Derivation of Equilibrium Strategy in 1st-price Auction? Hey, everyone. I desperately need help understanding some math in auction theory.  I have been writing a paper as an undergrad for auction theory, and after all of my research I just cannot understand one step the author of a book uses to derive the equilibrium strategy for first-price auctions under independent private value conditions.
The steps in the derivation process essentially follow this:
Let x = the chosen player's personal valuation, G(x) = the probability that the highest opposing bid, an order statistic, Y_1, is &lt; x.
\begin{equation}
\beta(x) = x - \int_0^x\frac{G(y)}{G(x)} \, dy \\
\end{equation}
Then without much explanation, other than "values are uniformly distributed on [0,1]" he just converts this equation to
\begin{equation}
\beta(x) = \frac{N-1}{N},
\end{equation}
Which makes sense and is usable.

The book I am using is Vijay Krishna's Auction Theory, and I would extremely appreciate someone helping me understand this last step.
I think you have a typo -- the equilibrium bidding strategy in the first-price auction you specify should be $$ \beta(x) = \frac{N-1}{N}x $$

Here's a hint that might help. The CDF of the order statistic, $Y_1$, is $$G(x) = x^{N-1}$$

To see why this is the case, notice that this is exactly the probability that some given quantity $x$ is greater than or equal to $N-1$ independently drawn random variables that are uniformly distributed on $[0,1]$.

I'll write something in more detail when I have more time tomorrow, but I think this should tide you over for now. (At the very least, it'll help you evaluate the integral.)