I'm trying to solve the following question:

"The seller wants to auction off a single item to two bidders, the valuation of each bidder is an iid draw from a uniform distribution on [0,1] where the seller sets a reserve price, r, and has zero valuation for the item.

I was able to solve for the second price auction, such that the equilibrium bidding function is to bid your valuation for v>r and to not enter if v<r.

For the first price auction, I usually go about solving for the bidding function by first calculating the probability that bidder i wins:

Pr(i wins)=Pr(v_i > r) * Pr(v_i > v_j).

Where I know that Pr(v_i > v_j) = (1-b^-1(b)) which I can then use to derive the bidding function as usual in an ordinary FPA (0,0).

My question is, how do I work the reserve price into the bidding function.

I'm trying to solve the following question:

"The seller wants to auction off a single item to two bidders, the valuation of each bidder is an iid draw from a uniform distribution on $[0,1]$ where the seller sets a reserve price, $r$, and has zero valuation for the item.

I was able to solve for the second price auction, such that the equilibrium bidding function is to bid your valuation for $v>r$ and to not enter if $v<r$.

For the first price auction, I usually go about solving for the bidding function by first calculating the probability that bidder i wins:

$\Pr(i\text{ wins})=\Pr(v_i > r) * \Pr(v_i > v_j)$.

Where I know that $\Pr(v_i > v_j) = (1-b^{-1}(b))$ which I can then use to derive the bidding function as usual in an ordinary FPA $(0,0)$.

My question is, how do I work the reserve price into the bidding function.