# Revenue Equivalence in Auction Theory: how does an English auction generate the same revenue as First Price Sealed Bid?

The theroem:

The revenue equivalence theorem states that, if all bidders are risk-neutral bidder and have independent private value for the auctioned items, then all four of the standard single unit auctions have the same expected sales price (or seller's revenue).The four standard single unit auctions are the English auction, the Dutch auction, first-price sealed-bid auction, and the second-price sealed-bid auction.

Given the preconditions, how is it possible for an English auction to genereate the same revenue as a first-price sealed auction?

Per my understanding, an English auction, when played optimally, generates the same revenue as a second price sealed auction (Vickrey auction).

On the other hand, a First Price Sealed Bid, when played optimally, results in a revenue equivalent to the highest bid.

So how can the two be revenue equivalent?

At first sight, if one person bids 10\$, another 20\$, and a third one 30\$, then it seems that first-price auction would result in 30\$ and second-price in 20\$. However, under different rules - that are of course known in advance - the participants would make different bids. E.g., under a second-price auction, you tend to bid more than what you are willing to pay because that's not what you are going to pay. Then again, you won't bid that much more and risk the second-price be above your original evaluation. This may explain how the "obvious" difference is not that obvious after all. But can we expect in this scenario that the second-highest price is much higher than what the second bidder estimates? Shouldn't they - by the same argument - bid more than 20, but only slightly more, perhaps 22\$ (so still way below 30\\$)?

The key part in the full theorem is also that it is about "privately-known signal[s] independently drawn from a common strictly-increasing, atomless distribution" (emphasis mine). So it is not the case that one participant inherently values the object more than the others. I didn't look up the proof though, but under these circumstances, it doesn't look that much infeasible any more.

• Hmm, I'm not sure I understand the last paragraph. How does that precondition make it possible for a an English auction to be revenue equivalent to a second price sealed bid? Jan 16 '21 at 18:50