WASHINGTON—Sen. Sherrod Brown is a baseball fanatic who once dreamed of turning his little league exploits into a career with the Cleveland Indians. Sen. Rob Portman is a bicycling and kayaking enthusiast who once fixed his own dislocated shoulder by slamming himself against a rock.

On Monday, the two Ohio senators and their staffs faced off in a friendly—but still fiercely competitive—game of softball, with a dash of politics of course. After all, this game pit “The Blue Collars” against the “Swing State Sluggers.”