A lively -- and expensive -- mayoral fight in Chicago

Wanted: a strong leader to administer complex, deficit-prone, $2 billion-a-year operation employing 41,000 workers. The city of Chicago, which is celebrating its 150th anniversary this year, is , of course, running no such advertisement. But the job of mayor in the nation's second largest city is for real and very much available.

Voters will do the ''hiring.'' Though the election is not until April, the choice in this historically Democratic city is almost sure to be made when that party's nominee is selected in the Feb. 22 primary.

Three candidates are vying for that honor in one of the liveliest and most expensive primary campaigns ever waged in this city. Two hope to replace incumbent Mayor Jane Byrne. They are Richard M. Daley, Cook County state's attorney and son of the late Richard J. Daley, Chicago's mayor for 21 years, and US Rep. Harold Washington, the first black considered a serious contender for the city's top job.

But Mayor Byrne, who won a surprise victory in 1979 over the Democratic organization's candidate, Michael Bilandic, with a mere $125,000 campaign, is fighting hard to keep her job in a $3 million to $4 million effort. Her opponents are expected to spend a combined $2 million.

Mayor Byrne is generally considered a strong leader and highly intelligent. But she has been criticized for a sometimes strident, impetuous personal style that has led to what some call a ''revolving door'' administration.

One of Mr. Daley's ads takes her to task - with the song ''I Apologize'' playing in the background - for having named four police superintendents, four comptrollers, and four budget directors in just three years.

Mayor Byrne's campaign ads and current behavior - confident, balanced, low-key - seem aimed at toning down that volatile image. She admits some mistakes, but claims to have matured in office.

Political analysts say they view Daley as the mayor's most formidable rival. Though he has earned a credible reputation as the county's chief prosecutor, much of his success may depend on how well he overcomes an image of inarticulateness in upcoming debates.

''His father didn't exactly say things in the king's English, but he overcame it with a blustery, straightforward personal style that managed to get his thoughts across,'' says James Nowlan, a University of Illinois political science professor. Daley ''has to present himself as a competent successor to his father. I think some voters aren't sure he has the same qualities and strengths - that he's really a 1983 chip off the old block.''

Despite the recent increase in the number of Chicago's registered black voters, Mr. Washington is not expected to snare all or even necessarily most black votes. Many analysts predict that the black vote will be split - possibly evenly - among all three. Mayor Byrne, who drew 60 percent of the city's black vote in 1979, has been working particularly hard to repair her relations with the black community. Those relations were strained last summer by her appointment of a white majority to the Chicago Housing Authority board and an increase in the number of whites appointed to the school board.

Washington, who has been critical of some local black ministers for supporting the other two candidates, is viewed by many analysts as handicapped by his late decision to enter the mayoral race and by his campaign's lack of money and organization.

For some voters, the congressman's 1972 conviction for failure to file income tax returns during four years in the 1960's looms as a drawback. But Michael Preston, a University of Illinois political scientist, argues that Washington faces a much larger challenge in successfully assuring city voters that he intends to represent all Chicagoans and not just the black community.

The infamous Chicago political machine, credited with delivering election after election to the Daley Democratic organization for so many years, will again be a factor.

''The machine has deteriorated,'' says Dr. Milton Rakove, a political scientist at the University of Illinois Chicago Circle campus.''It's not dead, and it's not alive - it's somewhere in between. Jane Byrne has some of it, but not all of it. And it can't deliver the electorate any more like it used to - voters have become very sophisticated.''