Sunny (literally; this is the brightest, squintiest film in months) and engaging, as far as it goes, “Battle of the Sexes” is a two-headed biopic reluctant to complicate its coming-out story with too many … complications.

This will not be a problem for most audiences. Collectively, the “Battle of the Sexes” team knows how to please a crowd. Its directing team of Jonathan Dayton and Valerie Faris made the wish fulfillment smash “Little Miss Sunshine.” The screenwriter Simon Beaufoy wrote the wish fulfillment smash “Slumdog Millionaire.” Emma Stone, who plays sports legend Billie Jean King, broke hearts all over the place in the wish fulfillment fantasy (bittersweet division) “La La Land.”

Stone’s co-star, Steve Carell, boasts a career born in comedy and now conversant in a wide range of seriocomic and dramatic projects. Here he plays the inveterate hustler and former tennis champion Bobby Riggs, who declared war on the “lib thing,” aka the women’s movement and women’s sports in particular, challenging King to a best-of-five match when she was up and he was down and the world was primed for a gender-war gimmick with teeth.

Deep inside the circus of the Sept. 20, 1973, “battle of the sexes” contest, watched by 50 million people, the stakes were high, and serious. The meaning and impact of the King-Riggs match far exceeded the boundaries of a tennis court. The whole time it was happening, King’s off-court life was consumed by a budding affair with a hairdresser, Marilyn Barnett, conducted while King’s marriage to her husband, promoter and business partner Larry King, had many years left to go.

This is the emotional focus of “Battle of the Sexes.” In the early scenes, King has conquered the 1972 U.S. Open, but the patriarchal U.S. Lawn Tennis Association (Bill Pullman oozes avuncular privilege as its figurehead, former tennis champ Jack Kramer) is throwing most of the prize money to the male players. King and the other women walk out and establish the Women’s Tennis Association.

Riggs, a compulsive gambler, gets King’s rival champ, Margaret Court, to play him in a match that came to be known as the “Mother’s Day Massacre.” Riggs prevailed that day. King, who’d turned Riggs down a few times, agreed to a match. If she could take care of Riggs, King figured a win might change the country’s thinking on a lot of fronts.

Beaufoy’s script lobs between King’s struggles and Riggs’ hustles. Andrea Riseborough plays Barnett, and as photographed by cinematographer Linus Sandgren, the scene where they meet in the salon chair is treated as an intoxicating vision of romantic promise. The subsequent relationship, a closeted affair for years, meant King lying to her husband, risking a career, potentially rupturing a bond with her traditional working-class parents and gambling more than Riggs ever did.

The movie’s determined not to demonize any of its major players. But there are tricky obstacles in “Battle of the Sexes.” The climactic match wasn’t much in terms of suspense (6-4, 6-3, 6-3 in straight sets), and to the degree this is a sports movie, it’s not a very exciting one.

So much has changed since 1973, and so much hasn’t. The lib thing is still a sticking point for about half of the nation, which is why the film’s both fun and a mild disappointment.

Michael Phillips is a Tribune critic