The quest for the ideal CoWorking space (CoWs) can be likened to the pursuit of a rare gem; it demands patience, discernment, and a keen understanding of your business requirements. It's a task that requires the application of the multi-attribute utility theory (MAUT), traditionally applied to complex decision-making scenarios. This theory postulates that any decision is influenced by multiple, often competing, factors. In this case, these factors could range from location, cost, networking opportunities, to the vibe of the CoWs. By quantifying and ranking these factors to suit your business needs, MAUT can lead you to the perfect CoWs selection.

Firstly, let's delve into the concept of CoWorking spaces. Defined as membership-based workspaces where diverse groups of freelancers, remote workers, and other independent professionals work together in a shared, communal setting, CoWs have emerged as a paradigm shift in the traditional workspace model. They offer a platform for collaboration, innovation, and networking - elements that are crucial for startups and small businesses.

The location of your CoWs is a key factor to consider, particularly in the context of agglomeration economics. This theory, first proposed by Alfred Marshall, argues that businesses clustered in an area can benefit immensely from shared resources and talent pool. Therefore, a CoWs located in a business district or startup hub would expose you to a potent mix of ideas, collaboration, and potential partnerships. Yet, the tradeoff is that such prime locations may come with a higher price tag.

Cost is indeed another MAUT factor. The Game Theory, another mathematical approach, can be instrumental in making this decision. Adopting a minimax strategy, where you aim to minimize your maximum possible loss (in this case, financial), can guide your choice. For instance, while a high-end CoWs with state-of-the-art facilities may seem attractive, your budget constraints may lead you to a more affordable, yet equally functional option.

Another consideration is the opportunities for networking. According to Granovetter's "strength of weak ties" theory, it's often the less-frequent, more casual connections that offer the most significant opportunities. A CoWs with a diverse member base offers a fertile ground to foster these "weak ties," leading to collaborations and partnerships that you may not have anticipated.

The vibe or culture of the CoWs is another factor. Just as Richard Florida's "creative class" theory posits the importance of an environment that fosters creativity, a CoWs with a positive, vibrant, and collaborative culture can enhance productivity and innovation.

Lastly, the facilities and amenities of the CoWs - such as high-speed internet, meeting rooms, 24/7 access, and kitchen amenities - should align with your business needs. Here, the Pareto Efficiency principle applies, where the optimal choice is one that benefits you without making anyone else worse off. For instance, a CoWs with a fully-equipped meeting room would be beneficial if you frequently host client meetings, but not as crucial if most of your work is done remotely.

The selection of a CoWs is a decision that carries immense weight for the growth and success of your business. By applying theories from various disciplines - economics, mathematics, social sciences - you can make an informed, objective choice that aligns with your business requirements. It's a quest that demands both intellect and intuition - a quest for the perfect gem in the world of workspaces.