American millionaires Marshall Cogan and Stephen Swid today abandoned their bitterly contested takeover bid for Sotheby's auction house, making a major profit in the process.

Their withdrawal after three months of acrimonious public debate makes it all but certain that the other American bidding for the company, Detroit shopping mall magnate A. Alfred Taubman, will succeed.

Sotheby's said Cogan and Swid had conditionally agreed today to sell their 29.95 percent holding in the 239-year-old company to Taubman, who holds a 15 percent stake.

The two New Yorkers, who built up a business empire selling carpet felt and furniture, will get 700 pence ($10.71) a share from Taubman.

This is considerably more than the $8.16 they offered when they were buying, or even their higher offer last week of $9.89 a share. Much of their stock was acquired last year when the price fluctuated between $4.54 and $7.94.

Cogan and Swid, in a statement released by their bankers, Morgan Grenfell and Co. Ltd., said: "The time, it seems, has come to respond to the reality of the situation in an appropriately pragmatic fashion . . . We have finally determined to sell our shares and withdraw our proposal."

Sotheby's directors, who fiercely opposed Cogan and Swid as men who knew nothing of the auction world, said today they were delighted at the news.

Taubman, who entered the bidding earlier this month as a "white knight" to save Sotheby's from Cogan and Swid, must now win the approval of the British Monopolies Commission and meet the requirements of the U.S. Hart-Scott-Rodino Antitrust Improvements Act.

Then Taubman, armed with almost 45 percent of the Sotheby's shares, will bid for the remaining 55 percent. "He's interested in nothing less than 100 percent," a Taubman spokesman said. He said Taubman would offer 700 pence a share for the remaining shares--an offer that values the auction house at 81.6 million ($124.8 million).

The agreement was the result of backroom negotiations between the rival bidders, and there were indications the British Monopolies Commission would now allow the Taubman bid to succeed, said Marcus Agius of Lazard Brothers and Co. Ltd., Taubman's bankers.

"They've been very understanding," Agius said of the commission.

The news sent shares in the Sotheby Parke Bernet Group up on the London Stock Exchange from $9.23 to $10.25 at one point in trading.

Taubman was recently listed by Forbes magazine as one of the 400 richest men in America.

Market analysts said Cogan and Swid's withdrawal appeared prompted largely by the prospect of a lengthy battle before the Monopolies Commission.

Other factors could be the profit they will reap from their brief involvement with Sotheby's by the sale to Taubman, and the almost solidly hostile response accorded them by the auction house.

At one stage in April, 133 full-time experts and middle managers at Sotheby's--virtually the entire European staff--threatened to resign if Cogan and Swid succeeded in their takeover bid.

Backers of the two men charged snobbery at the upper-crust auction house, but Sotheby's said its concern was that Cogan and Swid would force radical, damaging changes to return the company to profitability.

The company, which has sold everything from Renoir paintings to Marilyn Monroe's bra, said it was also afraid that Cogan and Swid would have to borrow too much to gain control of the company.

One art expert charged that Cogan and Swid "would trivialize us and destroy this business in pursuit of profits."

Sotheby's, badly hit by the slump in the international art market and an over-extension of its branches, lost almost $4.6 million in the year ended last August. It was the first trading loss in its history.

Taubman, a property developer and trustee of New York's Whitney Museum, has said he plans no upheavals at Sotheby's.