Mathematical Models & Movies: A Sneak Preview Ron Buckmire Occidental College Los 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 Dynamics Important 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

EBM Parameters

Dimensionless EBM where

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)A p (t) which satisfies Where A p is the revenue per screen if everyone who wanted 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 A p Derivation: Doing The Math

Modified EBM

Comparing Original EBM to Modified EBM

Numerical Calculations Analyzed119 movies from (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 Parameters The Expendables (2010)

Numerical Results: Using Global Parameters Taken (2009)

Numerical Results: Using Global Parameters The Love Guru (2008)

Numerical Results: Using Global Parameters Spider-Man 3 (2007)

Numerical Results: Using Global Parameters Open Season (2006)

Numerical Results: Using Chosen Parameters The Expendables (2010)

Numerical Results: Using Chosen Parameters Open 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, A 0, 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