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The general scenario is that slaves or submissives are presented on a stage or presentation by their tops or masters, and the latter would list (or have the former demonstrate) their skills and credentials, as well as what sorts of BDSM activities they have to offer for the auction. Usually a short scene of spanking, flogging etc (in rarer cases also including sexual activity), accompanies the auction process to promote the submissive or slave.
There are many variations of this practice in terms of location, dress codes, etc. Typically, a slave auction is combined with other types of kinks that provide a context for it to take place. For example, if the participants of the auction enjoy public nudity, then the auction can be performed outdoors with the slaves scantily clad or even nude. Other times, the auction can be indoors, for example in an office setting, with all participants dressing like professionals. A slave auction can also be just another form of "punishment" by the dom for a slave's unsatisfactory performance. In this case, the slave is punished (or even banished) by being sold off to another dom.
Then people in the audience bid on their offer, usually using play money purchased with a donation to a local charity. The winner then makes arrangements for the "scene" with the person who bid their services. No actual exchange of property takes place, owning people is illegal in most countries. Technically, no solicitation or prostitution takes place since the auction is just for time, but not for a particular service or act. The auction is supposed to be for fun and consensual enjoyment, but no monies exchange hands between the parties involved.
The internet has expanded the possibilities of the traditional slave auction. Mistresses or masters now auction slaves on their own sites. The internet allows the mistress or master to reach a far greater audience and the auction can last for as long as the site's owner wishes to maintain the page.
From a slave's point of view, the pleasure gained not only stems from the humiliation he or she receives from his or her top (by being "owned" and "sold"), but also from being able to fulfill certain sexual fantasies during the course of the auction. Sometimes a master or mistress, by auctioning their slave, makes the auction process a "scene" (encounter) in itself, and has the opportunity to show off his own performance as well as his 'possessions'.
It is also a popular way for people who are new to the scene to demonstrate what they have to offer, or for the newly single to advertise their availability, much like a vanilla "bachelor auction" would be.ja:奴隷市場 (性風俗)