Paul Brinkley is the head of a special American task force that aims to bring lasting peace to Iraq using the tools of capitalism. He represents a new approach to waging war, where the economic experts come in with the ground troops. And it's working.

Nowadays, Baghdad is a place where two very different things can be happening at the same time. While a special Iraqi police unit is engaged in a gun battle with Sunni militias in the Fadhil neighborhood, old men and young women play tennis in the relatively tranquil surroundings of the Alwiya Club on Firdaus Square, where a statue of former dictator Saddam Hussein was toppled six years ago.

This is the situation in these two Baghdad locations on the last Saturday in March. Under a bright blue sky, some...