Initiating a home improvement project involving a deck often necessitates the expertise of a deck staining company. However, the process of hiring the right company can seem like a daunting task, especially when it comes to understanding the nuances involved in the selection process. Let us delve into the particularities of making an informed choice, leveraging concepts from various fields such as economics and statistics.

Firstly, consider the "Principal-Agent Problem," a concept from economics. In this situation, you (the principal) are hiring a deck staining company (the agent) to perform a task. However, you may not be fully equipped to evaluate the quality of the work, creating an information asymmetry that the agent could potentially exploit. Therefore, it is crucial to gather sufficient information about the company to mitigate this risk.

A good starting point is to investigate the company's reputation. This entails examining online reviews and ratings, as well as soliciting feedback from previous clients. If we draw from the field of statistics, we can consider these reviews as a random sample from the population of all the company's projects. If the sample size (number of reviews) is large enough and the mean rating is high, we can be relatively confident that the company consistently delivers quality work.

However, it's important to be aware of "Selection Bias," another statistical concept. People who had an exceptionally good or bad experience are more likely to leave a review. This could skew the average rating and not provide an accurate representation of the company’s typical performance.

In addition to reputation, consider the company's experience and specialization. A company that has been in the business for a long period has likely encountered a broad range of deck conditions and can handle unforeseen issues that may arise during the staining process. Moreover, a company specializing in deck staining would be more knowledgeable about the latest products, techniques, and industry standards, compared to a general home improvement company.

When it comes to cost, it's best to adopt the mindset of a 'rational consumer' from economics – one that seeks to maximize utility (satisfaction) given a budget constraint. In this context, the utility derives from the quality of the deck staining job. Thus, while cost is a crucial factor, it should not be the sole determinant. Instead, balance it against factors like reputation and experience to ensure you are getting the best value for your money.

After shortlisting a few companies based on the above criteria, request for quotes and scrutinize the details. A comprehensive quote should include the cost of materials, labor, and any additional services such as deck cleaning or sealing. It should also specify the type of stain to be used. Some stains, like oil-based ones, offer better penetration and durability but may be more expensive and environmentally unfriendly compared to water-based stains. Therefore, the choice of stain involves a tradeoff between performance, cost, and environmental impact.

Finally, consider the contract terms before making a decision. A well-drawn contract should stipulate the scope of work, cost, payment terms, project timeline, and guarantees or warranties. If possible, have it reviewed by a legal expert to ensure your interests are adequately protected.

In conclusion, hiring the right deck staining company is a multi-faceted decision that involves evaluating the company's reputation, experience, specialization, cost and contractual terms. By applying principles from economics and statistics, and giving due consideration to legal aspects, you can make an informed choice that ensures your deck is beautifully stained and well-protected, enhancing the value and aesthetic appeal of your home.