NEW YORK -- Robert W. Seijas, sole auctioneer of the shares of Cooper Industries Inc. on the floor of the New York Stock Exchange, cocks his ear toward the stockbroker standing next to him and ponders whether to buy his wares.

The broker, from Bear Stearns & Co., wants to sell Seijas 40,000 shares of Cooper, a Houston company that makes Champion spark plugs and other electrical equipment. His asking price: $41 a share.

That doesn't fly with Seijas. He already has 12,000 shares that he would like to unload himself, and the price has been slipping today. He shakes his head "no."

"What's a good price?" the broker persists, knowing that exchange rules oblige Seijas to buy the shares at some price to fulfill his responsibility to maintain a marketplace for Cooper's shares.

"You're on your own, for now," Seijas replies. That signals that the broker should wait and see what develops, rather than accept a lower price from Seijas to make the sale right away.

"Okay, Bobby, if you see any demand, let me know," the broker says cheerfully, and hurries off to handle another order.

Twenty minutes later, just as Seijas had hoped, a buyer shows up. Seijas uses a hand-held paging device to call back the Bear Stearns broker, who after a small amount of negotiation reaches a deal to sell a quarter of his shares to a broker from Smith Barney, Harris Upham.

"Ten thousand Cooper trades at one," Seijas declares, meaning that 10,000 shares were sold at $41 even.

Here on the floor of the nation's premier financial market, beneath $400 million of glowing computer screens and other electronic hardware, the bargaining still takes place face-to-face, person-to-person, much as it did when merchants hawked their goods in the agoras of ancient Greece 2,500 years ago.

Although Seijas employs four computer screens to keep track of trading on the exchange floor, his most important tool is his voice. He spends much of his day answering questions about Cooper and three other stocks assigned to him, and encouraging buyers and sellers to make the little compromises that result in deals.

In the eyes of rival stock markets and of many independent critics, the trading system used by the New York exchange is unnecessarily costly and ludicrously out-of-date. The New York floor is "a roller derby kind of setting," sniffs an official at the National Association of Securities Dealers, which uses a highly automated computerized system for trading stocks on a market called Nasdaq.

Generally, the companies whose stocks are traded this way are much smaller and often less prestigious than the blue-chip stocks of companies like International Business Machines Corp. and American Telephone & Telegraph Co. that gravitate toward New York.

But rising competition from more automated markets, both in the United States and overseas, has forced the New York exchange for the first time to consider seriously whether to introduce a method of trading that would take some decision-making and price-setting away from humans and give it to a vast computer network that would link big brokerage houses and investment firms. Exchange Chairman John J. Phelan Jr. recently urged launching an experimental program later this year that would provide for stocks to be traded via computer after the floor shuts down at the 4 p.m. closing bell.

Such plans draw a decidedly mixed response from Seijas and other traders who have built their careers mastering the organized turmoil that prevails on the exchange floor. While they welcome the arrival of new technology that increases the market's ability to handle big volume, they maintain that the current system provides valuable flexibility that saves the public money and offers a critically important guarantee that the market will remain open even in times of stress.

Seijas, who allowed a reporter to watch his work last Wednesday and on one earlier occasion, pointed to the Bear Stearns broker as an example of someone who benefits from the personal give-and-take among traders who work with each other every day.

"In an automatic system, he wouldn't have the option to wait in hopes of getting a better price" than the one available when he first proposed to sell, Seijas said. "My job is to bring buyers and sellers together."

Seijas, trim in a dark pinstripe suit and white shirt, is one of 429 auctioneers, called specialists, on the floor of the exchange. For incomes that often exceed more than $1 million a year, these people are responsible for handling all the trading in the stocks of the companies assigned them. Consequently, their role is central to the Big Board's system.

Brokers continuously bombard the specialists with orders to buy or sell stock. The brokers hustle back and forth between their telephones at the edge of the floor, where they receive orders from their firms and clients, and the specialists' stations on the rims of the large, circular trading desks that dominate the floor of the exchange.

The specialists are generally partners in private firms that pool trading capital and hire clerks and other staff for several auctioneers. Seijas is a partner of Lasker Stone & Stern.

For the privilege of holding the franchise over the auctioning of a stock like Cooper, Seijas must maintain an orderly market. That means, above all, that he must step in with his own firm's money to buy or sell Cooper stock when nobody else is willing, in order to guarantee that price changes are as gradual as possible.

New York exchange officials point to this as a principal distinguishing characteristic between the trading method used on the New York exchange, and the Nasdaq method.

With Nasdaq, there are several firms "making markets" in any given stock, meaning they will step in to buy or sell its shares. The advantage, Nasdaq says, is that the competition among market-makers pushes down the cost of carrying out transactions.

The disadvantage, as the New York exchange is anxious to point out, is that none of these firms is required to cushion the stock from jumps or plunges in its price.

And as Big Board officials repeat endlessly -- during the "Black Monday" Oct. 19, 1987, stock market crash, it was impossible to get through by telephone to many Nasdaq market-makers to sell stocks. The New York exchange boasts that it never closed its doors during the crisis, although it came close.

"What we sell here is liquidity," Seijas said. "In a crisis, I can't hide. I'm here, and as long as this market is open, I have to keep it going. When you had a telephone market, and things were crashing, dealers didn't answer the phone."

Last Wednesday morning, the specialist's obligation to deal had unpleasant consequences for Seijas. After only 90 minutes of trading, Seijas estimated that he was down $30,000. The reason: Both of his principal stocks, Cooper and Sysco Corp., a Houston food distributor, were moving against him.

Seijas had begun the day "long" on Cooper, meaning he had shares in his inventory. But more brokers were interested in selling than buying, pushing the price down and forcing Seijas to keep buying to guarantee a steady market.

In contrast, Seijas had started out "short" in Sysco, meaning he had sold shares that had been borrowed temporarily for trading purposes. Sysco's price was rising, however, and he had to sell more -- that is, increase his short position -- to meet demand.

"I'm getting hammered today. I'm long Cooper, and it's going down, and I'm short Sysco, and it's going up, in the same market," said Seijas, shaking his head. "That's the downside of being forced to lean against the wind."

But Seijas knew he would make back the money eventually. Traders are willing to be specialists because of the money they make whenever the market turns. Since they are continually working against the market's trend, they normally find themselves buying stocks at the lows and selling at the highs.

Sure enough, the market turned Seijas's way by early afternoon. Prices stabilized for both Cooper and Sysco, and he was able to begin getting out of his losing positions.

The billion-dollar question that will determine the specialists' future is whether the public will continue to be willing to pay the price, in added transaction costs, for the liquidity that the specialists provide.

"Anybody in a monopoly position extracts rents, and a specialist is no different from that," said Bruce N. Lehmann, a professor of finance and economics at Columbia University's Graduate School of Business. "They're taking off some fraction, and the cost of trading on the exchange is larger than it would be otherwise."

Such costs are difficult to calculate because they occur in tiny fractions of a dollar charged in millions of transactions in the heat of the marketplace.

Another important question is whether the specialist's exclusive role as auctioneer, and the importance of human interaction in the marketplace, constitute an invitation for abuses.

The exchange is sensitive to this danger and has tightened surveillance since the insider trading scandals of the last decade. Every trade must be recorded, on slips of paper, with the stock, price, volume and time, providing a trail for investigators to pursue if there are signs of wrongdoing.

Said Gregory S. Winski, a broker for C.J. Lawrence, Morgan Grenfell Inc.: "The exchange has such strict rules that if there's any little thing, even involving 100 shares, surveillance comes down and asks about it."