Expected payment in a second price auction with affiliation

Question

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.

Answer 1

I see nothing wrong with Krishna's expression. If I had to take a guess, it would be that either Menezes and Monteiro define their $f_{Y|X}(y|x)$ differently or they simply forgot to adjust the density on $Y_1 < x$. Both expressions are supposed to be the same.

If you have some random variable $Y$ with cdf $F$, density $f$ and support $[a,b]$ , the following two notations say the same thing $$\int v(y) f(y) dy = \int v(y) d F(y) \quad \mbox{for any function } v(y).$$ If you condition $Y$ on being $Y<x$, you have to adjust the cdf because the new support only goes up to $x$, $$ F_{Y|Y<x} (y) = \begin{cases} 0 \quad &\mbox{if } y <a, \\ \frac{F(y)}{F(x)} \quad &\mbox{if } y \in [a,x],\\ 1 \quad &\mbox{if } y >x. \end{cases}$$ That way you have an expression that is one for $y=x$.

Hence, $$E[v(Y_1,Y_1)|X_1=x,Y_1<x]= \int_0^x v(y,y)d\cfrac{ G_{Y|X}(y|x)}{G_{Y|X}(x|x)} = \int_0^x v(y,y) \cfrac{ g_{Y|X}(y|x)}{G_{Y|X}(x|x)} d y. $$