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.

  • $\begingroup$ I feel like Krishna is correct here and M&M forgot to condition on Y<x, i.e., they did not divide the density by $G_{Y|X}(x|x)$. Otherwise, their expressions are equivalent (but you need do remove a $dy$). $\endgroup$ – Bayesian Apr 4 '19 at 8:24
  • $\begingroup$ @Bayesian Can you please explain how term $G_{Y|X}(x|x)$ comes in the denominator. $\endgroup$ – superhulk Apr 4 '19 at 10:51
  • $\begingroup$ In M&M, they compare the expected payment in SPA, and the expected payment in FPA. The expected payment in FPA as per M&M is given as $\beta(x)*F_{Y|X}(x|x)$. Vijay Krishna compares the expected payment in SPA with the payment in FPA(according to him, the payment in FPA us exactly equal to his bid, he does not consider the expected payment in FPA as such). $\endgroup$ – superhulk Apr 4 '19 at 11:01
  • $\begingroup$ I personally think that M&Ms expression for expected payment is more relevant, as if we consider the expected payment in FPA with IPV, we get the expression as $F_{Y}(x)*\beta(x)$, which is the distribution of the highest ordered statistic in $n-1$ bidders. This same expression is given in Vijay Krishna's book(3rd Chapter). $\endgroup$ – superhulk Apr 4 '19 at 11:05
  • $\begingroup$ From M&M, I can also see that for affiliated values, $\int_0^x v(y,y)g_{Y/X}(y/x) dy$$\geq$$G_{Y}(x|x)*\beta(x)$. This equation (I think) can further be written as $\int_0^x v(y,y)\frac{g_{Y/X}(y/x)}{G_{Y}(x|x) }dy$$\geq$$\beta(x)$.Is the above reasoning correct? $\endgroup$ – superhulk Apr 4 '19 at 11:16

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

  • $\begingroup$ You're reasoning is correct (although I'm not much familiar with probability theory, so I'm assuming this holds true). However, I'm still confused over the use of terminology. Consider the example in M&M(Chapter 4, Example 9). The expected payment by the bidder in SPA is calculated using $\int_0^x v(y,y)g_{Y/X}(y/x) dy$. Now, if I consider the expression given in Vijay Krishna's book to calculate the expected payment by bidder in SPA, i.e, $\int_0^x v(y,y)dK(y|x)$, I'll get a completely different answer. Am I missing something over here? $\endgroup$ – superhulk Apr 4 '19 at 20:09
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    $\begingroup$ I think now I see the point! The two books seem to have different interpretations of "expected payment". Krishna considers the transfer the winner has to pay CONDITIONAL on winning. M&M consider the unconditional expected payment of a buyer which is zero in case of losing and Krishna's term in case of winning. So that is why they multiply it by $G_{Y|X}(x|x)$, the probability of winning. $\endgroup$ – Bayesian Apr 5 '19 at 7:56
  • $\begingroup$ This makes sense. Thanks! $\endgroup$ – superhulk Apr 5 '19 at 10:48

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