I don't have a lot of time for anything gambling-related besides the NFL pick 'em league this year, but I might have to make an exception for this:
To put it simply, create an account, join a league, draft a team of real U.S. Members of Congress and have fun as you compete to score as many points as possible. As the Members of Congress you drafted put real legislation through the lawmaking process they will score points for your team.
Like other sports, Fantasy Congress requires three things: a team, a system of scoring points, and a framework within which players can score. In this game, your team is a group of U.S. National Congress legislators, the legislators score points by pushing bills through the steps to create a law, and the framework within which they play is Congress itself
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At the end of the season, the Citizen with the most points wins. To accrue the highest number of points, use your political savvy to decide which when MCs should be playing and when they should be benched.
This will likely validate my theory that the best way to succeed at fantasy sports is to not have an emotional attachment to it. The most money I've ever made in a fantasy endeavor was an NBA league, because a) I dont like basketball and b) I had no compuction about drafting and/or trading for players with felony assault and attempted murder on their records if they could strengthen my roster.
I have more attachment to the NFL and MLB, and as much as I try to avoid it, I let personal allegiances enter into my decision-making. That's what makes Fantasy Congress so appealing. Considering that the effective members are likely to be the most corrupt and those that have been seated for the longest time, it definitely helps to be able to disassociate yourself from that sensation of utter distaste when adding Don Young or Jim Bunning to your roster.
This also might be one of the few times you miss Tom DeLay.