When David Einhorn, the founder of Greenlight Capital, plays host to his investors at the American Museum of Natural History in January, his guests will sip cocktails and dine under a 94-foot blue whale in the Milstein Hall. William A. Ackman will hold court one week later, at the New York Public Library at Bryant Park, where investors in his Pershing Square Capital Management will mingle in the historic halls of marble, wood and gold. The lavish settings will be the same as in years past, but the circumstances will be strikingly different: Both billionaire hedge fund managers, and many...