VCG mechanism for a reverse auction

Question

Consider a simple reverse auction for some work. Bidder (potential worker) $i$ submits a cost $b_i$ for the work based on their valuation (true cost) $\theta_i$ of it. Let $x_i(\mathbf{b}) \in \{0,1\}$ denote whether the bid is successful, and $m_i(\mathbf{b})$ the monetary compensation.

Bidder $i$ tries to maximize the payoff function $$ u(b_i) = x_i(b_i, \mathbf{b}_{-i})\theta_i + m_i(b_i, \mathbf{b}_{-i}) $$

As I understand (which is probably wrong), the VCG mechanism involves finding $\mathbf{x}$ which maximizes $\sum_i x_ib_i$, subject to $0 \leq \sum_i x_i \leq 1$, supposing only one piece of work is available.

However, given that $\theta_i$ are expected to be negative, this would generally result in $x_i = 0$ for all $i$, which does not seem to make sense.

What is the proper social choice function in this reverse auction setting?

Answer 1

I still haven't managed to find an official answer for this, as there appears to be various different treatments in the literature. However, I suppose a natural answer would be to define "welfare" to be the welfare of both the bidders' welfare and the auctioneer's. In this case, welfare would be defined as \begin{equation} \sum_i x_ib_i + W\sum_i x_i \end{equation} where $W$ denotes the welfare of the work to the auctioneer. This is similar to the situation in a forward auction when the auctioneer sets a reserve price, in which case we have $W < 0$. The scenario with no reserve price is thus equivalent to setting $W=0$.