# VCG mechanism for a reverse auction

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?

## 1 Answer

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 $$$$\sum_i x_ib_i + W\sum_i x_i$$$$ where $$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$$.