The two houses will do almost anything to outmaneuver each other, and the friction between them will likely only increase under new CEOs.

This May, after billionaires had outbid billionaires in New York’s contemporary art auctions, something became immediately clear: Christie’s had just clobbered Sotheby’s with a gavel.

Over four days, Christie’s sales totaled $1.7 billion, its biggest week ever. On one of those evenings, frantic bidding inside its Rockefeller Center salesroom enabled the auction house to sell $706 million of art spanning the 20th century in less than two hours. An anonymous bidder even plunked down $179.4 million for Pablo Picasso’s Les Femmes d’Alger, smashing the record for the most expensive work ever sold at auction. “We’re in a fantasyland,” proclaimed collector Michael Ovitz, the former president of Walt Disney Co., as he left the room...