Emma Stone played brilliantly in a movie subtly depicting the nuances of what game theorists call a battle of the sexes. More recently, she played Billie Jean King, who famously defeated Bobbie Riggs in a 1973 match in the Houston Astrodome. Both movies have game theory implications. The titles keep us in suspense.
La La Land:
Everyone knows La La Land. If you don’t, or if you don’t think it’s flawless (check), you might just get arrested. See here. Aspiring singer and actor, Mia Dolan (Emma Stone), meets aspiring Jazz musician, Sabastian Wilder (Ryan Gosling), in her own thankfully mild road rage incident. After yet another stumbling start, the two find themselves falling in love. They also find themselves pushing each other to pursue their respective dreams, one that will take Mia to England, and lead Sebastian to quit a financially lucrative Jazz fusion gig featuring his friend Keith (John Legend), that is too pop for Sebastian's taste. (Still, the question remains who saved Jazz. See here.) The tension builds over whether the lovers will remain together, with either sacrificing dreams of stardom in favor of the other, or if they will be pulled apart by their mutual drives for success. The set-up is a perfect battle of the sexes, but with a twist.
The classic (if sexist) set up involves a husband and wife deciding whether to go to a football game, which the husband prefers, or to the theater, which the wife prefers. Despite the different tastes, the happy couple enjoys each other’s company, so much so that each prefers joining in the other’s chosen activity to attending her or his own preferred activity alone.
Imagine four ranked payoffs: 10, the highest; 7, the second highest; 5 the second lowest, and 3, the lowest. If the husband and wife together attend either football or theater, the spouse favoring the activity gets the highest payoff, 10, and the other spouse gets the second highest payoff, 7. Named for Nobel Laureate John Nash, who developed the concept, those two outcomes are pure-Nash equilibria, meaning that neither spouse could do better by changing strategy alone given the spouse’s strategy. By contrast, if the husband and wife each pursued his/her preferred activity alone, each receives the second lowest payoff, 5. If each attends the other spouse’s preferred activity alone, each spouse receives the lowest payoff, 3.
The better results for each spouse arise when they join together, either watching football or theater. As I tell my own students, happy couples shouldn’t worry too much about such things; the game hopefully gets played over and over, so just learn to take turns! One student quibbled about the positive payoff, 3, when each attends the other’s first choice activity, arguing the payoff should be a negative. How unromantic is that? Consider The Gift of the Magi: the wife crops her beautiful locks of hair, then sells them to buy her husband a chain for his watch; the husband, meanwhile, sells his watch to buy his wife a comb for her beautiful hair. The poor couple each favored the other’s preference, but they are so in love!
Like The Gift of the Magi, La La Land doesn’t end pure-Nash. Mia and Sebastian go their separate ways, but not for want of deep mutual affection. Instead, each relentlessly pushes the other to pursue her/his life dream. Only years later, joined by her husband and young daughter, the star, Mia, stumbles upon a new, yet familiar Jazz bar, whose logo she recalls, having designed it herself years before. Upon seeing her, Sebastian plays their song. When their eyes catch, each smiles, dreaming of what might have been, yet knowing what they helped each other other to become.
Battle of the Sexes:
Despite the title, Battle of the Sexes evokes a different game. Billie Jean King (once more, Emma Stone) learns that the prize money for women in a major tennis championship is one eighth of that for the men, despite the equal audience draws. She and Gladys Heldman (Sarah Silverman), the founding editor of Tennis World Magazine, lead a revolt, with the women accepting $1 each to sign a contract with the soon fledgling Virginia Slims Tour. This will compete with the existing professional tennis association, minus the perks, such as they were for the women who played. Bobbie Riggs (Steve Carrell), a former tennis champion and present gambling addict, is always angling, and thinks he has found one. If the 55-year-old Riggs can get the 29-year-old King to play in a sex-challenge tennis tournament, with a $35,000 prize, he will regain his former sexist glory. King recognizes the publicity stunt for what it is, and declines. As this is happening, King struggles with two transformative events. First, although married to the seemingly saintly Larry King (Austin Stowell) (not to be confused with CNN’s Larry King), she realizes that she is rapidly falling for her hairdresser, Marilyn Barnett (Andrea Riseborough). (In real life, Barnett, who became King’s personal assistant, eventually sued unsuccessfully for palimony, thereby outing King, and costing her considerable endorsements. See here.) Second, King is about to play a match against Australian Woman’s pro, the conservative, Margaret Court (Jessica McNamee).
As presented in the movie, at least by Riggs, if Court defeated King, that would make Court the number 1 woman tennis player. After being rebuffed by King, Riggs then offers Court a shot at the $35,000 prize, assuming she’ll defeat King. Court defeats King, and proceeds to enter what, due to the date, became known as the Mother’s Day Tournament. King informs Court that winning matters deeply to women’s equality, but Court, conservative to the core, isn’t game. She loses badly, at which point, King decides, with a midnight call to Larry, to accept Riggs’s now higher offer, a $100,000 winner take all prize.
We generally assume that individuals rank preferences over options in a seemingly rational way: If I prefer (as I do) coffee ice cream to chocolate ice cream, and chocolate ice cream to vanilla ice cream, it is reasonable to infer that I also prefer coffee ice cream to vanilla ice cream. And I do. Although the transitivity assumption doesn’t always hold for individuals, see here, here, and here, demonstrating that the assumption of rational orderings sometimes breaks down is more intuitive with groups.
Consider, once more, the tennis matches. Court, having defeated King, is declared the better player. Riggs, having defeated Court, is declared the better player. From this, we might infer that Riggs (having defeated Court who defeated King) is also a better player than King. On this reading, the whole “Battle of the Sexes” game proves pointless. Or does it? Groups sometimes prefer coffee to chocolate to vanilla to coffee, a phenomenon known as a cycle, or empty core game. (Imagine three friends ranking their ice cream preferences as follows: Max: Coffee, Chocolate, Vanilla; David: Chocolate, Vanilla, Coffee; Paula: Vanilla, Coffee, Chocolate. There’s no first-choice winner, and if you run binary comparisons, you’ll expose the ice cream flavor cycle yourself.). So too with the King-Riggs match: Riggs defeats Court, Court defeats King, but King defeats Riggs.
Who’s the better player? Who knows? Despite its game theoretical intrigue, what makes the movie so interesting isn’t the cycle; it’s the nuanced exploration of each player's very human frailties and vulnerabilities. King was distracted by discovering new love, and realizing her sexuality, disadvantaging her in the match against Court. Court failed to appreciate, or care about, the larger symbolism of the Mother’s Day match against a backdrop in which women were routinely infantilized (witness the nauseating, and ever so relevant, clip of Howard Cosell with his arm quite literally putting down tennis star commentator Rosemary Casals (Natalie Morales), his female counterpart), thereby letting the older Riggs put Court down as well. The ever so reckless and bombastic Riggs, despite preparing meticulously against Court, took the King match for granted (and then wore that idiotic Sugar Daddy jacket!)
Back in La La Land, at a major plot turn, Mia ditches her boyfriend and another couple to join up with Sebastian for her double-booked movie date. The movie: Rebel Without a Cause. In that classic, Buzz Gunderson (Corey Allen) and Jim Stark (James Dean) settle their differences, over a woman of course, by playing a Game of Chicken. Each drives toward a cliff, the last one to stop declared the victor. Except the brakes fail in Buzz’s car, and as he tries to jump out the window, his shirt sleeve catches the steering wheel, sending him over the cliff to his demise. Not all games are fun.
Emma Stone is truly gifted. She can sing. Dance. Play tennis. (And even out-lip-sync Jimmy Fallon. See here). I don’t quite see her leading in a remake of the James Dean classic, this time with women playing chicken. But I have another, better idea involving cars. Emma as Danica Sue Patrick! With that, our game theoretical actor could also help us explore the Driving Game!
Special thanks to David and Paula, whose ice cream preferences I don’t actually know, for inviting my wife and me to join them for a fun dinner and movie that we all easily agreed on!
Your comments are, as always, most welcome.
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