# Third price auction from Auction Theory by Krishna, Order statistics

Notation:

$$Y_1$$: Highest order statistics of $$(N-1)$$ players' valuation.

$$F_n^M:$$ The distribution function of the highest $$n$$th order statistics of $$M$$ players.

$$f_n^M:$$ The density of the highest $$n$$th order statistics of $$M$$ players.

In Vijay Krishna's book "Auction theory", page $$31$$. It said that the density of the second highest valuation of $$(N-2)$$ players is

$$f_2^{(N-1)}(y|Y_1

Ask yourself: What is the CDF of $$Y_2^N$$, the second highest valuation of $$N$$ iid draws? It is the probability that the realization of random variable $$Y_2^N$$ is less than some $$x$$. Next, ask yourself two auxiliary questions: What is the probability that the highest value is less than $$x$$? (It's $$F^N(x)$$) What is the probability that exactly one of $$N$$ values is higher than $$x$$ and $$(N-1)$$ values are lower than $$x$$? (It's $$N (1-F(x))F^{N-1}(x)$$ as there are $$N$$ candidates to be the one above $$x$$!) Once you have answered these two questions, you can think about the first question again: What is the probability that the second-highest value is at most $$x$$? Exactly, the CDF is the sum of the answer to the earlier questions: $$F_2^N (x) = (F(x))^N + N (1-F(x)) (F(x))^{N-1}= N F^{N-1}(x) - (N-1)F^N(x),$$ and the derivative is $$f_2^N(x) = N(N-1) f(x) (F^{N-2}(x) - F^{N-1}(x)) = N(N-1)f(x) F^{N-2}(x)(1-F(x)).$$ This is the density of the second-highest value of $$N$$ draws. You can rewrite this as $$f_2^N(x) = N (1-F(x)) [(N-1)F^{N-2}(x)f(x)] = N (1-F(x)) f_1^{N-1}(x).$$ You can learn more about order statistics in Appendix C of Krishna's book and the references therein.
Why is this not exactly your equation with another $$N$$? Because your equation gives us something else. It is the density of the second-highest of $$N-1$$ iid draws conditional on the event that the highest of those draws is at most $$x$$.
The first part in the equation corrects for this event. Appendix A or any book on probability theory should give you an idea on conditional CDFs and densities. The second part is the analogue to $$N(1-F(x))$$ with another $$N$$ and also corrected for $$Y_1. The third part is also analogous to the final part of my equation above.