Why are second price auctions preferred if they don't maximize expected revenue? The reasoning usually given for why you would use a second price auction (SPA) as opposed to a first price auction (FPA) is that FPA encourage underbidding while SPA encourage truthful bidding. That is, bidding your true value is a weakly dominant strategy in SPA, whereas in FPA, bidding below your true value is the dominant strategy. See this question.

My question is: why does it matter to the seller whether people bid truthfully or not? Initially, I thought that since SPA induce truthful bidding, they also maximize expected revenue for the seller, which is why a seller would use SPA.  But, according to the revenue equivalence theorem, FPA and SPA generate the same expected revenue. So, shouldn't it not matter to the seller what auction they choose? Why would a seller prefer SPA -- which are generally harder to explain -- if it doesn't generate higher expected revenue?

I can think of three possible explanations:


There's a mismatch between theory and the real world. Maybe in theory SPA and FPA should generate the same revenue, but in practice they don't. If so, is there any empirical research that shows this?
The seller's goal is to collect information rather than to maximize revenue. Maybe the seller has a product and they don't know how much to sell it for. In that case, an SPA could help them generate an accurate demand curve.
To benefit the buyer. If there's no cost to the seller in terms of revenue, but a benefit to the buyer, a seller might want to use SPA. But this leads to the question: why does the seller care about helping the buyer? Maybe in order to encourage more buyers to participate in the auction? And, possibly encouraging more buyers to enter the auction would raise the expected revenue from the auction, if those new buyers were higher bidders.


Since SPA are so common in online advertising (google adwords) and eBay, they might be the most widely used auction there is today. So, I'm wondering why sellers prefer them to FPA. Is any of these explanations on the right track?
I don't think sellers would prefer SPA over FPA. In fact, the SPA is riskier than the FPA, if we look at it from a seller's perspective. The reason is because the distribution of prices in case of the SPA is a mean preserving spread of the distribution of prices in FPA(you can refer to Vijay Krishna's Auction Theory for the proof). 

However, the VCG mechanism finds a lot of applications in real life scenarios(like in case of network routing algorithms). It just so happens that the VCG mechanism can turn out to be the second price auction, given that certain conditions are met. 

The Generalized SPA is another such example, that draws its basis from the standard SPA. Please see https://en.wikipedia.org/wiki/Generalized_second-price_auction .

As far as the English Auction is considered, it is equivalent to the SPA only in the case of independent bidders. If the values realized by the bidders depend upon the the values realized by others(meaning the valuations are now interdependent), the English Auction no longer remains equivalent to the SPA(interdependence is something that on can expect to be more prevalent in real life cases).

Another thing, the RET states that the 'expected revenue' is same for standard auctions with symmetric bidders. The payment received by the seller is just the highest bid. Thus, receiving a higher payment in either auction(FPA or SPA) depends on the values realized by the bidders(and the bidding strategy).