Say there are three roommates moving into an apartment with three rooms. Two of the apartment's rooms are identical, but the third one is valued higher by all three parties (say it's bigger and has a private bathroom). To decide which one gets the big room, the roommates decide to hold an auction.

If the apartment's monthly rent is $p$ dollars in total, the idea behind the auction is to find a number $\delta$ such that the winner of the auction pays $\frac{p}{3} + \delta$, while the two losers pay $\frac{p}{3} - \frac{\delta}{2}$. In other words, the winner of the auction (who gets the big room) pays a bigger share of the rent compared to the two losers.

Does this type of auction—where the two losers benefit from the value of $\delta$ being large—differ from traditional auctions, where it doesn't concern the losers what the winner pays? For example, does it still apply that if the price is decided using an English auction, that each bidder $i$ is incentivized to never bid higher than their value $v_i$? Intuitively, you might think that if the bidder $s$ with the second highest value $v_s$ for $\delta$ is reasonably confident that the to-be winner $w$ has a much higher value $v_w$ for $\delta$ they might bid higher than their own value $v_s$, in hopes of raising $\delta$ and lowering the rent that they have to pay.

by definition, the losers neither make nor receive any payments. $\endgroup$