We introduce the following consensus estimate problem. Several processors hold private and possibly different lower bounds on a value. The processors do not communicate with each other, but can observe a shared source of random numbers. The goal is to come up with a consensus lower bound on the value that is as high as possible. We give a solution to the consensus estimate problem and show how it is useful in the context of mechanism design. The consensus problem is natural and may have other applications. Based on our consensus estimate technique, we introduce Consensus Revenue Estimate (CORE) auctions. This is a class of competitive revenue- maximizing auctions that is interesting for several reasons. One auction from this class achieves a better competitive ratio than any previously known auction. Another one uses only two random bits, whereas the previously known competitive auctions on n bidders use n random bits. Furthermore, a parameterized CORE auction performs better than the previous auctions in the context of mass-market goods, such as digital goods.