Facing the worst labor shortage in decades, Japan's carmakers have seen the light: hire more women.

The problem is they're all having the same epiphany at the same time, and so now find themselves fighting over the world's smallest pool of female engineering graduates.

The story of Yui Mitsuhashi illustrates the bind. Captain of a team that builds race cars at prestigious Osaka University, the 24-year-old engineering student would be among the most prized recruits for Toyota Motor Corp. or Nissan Motor Co., if only she wanted to work for them. But as much as she loves cars, she isn't sure she wants to spend her life in the industry — which has a reputation for long hours and big gender imbalances.