Mathematical Models & Movies: A Sneak Preview

Ron Buckmireron@oxy.edu

Occidental CollegeLos Angeles, CA

Outline• Introduction to Cinematic Box-Office Dynamics

– Important variables and concepts– Graphs of typical movie data

• Presentation of Edwards-Buckmire Model (EBM)

• Drawbacks of EBM• Derivation of Modified EBM• Numerical Results Using Modified EBM• Future (and Past) Work

– The Holy Grail: A Priori Prognostication– Sequels: Parent-Child Relationship

• Conclusions• References and Acknowledgements

Introduction: Cinematic Box-Office DynamicsImportant variables

• G(t) : cumulative gross receipts of the movie

• S(t) : number of screens movie is exhibited• A(t) : normalized weekly revenue ($ per

screen average)• t : time in number of weeks

Important concepts• A and S have quasi-exponential

profiles•

Actual Movie Data: The Expendables (2010)

Actual Movie Data: Taken (2009)

Actual Movie Data: The Love Guru (2008)

Actual Movie Data: Spider-Man 3 (2007)

Actual Movie Data: Open Season (2006)

The original Edwards-Buckmire model SA

dt

dG

max* A

ASS

dt

dSS

AGHPMS

S

Ddt

dA A

2

%)1(

0)0( AA

0)0( SS

0)0( G

EBM Parameters

Dimensionless EBM SA

dt

dG AS

dt

dS

AGS

S

dt

dA

~

~~

where

*

~S

M )1(

1~

DS

A

SP

ASH

max*

2%~

Typical solution curves

Drawbacks of EBM• H% varies with time

• Parameter ( ) estimates are difficult to make and somewhat arbitrary

• Most movies have a contract period in which screens is constant, i.e.S’=0

• S and A actual data more erratic than first thought; G is relatively smooth

Modifying the EBM (J. Ortega-Gingrich)• Uses an Economics-inspired

“demand” model• Incorporates fixed contract

periods when screens are constant

• Greatly modifies the A equation• Both versions of EBM have 3

unknown parameters

Deriving the new A equation

Consider a Demand function D(t)=S(t)Ap(t) which satisfies

Where Ap is the revenue per screen if everyone whowanted to see the film, saw it, i.e. “A potential”

Recall that G’=SA and assume that G could satisfy the IVP

Which leads to

and

The selected form of μ(S) used is given below (a=1/T),

T is total number of movie theaters in North America (~4,000)

The function μ(S) should satisfy the following conditions

We apply the product rule to A and Ap

Derivation: Doing The Math

Modified EBM

Comparing Original EBM to Modified EBM

SAdt

dG

ASdt

dS

AGS

S

dt

dA

~

~~

Numerical Calculations• Analyzed119 movies from 2005-2010

(minimum final gross $50m)• All dollars adjusted for inflation to 2005• Used Mathematica to generate

numerical solutions to the modified EBM

• Attempted to find “global” values of parameters that would minimize std. dev. in difference between computed G∞ and actual G∞ while minimizing error

Numerical Results: (N=119)

Distribution of G Computed/G Actual as Histogram

mean=1.0389, std. dev.=0.158

Numerical Results: (N=119)

Numerical Results: Using Global ParametersThe Expendables (2010)

Numerical Results: Using Global ParametersTaken (2009)

Numerical Results: Using Global ParametersThe Love Guru (2008)

Numerical Results: Using Global ParametersSpider-Man 3 (2007)

Numerical Results: Using Global ParametersOpen Season (2006)

Numerical Results: Using Chosen ParametersThe Expendables (2010)

Numerical Results: Using Chosen ParametersOpen Season (2006)

Future Work“The Holy Grail”: Predict the

opening weekend gross before the movie is released

The sequel problem: predict the gross of a sequel based on the parent’s characteristics

The Sequel Problem• Considered a subset of the a priori

prediction problem with (possibly) more known information

• Main assumption is opening weekend revenue, A0, must depend on awareness of film (which probably depends on marketing, M)

Conclusions• Predicting the final accumulated

gross of any given movie before it is released is a hard problem

• The original EBM should probably be modified to be less movie-specific and the modified EBM changed to be more movie-specific

Acknowledgements• Joint work with Occidental

College students Jacob Ortega-Gingrich ’13 and Rohan Shah ’07

• Many thanks to David Edwards and the staff and faculty of University of Delaware