Two-player first-price auction with resale Exercise 2.5 in Vijay Krishna's Auction Theory (slightly modified):


  Consider a two-bidder ﬁrst-price auction in which bidders’ values
  are distributed according to $F$. Let $\beta$ be the symmetric equilibrium, i.e., $$\beta(x)=E[Y\mid Y&lt;x],$$ where $Y$ is drawn from $F$. 
  
  Now suppose that after the auction is over, both the losing and winning bids are publicly announced. In addition, there is the possibility of postauction resale: The winner of the auction may, if he so wishes, offer the object to the other bidder at a ﬁxed “take-it-or-leave-it” price of $p$. If the other bidder agrees, then the object changes hands, and the losing bidder pays the winning bidder $p$. Otherwise, the object stays with the winning bidder, and no money changes hands. The
  possibility of postauction resale in this manner is commonly known to
  both bidders prior to participating in the auction. Show that $\beta$ remains an
  equilibrium even if resale is allowed. In particular, show that a bidder with
  value $x$ cannot gain by bidding an amount $b&gt;\beta(x)$ even when he has the
  option of reselling the object to the other bidder.


Call the two bidders Jane and Mike. Given Mike's bid, and assuming that Mike follows the equilibrium strategy $\beta$, Jane can determine Mike's exact value. This implies that if Jane wins the item but realizes that she has lower value than Mike, she will resell it to Mike at exactly Mike's value.

We should calculate Jane's expected payoff when bidding $\beta(z)$ with $z\geq x$ when her value is $x$, and show that the expected payoff is maximized when $z=x$. 

Jane wins with probability $F(z)$. If she loses, her payoff is $0$. If she wins and her value is higher than Mike's value, her payoff is $x-\beta(z)$. If she wins and her value is lower than Mike's value, $Y$, her payoff is $y-\beta(z)$. As a result, her expected payoff is $$-F(z)\beta(z)+F(x)x+(F(z)-F(x))E[Y\mid x&lt;Y&lt;z].$$

The term $F(x)x$ does not depend on $z$, so we are left to maximize $$-F(z)\beta(z)+(F(z)-F(x))E[Y\mid x&lt;Y&lt;z].$$

How do we maximize this?
Let's look at this with backward induction. Let $v_{i}$ be my valuation and $v_{-i}$ be your valuation. Suppose I've won the item. Then $v_{i} \geq v_{-i}$. If I sell you the item at price $v_{-i}$, then my utility is $v_{-i} - b_{i} \leq v_{i} - b_{i}$, where $b_{i}$ is my bid according to the symmetric equilibrium bidding strategy (which may not necessarily be $\beta$). Notice that I do not gain $v_{i}$ by selling you the item, as I do not possess it.

We see that I will never have incentive to sell you the item. So I should bid as if I'm not going to sell you the item. We have an equilibrium bidding strategy for this anyways, given by the first price auction.