When the auction involves a single item for sale and each participant has as an independent private value for the item auctioned, the expected payment and expected revenues of an English auction is theoretically equivalent to that of the Vickrey auction, and both mechanisms have weakly dominant strategies. Both the Vickrey and English auction, although very different procedurally, award the item to the bidder with the highest value at a price equal to the value of the second highest bidder.
In an English auction, the object goes to the highest bidder who pays at the price they proposed.
In a Vickrey auction, the object goes to the highest bidder, who pays at the second-highest price.
Given this, how are they "equivalent" to each other? The only thing common between the two, per my understanding, is that the highest bidder always wins.
My guess is that, in an English auction, when the second-highest bidder eventually places a bid (say
b0) corresponding to their max affordable value, the highest bidder can place another bid ever so slightly higher (say
b1). The object is then sold at
In a Vickrey auction, the highest bidder would pay
b0 is almost equal to
b1, I imagine that's where the equivalence comes from.