The pursuit of an optimal solution in the sphere of Field Service Management (FSM) is an elaborate exercise that involves various considerations, including the necessity for high-level expertise and profound comprehension of the underlying technological framework. Engaging an FSM software provider necessitates an understanding of the provider's capabilities within a larger context that encompasses diverse aspects such as service responsiveness, the extent of automation, and integration capabilities with existing systems, among others.

Firstly, the existential theory of service extends to FSM software, positing that service is a fundamental and existential business activity. Consequently, service level agreements (SLAs) play an imperative role in the engagement of an FSM software provider. The SLA serves as an explicit contract between the client and the provider, outlining the nature of the service, the quality, availability, and responsibilities. It is therefore critical to assess the provider's SLAs for their ability to meet specific business requirements.

Secondly, the provider's system should possess a high degree of automation to improve efficiency and reduce the reliance on manual processes. Advanced FSM software often applies Artificial Intelligence (AI) and Machine Learning (ML) algorithms to automate processes such as work order creation, scheduling, dispatching, and invoicing. The Pareto Principle, also known as the 80/20 rule, may be applicable here - an effective FSM system should automate approximately 80% of processes, allowing human intervention mainly in the remaining 20% of exceptional cases.

Thirdly, in the realm of systems theory, a crucial factor in considering an FSM software provider is its ability to integrate with existing systems. Seamless integration with Enterprise Resource Planning (ERP) systems, Customer Relationship Management (CRM) software, and other relevant systems ensures data coherence and streamlined operations. Interoperability, defined as the ability of diverse systems to work together, is therefore a key metric in the evaluation of an FSM software provider.

Beyond these considerations, the FSM software provider's capabilities to provide real-time tracking and reporting functions are essential. When viewed through the lens of information asymmetry theory, real-time tracking and reporting abilities can significantly reduce the information gap between the provider and the client, leading to better decision-making and planning.

FSM software is not static; hence, the provider's commitment to continuous improvement and software updates is a critical determinant in provider selection. This is underscored in Schumpeter’s theory of creative destruction, where innovation is an essential factor in the competitive process. In this context, the provider must consistently innovate to improve efficiency, service delivery, and the overall user experience.

Finally, the provider's industry-specific experience and expertise should be a significant consideration. The Nash Equilibrium, a fundamental concept in game theory, posits that the best outcome comes from everyone in the group doing what's best for themselves and the group. In this case, an FSM provider with deep industry knowledge and experience would be best for both their business growth and the client's service delivery improvement.

In conclusion, the selection of the right FSM software provider is a complex task that calls for a multi-dimensional approach. The combination of existential theory, systems theory, information asymmetry theory, Schumpeter’s theory of creative destruction, and the Nash Equilibrium provides a robust framework for evaluating the suitability of a provider. This selection process, while sophisticated, promises a rewarding outcome: a partnership that elevates the efficiency, effectiveness, and competitiveness of field service management.