Math Reveals the Formula for a Hollywood Blockbuster

What was the last movie you saw in theaters? I rarely go to the movies anymore, partly because I prefer TV series—particularly action shows with spies (recommendations welcome!). But I do go to the movies for real blockbusters, such as the Dune films. I have to admit, seeing an epic story on the big screen is still a special experience.

In 2020 cinema-going plummeted amid the COVID pandemic. In the years since, attendance has been on the rise in many parts of the world but is still below prepandemic levels. Inflation, the rise of streaming services such as Netflix and a shift towards prestigious TV series may all play a part in these trends. The film industry is therefore under more pressure than ever to attract people to movie theaters. Math might be able to help.

In 2020 British data scientist Ganna Pogrebna and her colleagues published a paper that analyzed the revenue, expenditure and popularity ratings of more than 6,000 different films. The findings mapped out story lines across movies and linked those narrative types with audience approval and money made.

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Storytelling: From Aristotle to Hollywood

The researchers’ study was based on an earlier analysis of literary works by researchers from the University of Vermont and University of Adelaide in Australia. Both studies built on the premise that the vast majority of stories can be mapped onto just a handful of narrative arcs.

This idea has been around for some time. Some 2,300 years ago, the polymath Aristotle was already thinking about the construction of stories and how they can move audiences. And in the 20th century, Kurt Vonnegut examined the emotional progression of well-known tales by charting out the sadness or joy each story conveyed from beginning to end.

Inspired by Vonnegut’s “story shapes,” the University of Vermont and University of Adelaide team fed 1,327 stories from Project Gutenberg’s fiction collection to an algorithm and identified six dominant emotional arcs across narratives (spoilers for classic movies and books ahead):

Rags to Riches

These stories start with a negative situation and a protagonist who must work their way out of it over time. A classic film example is Groundhog Day, in which the main character finds himself in the strange predicament of reliving the same day over and over, and his circumstances improve as he learns more and more from his situation.

Tragedy (or Riches to Rags)

This is the reverse arc. In tragedies, things starts positively and end negatively. Think of Romeo and Juliet: the couple are happy in their love at the beginning, but by the end, both protagonists are dead.

Man in a Hole

In these stories, the protagonist is doing well, and then things take a negative turn, but by the end they are often in a more favorable situation than at the beginning. A classic example is The Godfather. The Corleone crime family is at the pinnacle of power at the start of the movie, but then the head of the family is seriously injured and his eldest son is murdered. The youngest son, the protagonist, then gains an unparalleled amount of power when he takes over the role of leader and is made the new godfather.

Icarus

This arc inverts the Man in a Hole storyline. Think of Leonardo DiCaprio’s character in Titanic: a poor man falls in love on a glamorous adventure but perishes of hypothermia at the end. The eponymous character of The Great Gatsby also follows what starts like a Rags to Riches success story and concludes with his murder.

Cinderella

Cinderella stories begin with a bad situation that initially improves for the protagonist, then hits an emotional descent but concludes with a happily ever after. Think of Babe, the little piglet from the eponymous movie who struggles to find his place on the farm but turns things around when he begins learning to herd sheep. Unfortunately, he then gets put in a dangerous situation while defending the flock. Happily, by the story’s end, everything works out, and Babe wins a herding competition.

Oedipus

The emotional arc of Oedipus is the reverse of Cinderella. Things start well, but then there’s a stroke of bad luck. The protagonist picks themselves up again only to ultimately end up in a bad situation. Examples of characters with these arcs include the doomed protagonists of Frankenstein, Moby Dick or Hamlet.

Pogrebna’s team wanted to find out whether feature films also follow these six emotional progressions and, if so, which structure pays off the most. The researchers downloaded the English subtitles of 6,174 films and then analyzed them—sentence by sentence—for their emotional content. They assigned each word a score based on sentiment: –1 for a negative, 0 for neutral and +1 for positive. Then the researchers assigned each sentence an emotional value based on the words and scaled the sentence values to fall between –1 and +1. In order to compare films of different lengths, they processed the data such that each story boiled down to 100 individual data points.

To group the narratives, the researchers needed a way to measure the distance between the emotional arcs of any two movies. According to the researchers, the difference between two arcs of tension with the respective sets of data points X and Y can be calculated as follows:

With that equation, the researchers were able to compare the difference between the emotional trajectories of two films at any time point (t) in the narrative. Because the absolute value in the square root is squared, the sign does not play a role in the comparison; it is therefore irrelevant whether one curve runs above or below the other—only their distance from each other matters.

Using this measure, the experts were able to group the films with similar arcs. In that way, they confirmed that all of the movies they studied fell into one of the six emotional arcs that the earlier study had identified.

Interestingly, movie genre often correlates with a story’s emotional progression, they found. For example, horror films usually have a tragedy structure, whereas comedies follow Man in a Hole or Cinderella narratives. Biographies are often Rags to Riches, and thrillers tend to have a Man in a Hole structure.

Which Films Are Most Successful?

Pogrebna’s team then collected estimated production and income data from a film industry data website called The Numbers, along with film ratings from the Internet Movie Database (IMDb).

First, the experts examined the domestic revenue generated by each film—this was the only information available for all works studied. The Man in a Hole category performed best, grossing an average of $37.48 million, compared with $33.63 million in the second most successful category, Cinderella.

People seem to go to the movies more for films in which the heroes suffer but ultimately receive happy endings. Intriguingly, the much simpler Rags to Riches arc generated the least revenue on average.

But maybe the big winners were simply the most expensive films? The answer to that question is nuanced. The team found no correlation between budget and success for Man in a Hole stories. These films apparently play big regardless of dollars spent or genre.

Tragedies, however, generate less money unless they have particularly high production costs (in the region of $100 million or more). “This may explain the financial success of large historical dramas such as The Last Samurai or survival epics like Life of Pi,” Pogrebna’s team wrote in the paper.

Which Films Are the Most Popular?

When the experts looked at user ratings on IMDb, a different picture emerged:

On a scale of 1 to 10, with one being least favorable and 10 being most favorable, films with a Man in a Hole plot performed 0.19 points worse than Rags to Riches stories. In fact, ranking the film categories by rating forms a list that is almost the opposite of the result obtained when ranking by revenues.

One reason the data are at odds could be that Man in a Hole tales draw more viewers and generally receive more ratings overall—and IMDb users are more likely to leave a negative review than a positive one. But the Metascore on IMDb—a score from review aggregator Metacritic that is based on a weighted average of critics’ reviews of a film—echoes the user reviews for this story category. “The Man in a Hole emotional trajectory does not produce the ‘most liked’ movies, but generates the most ‘talked about’ movies,” the researchers wrote.

Meanwhile tragedies fare best in reviews from critics, who seem to favor “serious films” rather than happy endings.

With this information, the film industry can make data-driven decisions. To attract as many people as possible to the movie theater, filmmakers can produce more movies with a Man in a Hole setup. For critical acclaim, heart-tugging tragedies remain the best. But they should keep in mind that the audience still loves the simple upward promise of a rags-to-riches journey.