One evening a week, a group of CEOs meets in a Manhattan psychiatrist's office and engages in an ancient ritual. Ostensibly, it is a support group. Inevitably, it becomes a battle for dominance.

"Whenever you put alpha males together, the most aggressive will overpower the others," says T. Byram Karasu, the veteran psychiatrist who has run the sessions for the past 23 years. The fighting is subtle, but it's vicious. "Even giving advice is geared toward lowering the others' self-esteem. Those at the lower end of the group...