ONE BREEZY afternoon last May, Sheik Mohammed Fassi blew in to, Washington from Miami with his flock of 60, a relatively small entourage by sheik standards. They needed a place to rest.

Of The Big Four resting places, The Madison, The Watergate, The Four Seasons and The Fairfax (now the Ritz-Carlton), Fussi chose the Fairfax.

The lobby looks like new money and smells like old money. Understated elegance. Clubby. Fassi said he would need seven suites and 20 rooms while here to donate $50,000 to the city.

No problem, said The Fairfax.

But wait.

Limousines snarled Embassy Row as white-gloved doormen unloaded the luggage. Mountains of it. The sheik stepped inside with his lieutenants to review the logistics.

No, no. On second thought, seven suites would never do. They would need 15. And 20 rooms weren't nearly enough. An entire floor was more like it.

"He started saying I want this and I want that and we just didn't have it," said the hotel's general manager, Paul Seligson. "It happens. You try your best, but sometimes . . ."

So the sheik's men piled back into their limos and drove seven blocks to The Madison, where they took over the entire top floor.

It cost $7,500 a night. Not including room service.

Another battle in the great Luxury Hotel Wars of Washington.

This time The Madison had triumphed.

It is politely referred to as "friendly competition" by all the proprietors of the high-visibility hotels--and it has intensified lately. Occupancy rates here were down 8 percent in 1982, according to the Washington Hotel Association, and next month Hilton's Vista International--a grand-scale, glass, steel and marble, structure--will be opening on M Street right behind The Madison. And The Mayflower, Embassy Row, Ramada Renaissance and Jefferson have recently begun or finished major upscaling of their facilities.

In this business, money talks and everybody runs.

"When a guest wants something," says one concierge, "he seldom questions the cost . . . The idea is to know when the guest has had enough of being on his own and wants to be waited on . . . If somebody drops something, there's a race to pick it up, and that's the way it's got to be."

The least expensive in luxury accommodations starts at about $100 for a single room, with suites averaging $500 to $1,200, and floors, like that occupied by the sheik, going for up to $20,000 with room service. In these hotels, this sometimes means whole roast pigs on a few hours' notice.

Everyday amenities in this league are not fresh ice down the hall and an After Eight mint on the pillow, but cognac before bed (The Ritz-Carlton), a fresh rose every night in the bathroom (The Four Seasons), daily homemade chocolates on the dresser (The Watergate) and heated towel racks (The Madison). Thick, cuddly terrycloth robes are standard bathroom equipment.

And all the hotels keep detailed guest histories on file cards or in a computer. After all, someone might not like cognac.

A manager, owner, concierge must never concede that he or she is the least bit tweaked by what his competitors are doing. No contest, they sniff. Just a dignified and rather quiet war to see who can do it better for the cream clientele of this planet.

Some of the hotels even have a special marketing agent on staff simply to woo the international market. And all of the managers work the social and political community as well as Washington's top defense lobbyists. It's part of the job.

What it gets down to is this: There is almost nothing one of the four top luxury hotels won't do to keep a monied guest happy.

Consider these examples:

* Barbara Walters prefers the junior suites at The Watergate and corner suites at The Madison. And, according to The Watergate, she also requires an extra phone line in her room. The Watergate installed a jack for an outside hookup in five of its junior suites so there would always be at least one suitable suite available whenever Walters visited.

* Frank Sinatra always takes a suite on the second floor at The Madison. He gets three phones installed even if it's only for an overnight stay.

* One Friday night a few years back, an Arab prince staying at The Madison decided Saturday would be perfect for a soccer game. The only trouble was that his team, or entourage, didn't have a soccer ball. Or shoes. Or uniforms. Or field. By evening's end The Madison's management had taken care of the little problem. They rented a field and purchased the uniforms. The $4,000 cost was tacked on to the prince's hotel tab.

* When well-known literary agent Irving (Swifty) Lazar wanted a circumcision kit for an African doctor, whom he had recently met, Ritz-Carlton concierge Christiane Juster promptly found a Brooklyn circumcision kit distributor. Once, she took a plane to New York to bring a forgotten mink to one of her guests who was leaving for Europe.

* The Four Seasons always has a baby grand ready for Van Cliburn's visits.

* The Watergate recently hired a new rooms manager. His first assignment: Spend a night in each of the other three hotels to make sure some precious amenity wasn't being offered that The Watergate didn't know about. Everything was in order.

"Small dogs stay with us, you know," explains Sharyn Thomas of The Four Seasons. She is sipping Earl Grey tea, one of the 100 varieties The Four Seasons keeps on hand. "So we have dog-sitters for our patrons. One night one of our dog-sitters got scared of the dog she was sitting for. So she locked herself in the bathroom and called the desk to come rescue her. Well, the point of the story is that we have phones in the bathroom. If we didn't have phones in the bathroom, no one would have known that poor dog-sitter was trapped . . ."

Over the years, each hotel has cultivated its special clientele and style. The Washington luxury hotel archetype, The Madison, owned by Marshall Coyne, has long been associated with oilmen and the Middle East. During the negotiations that led to the Camp David Accords, both the Israelis and Egyptians stayed there--riding on separate elevators. When the Rolling Stones were here, Coyne let them take over the entire Dolley Madison, a mini-version of The Madison just across the street.

Texans seem to have a penchant for The Ritz-Carlton, young wealthy Californians gravitate toward The Four Seasons, and performers tend to stay at The Watergate because it is near the Kennedy Center.

And no matter what anyone says, all the hotels want the Arabs.

Usually here for business or politics, the Arabs tend to travel in large contingents, stay a long time, and require different and costly room service. The Saudi Embassy, for one, uses The Ritz-Carlton when its guests visit. "Except for the royal family, of course . . . They stay with us," says the ambassador's social secretary, Helen Smith.

"First of all, they usually come in large groups and spend a lot of money," says Alexander Braune, general manager of the new Vista International, who has spent nearly three decades in the international hotel business. "You have got to be prepared to receive them and get the food they want. When they pick up the phone, you have got to be ready . . . I dealt with them in London and it's worth it."

"Arabs are associated with well-spending people," says Peter Buse, graduate of The Madison and The Watergate and now general manager of The Embassy Row. "And a deluxe hotel should associate itself with those kinds of people and contacts."

An Arab prince, here for medical treatment recently, occupied a suite on the top floor of The Ramada Renaissance Hotel, running up a $1,500 room bill each day. "He wanted goat milk, we got him goat milk; he wanted lamb, we got him lamb," says hotel manager Klaus Henck. "We had all his meals prepared at Middle Eastern restaurants and brought it in."

Hot rumor on the hotel circuit: The Arabs have abandoned The Madison in favor of The Four Seasons.

"The Arabs have been staying at The Madison for years--it's just had that reputation," said one Washington lawyer who knows about these things. "Just recently, several medium-level Kuwait officials were booked to stay there and Coyne put them in The Dolley Madison. They were offended and checked into The Four Seasons . . . I think a lot of that has been happening."

Fueling this rumor was King Hussein's visit here a few weeks back. He and his entourage took the top floor of The Four Seasons for $7,000 a day. Hussein has stayed at The Madison during at least two earlier visits. "He likes hamburgers," says Four Seasons general manager Seamus McManus.

What has also helped The Four Seasons' Arab draw is Desiree, the private, dimly lit disco that has become a hangout for young, wealthy Arabs, both Washington residents and those passing through.

Coyne dismisses the notion that the Arabs have moved away from his hotel, or that he ever catered predominantly to them.

"We get a cross section of all business," says Coyne. "We probably get less Arab business than our share."

As for The Four Seasons, McManus takes a cold-blooded business approach to his newly found Arab trade.

"The total Middle Eastern market has grown . . . I'm not so sure that we have picked up that market. We don't want to get into a predominance of any one market because then you have to service it . . . If you lean service towards it, then you exclude others . . . If Australians took over, then before you know it I'd have to put little kangaroos on the menu."

Luxury hotel competition in Washington hasn't always been this fierce. There was a time when The Hay-Adams was considered Washington's finest hotel. That was before Marshall Coyne entered the market in the early '60s.

It was then that Coyne, a wealthy real estate developer and contractor, decided Washington needed a hotel for the wealthy and began traveling through the United States and Europe looking for role models. In a confidential pamphlet to employes published in 1962, a year before the hotel opened, the staff was sternly warned: "The Madison recognizes that its guests are persons of refinement and taste . . . You will be required to learn the name and title (if any) of every guest immediately after his or her arrival . . ."

One former employe says Coyne has never been worried by the competition, and in fact, is not as aggressive in sales as is the management of other hotels. "He should be," said the former employe. "It's catching up to him."

But The Madison, observers note, is Coyne's hobby, his "toy." "Everyone who builds a hotel tries to make it like the Madison," said Coyne. "We have an image and everyone tries to imitate it."

Last week, Coyne denied talk around town that The Madison was for sale. "It's absolutely not true," he said. Since Coyne has always said that the hotel is nowhere near his primary source of income, service has always been a priority over profit. Thousands of guest histories are kept meticulously. If someone needs an umbrella, he gets an umbrella. Never mind the room service menu. Order what you want and it'll be there. Then there are the fully equipped bars in all the rooms, Godiva chocolates and anything, simply anything, money can buy for the guest.

But the whims of the wealthy exact their price for those who cater to them.

"The Watergate scandal killed us," said Nicholas Salgo, chairman and owner of The Watergate. "People took everything movable home with them if it had a 'W' on it. We lost $700,000 . . ."

Last year, Coyne was fined $5,000 by a federal judge in Baltimore after being accused of illegally buying Canada geese to be served as pa te' in the hotel's posh Montpelier Restaurant.

When Ritz-Carlton owner John Coleman purchased the hotel in 1977, he sank $7 million into refurbishing and renovation. The Four Seasons spends $72,000 a year to keep its rooms and lobby in floral splendor. And the new Vista won't be outdone: Hubert de Givenchy has personally designed eight of its penthouse suites.

According to a Watergate manager, when The Who stayed there several years back, several members of the rock group wrecked a suite. They were asked to leave and promptly came to the desk with $5,000 cash to cover that damage and any additional wreckage. They were permitted to stay and the hotel looked the other way.

"There is only a small sector of the clientele that stays at luxury hotels, and they are very discriminating," says Coyne. "They live that way in their office and in their home. We can't do something by advertising. You do it by performance . . . You can never fluctuate standards, and you have to spend money to do it or it's not going to get done. You can't do it by being concerned with cost, and you can't cut corners. You have to go all the way."