In the world of communication, the ability to articulate your thoughts effectively is paramount. This is where the role of a public speaking coach comes into play. A profound understanding of the nuances of public speaking, from voice modulation and body language to content structure, can be instrumental in empowering an individual or organization to communicate with an audience powerfully and persuasively.

Before diving into the process of hiring a public speaking coach, let’s delve into the concept of public speaking itself. Deriving from the Greek word 'rhetoric', public speaking is the art of using language to persuade or motivate an audience. In Ancient Greece, this skill was considered necessary for advocates and politicians. Fast forward to our modern era, the ability to communicate effectively has transcended the sphere of politics and advocacy, becoming an indispensable competency in a wide array of fields, be it business, social activism, or academia.

The selection of a public speaking coach, therefore, is a decision of immense importance. This process requires a meticulous analysis, a comparison of alternatives, and the formulation of a choice - all conducted within a framework reminiscent of a multi-criteria decision analysis (MCDA). The MCDA is a sub-discipline of operations research that explicitly considers multiple criteria in decision-making environments.

The 'coach selection' MCDA would consist of various stages, each with its own set of criteria. These stages could broadly be identified as: identifying potential coaches, evaluating their credentials and experience, conducting preliminary interviews, and finally, making a choice based on a numerical rating system.

The first stage - identifying potential coaches - is contingent on a number of variables. Chief among them is the issue of proximity. In today's digitized world, engagements can easily happen remotely via technology platforms like Zoom or Skype. However, some individuals may prefer a more traditional, face-to-face interaction. Here, the principle of trade-offs applies: remote coaching might offer more flexibility, while in-person coaching might provide a more personalized experience.

The evaluation of credentials and experience, the second stage in our MCDA, is a critical one. At this juncture, the Pareto Principle, or the 80/20 rule, could potentially be applied. This economic theory, named after the Italian economist Vilfredo Pareto, posits that roughly 80% of consequences come from 20% of the causes. In the context of hiring a public speaking coach, it suggests that a significant proportion of a coach's effectiveness is likely to come from a small subset of their overall skills and experience.

The preliminary interviews could be likened to a series of mini-experiments, where each potential coach is given an opportunity to demonstrate their coaching style and methodology. This stage is influenced by the principles of game theory, where each 'player' (in this case, the coach) seeks to present themselves in the best light to win the 'game' (i.e., get hired).

The final stage - making a decision - could be based on a numerical rating system informed by a Likov scale. Developed by the psychologist Rensis Likov, this scale allows for a subjective evaluation of various factors, ranging from a coach's feedback style to their interpersonal skills.

In conclusion, the process of hiring a public speaking coach is a complex one, shaped by economic principles, psychological theories, and operational research methodologies. It combines quantitative analysis with qualitative judgment, all aimed at selecting a coach who can provide the client with the most value in terms of developing their public speaking prowess. The ultimate goal is not just to hire a coach, but to forge a partnership that will enhance one's communicative abilities and consequentially, their influence over the audience.