How the Avengers assemble: Ecological modelling of effective cast sizes for movies

TL;DR

This paper introduces an ecological diversity-based metric to quantify and compare the cast sizes of movies, providing insights into franchise success and character relationships, with applications in classification and recommendation systems.

Contribution

It proposes a Shannon-entropy and Jensen-Shannon divergence-based metric for character analysis in movies, offering a novel approach to understanding cast complexity and inter-movie similarities.

Findings

The metric effectively characterizes movie cast diversity.

It predicts film success within the MCU.

It reveals relationships between story arcs.

Abstract

The number of characters in a movie is an interesting feature. However, it is non-trivial to measure directly. Naive metrics such as the number of credited characters vary wildly. Here, we show that a metric based on the notion of "ecological diversity" as expressed through a Shannon-entropy based metric can characterise the number of characters in a movie, and is useful in taxonomic classification. We also show how the metric can be generalised using Jensen-Shannon divergence to provide a measure of the similarity of characters appearing in different movies, for instance of use in recommender systems, e.g., Netflix. We apply our measures to the Marvel Cinematic Universe (MCU), and show what they teach us about this highly successful franchise of movies. In particular, these measures provide a useful predictor of "success" for films in the MCU, as well as a natural means to understand the relationships between the stories in the overall film arc.

Tables

Table 1. Table 1: Facts and figures for the MCU movies.

Title	Type	Credited	Named	Budget ($M)	Box Office ($M)
Ant-Man	origin	67	30	130	138.0
Ant-Man and the Wasp	sequel	61	27	162	NaN
The Avengers	team-up	53	23	220	623.3
Avengers: Age of Ultron	team-up	67	34	191	429.1
Avengers: Infinity War	team up	56	42	321	665.0
Black Panther	origin	66	22	200	501.1
Captain America: Civil War	team-up	103	25	250	408.1
Captain America: The First Avenger	origin	95	25	140	176.6
Captain America: The Winter Soldier	sequel	73	27	170	228.6
Captain Marvel	origin	51	30	152	NaN
Doctor Strange	origin	31	22	165	232.6
Guardians of the Galaxy	team-up	72	26	170	270.6
Guardians of the Galaxy Vol. 2	sequel	48	34	200	389.8
The Incredible Hulk	origin	72	16	150	134.5
Iron Man	origin	63	26	140	318.3
Iron Man 2	sequel	61	30	200	312.1
Iron Man 3	sequel	110	39	200	409.0
Spider-Man: Homecoming	origin	64	37	175	334.2
Thor	origin	50	29	150	181.0
Thor: The Dark World	sequel	39	28	170	206.4
Thor: Ragnarok	sequel	43	18	180	315.0

Table 2. Table 2: Size and completeness of datasets, including the numbers of transcribed conflicts and lines of dialogue, and whether the dialogue file was complete, partially incomplete or largely incomplete.

Title	Run Time (m)	No. of conflicts	No. of lines	Status
Ant-Man	117	82	865	complete
Ant-Man and the Wasp	118	110	180	incomplete
The Avengers	143	237	830	complete
Avengers: Age of Ultron	141	302	975	complete
Avengers: Infinity War	149	278	991	complete
Black Panther	134	117	728	complete
Captain America: Civil War	147	263	982	complete
Captain America: The First Avenger	124	99	619	complete
Captain America: The Winter Soldier	136	149	822	complete
Captain Marvel	123	165	686	partial
Doctor Strange	115	112	159	incomplete
Guardians of the Galaxy	121	157	576	partial
Guardians of the Galaxy Vol. 2	136	127	956	complete
The Incredible Hulk	112	92	63	incomplete
Iron Man	126	81	124	incomplete
Iron Man 2	124	112	1006	complete
Iron Man 3	130	121	571	partial
Spider-Man: Homecoming	133	58	1558	complete
Thor	115	95	873	complete
Thor: The Dark World	112	106	732	complete
Thor: Ragnarok	130	197	970	complete

Table 3. Table 3: Table showing results for three cases with two alternative views

	Dataset 1		Dataset 2
Title	No. of Conflicts	$\hat{N}$	No. of Conflicts	$\hat{N}$
Iron Man	81	6.59	36	4.74
The Avengers	237	14.08	92	13.41
Avengers: Age of Ultron	302	14.77	180	12.95

Equations

$$\hat{N}(p_1,p_2)=\hat{N}(p_2,p_1).$$

$$\hat{N}(p_1,p_2,0)=\hat{N}(p_1,p_2).$$

$$\hat{N}(p_1,p_2)<\hat{N}(p_1,p_2^{\prime },p_3^{\prime }),$$

$$H(\mathbf{p})$$

$$\hat{N}(\mathbf{p})$$

$$D_{KL}(P||Q)=-\sum _{x\in X}P(x)\log \frac{Q(x)}{P(x)},$$

$$D_{JS}(P||Q)=\frac{1}{2}\left[D_{KL}(P||M)+D_{KL}(Q||M)\right],$$

$$D_{JS}(P|\!|Q)=H\left(M\right)-\frac{1}{2}\big{[}H(P)+H(Q)\big{]}.$$

$${\bar{D}}_{JS}=2^{D_{JS}(P||Q)}-1.$$

$${\bar{S}}_{JS}=1-{\bar{D}}_{JS}.$$

$$D_{\mathrm{effective}}(P||Q)$$

$${\bar{D}}_{\mathrm{effective}}(P||Q)=\frac{D_{\mathrm{effective}}(P||Q)}{(\hat{N}(P)+\hat{N}(Q))/2}$$

Full text

How the Avengers assemble: Ecological modelling of effective cast sizes for movies

Matthew Roughan, Lewis Mitchell, Tobin South

ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS),

School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, SA, Australia {matthew.roughan,lewis.mitchell,tobin.south}@adelaide.edu.au

Abstract

The number of characters in a movie is an interesting feature. However, it is non-trivial to measure directly. Naive metrics such as the number of credited characters vary wildly. Here, we show that a metric based on the notion of “ecological diversity” as expressed through a Shannon-entropy based metric can characterise the number of characters in a movie, and is useful in taxonomic classification. We also show how the metric can be generalised using Jensen-Shannon divergence to provide a measure of the similarity of characters appearing in different movies, for instance of use in recommender systems, *e.g., *Netflix. We apply our measures to the Marvel Cinematic Universe (MCU), and show what they teach us about this highly successful franchise of movies. In particular, these measures provide a useful predictor of “success” for films in the MCU, as well as a natural means to understand the relationships between the stories in the overall film arc.

Introduction

The Marvel Cinematic Universe (MCU) is arguably the most successful movie franchise of all time. It has grossed more revenue from the films alone111The MCU includes TV series series, one shot short movies and other media, but here we regard only the canonical series of 22 released movies (as of May 2019). See later for a precise list. than the next two most successful (Harry Potter and Star Wars).

What makes the MCU so successful? Clearly, there are many factors: the deep and expansive nature of the source material they draw on, the clever adaptation to movie format, the talented actors and directors, and the number of pre-existing fans. However, there are other similar attempts to translate graphic novels to the big screen, none as large or successful as the MCU.

The cast of a movie has a very important impact on its success. Most obviously, “star power” can attract an audience, and ultimately the talent of the actors and director are critical. However, underlying this is a question of just how big the cast is. On a superficial level, this seems trivial to answer, you just count. But that is naive. Movie credits are not a uniformly defined source of data: the onscreen credits of Closer (2004) list 6 cast, whereas Stuck on You (2003) credits 383 actors. These numbers hide the fact that the realistic count of meaningfully contributing characters in these movies is similar.

Further, not all characters in a movie are equal. Most obviously, there are named and unnamed characters. The first are those who have the importance to require a proper noun designation, the second require a designation but are not important enough to warrant a real name and can include characters without lines such as those in the background of a scene. Counting these so-called extras is problematic because many of them are unlikely to be mentioned by name in the credits. And more importantly, should a profusion of extras carry the same weight in a metric of cast size as the number of named characters? Ben-Hur (1959) won a reputation as an epic in part because of its reputed 10,000 extras, but should these each count as much as its star Charlton Heston?

We argue in this paper that an effective measure of the cast size is useful, and such theoretically appealing metrics can be derived from those used to measure ecological diversity222The term used in ecology is diversity (or biodiversity), with reference to species, genera or similar, but the term diversity in analysis of media usually refers to gender or race, which is not the topic of this paper so we prefer to use alternative nomenclature. [1]. Such a metric could be useful in

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better predictors for movie success;

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taxonomic classification of movies for subsequent analysis; and

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in creating features for recommender systems.

The metric we focus on here is a effectualisation of Shannon entropy. However, entropy (and other metrics) are derived from a probability distribution. The second major platform of this paper is that the probability distribution underlying the metric is of great interest: we investigate two here, one based on dialogue and the other on the relative prevalence in conflicts within the film.

We will show that the Shannon effective cast-size metric can measure the size of a cast in a way such

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that it have an intuitive meaning (*e.g., *meaningful units);

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that it be built on firm mathematical foundations;

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that it be practically measurable, and that it be insensitive to noise in the data collection process; and

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that it is effective, *i.e., *numerically comparable across movies, and in particular useful for prediction or classification tasks.

Thus we can think of the metric as an effective population size metric, where population refers to the number of characters in a movie. Moreover, this metric shows a useful relationship in classifications of the movies both into generic types, and into classes such as dialogue-based vs action-based.

Related work

Quantitative analysis of stories in most cases necessarily begins from a content analysis of the text of the story itself. Studies can be broadly categorised into two main approaches: Temporal studies tend to focus on aspects internal to the progression of the story, such as the ‘arc’ of the narrative, while structural studies focus on the relationships between characters, often through social network analysis. The present work fits most closely with the structural approach as it focusses on measuring the effective number of characters, however we borrow some techniques from information theory that are most typically associated with the temporal approaches.

Temporal analysis of stories — While analysing plot structure has an extremely long history, arguably dating back to Aristotle (the Poetics), in modern (Hollywood) filmmaking the notion of the “three act structure” characterised by rising tension followed by resolution is codified/popularised by Syd Field [2]. Taxonomies of stories more generally have been debated throughout the 20th century, with characteristic numbers of plots/stories found by authors apparently decreasing over time, from thirty-six in 1916 [3], to twenty in 1993 [4], to seven by 2004 [5]. Entropy has been extensively used in analysis of language, *e.g., *see [6]. Indeed, Shannon’s original paper introducing information theory discusses redundancy in Joyce’s “Finnegan’s Wake”, and later non-parametric entropy estimators have been employed to quantify entropy in literature [7], plays and poems [8], and online social media [9]. Perhaps most similar to the methodological approach in this work are the studies on vocabulary [10] and on placing information-theoretic bounds on human language acquisition [11], where entropy is used as a measure of language capacity. Such approaches use information-theoretic techniques to estimate an “effective” vocabulary size, in a similar manner to how we estimate the effective cast size here.

At a more fine-grained level of temporal analysis over the course of a story, Vonnegut proposed studying the “shapes of stories” by charting “ill fortune-great fortune” along a “beginning-end” axis [12]. This notion was made quantitative by analysing story “arcs” (which we distinguish from “plots” here) by applying sentiment analysis techniques to the text of written works. Notable examples here include the body of work by Jockers and coauthors [13, 14, 15, 16], as well as Reagan et al., who find six characteristic emotional arcs in novels [17]. For films, there is some suggestion that these emotional arcs relate to the eventual success of the film, as measured through box office returns [18]. In this paper we find relationships between statistics of MCU films and box office returns as well, but through our analysis of effective cast size.

Structural analysis of stories — The present work is more closely related to the literature on analysing structural properties of stories using networks, either between characters or portions of text. Authors have studied the complex networks encoded in stories from throughout history, from ancient myths and sagas [19, 20], to Romantic novels [21, 22], to films [23]. Of particular relevance here are recent social network analyses of the Marvel (comic) social network [24] and some social networks from superhero movies including Wonder Woman and Thor [25]. Edwards et al.[26] compare different ways of constructing social networks based on scripts, focussing on the television series Friends.

To some extent this paper unifies the two major approaches to the quantitative analysis of stories by using entropy, which is typically associated with temporal analysis, to quantify something structural about the films we study, namely the effective cast size.

Materials and methods

The anatomy of a cast list

We are interested here in the cast of a movie, its dramatis personæ so to speak, but we are interested in the story, not its means of implementation, so we are interested in the number of characters (or parts or roles), not the number of actors (the two are not identically the same). The most obvious place to access the list of characters is through the credits. A movie is constructed by a long list of creative artists, actors and technical experts. Their contribution to the movie is generally acknowledged through the credits, which historically have been displayed on screen at the beginning of a movie in the title sequence and near the end in the end credits. The credits are incredibly important to today’s movie participants because they effectively form their curriculum vitae.

However, there is no unique definition of how a credit list is constructed, and it has varied historically and regionally in level of detail, and the types of activity that are credited333For example, since Toy Story (1995), lists of “Production Babies” born to crew members have been increasingly included in movie credits.. In the United States, the standardisation of credits starts around the 70s444Superman (1978) had the longest credits at its release, setting a trend for superhero movies, whose credits can now include many thousands of participants., but has evolved continuously.

In the context of this paper the credits are important because one of their key components is a cast list. However, credited cast lists are often incomplete and sometimes misleading. There are many reason for this, and discussion requires some understanding of the different types of roles within a cast. Definitions for such vary (often base on local union specifications) but losely we speak of

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Main roles: these are the primary roles in a movie. There are usually only a small number of such roles, and these characters perform the majority of action and dialogue.

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Bit parts: these are smaller roles, often defined by the number of lines of dialogue the character has (less than 6 lines seems a common standard).

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Extras: (or background talent, or atmosphere, or supernumeraries) these are much smaller roles, often as simple as providing background in a scene. The number of extras varies tremendously, potentially into the thousands. Extras are usually “silent” roles, though in the UK they are allowed to speak up to 12 words).

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Cameos: these are small roles, usually played by well-known actors or personalities (Stan Lee performed a cameo in all of the Marvel movies).

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Doubles: stunt, body or costume doubles for a main actor fill the same character role with a different actor for various reasons.

The categorisation is important because credit lists of the cast of a movie focus on the more important roles. Extras, cameos and doubles may often be omitted for various reasons [27, 28]. Actors may not want to be listed for artistic reasons (to enhance anonymity of a character where that is important) or personal reasons, and movie producers may not want spoilers to be disclosed through a cast list (these days cast lists are often available before the movie release). Producers may also not wish to credit actors because this may have implications for employment costs, or simply because it dilutes the meaning of being given credit. There are also many new types of participation modes for actors (*e.g., *voice only, or motion capture), and movies appear in multiple formats (with different cutting) and it is ambiguous how these might be credited, *i.e., *should a character who only appears in a deleted scene be credited?

IMDb collects cast lists for movies, somewhat provided by fans identifying actors on screen, or actors self-identifying, and although such contributed information can have inaccuracies, it allows a window into how many roles are credited or uncredited [27, 28]. IMDb also lists additional attributes for roles: uncredited, voice, motion capture, archive footage, scenes deleted, or credit only [29]. Thus the data is valuable, but it still has many issues. Most notably, consistency is not enforced between movies, so a character (or even an actor) might be listed under different names (see below for more discussion).

Within a cast list, we notionally divide the characters into subclasses based on these notions, and the extra notion of “naming”, *i.e., *we break the characters into the classes:

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Named: These are characters that are assigned a proper noun designation, e.g., “Iron Man”.

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Unnamed: These are characters that are only indirectly names, *e.g., *“Tony Stark’s secretary”, or given a job name or other designation, e.g., “SHIELD agent”.

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Minor roles: These are characters that only appear as “uncredited” in lists such as that of IMDb, or are not noted on any list. Even as such, they may play a role, *e.g., *as the victim of another character’s actions.

In general we use IMDb to build such lists primarily as a comparison point to demonstrate the difficulty of using such for assessing cast size, and to illustrate the issues of data quality towards assessing cast size. Our main data will be informed by the content of movies, explicitly to avoid such issues.

How not to measure cast size

In ecological terms a “richness” metric just counts numbers of species. Here it would be a direct count of the characters in a movie. In ecological settings this is seen to be naive [1]. In this section we demonstrate that such counts are naive here also.

The numbers of credited and named characters for each movie are given in Table 1. There are many notable defects that one observes, for instance with regard to the number of credited characters:

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Iron Man 3 apparently has the largest cast. The movie, however, is a relatively straightforward linear plot with the lead character (Tony Stark/Iron Man) being dominant in terms of scene time and action.

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On the other hand, Avengers: Infinity War, which has a cast that includes almost every555A notable exception is Hawkeye / Clint Barton. prior superhero from the franchise has a rather intermediate credited cast.

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Similarly, Thor: The Dark World also has a rather small list cast, given it takes place over three settings (Asgard and Earth and the eponymous Dark World) with three corresponding sets of cast members.

There are also issues with regard to the number of named characters:

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Thor: Ragnarok is listed as having the second smallest set of named characters. As with the other Thor movies this takes place over several settings, each with its own cast, and it involves a team-up between Thor (and his usual team) and the Hulk.

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Spider-Man: Homecoming is listed as having one of the largest sets of named characters. In this case, there are indeed quite a few such, but they play very little if any role in the plot.

However, a more important problem with this metric is that it does not provide much separation between movies. The majority of movies all lie in a narrow band in the middle. This lack of differentiation means the metric is not terribly useful.

Measuring a character’s participation

Ideally, we would like to measure each actor’s contribution to a movie. However, this is difficult. Measures such as screen time are difficult to measure, and in any case don’t necessarily take account of whether a character contributes meaningfully during this time. Thus we need proxy measures of each actor’s scale of contribution.

Many movies are driven by dialogue: what the characters say informs us of the plot and the attitudes and emotions of the characters. Other aspects are important, their actions, the setting, the background music and so on, but the dialgue is the platform around which these aspects revolve. Hence, one approach to measuring the participation of a character in the movie is to measure the number of lines of dialogue they speak.

This is not the only metric, or even uniquely the best. Some characters may be laconic, or overly verbose. And the importance of any given line may vary. At a deeper level, however, not all movies are driven by dialogue. A musical or opera is driven by the music (the dialgue and plot are often only there to provide connections between the songs). Pornographic movies are driven by sex, with plot and dialogue again providing (usually extremely minimal) connections. Here we are concerned primarily with the MCU, which falls into the specific superhero genre, and more generally into the action movie genre. And action movies are often driven by conflict. Thus measuring the number of conflicts in which a character participates will give us another measure of their level of participation in the movie.

Effective population size measurement

Details often make a character, and so smaller cast of more carefully constructed characters might be better; or perhaps volume carries weight? We need a way to measure cast size in order to consider which of these hypotheses is more true.

Our goal is to find an effective metric for the “population size” of the cast of a movie. There are several goals in forming such a metric, described in the introduction. Luckily the problem has been considered extensively in the context of measuring ecology diversity of a habitat (for a review see [1]), and there are several parallels. In ecology there is a need to understand not just how many members of each species are present in a habitat, but to capture what this means in terms of diversity. The problem is complicated by the noise in estimating species numbers, and the underlying variation in those numbers. For instance, a naive count of the number of different species can be misleading if most of those species are near extinction.

However, there are several ecological metrics, and their purpose is different, so rather than choosing one without care, we will consider here the goals of such a metric in axiomatic terms [30]. That is, we shall describe (mathematically) the properties that such a metric should have in our specific context, and thereby derive a suitable metric.

First, such a metric should be based on the proportion of contribution of each actor, not other features (such as their name). Hence, we can write the effective number as a function of the proportional contributions of each character to the story, *i.e., *if we write the effective number as $\hat{N}$ then it is a function $\hat{N}(p_{1},p_{2},\ldots,p_{m})=\hat{N}({\mathbf{p}})$ where the proportional contribution of character $i$ is $p_{i}$ and there are $m$ characters to consider. Given this definition we have the axioms.

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Non-negative: $\hat{N}(\cdot)\geq 0$. It makes little sense to report a negative population size.

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Continuity: $\hat{N}(\cdot)$ is a continuous function of the variables: small changes in the data should result in small changes in the metric.

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Symmetry: Reordering the actors, for instance such that $p_{i}$ and $p_{j}$ are swapped, should not change the metric. For instance

[TABLE]

This condition arises because we want a metric in which characters are interchangeable, in order to be able to make comparisons, for instance, between different movies.

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Normalisation: If there are $M$ equally contributing characters, i.e., $p_{1}=p_{2}=\cdots=p_{m}=1/M$, then we require that $\hat{N}=M$, and this should be maximal. This condition is imposed to scale the values intuitively, so that a value of $\hat{N}$ can be associated to an actual number of characters.

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Zeros: Adding a character that makes no contribution should not change the metric, e.g.,

[TABLE]

A net result of this and the prior condition is that $\hat{N}(1,0)=1$. This condition arises both for technical reasons, and because we don’t want noise (small errors) to lead to instability in the metric. For instance, introducing a character who makes very little contribution should not change the metric by much.

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Monotonicity: If the number of participating characters increases, then the metric should increase, *e.g., *assuming all participation is non-zero

[TABLE]

where $p_{2}^{\prime}+p_{3}^{\prime}=p_{2}$ where $p_{2}^{\prime},p_{3}^{\prime}>0$. The intention is that larger casts should report a larger metric.

Additionally we expect the metric to be sub-additive in the sense that if one considers two movies jointly, then the effective cast size of the joint movie should be no larger than the sum of the casts of the two components.

This list of axioms has redundancies, but more importantly it does not lead to a unique metric. There are several alternatives that are valid. We use the method based on Shannon Entropy $H({\mathbf{p}})$ [30], which is defined as follows:

[TABLE]

with the common convention that $0\log 0=0$. The metric $\hat{N}({\mathbf{p}})$ has sometimes been called the perplexity of a distribution (because of its relationship to one’s ability to predict the outcome of a random event with this distribution) and has been used in natural language processing [6].

There are other metrics that satisfy these axioms666These roughly correspond to axioms used to derive Shannon entropy, but there is a key grouping/partitioning/recursivity axiom missing here as it appears hard to justify in the context. [30], but the metric we have chosen has several advantages: (i) Shannon entropy is easy to calculate, (ii) it is used in many other fields (*e.g., *physics and information sciences) and so has common interpretations, (iii) the maximum entropy principle naturally leads to models, and (iv) entropy has many generalisations, for instance to distance metrics – this is a key advantage here in that we can adapt the same ideas for comparisons between movies.

The metric leaves open the means by which we estimate the $p_{i}$, the proportional contributions of each character. We test the use of dialogue or conflict.

Data

Public data sources

Public databases, e.g., IMDb and OMDb, are used here to provide cast lists, revenue and cost figures, and ratings. Some considerable effort has gone into cleaning the cast lists in particular. The most notable problem is aliasing of character names, *i.e., *a single character may appear under different names in different movies. This problem is particularly prevalent in superhero movies. Alias lists were construct by hand for all significant characters in the movies.

As a source for dialogue, publicly available movie scripts were used. This has been a valuable data source in previous work [31], but has been focused in constructing dialogues for use in computational linguistics [32]. Here we are not analysing the text of the speech, but rather the speakers.

No single source has scripts for all of the MCU movies. We used two sources:

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The community run Transcripts Wiki on the website fandom.com [33]. These webpages were downloaded and processed to find appropriate lines of dialogue and its speaker. The scripts come in a number of formats, each requiring different methods of parsing. Some scripts had no consistent format or incomplete scripts (see below for more discussion).

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PDF documents of movie scripts were sourced from Script Slug [34] if they were unavailable or incomplete on the Transcripts Wiki. The text from the PDFs was extracted and parsed to collect the names of each speaker of dialogue. Individual care was put into each PDF parsing as formats varied.

Transcripts, especially fan transcripts, often lack the depth of information desired for more complex analysis [35], but the extraction of speakers in dialogue sequences is relatively reliable. However, as noted, incompleteness was a problem.

Even with two data sources we did not have complete scripts for all movies. Only 14 movies have complete scripts. Three have mostly complete scripts that may miss a scene or have minor uncorrected issues throughout the script. The four remaining movies had only partial scripts. Details of the movies and numbers of lines of dialogue available in their transcription are shown in Table 2.

Conflict transcription

Underlying this work is the desire to form a theory of conflict in narrative analysis. Much of narrative concerns conflict, for instance, between a hero and villain. The development of this component of a narrative is particularly obvious in the action movie genre, in which plots of movies are predominantly carried by a sequence of interdependent conflicts. Indeed, there are famous cases of movies whose main actor spoke almost unintelligibly, but carried the film through charisma and physical presence, which indicates the importance of “action” to the genre.

Mathematically, we can think of a story as having a set of characters ${\cal C}$. A conflict is a mapping from a pair of these characters to an outcome, *i.e., *it is a function $f_{i}(a,b)\rightarrow\{a,b,d\}$, where $a,b,\in{\cal C}$ and $d$ means a draw or an indeterminate outcome. A scene is made up of a sequence of such conflicts $i=1,\ldots,n$. For instance, a movie scene often follows a backwards-forwards motif oscillating between minor victories for the villain and hero. Alternatively, the hero might be the underdog, physically over-matched by the villain, suffering repeated setbacks, but eventually, the hero wins by adopting a clever stratagem. We are not concerned in this paper with the temporal structure of this process. Here we are only interested in the proportionate contributions of each character to the action of the story, but future work aims to tease out such motifs from this data.

It is not always obvious from observation how to divide a story into a sequence of such mappings. Technical considerations aid in the decision: (i) if conflicts are pairwise, then we must divide conflicts involving more than two parties into components small enough to be modelled as pairwise, and (ii) estimates based on the data will be limited by the resolution of the data. Both of these considerations push towards as fine a division of conflicts as possible. On the other hand, there is a natural desire that the outcome $d$ be in the minority, *i.e., *that most conflicts have a winner and loser, and this means that we cannot meaningfully break a scene into individual frames. Additionally, the overhead of a frame based approach is impractical for manual transcription, not to mention that individual frames are often meaningless without being seen in context.

We transcribed the conflict data manually by viewing and annotating the movies. Transcription was performed by pausing the movie (frequently) so that notations could be made immediately, avoiding problems of memory. In transcribing we are aiming to balance the alternative pressures: to subdivide scenes at a fine granularity vs the desire for conflicts to have determinate outcomes. There is inherent subjectivity in any such transcription process. How finely to subdivide conflicts, and even who is victorious is often straight-forward, but not always. Ideally we would involve multiple transcribers, and using a metric of inter-coder reliability determine a consistent set of data. However, (i) the movies represent a large corpus of material; (ii) conflict data does not fall into a standard model for inter-coder concordance in any case777For instance, commonly used correlation coefficients are designed to assess agreements amongst ratings, not between choice of subdivision of a dataset., and (iii) we acknowledge that the theory of conflicts described so far is not fully formed at present. Instead, we aimed here to at least provide a dataset that was consistently transcribed, and to build a metric of effective cast size that was robust to transcription variations.

However, in order to reduce inconsistency, all movies were examined and transcribed by one observer. In addition, the order of transcription was randomised (constrained by movie releases) to avoid systematic biases. Transcription was performed over as short a period as practical for the same reason888The main body of the MCU was transcribed over a few weeks, but transcription of Captain Marvel occurred somewhat later when it was released outside theatres. Avengers: End Game has not yet been transcribed as it only has a theatrical release at present., again with the aim of applying the same standard to classification.

A more difficult issue to address is that as transcription progressed, there was a parallel learning process about how to most accurately and consistently transcribe events. In particular, the level of subdivision, and the degree of inclusion of conflicts between minor parties increased in later transcriptions as their importance was realised. In order to address this drift in transcription we reprocessed the first three movies. The second transcription of these movies was substantially different from the first, which contains a much courser breakdown, and hence many fewer conflicts. Fortunately, this data provides a comparison that allows us to test the sensitivity of our metric to the collection methodology (see below).

In general, releases of movies in different regions can be cut slightly differently, for instance to fit into a particular ratings scheme. The movies transcribed were all the standard Australian release. We did not analyse “director’s cuts” or other such modified releases. We have not seen any evidence that the MCU movies released in Australia differed from the US release, or any other worldwide releases.

Data cleaning

It’s often said that 80% of the effort in a data analysis is spent on data cleaning, the process of getting the data ready to analyse.

Hadley Wickham [36]

The conflict data described above was collected as expeditiously as possible. It was quickly typed into a spreadsheet, resulting in typographic errors. Likewise fan transcripts contained some errors. These errors were detected by parsing the data, and validating against a set of criteria, *e.g., *time sequence monotonicity, name frequency, and simple matching algorithms; as well as matching to external data sources. Errors were corrected manually, with reference to the original movie where the mistake was not obvious (typos were generally very obvious once discovered).

A more critical issue lies in character names. Many characters appear in multiple MCU movies. However superhero names are not standardised. Characters introduced in one film under a given name are sometimes reintroduced under another (*e.g., *Bucky Barnes v the Winter Soldier). Also naming conventions in movie credits are not standardised, and superhero movies add a layer of complexity because superheroes often have aliases. For instance Natasha Romanoff is known as the superhero Black Widow, but also appears in credits as “Agent Romanoff” and “Natalie Rushman”, and abbreviations or combinations of these. Characters also sometimes appear in alternative forms, *e.g., *“Young Gamora,” or are known by nicknames. The dialogue in scripts also uses variant forms of these names. An aliases file was created in order to disambiguate characters and convert all appearances to a common format. Extensive use was made of “canonical” sources of data on the movies, in particular [37] in order to ensure accuracy of the alias lists999The listing is available at https://github.com/mroughan/AlephZeroHeroesData/tree/master/MarvelCinematicUniverse.. Nearly 200 characters are listed in this data, commonly with two aliases, but sometimes with significantly more. For example, The Winter Soldier, James Buchanan ‘Bucky’ Barnes has six.

Unnamed characters do not use the alias list, and some generic characters may be grouped together by a credits lists or the names used in scripts. For instance, two separate characters may be listed as “SHIELD Agent”. The number of such, and the number of conflicts and/or lines of dialogue that such characters appear in is small, but may result in a marginal decrease in the effective cast size. A key advantage of the approach proposed here is a high degree of robustness to such noise.

All data and code will be released with the publication.

Types of movies

In order to better understand results, the movies are classified into types within the MCU based on the their characteristics and relationship to other movies in the franchise. We classify movies as

• •

origin movies – these are the first major appearance of a character, and often explain some of that characters back story;

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sequels – these are movies that follow on directly from another with a large overlap in cast and plot elements; and

• •

team-ups – these movies involve a group of superheroes forming into team for a larger purpose.

The classification is soft in the sense that many of the movies have aspects of more than one type of movie: for instance, most movies in the MCU are sequels in some sense. Other movies such as Guardians of the Galaxy is both a team-up and an origin movie for the characters in the team. Here we identify the primary role of the movie. The classification is given in Table 1.

Results

Effective (conflict) cast size

The movies considered are listed in Table 1. We consider first the effective cast sizes obtained from the conflict data because of the completeness of this data. Effective cast sizes are shown in Figure 1 with the type of movie indicated by the marker, and specific sub-sequences of the overall set of movies are indicated by dashed lines. There are many interesting features of this plot.

When considered by class we see notable features: most origin movies have a small effective cast, which grows in sequels. The exceptions to this rule are the Thor and Guardians sequences.

The Thor movies have a more stable cast size (though the actual cast, and even the distribution varies). The movie Thor is unusual as an origin movie because it takes place across two major settings (Earth and Asgard) each with their own somewhat disjoint casts. It is therefore not surprising that the effective cast size is approximately double that of other origin movies. The Guardians of the Galaxy movies start with a team-up movie without the typical origin movies for team members, making these movies an outlier within the MCU, but leading to consistent casts sizes when considered by type.

Another notable feature is a general level of inflation in the cast sizes with time. This is particularly the result of an increasing number of team-up movies which have larger casts, and Infinity War’s huge cast (this movie is a team-up of teams).

None of these results are terribly surprising. Most fans or movie critics might have drawn a plot with a similar shape given a little thought, but deriving this from actual cast lists is hard as we showed earlier. It is a measure of the success of the metric that it should match with intuition about the movies.

We can also consider the overall cast size of the MCU. If we combine the movies by taking averaged proportional contributions from the full set of characters, then the effective cast size is approximately 119.6 characters. This is an extraordinarily large number for a franchise, and reflects both Marvel’s willingness to develop multiple streams of movies building up to the central Avengers movies, and the fact that the action is well-distributed amongst this cast, *i.e., *there is no single, central character that dominates the movies. Compare, for instance, to the James Bond franchise which has a single, extremely dominant character, the eponymous James Bond.

On that topic, the top-5 characters that make the largest contributions to the overall metrics are in order:

Iron Man / Tony Stark; 2. 2.

Captain America / Steve Rogers; 3. 3.

Thor Odinson; 4. 4.

The Hulk / Bruce Banner; and 5. 5.

Black Widow / Natasha Romanoff.

Again, there is no surprise here, but the relatively equal contributions from the top-5 characters is somewhat unique in film franchises.

The large cast size of 119.6 characters is also interesting in comparison to a simple sum over the effective cast sizes of the movies (248.5). In ecological terms this disparity would be described in terms of $\alpha$-diversity (the within habitat biodiversity, $\beta$-diversity (the between habitat biodiversity) and $\gamma$-diversity (the overall biodiversity of a set of habitats) [38]. The MCU sits in an interesting place where a significant part of its diversity occurs inside each movie ($\alpha$), and a significant part between movies ($\beta$) but there is also a strong overlap (which we will consider further later in the paper).

Prediction

I win my awards at the box office.

Cecil B. DeMille

A metric is as useful as its uses. In the context of movie production a key performance indicator is how profitable the movie is, *i.e., *the ratio of the movie’s box office receipts101010There are many other facets to the success of a movie: DVD sales, for instance, and Star Wars is legendarily famous for its merchandising, but Box Office receipts is (i) a somewhat easier number to obtain, and (ii) is possible to obtain more quickly than other metrics, which is important in order to compare recent movies. to its cost111111Exact costs are notoriously hard to obtain – here we use the budget as stated in IMDb.. Some effort has gone into developing predictive models for movie performance, *e.g., *see [39]. Amongst many factors they consider (audience-based, release-based, and content-based) the content includes who is in the cast (that is the actor, not the character), and the actors’ popularity. They also note that features of the acting team can contribute, but again they are considering actors not cast, and they consider semantic features of the group (*e.g., *team chemistry) rather than purely quantitative measures. Instead, we test the effective cast size as a predictor: this is much simpler to measure, as it is a feature of the movie itself, not the various qualities and relationships of the actors.

In Figure 2 we compare the effective cast number to the profitability ratio. Of note in the MCU is that all but one movie have ratio greater than 1, the exception being The Incredible Hulk, which was widely considered a failure (its lead actors was one of the very replaced in later movies). However, its role in establishing one of the key characters of the franchise should not be underestimated.

We also plot on Figure 2 a least-squares fitted line121212The fit is is through 0, the argument being that a movies with no cast, would have profitability zero.. The $p$-value for this fit is below $10^{-8}$ indicating strong significance to the relationship, and Pearson’s correlation coefficient is 0.184 indicating a non-negligible correlation between the two statistics.

There is considerable variation around the line. There are quite a few movies that lie well above it: for instance Iron Man, which has been hailed as one of the best movies in the franchise, and as one of the key initiators of the cinematic universe. Quality of acting and direction, cast “star power”, timing and other factors cannot be discounted as important to the overall success of a movie. However, the effective cast size also appears to influence the profitability of a movie.

This is a fact that does not seem to be missed by the producers of the MCU franchise. As we noted earlier, larger casts seem to be becoming more common.

Why should this be? Do audiences really prefer a larger cast? We have to be very careful about this conclusion because the movies with larger casts tend to be team-ups. Dedicated fans might watch all of the MCU movies, but other fans may choose to watch only a single sub-sequence. When these culminate in a team-up, this draws in audiences from the multiple strands.

Guardians of the Galaxy is a useful data point here. This movie is ostensibly a team-up, but lacks the usual prequel origin movies. Although the movie was widely praised, it wasn’t a profit generating engine compared to several other team-up movies.

The fact that the MCU is constructed not of individual movies, or individual sequences is one of its chief beauties. There appears to be an understanding in the producers of the franchise that the smaller scale origin movies are a necessary pre-condition for the profit generating team-up movies.

We can also look at the relationship between ratings and cast size (see Figure 3) and we see the same type of pattern. Larger effective cast sizes are correlated with higher ratings, though once again we must take care to note that this is a correlation, not necessarily a causative relation.

Robustness

A key concern in any such metric is the impact that noise in the underlying data might have on the results, particularly where there is an element of subjective judgement in the transcription process. To understand how sensitive the effective cast size is to transcription variations, we performed two separate transcriptions of three of the MCU movies. The two transcriptions were very different: the second set were performed at a much courser level. Table 3 shows the results. The number of conflicts found in each dataset is listed in order to show just how different these two transcriptions were. The effective cast sizes for each case are also reported. The notable feature of effective cast size is that it varies by a much smaller amount than the input data. There are some changes, mainly because in the courser grained analysis, some characters do not appear at all, and hence it makes sense that the effective cast size would decrease, but this is a comparitively small decrease!

This robustness to input data variations is a key ingredient in any good metric.

Effective (dialogue) cast size

So far we have only considered the conflict-based metric for cast size. In this section we consider the related dialogue metric – as a reminder this uses the same formulation but an alternative estimate of the proportional contribution of each character to the story.

Figure 4 shows a scatter plot of the two alternative metrics. The plot also shows two reference lines: the first a fit (through zero) to the data, and the second a 1:1 line. The former shows that overall the dialogue measures of cast size report very slightly larger numbers than the conflict based metric. However, they are remarkably close given the fact the two use completely different views of the movie.

There are additional patterns of note. We can understand that movies that sit above the reference line contain more dialogue-based participation, and less conflict, and in turn, those below the line entail more conflict. Extreme examples are Spider-Man and Captain America: The First Avenger.

Origin movies usually lie above the line (or close to it). Origin movies often use extensive dialogue to show character development from “zero to hero.”

Sequels seem, on the other hand, to lie close to or below the line, thus involving more action than dialogue. Team-ups often lie quite close to the line: the formation of a team often requires the members to meet and talk, but these movies use action to display the abilities of the entire team.

The location of a movie in this plot represents subtle changes in style in screenplay and direction. It is interesting that a level of detail that one might expect only to be exposed by detailed semantic analysis of content can be seen in a single pair of metrics.

Movie comparison results

An additional task for which cast metrics are useful is comparison of movies. That is, we would like to have a metric for how similar or dis-similar two movies are. For the same reasons that we consider above, it would be useful to have a metric that is measured in units that correspond to effective cast size, *i.e., *it would be appealing to say that the difference between two movies is $X$ effective cast members. Thus we should have that two disjoint movies $A$ and $B$ would have an distance that is directly related to $\hat{N}(A)+\hat{N}(B)$, and that two movies with identical character contributions would have a distance of 0.

There are many formal distance metrics that can be applied to probability distributions. A survey of metrics and their properties can be found in [40]. Typical distance metrics between probability distributions either

assume the same support, *i.e., *, that the two have positive support on an identical set of elements (*e.g., *Kullback-Leibler or $\chi^{2}$ divergences), or 2. 2.

they ignore the measures on the elements and simply use counts of the size or relative size of overlaps of support (*e.g., *Jaccard, Bray–Curtis or Dice distances/dissimilarities).

Neither of these two conditions is applicable in our domain. We need a metric that takes into account the probabilities, but also allows the supports to be different as different movies have intrinsically different casts.

There are two approaches we can adopt: one formal, and the other more intuitive. The formal approach is to adapt the Jensen–Shannon divergence (JSD) [41, 42] (also called the total divergence to the average or the increment of the Shannon entropy). This is a symmetrised, smoothed version of the Kullback–Leibler divergence (KLD). It can be defined in terms of the KLD

[TABLE]

as

[TABLE]

where $M=(P+Q)/2$ is short-hand for the distribution formed from the averages of the probabilities of the two distributions or equivalently in terms of entropy

[TABLE]

It has the advantage of being theoretically understood [41, 42], *e.g., *we know that $0\leq D_{JS}(P|\!|Q)\leq 1$. In linguistics [41] this has been used by taking an exponent, the natural such for us being to take a power of 2, i.e.,

[TABLE]

We subtract 1 because we should like our measure of dissimilarity to be 0 when comparing a movie to itself. We can form a measure of similarity then by noting that the maximum value of $D_{JS}$ is 1, and taking

[TABLE]

The approach above is theoretically appealing, consistent with our use of entropic measures, and useful, but it lacks an interpretation in terms of effective cast size. That is, we can only interpret $\bar{S}_{JS}$ as describing the proportion of cast members in common.

An alternative measure is to proceed with the calculation we might pursue in terms of measuring the effective increase in cast size of the combination of two movies over the average size of the casts by taking

[TABLE]

This expression looks very similar to that for $\bar{D}_{JS}$ but it has units of characters.

The two appear similar, and in fact are very closely related if we normalise $D_{\rm effective}$, by taking

[TABLE]

and as before we form a similarity measure $\bar{S}_{\rm effective}=1-\bar{D}_{\rm effective}$.

The two have very similar results, but the latter is can be smaller (as one might expect from Jensen’s inequality). Figure 5 shows a comparison between the two. The effective measure is highly correlated with the Jensen-Shannon measure, but has the advantage that the large mass of cases with $\bar{D}_{JS}=1$ are spread out over a range of values of $\bar{D}_{\rm effective}$ providing a little more information. We found minor improvements in clustering with the latter metric, and so shall use this in what follows.

In practical terms the metric performs exactly as you might expect. Figure 6 shows a heat-map of the similarities $\bar{S}$. Obvious clusters emerge between directly related movies, *e.g., *the Thor sequence.

We can draw out these clusters, for instance, using a hierarchical clustering technique. The dendrogram for the method used is presented in Figure 7. If we set the number of clusters to be 10 then we match the obvious groupings of the movies by sub-sequence.

We illustrate this clustering by performing (classical) MDS on the distance matrix $D$, to obtain projection of the movies into local coordinates in a 2D Euclidean space. The coordinates are shown in Figure 8, which also shows the clustering results. The original metric space being embedded is highly non-Euclidean (many of the distances are near 1), and so the 2D embedding is a poor approximation in some ways, but it does allow visualisation of the clusters.

Of note:

• •

the upper-right quadrant is the Thor/Hulk quadrant (remember that the Hulk plays a major role in Thor: Ragnarok.

• •

the upper-left quadrant is an Avengers/Iron Man/Captain America cluster. These movies are tightly wound together so it is no surprise that they appear together in this quadrant.

• •

The lower half holds a more miscellaneous collection of movies, though the two Guardians movies cluster tightly (again unsurprisingly).

Note that although the Captain America movies are in a separate cluster with the Iron Man/Avengers cluster, they appear to overlap. This indicates the limitations of projecting into a 2D space, the 3D projection more clearly separates these clusters, but is hard to illustrate here.

Informally, we can see in the plot a left-to-right division between the more technology-derived superheros (Iron Man and Captain America) vs those that come from a magical or alien background (Doctor Strange, Thor and the Guardians). The placement of the Hulk and Ant-Man towards the right is somewhat of a surprise from this point of view, but perhaps reflects that sometimes superpowers are really magic explained explained as technology.

It is tempting to speculate how other movies will embed in this space. We might expect Avengers: End Game to hold a similar or even more central position than Avengers: Infinity War. Spider Man: Far From Home is not released, but there is a version of its cast list available, and it suggests it will sit close to the existing Spider-Man movie. We might speculate on hypothetical sequels to Black Panther, Captain Marvel, or the Guardians (likely close to the existing movies).

Conclusion and Future Work

Measuring the number of characters in the a movie or franchise such as the Marvel Cinematic Universe is an interesting and novel challenge. While naively seen as easy, there is a need for a robust metric of character count that incorporates character participation. Such a metric is less sensitive to noise in the input data, and harder for a commercial entity to “game.”

The use of Shannon-entropy based measures built from frequency distributions of participation provides a metric based on “ecological diversity,” which is an intuitive, robust and useful metric. The robustness is shown through the comparisons of the metric under constructions from dialogue and visual conflicts separately, and alternative transcriptions of the same films. The usefulness of this metric is explored with respect to the success of the movies measured via their profitability and shown to have a strong correlation. A generalisation of this metric using Jenson-Shannon divergence allows a measure of similarity between movies that gives rise to a clean clustering of movies, supporting the notion of usefulness of this metric as a tool for the taxonomy of movies.

The usefulness of this metric does not end in the Marvel Universe and possible extensions of this work are widespread. The presented metric and its generalisation could be explored in relation to other cinematic universes, or TV shows, where data is available. Further, the metric has been applied to the Shannon-entropy of frequency distributions, however, this metric could be applied to a variety of other sources with relevance to character sentiments, transitions between evil and good characters, or the entropy of the speaker sequence as a Markovian process.

Supporting information

S1 Fig.

The numbers of credited characters in the MCU movies.

S2 Fig.

The numbers of named characters in the MCU movies.

S3 Fig.

Jenson-Shannon similarities (compare to Figure 6).