The symmetric bidding strategy in a second price auction with affiliation is given by $\beta(x)=v(x,x)$, where $v(x,y)=E[V_1|X_1=x,Y_1=y]$(here $Y_1$ is the highest ordered statistic among the remaining $n-1$ bidders, and bidder 1 is assumed to be the winner).

Given this, the expected payment by the bidder is given as $E[v(Y_1,Y_1)|X_1=x,Y_1<x]$. This, I suppose, should be equal to $\int_0^x v(y,y)g_{Y/X}(y/x) dy$. The same expression is given in Introduction To Auction Theory by Menezes and Monteiro. However, Vijay Krishna in his book writes $E[v(Y_1,Y_1)|X_1=x,Y_1<x]=\int_0^x v(y,y)dK(Y/X) dy$, where $K(Y/X)=\cfrac{ G_{Y/X}(y/x)}{G_{Y/X}(x/x)}$.

My question is, are the two expressions same?

Note: There is a notational difference in the two books, while Vijay Krishna uses $g(.),G(.)$ for the density and the distribution, Menezes and Monteiro use $f(.),F(.)$, respectively.

The symmetric bidding strategy in a second-price auction with affiliation is given by $\beta(x)=v(x,x)$, where $v(x,y)=E[V_1|X_1=x,Y_1=y]$  (here $Y_1$ is the highest ordered statistic among the remaining $n-1$ bidders, and bidder 1 is assumed to be the winner).

Given this, the expected payment by the bidder is given as $E[v(Y_1,Y_1)|X_1=x,Y_1<x]$. This, I suppose, should be equal to $\int_0^x v(y,y)g_{Y|X}(y|x) dy$. The same expression is given in Introduction To Auction Theory by Menezes and Monteiro. However, Vijay Krishna in his book writes $E[v(Y_1,Y_1)|X_1=x,Y_1<x]=\int_0^x v(y,y)dK(y|x)$, where $K(y|x)=\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)}$.

My question is, are the two expressions same?

Note: There is a notational difference in the two books, while Vijay Krishna uses $g(.),G(.)$ for the density and the distribution, Menezes and Monteiro use $f(.),F(.)$, respectively.