The Third Party Game refers to all the ways the major parties use or manipulate third parties for their own advantage. The Republicans are far better at it than the Democrats are.
One way that it is played is to make donations to third parties that rival the opposition party. Democrats are still complaining about how Ralph Nader "stole" the 2000 election from Al Gore, but what made that "theft" possible was Republican donations to Nader. It is no secret that many of the donations for Nader's campaign came from Republicans.
If someone were to suggest to a Democrat that a donation be made to, say, the Constitution Party, the response would be shock and horror. "Oh no, they're evil, they want to destroy everything I believe in, I could never donate to them." Republicans didn't donate to Nader in order to advance Nader's agenda, but their donations weren't intended to advance Nader's agenda.
The other way that the Third Party Game is played is to nullify threats. Republicans also are better at this, sending Patrick Buchanan into the Reform Party, Alan Keyes into the Constitution Party, and sending Bob Barr into the Libertarian Party. Patrick Buchanan was able to destroy the Reform Party through an internal civil war. Alan Keyes didn't get the Constitution Party nomination, but did get the ballot line in California and thus lowering the nation-wide totals for their candidate Chuck Baldwin. Bob Barr (who has since endorsed Newt Gingrich) and Wayne Root (who has since endorsed Mitt Romney) alienated a sufficient portion of the Libertarian Party base that many wrote in Ron Paul, and thus lowered the totals for the Libertarian Party as well.
These tactics may be underhanded, but there are no rules saying not to do either of those actions with regards to third parties. What is interesting is that the Democratic Party leadership does such a poor job of doing the same thing. Perhaps it is a lingering sentiment of there being "no enemy to the left" so they cannot bring themselves to label parties such as the Green Party as enemies. Republicans have no such compunctions holding them back. Democrats are quite willing to share a stage at allies with various independent groups of a shared platform; the Republicans did a deliberate take over of the Tea Party when it appeared that a stage might be shared.
It will be interesting to see how the Third Party Game might be shared in the 2012 election cycle.
It is also possible, but somewhat far fetched, to suggest that occasionally Republicans support a party or candidate outside their own party that they would normally squash. This would be done to prevent a different third party from gaining prominence. Although Ross Perot did much to spoil George Bush's chances at re-election, he also gathered up all of the internal dissent that might have gone to more established third parties and catapulted them to prominence. Although John Anderson did not threaten Reagan, he did absorb much of the third party protest vote without even having much to offer in the way of concrete ideas. Perhaps Republicans are willing to throw an election in order to preserve the status quo of two parties interchangeable.