A Distributed kWTA for Decentralized Auctions

Abstract

A distributed k-Winner-Take-All (kWTA) is presented in this paper. Its state-space model is given by

$$\begin{aligned} \frac{d}{dt}x_i(t) = - \left[ h(x_i(t)+u_i) - \frac{k}{n} + \beta \sum _{j \in \mathcal{N}_i} (x_i(t) - x_j(t)) \right] \\ z_i(x_i(t)) = h(x_i(t)+u_i) = \left\{ \begin{array}{ll} 1 &{} \text{ if }\, x_i(t) + u_i > 0 \\ 0 &{} \text{ if }\, x_i(t) + u_i \le 0 \end{array} \right. \end{aligned}$$

for $i = 1, \cdots , n$. Here, $u_i$s and $z_i$s are the inputs and the outputs; $\beta$ is a positive constant and $\mathcal{N}_i$ is the neighbor set of the $i^{th}$ node. If $\beta \rightarrow 1$, both $x_i(t)$ and $z_i(t)$ converge in finite-time; $z_i = 1$ if and only if $u_i$ is one of the k largest inputs; and $x_i \rightarrow -u_{\pi _{n-k}}$ (resp. $-u_{\pi _{n-k+1}}$) if $x_i(0) \ll -1$ (resp. $x_i(0) = 0$). Accordingly, our kWTA is the best algorithm to be applied as a decentralized mechanism for sealed-bid first (resp. second) price auction if $\beta \rightarrow \infty$, $k = 1$ and $u_i$s are the bid prices. Compared with the conventional mechanism, our novel mechanism is able to protect the privacy of the bidders. Auctioneer can be ripped out. Bidders can only know the value of the payment and losing bidders cannot know who is the winner.