The two men attempting to forge a deal between the Metropolitan Transportation Authority and its largest union are separated by vast differences in both bargaining position and personal history.

One, John Samuelsen, is the fiery leader of Transport Workers Union Local 100, one of the city's most militant unions. He has spent most of his career underground, inspecting track in the dark while dodging trains and rats. He wants raises and a guarantee that the MTA won't resort to layoffs.

Across...