<p>For questions like this, it can help to first consider the CDF. We can then take the derivative and have our density. I'll give you some hints and you need to fill in the gaps.</p>
<p>Ask yourself: What is the CDF of <span class="math-container">$Y_2^N$</span>, the second highest valuation of <span class="math-container">$N$</span> iid draws? It is the probability that the realization of random variable <span class="math-container">$Y_2^N$</span> is less than some <span class="math-container">$x$</span>. Next, ask yourself two auxiliary questions: What is the probability that the highest value is less than <span class="math-container">$x$</span>? (It's <span class="math-container">$F^N(x)$</span>) What is the probability that exactly one of <span class="math-container">$N$</span> values is higher than <span class="math-container">$x$</span> and <span class="math-container">$(N-1)$</span> values are lower than <span class="math-container">$x$</span>? (It's <span class="math-container">$N (1-F(x))F^{N-1}(x)$</span> as there are <span class="math-container">$N$</span> candidates to be the one above <span class="math-container">$x$</span>!) 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 <span class="math-container">$x$</span>? Exactly, the CDF is the sum of the answer to the earlier questions:
<span class="math-container">$$F_2^N (x) = (F(x))^N + N (1-F(x)) (F(x))^{N-1}= N F^{N-1}(x) - (N-1)F^N(x),$$</span>
and the derivative is
<span class="math-container">$$f_2^N(x) = N(N-1) f(x) (F^{N-2}(x) - F^{N-1}(x)) = N(N-1)f(x) F^{N-2}(x)(1-F(x)).$$</span>
This is the density of the second-highest value of <span class="math-container">$N$</span> draws.
You can rewrite this as
<span class="math-container">$$f_2^N(x) = N (1-F(x)) [(N-1)F^{N-2}(x)f(x)] = N (1-F(x)) f_1^{N-1}(x).$$</span>
You can learn more about order statistics in Appendix C of Krishna's book and the references therein.</p>
<p>Why is this not exactly your equation with another <span class="math-container">$N$</span>? Because your equation gives us something else. It is the density of the second-highest of <span class="math-container">$N-1$</span> iid draws conditional on the event that the highest of those draws is at most <span class="math-container">$x$</span>.</p>
<p>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 <span class="math-container">$N(1-F(x))$</span> with another <span class="math-container">$N$</span> and also corrected for <span class="math-container">$Y_1&lt;x$</span>. The third part is also analogous to the final part of my equation above.</p>

CC BY-SA 4.0
2023-04-04T19:40:48.110Z
2023-04-04T19:40:48.110Z
Bayesian
<p>Notation:</p>
<p><span class="math-container">$Y_1$</span>: Highest order statistics of <span class="math-container">$(N-1)$</span> players' valuation.</p>
<p><span class="math-container">$F_n^M:$</span> The distribution function of the highest <span class="math-container">$n$</span>th order statistics of <span class="math-container">$M$</span> players.</p>
<p><span class="math-container">$f_n^M:$</span> The density of the highest <span class="math-container">$n$</span>th order statistics of <span class="math-container">$M$</span> players.</p>
<p>In Vijay Krishna's book &quot;Auction theory&quot;, page <span class="math-container">$31$</span>. It said that the density of the second highest valuation of <span class="math-container">$(N-2)$</span> players is</p>
<p><span class="math-container">$f_2^{(N-1)}(y|Y_1&lt;x)=\frac{1}{F^{(N-1)}_1}(x)(N-1)(F(x)-F(y))f_1^{(N-2)}y$</span></p>
<p>Would you please help me explaining this equation? And may I ask what books or materials should I learn to understand it?</p>
<p>Thank you very much for your time!</p>

<auctions><probability>
Third price auction from Auction Theory by Krishna, Order statistics
Longye Tian
# Third price auction from Auction Theory by Krishna, Order statistics

(Asked by: username_0 on 2023-02-10)

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

Would you please help me explaining this equation? And may I ask what books or materials should I learn to understand it?

Thank you very much for your time!

**Tags:** auctions, probability


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## 1 answers

### Answer by username_1 on 2023-04-04. Score: 3 | Accepted answer

For questions like this, it can help to first consider the CDF. We can then take the derivative and have our density. I'll give you some hints and you need to fill in the gaps.

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.


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