The Green Party has been pushing "Instant Runoff Voting" (IRV) for years. In theory, Instant Runoff Voting allows third parties to run candidates without spoiling elections. In practice, IRV is just as broken as the current system, and several much better methods exist.
How Instant Runoff Voting Works
Voters rank all their choices, in order. A Green Party member in 2004 might vote:
1. Camejo
2. Kerry
3. Bush
4. Roy Moore
In the first round, the total is 2% Roy Moore, 4% Green, 46% Kerry and 47% Bush. Roy Moore has the fewest votes, so he gets eliminated. Not surprisingly, all of his voters listed Bush as a second choice, so all his votes transfer to Bush.
In the second round, the total is 4% Green, 46% Kerry, and 47% Bush. The Green candidate is eliminated, and all the votes go to Kerry.
The total is now Kerry 50% and Bush 47%. Kerry wins the election, and the Democrats and Greens are reasonably happy.
How IRV Breaks, Badly
Now, let's pretend that the Greens become quite popular (the plurality party, actually) and the Democrats move somewhat to the right. Together, the Greens and the Democrats hold a huge majority, but IRV may still elect the Republican because some Democrats will list Bush as their second choice.
In the first round scores are 36% for Camejo, 30% for Kerry, and 34% for Bush. Kerry is eliminated, and his now center right votes go to--watch for it--20% go to Bush, and 10% to Camejo.
In the second round, the vote is 46% Camejo, 54% Bush. Despite 66% of the electorate prefering either a Green or a Democrat, the Republicans win by a landside.
To avoid this, the Greens need to list Kerry as their first choice, and Camejo as their second. This way, Kerry gets 66% of the first round, and the election is over. But what if some of the Democrat-Green voters tried to throw their votes to Camejo instead, because he had the lead?
Major yucko. When the three parties are roughly tied, the mathematics become deeply unstable and the outcome becomes highly unpredictable.
Better Methods
In approval voting, you check off every candidate you could reasonably live with (given the alternatives) and the candidate with the most votes wins. Greens, for example, would check off both Camejo and Kerry. Hopeless centrists would check off Kerry and Bush. Right wing nuts would check off Bush and Roy Moore. Whoever gets a thumbs-up from the largest chunk of the electorate wins. This system has the advantage of ridiculous simplicity and no major theoretical flaws. This is how the UN picks the Secretary General.
Condorcet voting is as close to a theoretically perfect election system as you can get (at least at the current state of theory). Like IRV, you list your preferences. Unlike IRV, Condorcet voting takes all your preferences into account when picking the winner, which prevents all sorts of weird stuff we saw in IRV. Unfortunately, the Condorcet rules are a bit hard to explain.
So there are two good choices: Dead-simple and theoretically solid approval voting, and slightly complex but close-to-theoretically-perfect Condorcet voting.
But if you continue to argue for IRV, I'll be arguing against you at every turn. It's a broken system.