A number of top Democratic strategists have wondered why, exactly, the Obama campaign let Mitt Romney launch the first attacks on Obama's handling of Medicare, rather than the other way around.

Among the explanations offered is the theory that the president's campaign wants to preserve its cash (which has been dwindling), that it is waiting until after the conventions to make this a major issue, and that it thinks the public so distrusts the GOP on the topic that it doesn't mind having the conversation center there, even if the Obama campaign is on the defensive.

That later point was bolstered in a Pew Research Center poll released on Tuesday, which showed that the public opposes the idea of changing "Medicare into a program that would give future participants a credit toward purchasing private health insurance coverage" by a margin of 49 percent to 34 percent. Moreover, 72 percent of respondents have heard a lot or a little about the idea, while just 29 percent say they heard nothing.

The margin between those opposing changing Medicare and those supporting a change is even greater among the elderly, with 55 percent of people 65 or older opposing the proposal and 24 percent supporting it. That is a large part of the reason that Romney and Ryan have pledged not to apply the changes to anyone 55 or older.

And yet, not all news is good news for the president on this front. While the public largely doesn't like a quasi-voucher system for Medicare, it also isn't entirely sure which politician has proposed it. As Pew notes:

At this point, most Americans do not associate Ryan with the proposal to change Medicare. Just 23% of those who have heard about the idea of shifting Medicare to a system of credits to buy private insurance identify it as Ryan’s. Nearly as many (17%) say Barack Obama proposed this, while 44% do not know who proposed it.

This post has been updated to clarify the Obama campaign's response to Romney's attacks.

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    US Republican presidential candidate and former Massachusetts Governor Mitt Romney (L) shakes hands with his running mate Wisconsin Representative Paul Ryan in front of the campaign bus prior to each boarding their own campaign airplanes at General Mitchell Airport in Milwaukee, Wisconsin, August 12, 2012. (Photo credit should read SAUL LOEB/AFP/GettyImages)

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    WAUKESHA, WI - AUGUST 12: Republican presidential candidate and former Massachusetts Gov. Mitt Romney and vice presidential candidate and Wisconsin native Rep. Paul Ryan (R-WI) (L) greet supporters during a campaign event at the Waukesha Expo Center on August 12, 2012 in Waukesha, Wisconsin. (Photo by Darren Hauck/Getty Images)

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    Republican presidential candidate, former Massachusetts Gov. Mitt Romney, left, listens to his vice presidential running mate, Rep. Paul Ryan R-Wis., as they campaign Sunday, Aug. 12, 2012, in High Point, N.C. (AP Photo/Mary Altaffer)

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    WAUKESHA, WI - AUGUST 12: Republican presidential candidate and former Massachusetts Gov. Mitt Romney and vice presidential candidate and Wisconsin native Rep. Paul Ryan (R-WI) (L) hug after speaking at a campaign event at the Waukesha Expo Center on August 12, 2012 in Waukesha, Wisconsin. (Photo by Darren Hauck/Getty Images)

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    Republican presidential candidate, former Massachusetts Gov. Mitt Romney, left and running mate Rep. Paul Ryan, R-Wis., talk in the campaign bus before an event at the Waukesha county expo center, Sunday, Aug. 12, 2012 in Waukesha, Wis. (AP Photo/Mary Altaffer)

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    WAUKESHA, WI - AUGUST 12: Republican presidential candidate and former Massachusetts Gov. Mitt Romney and vice presidential candidate and Wisconsin native Rep. Paul Ryan (R-WI) (L) greet supporters during a campaign event at the Waukesha Expo Center on August 12, 2012 in Waukesha, Wisconsin. (Photo by Darren Hauck/Getty Images)

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    Republican presidential candidate, former Massachusetts Gov. Mitt Romney, right, and his newly announced vice presidential running mate, Rep. Paul Ryan, R-Wis., left, greet supporters, during a campaign rally in Manassas, Va., Saturday, Aug. 11, 2012. (AP Photo/Pablo Martinez Monsivais)

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    Republican presidential candidate, former Massachusetts Gov. Mitt Romney, right, and vice presidential running mate Rep. Paul Ryan R-Wis. are joined by Ronmey's granddaughter Chloe while talking to reporters on the charter flight between Virginia and North Carolina, Saturday, Aug. 11, 2012. (AP Photo/Mary Altaffer)

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    WAUKESHA, WI - AUGUST 12: Republican vice presidential candidate and Wisconsin native Rep. Paul Ryan (R-WI) (L) wipes away tears as he and presidential candidate and former Massachusetts Gov. Mitt Romney greet supporters during a campaign event at the Waukesha Expo Center on August 12, 2012 in Waukesha, Wisconsin. (Photo by Darren Hauck/Getty Images)