Which is good for Thompson since it takes out his conservative competition:

Sensing weakness, Sen. John McCain and Rudy Giuliani have formed an unspoken alliance to try to torpedo Mitt Romney just as many voters are tuning in to the Republican presidential race.
[…]
McCain has been running a mostly positive race, even refusing at one point to read a text by his aides that included attacks on Clinton. So his joint barrage with Giuliani is enough of a departure that it is even sparking GOP speculation about whether they might form a future ticket.
The two are friends and Giuliani said that if he weren’t running, he’d support the senator from Arizona. If Giuliani were the nominee, though, he’d need someone to help him turn out the Republican base, and McCain wouldn’t be much help there.
[…]
In New Hampshire the next day, McCain uncharacteristically dumped on Romney by packing many of Romney’s vulnerabilities into one brutal paragraph: “When Governor Romney donated money to a Democratic candidate in New Hampshire, I don’t think he was speaking for Republicans. When he voted for a Democratic candidate for president, Paul Tsongas, I don’t think he was speaking for Republicans. When he refused to endorse the Contract with America, I don’t think he was speaking for Republicans.”

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Romney’s reply writes itself, “When McCain worked with Kennedy to write the immigration bill, I don’t think he was speaking for Republicans. When McCain opposed Bush’s tax cuts and supported raising Social Security taxes, I don’t think he was speaking for Republicans. And when he voted to muzzle political speech, I don’t think he was speaking for Republicans.”
Having McCain on the ticket is redundant. McCain will do nothing to attract conservatives since we know that he isn’t one of us.