Ilan Goldenberg reports from Cato, where former Special Envoy to Places You'd Be Afraid to Visit James Dobbins made this point:
Dobbins argued that in every case of trying to fix a failed state the neighbors play a critical role. They have serious national interests because they are the ones who have to deal with the refugees, violence, crime, economic shocks and all the other wonderful things that happen as a result of a total meltdown on your border. They simply are not going to sit on the sidelines.
All of the neighbors have an interest in maintaining stability. To do this they search for proxies who will carry out their agenda. Paradoxically, this proxy strategy only ends up exacerbating the situation by strengthening various warring parties and creating greater potential for broader regional conflict. The only way around this, is to create a regional dialogue that forces all the neighbors to come together and coordinate their strategies. Instead of a zero sum game they should be working towards the same greater goal of keeping Iraq from totally falling apart.
I think this is missing a piece, however. Iran, Syria, Saudi Arabia, and the rest surely recognize that they need to coordinate in order to achieve stability in Iraq. They don't need America to explain that to them. At the moment, however, none of the regional partners actually want a stable, functioning Iraq -- because currently, a stable, functioning Iraq would be occupied and effectively controlled by America. And Iran doesn't want an American vessel next door.
So their incentives work basically like this: Amass proxies who they can use to keep some sort of control over the nation and, if necessary, drive us out. Once that's done, begin working to use those proxies to stabilize the nation. So long as we're occupying the place, however, it's decidedly not in Iran or Syria's interest to stabilize the country -- from their perspective, they'd be stabilizing a military platform that Washington can eventually use for its long-stated goals of regime change. So their proxy strategy is not, I'd suggest, accidentally working against stabilization, but rather, working to ensure Iranian influence after we leave and stabilization becomes a non-threatening goal for them.