Economic theory has produced a number of cute mechanisms that have been proven to satisfy various desirable properties. However, and in some cases surprisingly, most of these mechanisms have never been used in practice.
There have been some recent attempts by computer scientists to make these mechanisms accessible to a broader audience. One of them, called Spliddit, developed by former TIH blogger Ariel Procaccia, provides implementations of five fair division mechanisms. Spliddit was revamped some days ago and now also offers the first(?) real-world application of the Shapley value.
Another such attempt that I developed with Christian Geist and Guillaume Chabin called Pnyx will be demonstrated at AAMAS in Istanbul tomorrow. Pnyx is a web-based tool for preference aggregation. We were inspired by the observation that most people use inferior mechanisms (such as plurality rule) and/or unsuitable tools (such as Doodle) when aggregating preferences in everyday situations. Depending on the desired output, Pnyx computes its outcome using the Borda count, Kemeny’s rule, and Fishburn’s maximal lotteries. We tried to keep Pnyx as simple as possible. A similar recently developed tool that offers many more voting rules is Whale3.
I am sure there are other websites or apps that implement well-studied economic mechanisms. Please let us know about these in the comments.
Turing's Invisible Hand 
Interesting post! Pnyx looks like an appealing service and I’m looking forward to playing around with it.
You asked about other websites or apps that implement well-studied mechanisms. One that I have used is CIVS, the Condorcet Internet Voting Service (http://civs.cs.cornell.edu/), which has been run since 2003 by Andrew Myers at Cornell University. It provides polls with ranked-choice voting, and has been used for thousands of polls and elections, according to the website. A discussion of the algorithms used by CIVS can be found at http://civs.cs.cornell.edu/rp.html and http://civs.cs.cornell.edu/proportional.html.
Can’t wait to teach my mechanism design class with these online mechanisms!
Thanks! CIVS indeed looks quite interesting.