The United States online auction software platforms bring a transformative shift in the way businesses and individuals transact in the digital space. Coupling the traditional auction concept with cutting-edge technology, these platforms are instrumental in creating an interactive, dynamic, and transparent environment for buying and selling a plethora of items online. To understand how they operate, one must peel back the layers to examine the intricate mechanisms that drive the auction process.

Online auction software platforms can be broadly classified into two categories based on the type of auctions they support: forward and reverse auctions. Forward auctions follow the traditional auction model where multiple buyers bid for a single seller's product, driving the price upward. Reverse auctions, on the other hand, represent an paradigm shift in the auction dynamic where multiple sellers compete to provide goods or services to a single buyer, inversely driving the price down.

The auction software platforms are designed to cater to both these models, often with the support for a mixed-mode auction environment. They utilize principles drawn from game theory, behavioral economics and computational complexity theory to create a thriving marketplace.

At a high level, every auction platform has three key components: the User Interface (UI), the Auction Engine, and the Payment Gateway.

The UI is the front-end of the platform that interacts with the users. It is designed to provide an intuitive and seamless experience to both buyers and sellers. The UI is also responsible for displaying real-time information about the auctions like the number of bids, the highest bid, time left for the auction to close, and so forth.

The Auction Engine is the core of the system. It is responsible for managing the auctions, tracking bids, and determining winners. The algorithms that drive the Auction Engine are largely based on the principles of game theory, more specifically, the concept of Nash equilibrium.

A Nash equilibrium represents a state of the system where no player can gain anything by unilaterally changing their strategy given that the strategies of the other players remain unchanged. In the context of an auction, it represents a state where no bidder can increase their utility (probability of winning the auction) by changing their bid given the bids of the others. The auction software employs a dynamic variant of this concept to handle real-time bids and maintain the equilibrium of the system.

Lastly, the Payment Gateway processes the financial transactions related to the auctions. It ensures secure and efficient processing of payments, often complying with complex financial regulations and standards like PCI DSS (Payment Card Industry Data Security Standard).

The technologies underpinning these components have their tradeoffs. For instance, in order to support real-time bidding, the auction software employs databases that offer high write speeds, like NoSQL databases. However, these databases often compromise on features like transactional integrity and complex querying capabilities that their SQL counterparts provide. Similarly, the use of secure cryptographic methods for payment processing comes at the cost of computational overhead and latency.

Navigating through these tradeoffs and striking the right balance is a Herculean task that the designers of these platforms grapple with. They often tackle these challenges by employing a layered, modular architecture that allows them to make independent decisions for different components of the system.

In conclusion, US online auction software platforms represent an amalgamation of traditional auction models, cutting-edge technology, and complex economic and game theoretic principles. As technology evolves and the digital marketplace grows, these platforms are poised to become even more sophisticated, complex, and essential to the online ecosystem.