innovation consultancy services is a multifaceted process fraught with complexities. By understanding the specifics of the consultancy services, the nature of the product, the cost structure, and the potential for economies of scale and scope, a business can create a comprehensive budget. Applying principles from game theory, statistical methods and scenario analysis further enhances the robustness of the budgeting process.

Despite the complexity, a well-structured and meticulously planned budget is a cornerstone for successful product innovation. It not only ensures financial sustainability but also fosters a conducive environment for creativity and innovation to flourish.