With that news, the number of major U.S. corporations that have two co-CEOs were just slashed by a quarter. Just three other companies in the S&P 500 -- Chipotle, Oracle and Torchmark Corporation -- have co-chief executives who share the company's most powerful role, according to data from S&P Global Market Intelligence. The unusual leadership structure is one that management experts say can work well in certain circumstances, but is also fraught with potential problems, from personal infighting to slow decision-making.

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Other companies have given up the dual CEO structure when facing headwinds. Less than a year after Martha Stewart Living Omnimedia named two co-CEOs in 2008, the company replaced the structure with one that was "leaner and meaner" in a tough economy where the media giant faced heavy losses. After failing to keep up with the iPhone juggernaut, BlackBerry's co-CEOs Jim Balsillie and co-CEOs resigned and were replaced by a single chief executive. In May, John Cryan became the sole CEO of Deutsche Bank after two co-CEOs announced their resignations in 2015; the bank reported a huge record loss last year and is attempting a turnaround.

That's why Whole Foods' change isn't terribly surprising. Tougher times necessitate the speed, urgency and clear communication that can be harder with two CEOs than one. "You need clarity, and clearly if you’re [struggling] what you don’t need is mixed signals coming out of the C-suite," said David Heenan, a visiting professor at Georgetown University's McDonough School of Business who wrote a book about co-leadership with the late leadership sage Warren Bennis. (A Whole Foods spokesperson declined to comment beyond the company's call with investors Wednesday.)

Heenan said that after he wrote his book with Bennis, a couple of co-CEOs remarked to him that the structure can be like a family, where employees play off mom and dad in an effort to get their way.

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"Unless they're totally on the same page," he said, employees will "work it. You're going to have problems. If the organization has been having difficulty, people understand it. And there are nefarious types who are going to use it to their advantage, which is exactly what you don't need" if the company isn't performing at its best.

Meanwhile, he says, it's a difficult setup for almost anyone with the typical psyche of a CEO who isn't a co-founder. Most people who reach those jobs, Heenan notes, "have scratched and clawed a long time to get to the corner office," he said. "Then all of a sudden you have to put a smile on your face and tell your spouse and friends 'gee, I'm going to share power with this guy'? There aren't many people who can stand up to that test. There is a Darwinian element to all this."

Yet there are times when companies find that the setup makes sense -- and may even provide value to investors. Stephen Ferris, the interim dean at the University of Missouri's Trulaske College of Business, found in a 2011 study that the stock market reacted positively to appointments of co-CEOs, and that they can be beneficial when the company has weak independent board members -- the two CEOs offer a check on each other -- or if there is little monitoring by large institutional investors.

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In an interview, Ferris said the structure can make sense if the CEOs have truly complementary skills of if there's extensive diversity in the company's product line or geography. His study also found it to be more common in the aftermath of a merger, typically as a temporary arrangement for three to four years. "It's not a popular long-term arrangement," he said.

At Whole Foods, Robb and Mackey had shared the top job for six years, longer than most. When Robb was promoted in 2010, Mackey, an outspoken libertarian who'd run into controversy the year prior for penning an article on health-care reform that prompted customers to threaten boycotts, told the Wall Street Journal he thought the two would be leading the company "together for another decade or so," noting "there's plenty of work for both of us to do."

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In an interview this February, Robb told the Journal that the two came to share the job because Mackey "didn't want to leave, and I wanted to be a CEO. We went for a walk and figured it out." He said the two settle disagreements "through email, at all hours," and praised the dual CEO system for bringing together their different skills. "Why wouldn't you have two CEOs? You have two perspectives, two folks who are deeply committed to the company. If it works, you get the best of both worlds," he told the paper.

Robb isn't leaving entirely, however. The 25-year Whole Foods veteran will remain a senior adviser and a member of its board after he leaves the co-CEO job at the end of this year. And he'll step down with some parting gifts: $10 million for agreeing to a noncompete arrangement, according to an SEC filing, and a lifetime discount at Whole Foods.

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