Examples of distributions that satisfy this condition include Gaussian, uniform, and exponential; some power law distributions also satisfy regularity.
[2] Consider a seller auctioning a single item to a buyer with random value
set by the seller, the buyer will buy the item if
is the expected revenue the seller would obtain by choosing
is the revenue that can be obtained by selling the item with (ex-ante) probability
[3] An important special case[note 1] considered by Myerson (1981) is the problem of a seller auctioning a single item to one or more buyers whose valuations for the item are drawn from independent distributions.
Myerson showed that the problem of the seller truthfully maximizing her profit is equivalent to maximizing the "virtual social welfare", i.e. the expected virtual valuation of the bidder who receives the item.
Thus a Vickrey auction can be used to maximize the virtual social welfare (with additional reserve prices to guarantee non-negative virtual valuations).
[4] Myerson's auction mentioned above is optimal if the seller has an accurate prior, i.e. a good estimate of the distribution of valuations that bidders can have for the item.
Obtaining such a good prior may be highly non-trivial, or even impossible.
Prior-independent mechanism design seeks to design mechanisms for sellers (and agents in general) who do not have access to such a prior.
Regular distributions are a common assumption in prior-independent mechanism design.
For example, the seminal Bulow & Klemperer (1996) proved that if bidders valuations for a single item are regular and i.i.d.
(or identical and affiliated), the revenue obtained from selling with an English auction to