US President George W Bush announced last week that he would settle for "nothing less than complete victory" in Iraq.
Bold words. But he knows the real battleground is public opinion in the US. Anything less than total confidence would have brought a serious setback.
How likely is a complete American victory, though?
More than 2,000 US troops have died since the invasion of Iraq
If this were a regular war, against an army in the field, the Americans would win it hands down. The US forces here in Iraq have the firepower to destroy any formal enemy outright.
But the qualities that make them the most effective fighting force in the world - ferocity, first-class training and superb logistical ability - don't necessarily help them win a low-intensity conflict like this.
In an all-out war, the loss of 10 US marines outside Falluja last Thursday evening would have been relatively insignificant in political terms. But in a war of public relations, at a time when the president was promising complete victory, it was an awkward blow.
Which was presumably why the Pentagon kept the news of the deaths secret for 24 hours. If it had come out in the usual way, Americans would have heard it on radio or television at pretty much the time they were reading about Mr Bush's confidence in complete victory in their morning newspapers. Not at all good for public opinion.
Victory is an elastic term. If Mr Bush can withdraw big numbers of troops from Iraq, cut down the American death-rate, and get the war off the news bulletins, then US public opinion might interpret that as victory of a kind.
Iraq would then be left to sink or swim, with a relatively small number of American soldiers working alongside Iraqi troops. Even if the Iraqis were not up to the job, and the insurgents either won the war outright or had to be brought into the government, the main American involvement would be long finished.
But all this depends on the ability of the Iraqi forces to fight a successful war. How good are they?
The Iraqi army of Saddam Hussein's time had mostly Sunni officers, many of whom had had the experience of fighting three wars, as well as crushing the Kurdish and Shia resistance. It was, relatively speaking, tough and disciplined - not up to fighting the Americans, but more than able to deal with an insurgency.
Paul Bremer, during his time as head of the provisional administration in Iraq, disbanded the army because it was too close to Saddam Hussein and the old Baath Party. He has defended his decision by saying the army had evaporated anyway. True - but the officer corps could easily have been lured back.
No-one knows how many recruits in the Iraqi army secretly support the insurgency, but there must be plenty
Gen David Petraeus, who had the task of training the Iraqi forces until earlier this year, argued that middle-ranking and junior officers from Saddam's army should be reinstated, because the Iraqi forces needed their expertise.
But it didn't happen. Gen Petraeus, a man of considerable vision - "America's last, best hope in Iraq," one American newspaper called him in 2004 - was reassigned to the United States.
Those who volunteered for the new Iraqi army did so for all sorts of reasons. Most joined because it was the only way they could earn decent money. Others were encouraged to infiltrate the army on behalf of particular political groups.
There are many entire units which are purely Shia or purely Kurdish - a real danger if civil war were eventually to break out.
No-one knows how many recruits in the Iraqi army secretly support the insurgency, but there must be plenty.
For all its firepower, the US military has failed to break the insurgency
Some soldiers show high morale, and face the daily dangers with remarkable cheerfulness. More do their job unenthusiastically. It's hard to generate much enthusiasm in your army if you don't pay your soldiers on time.
And if you don't give them proper protection. American soldiers drive around in heavily armoured vehicles, and wear body armour. Iraqi soldiers have little to shield them against ambush and roadside bombs, and their rate of death and injury is far greater.
Some soldiers sell their ammunition, presumably to people who pass it on to the insurgents.
In September, according to the senior US ground commander in Iraq, only one Iraqi battalion out of 120 was capable of operating independently of US forces.
And the problem the Iraqi soldiers face is arguably worse than ever. The US military, despite its ferocious tactics, has not succeeded in neutralising the insurgency in any of the major population centres in the Sunni heartland.
That includes Falluja. It also includes Ramadi, where a group of insurgents paraded in front of the cameras of the APTN and Reuters agencies last week, patrolling one or two streets in broad daylight and firing their rockets.
It certainly does not mean the insurgents control the city; they don't. But it also means the Americans don't either.
For the insurgents, too, this is a war for public opinion. The history of this kind of conflict, which includes Algeria's war of independence and Vietnam, strongly favours the insurgents.
Last week, despite Mr Bush's confident words, the insurgents certainly seemed to have the better of it.
Do you agree with John Simpson's views? Who will win the battle for public opinion on Iraq? Send us your comments.