The Kevin Bacon Game: Properties of the Kevin Bacon Absorbing Set

D. D. Smith
Date: 12 November 1996

Abstract:

We develop formal rules for the heretofore largely ill-defined Kevin Bacon Game. What develops from these rules is akin to an absorbing set in stochastic theory which contains actors and actresses from the early days of film to movie releases far into the future. We examine topological properties of the Kevin Bacon Absorbing Set and its complement. We consider criteria for losing the Kevin Bacon Game, plus several examples of the elements in this set.

Kevin Bacon and the Kevin Bacon Game

Kevin Bacon is a prolific actor who began his film career in 1978 when he played Chip Diller in Animal House. After several so-called "B-movie" appearances (including the original Friday the 13th), he gained notoriety in 1984 with Footloose. Since then, he has gone on to work with some of the most highly-paid and best-recognized actors and actresses in the film industry. See Figure 1.

**Figure 1:** Kevin Bacon

A certain game has inexplicably grown up around Kevin Bacon. The rules vary widely, but the premise remains the same:

1.: Choose any movie actor or actress. Denote this actor or actress as A₀.
2.: Consider the movies in which A₀ appears. If A₀ has appeared in a film with Kevin Bacon, and you can recall the film, then you've won the game. Otherwise go to 3.
3.: Consider the other films in which A₀ has appeared. Let {A₁} denote the actors or actresses with whom A₀ has appeared in films. Let {A₂} denote the actors or actresses with whom every element in {A₁} has appeared, etc. Note that A₀ is an element of {A₂}. If there exists some i in which Kevin Bacon is an element of {A_i}, and you can recall the sequence of films and elements of {A₁}, {A₂}, ..., {A_i}, then you've won the game.

Intuitively, the player attempts to build a chain of actors/actresses and films such that the chain leads to Kevin Bacon. Each node of the chain is an actor or actress, and each node is linked by a film in which adjacent actor nodes have each appeared.

We present the following simple example. Let A₀ equal Woody Allen. Then the following chain may be constructed

where KB is Kevin Bacon.

Formalization of the Rules

As we have previously mentioned, the rules of the Kevin Bacon Game are subject to many modifications and adaptations depending upon the skill of the players. Some sources, such as [1], have attempted to explain the rules while not actually considering them from a scientific standpoint. In this paper, we attempt to present a formal treatment of the rules and the consequences of such rules.

Rule 1

A₀ must be selected from a film produced by an American production company.

Rule 2

A link between nodes must consist of films in which the adjacent actor/actress nodes have appeared. There is no restriction upon country of origin or the language of the films appearing in the links between nodes. The roles of each adjacent actor/actress must be credited roles.

These rules are often enforced implicitly. One can begin to sense the elementary nature of the Kevin Bacon Game and the ease of winning it.

Rule 3

Film refers to projected moving pictures, whose process was poineered around 1895 by the The Lumière Brothers [2]. Media such as television programs, made-for-TV movies, Broadway shows, stage and theatrical works, pornographic movies, interviews, news reports and television documentaries, movies that appear on videotape only, and all other media that don't conform to widely-released theatre-viewed motion pictures are excluded from possible links between nodes.

While these restrictions may seem harsh, they prevent rather unsubstantial claims of association between Kevin Bacon and unsporting figures. What is not restricted in Rule 3 are voice actors, who lend their voices to animated features. Also, we place no restriction upon the size of the production company, hence B-movies and independent films are allowable for links.

We consider the consequences of these rules in the following sections.

Some Consequences of the Rules

Rule 1 states that the initial A₀ must be selected from an American film. That is not to say that the film must be set in America or A₀ must be an American actor or actress. Many other countries have thriving film industries, e.g., Italy and Japan. However, they often tend to cast actors and actresses from that particular country who are adept at the film's language. So an actor like Diego Abatantuono (Turné, etc.), while prolific, may not have an opportunity to appear in an British or American film. It is rare when such actors have a chance at forming links to Kevin Bacon.

Rules 2 and 3 clarify what constitutes a link and which media form acceptable links. There are some notable results from these rules.

There is no restriction on the year that the film was released. Kevin Bacon's career extends from 1978 to at least 1998, when Wild Things will be released. However, there are hundreds of actors and actresses who are not contemporaries of Kevin Bacon, and yet may be linked to him. For example, let A₀ equal Charlie Chaplin. Then

KB <> Charlie Chaplin via Burt Lancaster

The Kevin Bacon game is not constrained in time in that an A₀ that is not be linked to Kevin Bacon today may perhaps be linked to Kevin Bacon in the future. Young actors, such as Elijah Wood, who are linked to Kevin Bacon, will continue to link other actors as their careers unfold.

Another example of the fact that a Kevin Bacon link may be formed in the future is the 1994 film Clerks, which was casted by largely unknown actors. It was proposed that this was an example where the Kevin Bacon Game couldn't be won. Shortly afterword, however, one of the actors appeared in another film with a well-worn link to Kevin Bacon. As Sutton noted, this is the effect of "... the future unfolding before your eyes. [3]"

$\begin{align*} ...eleft Clerks \vartriangleright \\ & \binom{\text{Cast of}}{Clerks}.\end{align*}$

There is no restriction on the nationality or language of the films in the intermediary links. If we let A₀ equal Hugh Grant (who has appeared in the American-produced Extreme Measures), the one possible path to Kevin Bacon would be

$\begin{align*}KB&\vartriangleleftA\,Few\,Good\,Men \vartriangleright\binom{\... ...and\,a\,Funeral\vartriangleright\binom{\text{Hugh}}{\text{Grant}}.\end{align*}$

The fact that Kristen Scott Thomas has largely appeared in French and British films is beneficial in constructing a link to Kevin Bacon in this case.

There is no restriction on the scope or genre of the film's production company, so long as it conforms to Rule 3. There are many B-movies and independent films with links to Kevin Bacon. Often, an actor or actress will appear in a few B-movies before being cast in an A-movie. Also, cameo appearences in B-movies by actors with a Kevin Bacon link will not only link the actors in that movie but many other B-movies as well. ... Joe Bob Briggs appeared in Hollywood Boulevard II in a cameo role. Briggs, however, has a short link to Kevin Bacon:

$\begin{displaymath}KB\vartriangleleft Sleepers\vartriangleright \binom{\text... ...Casino \vartriangleright\binom{\text{JoeBob}}{\text{Briggs}}.\end{displaymath}$ B-movie and independent film actors and actresses are much more challenging to link with Kevin Bacon. However, there are many instances of entire independent production companies being linked to Kevin Bacon through a cameo or a single actor or actress who later appeared in an A-movie.

Animated features are also not excluded from links. A prolific voice actress like Tress MacNeille may use animated films to build a link to Kevin Bacon:

$\begin{align*}KB & \vartriangleleft JFK\vartriangleright \binom{\text{Tommy}... ...angerfield\vartriangleright\binom{\text{Tress}}{\text{MacNeille}}.\end{align*}$

The set of nodes linked to Kevin Bacon have some interesting topological properties. Let {KB} be the set of all actor nodes linked with Kevin Bacon, either now or in the future. We refer to this as the Kevin Bacon Absorbing Set, because of it being akin to an absorbing set in stochastic theory. It is clear that any two elements of {KB} communicate with each other, that is, a link may be formed between any two elements of {KB} via Kevin Bacon. Consider any element A₀ in {KB}. Define a neighborhood of A₀ as those elements in a non-empty {A₁}. With this definition of neighborhood, we examine some of the topological properties of {KB}.

Theorem 1

{KB} is an open set.

PROOF Choose any A₀ in {KB} and a non-empty neighborhood U of A₀. Choose any A₁ in U. A₁ is an element of {KB} since A₀ is an element of {KB}, and hence U is an element of {KB}.

Theorem 2

{KB}^c is an open set.

PROOF Choose any A₀ in {KB}^c and a non-empty neighborhood U of A₀. Choose any A₁ in U. A₁ is a element of {KB}^c since A₀ is not contained in {KB}. Hence, U is an element of {KB}^c.

Theorem 3

{KB} is dense.

PROOF It is sufficient to show that every point in {KB} is a limit point. Choose any A₀ in {KB}. Choose any non-empty neighborhood U of A₀. Then every A₁ in U is contained in {KB}, since A₀ is in {KB}.

Corollary

There are no isolated points in {KB}.

PROOF There exists no node A₀ in {KB} such that A₀ is not a limit point of {KB}.

Theorem 5

It is possible to contruct a point in {KB}^c that is an isolated point.

PROOF Construct an A₀ not equal to Kevin Bacon such that A₀ is the only actor or actress in an American produced film. Suppose that A₀ only makes one film in accordance with Rule 3. Then A₀ is isolated since there are no neighborhoods which contain other nodes. Independently-produced films with a cast of one or films of stand-up comedy, similar to that in Eddie Murphy: Raw, are examples of constructions in {KB}^c that could potentially be isolated.

Strategies for Losing the Game

Even at its most restrictive, the Kevin Bacon game is easy to win. Suppose that a player has total recall of all links that define {KB}. Then that player will always win the game if A₀ is in {KB}. A much larger challenge is whether one can discover an A₀ such that no chain exists between A₀ and Kevin Bacon. That is, contruct an A₀ such that A₀ is in {KB}^c. and subsequently lose the game.

The fact that this is difficult is a credit to the organization of the American film industry. At each node, the entire body of the actor or actress is a potential link closer to Kevin Bacon. A single intersection between one's body of work with another's body of work is sufficient for a link to form. So it is not surprising that the network of links is so extensive.

If one wants to lose the Kevin Bacon Game, then it is somewhat advisable to choose A₀ such that A₀ is not in an A-movie. There are far too many links already established within the large production companies, and chances are excellent that it is just a matter of time before A₀ is absorbed into {KB}.

One strategy for losing is to pick A₀ from

older B-movies e.g., The Creature from Galaxy 27
obscure films or films by new film makers e.g., Palookaville or Boyz in the Hood.
films that are based on a television program e.g., Star Trek: Generations
actors or actresses who have appeared in very few films e.g., Holly Robinson.

Many low-budget "exploitation films" were widely released in the 1950's and 1960's via the drive-in circuit. Although there were many independent production houses at that time, few achieved any sort of notoriety. The actors and actresses generally consisted of friends or relatives of the film makers or other cheap talent. This is one of the densest areas of {KB}^c.

Television actors are occasionally resigned to do television for their entire career except for an occasional film or so. One example is Holly Robinson, who was only in one film, Howard the Duck. Robinson has spent the rest of her acting career in television. Robinson would have been a losing A₀ if it weren't for Tim Robbins and Lea Thompson; unfortunately, Howard the Duck was not obscure enough.

$\begin{align*}KB& \vartriangleleftEnd\,of\,the\,Line \vartriangleright \bino... ...re\,III \vartriangleright \\ &\binom{\text{Lea}}{\text{Thompson}};\end{align*}$

$\begin{displaymath}KB\vartriangleleft Flatliners\vartriangleright \binom{\te... ...o\,Wear \vartriangleright \binom{\text{Tim}}{\text{Robbins}}.\end{displaymath}$

Large screen adaptations of television programs may have potential for containing actors and actresses in {KB}^c. Star Trek: The Next Generation became Star Trek: Generations on the big screen. There were only two notable film actors in the original TV series: Patrick Stewart (Dune, etc.) and LeVar Burton (Roots, etc.) Most of the other actors and actresses in the TV series spent most of their careers in TV. These actors may have remained in {KB}^c for a long time (working on minor film projects), except that Star Trek: Generations came along. This film featured Whoopi Goldberg, Malcolm McDowell and William Shatner, each of whom have been assimilated into {KB}:

$\begin{align*} \binom{\text{WhoopiGoldberg}}{\text{William Shatner}}.\end{align*}$

One such TV series that has been made into a movie and yet still contains actors in {KB}^c is Mystery Science Theater 3000 and the movie by the same name. The three principal actors, Michael J. Nelson, Trace Beaulieu, and Kevin Murphy, have not become linked with Kevin Bacon as of this writing, but it is unknown if they will become linked as their careers unfold.

Further Research

Although we have tried to define and perhaps restrict the influence of the Kevin Bacon Absorbing Set, it is not known each year what the rate of absorption, or the number of new elements in {KB}, is. There are many issues to consider, namely, how does one collect the names of the releases from all production companies in America? This may be similar to a stratification problem in sampling theory.

If one could find the rate of absorption into {KB}, there is a possibility that models could be constructed to predict the number of years until, say, 99% of all {KB}^c is absorbed into {KB} Markov or other stochastic models may be appropriate for such predictions.

Finally, there have been claims [1] that smallest number to sufficiently link any A₀ in {KB} is six. This seems to be true in for many examples, but it is not known whether this is true in general. Certainly many B-movie actors/actresses who are contained in {KB} must exert much effort (exhaust many links) in order to communicate with Kevin Bacon. There may exist counterexamples of this conjecture by considering an A₀ linked solely from already-lengthy-links, like Charlie Chaplin's, and extending from there.

ACKNOWLEDGEMENTS The author wishes to thank the staff of the Internet Movie Data Base, Thomas E. Shakow, Yale University, and Michael L. Woodruff for his corrections.

References

1: Mike Ginelli, Brian Turtle, and Craig Fass.; Six Degrees of Kevin Bacon.; Penguin Publishing, New York, 1996.
2: Louis Lumière.; The cinematograph, la nature, 12 october 1895.; In Auguste and Louis Lumière. (Jacques Rittaud-Hutinet, ed.) Letters, London, 1995. Faber and Faber.
3: Richard K. Sutton.; Personal communication, 1995.