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Sliding-Scale Contracts in Movie Distribution: A Theoretical

Rationale

Journal: Management Science

Manuscript ID: MS-11-00466

Manuscript Type: Marketing

Keywords: Marketing : Channels of distribution, Marketing : Competitive strategy, Marketing : New products

Abstract:

We propose a theoretical rationale – using a game-theoretic model – for sliding-scale contracts in movie distribution. Sliding scale contracting refers to the commonly observed practice of the distributor (studio) taking a much larger (usually 70%) share of box-office revenues than the exhibitor (theater) in the week of a

movie’s release, with the exhibitor’s share increasing, in gradual steps, over subsequent weeks. Using a two-period game-theoretic model that involves two studios and a theater (which is assumed to have sufficient screen capacity to display both movies in both periods), as well as utility-maximizing consumers who choose between the available movies and the outside good in each period, we show that under a flat-scale contract (i.e., studio’s share of the box-office revenues is invariant over time), it is optimal for both studios to release their movies in period 1. Under a sliding-scale contract, however, one of two sets of outcomes is optimal: (1) Steep Sliding Scale: Both studios release their movies in period 2, or (2) Shallow Sliding Scale: One studio releases its movie in period

1, while the other does in period 2. Both of these cases lead to improved profit outcomes for the studios and the theater (“win-win”) compared to the base case of flat contracting, for the following reasons. Under case (1), the theater’s display costs greatly decrease, which increases the ability of both studios to demand a much higher share of the box-office revenues during the release period. Under case (2), one studio finds it viable to open its movie early in order to insulate itself from competition from the other studio (and this decreased competition also benefits the second studio that releases its movie later); each studio’s share of box-office revenues during the release period goes down to counter the theater’s increased display costs from screening in both

periods. We show that our key results are robust to the relaxation of various modeling assumptions. We also find empirical support for the key implications of our theoretical model.

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Management Science

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Sliding-Scale Contracts in Movie Distribution: A Theoretical Rationale

Abstract

We propose a theoretical rationale – using a game-theoretic model – for sliding-scale contracts in

movie distribution. Sliding scale contracting refers to the commonly observed practice of the distributor

(studio) taking a much larger (usually 70%) share of box-office revenues than the exhibitor (theater) in

the week of a movie’s release, with the exhibitor’s share increasing, in gradual steps, over subsequent

weeks. Using a two-period game-theoretic model that involves two studios and a theater (which is

assumed to have sufficient screen capacity to display both movies in both periods), as well as utility-

maximizing consumers who choose between the available movies and the outside good in each period, we

show that under a flat-scale contract (i.e., studio’s share of the box-office revenues is invariant over time),

it is optimal for both studios to release their movies in period 1. Under a sliding-scale contract, however,

one of two sets of outcomes is optimal: (1) Steep Sliding Scale: Both studios release their movies in

period 2, or (2) Shallow Sliding Scale: One studio releases its movie in period 1, while the other does in

period 2. Both of these cases lead to improved profit outcomes for the studios and the theater (“win-win”)

compared to the base case of flat contracting, for the following reasons. Under case (1), the theater’s

display costs greatly decrease, which increases the ability of both studios to demand a much higher share

of the box-office revenues during the release period. Under case (2), one studio finds it viable to open its

movie early in order to insulate itself from competition from the other studio (and this decreased

competition also benefits the second studio that releases its movie later); each studio’s share of box-office

revenues during the release period goes down to counter the theater’s increased display costs from

screening in both periods. We show that our key results are robust to the relaxation of various modeling

assumptions. We also find empirical support for the key implications of our theoretical model.

Keywords: Movie Distribution, Sliding-Scale Contracts, Movie Release Timing.

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INTRODUCTION

Movies represent a significant portion of the US economy. Hollywood movies in 2010 constituted

an $9b industry (www.the-numbers.com/market/). A commonly observed institutional practice in this

industry is that the distributor (studio) takes a very high share of the box-office revenues, relative to the

exhibitor (theater), during the week of release of the movie. (Daniels, Leedy and Sills (2006)). The share

of the theater increases, in gradual steps, in subsequent weeks. For example, the theater’s share of the

box-office revenues would go up from 10% in week 1 to 30% in week 2 to 40% in week 3 etc. For this

reason, the channel contract in the movie industry is referred to as a sliding-scale contract. Despite the

emergence of a rich body of recent research on the movie industry over the past decade, there is a

surprising dearth of research on why contractual arrangements between studios and theaters follow the

sliding-scale (as observed recently by Eliashberg, Elberse and Leenders (2006) in their review article,

where they explicitly call for future research on this issue). The goal of this paper is to address this gap in

the literature. We provide a theoretical rationale – using a game-theoretic model – for sliding-scale

contracts in movie distribution. 1

Concession sales -- which include sales from food (such as popcorn and soda), as well as arcade

income from electronic games or other machines -- remain with the theater and are never shared with the

studio, and constitute the lifeblood of theaters (Friedberg 1992, Durwood and Rutkowski 1992). In fact,

the operating margins from concession sales vary from 50-75%, which is why concessions are generally

understood to be the reason for the willingness of theaters to accept sliding-scale profit contracts that

disproportionately favor studios, as well as represent the core business model of theaters (Sawhney and

Eliashberg (1996), Vogel (1998)). Given this, our theoretical analysis of sliding-scale contracts explicitly

accommodates the role of concession profits of theaters.

We employ a two-period game-theoretic model that involves two studios and a theater (which is

assumed to have sufficient screen capacity to display both movies in both periods), as well as utility-

maximizing consumers who choose between the available movies and the outside good in each period.

The reason for using a two-period model is to show the relative viability, from a profitability standpoint,

of a delayed / limited-time release (i.e., second-period only) versus an early / extended-time release (i.e.,

both periods) for both the studios and the theater. We show that under a flat-scale contract (i.e., studio’s

share of box-office revenues is invariant over time), it is optimal for both studios to release their movies

in period 1. Under a sliding-scale contract, however, one of two sets of outcomes is optimal: (1) Steep

Sliding Scale: Both studios release their movies in period 2, or (2) Shallow Sliding Scale: One studio

releases its movie in period 1, while the other does in period 2. Both of these cases lead to improved profit

1 Prior research by Filson, Switzer and Besocke (2005) provides an alternative explanation for sliding-scale contracts. We discuss in the next section why that explanation is less compelling than ours.

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outcomes for the studios and the theater (“win-win”) compared to the base case of flat contracting, for the

following reasons. Under case (1), the theater’s display costs greatly decrease, which increases the ability

of both studios to demand a much higher share of the box-office revenues during the release period.

Under case (2), one studio finds it viable to open its movie early in order to insulate itself from

competition from the other studio in one period; the other studio also does not simultaneously open its

movie early because it benefits from having consumers who have watched the competing studio’s movie

earlier to watch its movie in the later period (instead of fighting head-on with the competing studio for

those consumers); each studio’s share of box-office revenues during the release period goes down to

counter the theater’s increased display costs from screening in both periods. To summarize, a sliding-

scale contract arises as an equilibrium that not only improves the studio’s profits (compared to a flat-scale

contract), but also is incentive compatible with the release times / lengths of the movies that improve the

theater’s profits (by decreasing the theater’s display costs). We show that our key results are robust to the

relaxation of our modeling assumptions, i.e., allowing for strategic (i.e., forward-looking) consumers,

endogenous movie closing times, and multiple (>1) theaters. We find empirical support for several key

implications of our theoretical model.

The rest of the paper is organized as follows. In the next section we discuss previous literature

and position our work relative to this literature. In section 3, we develop the primitives of our theoretical

model. In section 4, we analyze the equilibrium implications of our theoretical model. In section 5, we

present some empirical findings that are consistent with some implications of our theoretical model. In

section 6, we conclude with opportunities for future research. In Appendix A, we present the proofs of

various propositions and claims that are presented in the model implications section. In Appendix B, we

show that our key model implications are robust to relaxing various modeling assumptions.

LITERATURE REVIEW

The only existing explanation in the academic literature for sliding-scale contracts is that

provided by Filson, Switzer and Besocke (2005). Their explanation is that movie sales are difficult to

forecast and that risk aversion of studios and the theater lead them toward sliding-scale contracts in order

to share their risks. This explanation is weak for the following reasons:

1. Demand forecasting for movies has become quite sophisticated in recent years. As noted by

movie industry insiders (see, for example, Dekom (1992), as well as Rosen (1993), as quoted

in Krider and Weinberg (1998)), the box-office revenues of a movie can be forecasted to a

good degree of accuracy – using computer-based forecasting models that employ demand

data for similar movies in the recent past -- after the production of the movie is complete, and

the pre-release advertising budget has been determined. For example, Zufryden (2000),

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Elberse and Eliashberg (2003) and Basuroy, Desai and Talukdar (2006) report R2 as high as

0.90, 0.92 and 0.86, respectively, for regression models that explain first-week box-office

revenues of movies using movie-specific exogenous variables. Prag and Casavant (1994)

obtain an R2 of 0.62 with a similar model even using very sparse and noisy data. Sawhney

and Eliashberg (1996) use a meta-analysis of past movies’ temporal sales patterns to forecast

the market potentials of new movies. Neelamegham and Chintagunta (1999) obtain high pre-

release forecast accuracies for movies’ box-office revenues using a Bayesian methodology

despite ignoring the role of advertising and production budget as predictors in the analysis

(also see Wallace, Seigerman and Holbrook (1993) and Jedidi, Krider and Weinberg (1998)

for good success in pre-release forecasting of movie revenues). Box office revenue of a

movie can also be forecasted using primary research data collected either using pre-release

market tests on consumers, as illustrated in Eliashberg and Sawhney (1994), or using

prediction markets, such as the Hollywood stock exchange (www.hsx.com), as discussed in

Elberse and Eliashberg (2003). While there are movies whose box-office revenues turn out to

be at odds with what studio executives anticipate a priori, they are clearly the exception

rather than the norm these days. In fact, Radas and Shugan (1998) observe that the release

date for “Speed” was chosen based on the studio’s pre-release forecast of surprisingly

positive word of mouth effects for the movie. Krider and Weinberg (1998), in their analysis

of optimal release times of competing movies, also assume that competing studios have

accurate forecasts of each other’s movie sales prior to release. Therefore, it would be

necessary to investigate whether one can explain the use of sliding-scale contracts even when

demand uncertainty is absent in the market. Under the framework of Filson, Switzer and

Besocke (2005), it is not possible to explain a sharing contract between a studio and a theater,

far less a sliding-scale contract, when forecasting uncertainty is absent in the market.

2. In situations when uncertainty in demand forecasting is deemed risky enough, studios

typically employ more efficient risk-mitigation strategies, such as co-financing (Goettler and

Leslie (2005), Daniels, Leedy and Sills (2006)), diversification (Squire (1992)), and short-

term contingencies in their contracts with the theater (De Vany and Eckert (1991), De Vany

and Walls (1996)). It seems unlikely, therefore, that the sliding-scale, which typically

characterizes all (as opposed to riskier) movie contracts, is used for the purpose of risk

mitigation. For example, sliding-scale contracts have been observed even for “very safe”

movies (from a demand forecasting standpoint) such as Star Wars: The Phantom Menace,

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Mission Impossible 2 etc.2 Therefore, a theoretical model that can rationalize the use of

sliding-scale contracts even for safe movies is necessary.

3. The reason for the sliding scale contract arising in the Filson, Switzer and Besocke (2005)

model lies in myopic (i.e., single-period profit maximizing) studios having to give a greater

share of temporally declining box-office revenues (a commonly observed empirical regularity

among movies) to the theater in order for the latter to be willing to display the movie in

subsequent weeks. 3 This does not explain why sliding-scale contracts have also been

observed for movies (such as My Big Fat Greek Wedding) whose sales increased over time

before beginning to decline. More importantly, it also begs the question of why studios do not

then plan a much simpler revenue-sharing contract with the theater that is based on the total

box-office revenues (summed over all periods of the movie’s run) of the movie. In fact,

Filson, Switzer and Besocke (2005) themselves argue that such an aggregate contract can

yield the same payoffs to the studios and the exhibitor as does a more complicated sliding-

scale contract (see Table 3 of their paper). Given that none of the studios use such aggregate

contracts in practice, it would be enormously useful to have a theoretical model that can not

only rationalize sliding-scale contracts, but also disallow aggregate contracts as being equally

effective for studios and the theater. It would also be useful for such a model to recognize the

fact that concession profits play an important role in the theater’s exhibition problem,

something that is ignored in the analysis of Filson, Switzer and Besocke (2005).

We propose an alternative explanation for sliding-scale contracts, which is not based on risk-

mitigation, in this paper. Our model addresses the weakness associated with the Filson, Switzer and

Besocke (2005) model, as discussed above. In other words, we assume that studios can forecast the

demand for their movies, and show that a sliding-scale contract, which endogenously arises as an

equilibrium strategy for each studio, would dominate an aggregate contract (which explains why the latter

is not observed in practice). Our explanation relies on the studio’s use of a sliding-scale contract to

increase its share of box-office revenues during the period of release, while also simultaneously

improving the theater’s welfare by shortening the length of time for which the theater displays the movie

(which directly goes to the theater’s bottom-line because of its decreased display costs). 4 In some

2 Sequels to successful movie franchises are generally understood to be the safest bets in the movie industry. 3 In fact, their model relies on the relative risk aversion of the theater being higher than that of the studio. There is no strong reason for one to expect this to hold in practice. Since theaters make most of their profits from concession sales, which depend on foot traffic from patrons for multiple (as opposed to a single) movies playing within the multiplex (the typical theatrical format in most US markets), one could argue that theaters are probably even less risk averse than studios! 4 Filson, Switzer and Besocke (2005) argue that sliding‐scale contracts give theaters the incentive to run movies longer. The empirical reality shows the opposite to be true, i.e., theaters do not run movies long enough to make

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situations (as will be shown in the next two sections), the sliding-scale contract also decreases

competition among studios by giving one studio the opportunity to release its movie early, while the other

studio releases its movie later. This latter effect is accentuated when the two movies have different levels

of attractiveness, with the movie of higher attractiveness releasing first, and the movie of lower

attractiveness releasing later. This is consistent with the movie release timing strategies derived by Krider

and Weinberg (1998) and illustrated using the example of “A Few Good Men” and “Hoffa,” both of

which were ready for release at the same time, but the former released first and the latter released much

later. To summarize, our explanation for sliding-scale contracts exploits dynamic incentives of competing

studios, which display their movies through a theater, to simultaneously manage both their revenue-

sharing arrangement with the theater, as well as the temporal display schedule of their movies (early /

extended time release, versus late / limited time release). While Filson, Switzer and Besocke (2005)

present compelling arguments for why asymmetric information between studios and theaters, as in

standard principal-agent models, cannot provide good explanations for why sharing contracts are used in

the movie industry, their explanation for why sliding-scale contracts are used appears to be unrealistic.

Our paper makes an advance in that direction. Given the abundance of papers on movie demand

forecasting in the marketing literature, our study fills the void created by a dearth of research on

contracting arrangements in the movie industry.

significant profits on each movie. These days, there are a large number of movies competing for limited screen space in theaters (Swami, Eliashberg and Weinberg (1999)).

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MODEL PRIMITIVES

Market/Game Structure

Consider two competing studios, j = 1, 2, with completed (and ready for release) movies of

attractiveness5 q1 and q2 (both of which are known to both studios) in a market with 1 theater6 and M

(assumed to be equal to 1, without any loss of generality) consumers. We consider these M consumers’

choices within a two-period framework, i.e., any among 0, 1 and 2 movies will be available to these

consumers for viewing in each of periods 1 and 2. Each studio independently negotiates a revenue-sharing

agreement – which says how box-office revenues will be shared between the studio and the theater during

each period after release of the movie -- with the theater (which is assumed to have sufficient screen

capacity to display both movies simultaneously) during period 0. This revenue-sharing agreement yields a

contract {θj1, θj2} for studio j, where θj1 and θj2 stand for studio j’s share of box-office revenues of movie j

in the initial period of release, and the subsequent period, respectively. Let θj = {θj1, θj2}. With the

revenue-sharing agreement in place, the studio then chooses the release time, {tj0}, for its movie, also

during period 0.7 Let t0 = {t10, t20}. For example, t10 = 1 and t20 = 2 would imply that movie 1 is released in

period 1, while movie 2 is released in period 2. We assume that once a movie is released, it is displayed

until the end of the second period. Therefore, in the above example, movie 1 will be displayed in both

periods, while movie 2 will be displayed in period 2 only.8 The reason for using a two-period model is to

understand the relative viability, from a profitability standpoint, of a delayed / limited-time release (i.e.,

second-period only) versus an early / extended-time release (i.e., both periods) for both the studios and

the theater. It is also consistent with the institutional reality that theaters typically write pre-release

contracts with studios for the first 2-3 weeks of release of a movie (Sawhney and Eliashberg (1996),

Radas and Shugan (1998)), as well as the fact that studios are facing more limited availability of screens

for their films, given the fast and furious pace at which new movies are being produced these days

(Swami, Eliashberg and Weinberg (1999)). We assume that each studio’s cost of producing its movie is

sunk, and that both studios incur no further costs (such as the marginal costs of displaying the movies,

which are fully borne by the theater). Figure 1 below shows the sequence of action choices and payoffs

5 Movie attractiveness, which is a function of movie characteristics, can be interpreted as a movie demand forecast which, as we have argued earlier, is generally known to the major movie studios prior to movie release. This assumption is also made by Krider and Weinberg (1998) in their analysis of release timing strategies of competing movies. 6 Our assumption about a monopolist theater reflects the reality that local movie markets in the US typically have a single (often, multiplex) theater. Later, we relax the monopoly assumption and show that our results are, in fact, robust to allowing for competition among theaters. 7 It is assumed that each studio, while choosing its release time, knows the other studio’s chosen revenue-sharing agreement with the theater. However, we will show later that each studio’s best-response function for its movie release time does not, in fact, end up depending on its competing studio’s chosen revenue-sharing agreement. 8 Later, we show that our results are robust to allowing for endogenous movie closing times.

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for all pla

by studios

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1 2

Pr .jijt

t

q

q q k

(3)

Suppose Dj (t; t0) denotes the demand for movie j in period t, given the chosen set of release times

t0. Under the utility primitives explained above, the demands for the two movies are derived below for all

possible cases of release times, t0.

Case 1: Both Movies Released in Period 1 (t10 = 1, t20 = 1)

11 0

1 2 1

22 0

1 2 1

2 1 11 0

1 2 1 1 2 1 2 1 1 2 2

1 2 22 0

1 2 1 2 2 1 2 1 2 2 2

1; ,

1; ,

12; * * ,

12; * * .

qD t

q q k

qD t

q q k

q q qD t

q q k q k q q k q q k

q q qD t

q q k q k q q k q q k

(4)

The first-period demand for each movie is derived based on all consumers choosing, according to the

MNL model of equation (3), between the two movies and the outside good of period 1. The second-period

demand for each movie comprises two terms. The first term captures the demand from consumers who

have watched the competing movie during the first period (and, therefore, are choosing between the focal

movie and the outside good of period 2). The second term captures the demand from consumers who have

watched neither movie during the first period (and, therefore, are choosing between the two movies and

the outside good of period 2).

Case 2: Both Movies Released in Period 2 (t10 = 2, t20 = 2)

1 0

2 0

11 0

1 2 2

22 0

2 2 2

1; 0,

1; 0,

2; ,

2; .

D t

D t

qD t

q q k

qD t

q q k

(5)

The first-period demand for each movie is zero since neither movie has released yet. The second-period

demand for each movie is derived based on all consumers choosing, according to the MNL model,

between the two movies and the outside good of period 2.

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Case 3: Movie 1 Released in Period 1, Movie 2 Released in Period 2 (t10 = 1, t20 = 2)

11 0

1 1

2 0

1 11 0

1 1 1 2 2

22 0

1 2 2

1; ,

1; 0,

2; * ,

2; .

qD t

q k

D t

k qD t

q k q q k

qD t

q q k

(6)

The first-period demand for movie 1 is derived based on all consumers choosing, according to the MNL

model, between movie 1 and the outside good of period 1. The first-period demand for movie 2 is zero

since it has not released yet. The second-period demand for movie 1 is derived based on consumers who

have not watched movie 1 in the first period (and, therefore, are choosing between the two movies and the

outside good of period 2). The second-period demand for movie 2 is derived based on all consumers

choosing between the two movies and the outside good of period 2.

Case 4: Movie 1 Released in Period 2, Movie 2 Released in Period 1 (t10 = 2, t20 = 1)

1 0

22 0

2 1

11 0

1 2 2

1 22 0

2 1 1 2 2

1; 0,

1; ,

2; ,

2; * .

D t

qD t

q k

qD t

q q k

k qD t

q k q q k

(7)

This is similar to case 3, except that the roles of movies 1 and 2 have been reversed.

To summarize, our demand primitives imply the following equations for the demand of movie j in the

two periods (where I (.) is an indicator function that takes the value 1 if the condition within parentheses

is satisfied, and 0 otherwise).

0

01 10 2 20 1

20 0

0 0 3 01 1 10 2 20 2 0 2

11; ,

1 1

2 22; 1 1; * 1; * .

2 2 2

j j

j

j j j j

j j jj j j

q I tD t

q I t q I t k

q I t q I tD t D t D t

q I t q I t k q I t k

(8)

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Payoffs

Since movie ticket prices at theaters are exogenous (i.e., not strategically chosen for a given

movie either by the studios or the theater), we normalize ticket prices to $1, so that the box-office

revenues for movie j are identical to the demand given in equation (8) above. The profits for studio j,

conditional on contract and release times (θj, t0), are given by

1 0 2 0 00

1 0 0

1; 2; , if 1,,

2; , if 2.j j j j j

j jj j j

D t D t tS t

D t t

(9)

The profits for the theater are given by

0 1 1 0 2 2 0, , , ,T t T t T t (10)

where Tj (θj, t0), j = 1, 2, stands for the profits from movie j and is given by

00 ( 1) 0 0( , ) (1 ) ; ( ),

jj j j t t j jt jt

T t r D t t C I t t (11)

where r stands for the profit per customer that accrues to the theater on account of concession sales

(which are not shared with either studio), and Cjt stands for the theater’s cost of displaying movie j in

period t. Given our earlier discussion in the introduction section about concession sales being the “raison

d’etre” for the running of theaters, including concession profit margins, r, within the theater’s payoff

function accounts for an important institutional feature of theaters. The total channel profit, therefore, is

given by

0 1 1 0 2 2 0 0, , , ,t S t S t T t (12)

which, using equations (9)-(11), becomes

2 2

0 0 01 1

1 , .j jt jj t

t r D t t C I t t

(13)

From the above equation, one can see that given the individual movies’ chosen release times, t10 and t20,

the total channel profit does not depend on either studio’s revenue-sharing contract with the theater, θ1 or

θ2. This is not surprising since the revenue-sharing contract must only influence the channel members’

relative shares, but not the absolute size, of the channel profit.

Equilibrium

With regard to the negotiation of the revenue-sharing agreement between each studio and the

theater, we assume that studio j makes a “take it leave it” offer of θj to the theater, which the theater then

accepts or rejects based on its participation constraint (which relies on the theater’s reservation profit

level djT > 0). We assume this decision sequence because it is well known that studios have much greater

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bargaining power than theaters in the movie industry (Vogel (1998), Swami, Eliashberg and Weinberg

(1999), Daniels, Leedy and Sills (2006)). Once the revenue-sharing rules have been decided, each studio

chooses the release time, tj0, for its movie. It is useful to note that each studio’s chosen revenue share (θj)

and release time (tj0) are based on its objective of profit maximization (which must take in to account the

participation constraint of the theater).

We formally define our model equilibrium, as well as the revenue-sharing contract, below.

Definition 1: Pure-Strategy Sub-game-Perfect Nash Equilibrium

An equilibrium is a set of sharing contracts θ* = {θ1*, θ2*} and release strategies t0*(θ) = {t10*(θ), t20*(θ)}

that satisfy the following three conditions:

D1.1: Incentive Compatibility of Release Times

For any set of sharing contracts, θ = {θ1, θ2}, for studios 1 and 2, the release strategies,

t0*(θ) = {t10*(θ), t20*(θ)}, form a Nash equilibrium, i.e.,

For any tj0 (θ) ≠ tj0*(θ),

Sj ({tj0*(θ), t3-j0*(θ)}, θ) ≥ Sj ({tj0 (θ), t3-j0*(θ)}, θ) (j = 1, 2).

D1.2: Theater’s Participation Constraint

0( *, *) dj jT t T (j = 1, 2).

D1.3: Optimality of Sharing Rules

The optimal sharing contracts, θ* = {θ1*, θ2*}, form a Nash equilibrium, i.e.,

For any θj ≠ θj* that satisfies the theater’s participation constraint (D1.2),

Sj ({t0*(θj*, θ3-j*), θj*, θ3-j*) ≥ Sj ({t0*(θj, θ3-j*), θj, θ3-j*) (j = 1, 2).

Definition 2: Revenue-Sharing Contract

Suppose φj = θj2 / θj1 (j=1, 2). The revenue-sharing contract, θj, for studio j, is then called

D2.1: Sliding-Scale Contract if θj2 < θj1 (i.e., φj < 1),

D2.2: Increasing-Scale Contract if θj2 > θj1 (i.e., φj > 1),

D2.3: Flat-Scale Contract if θj2 = θj1 (i.e., φj = 1)

Definition 1 says that the set of optimal release times of the two studios for any given set of

sharing contracts forms a Nash equilibrium (D1.1), and that the optimal set of sharing contracts among

studios also forms a Nash equilibrium (D1.3). In other words, the equilibrium in sharing contracts is sub-

game-perfect. Definition 2 implies that lower values of φj correspond to steeper sliding-scale contracts

and that a studio’s contract can be expressed as θj = {θj1, φj θj1}.

We make the following additional assumptions before solving for the equilibrium.

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A1. Both movies are displayed profitably (absent channel coordination problems), i.e.,

0 0 3 ,0, 3 0,jt o t jo jMax t Max t t

where t3-j, 0 = 3 implies that the release time for movie (3-j) is after the last (i.e., second)

period (which can be used to calculate the total channel profit that will accrue when only

movie j is released).

A2. Profits to the theater from concession sales are not large enough to cover the theater’s

costs (or, profits from movie ticket sales matter to the theater), i.e.,

2 2 2 2

0 0 01 1 1 1

, .j j jt jj t j t

r D t t I t t C I t t

A3. The increase in the total channel profit from both movies j and (3-j) being displayed by

the theater is larger than the theater’s reservation profit level for movie j, i.e.,

00 0 3 ,0, 3 .j

dj t j jT t Max t t

Assumption (A1) is used to rule out the uninteresting case of only one movie being displayed in

equilibrium, and allows us to focus on the more interesting (and realistic) case of competing studios /

movies. Assumption (A2) reflects the institutional reality that theaters incur costs that are not easy to

overcome using concession sales only. For example, the annual financial statement of Cinemark Holdings,

Inc. (the third largest theater chain in the US and Canada) for fiscal year 2008 states that the concession

sales were $535 million, while its operating costs – even excluding movie rental costs and advertising

costs – were $699 million. Assumption (A3) ensures that the theater will not reject a profitable movie in

equilibrium.

MODEL IMPLICATIONS

Equilibrium Solution Procedure

The sub-game-perfect equilibrium of the proposed game-theoretic model is arrived at using

backward induction, as follows: first, taking the revenue-sharing contracts of both studios with the theater

as given, we solve for the equilibrium release times of both movies in the second stage of the sequential

game (STAGE 2 SOLUTION); next, we solve for the equilibrium revenue-sharing contracts of the first

stage, explicitly accounting for the effects of the chosen strategies on the second-stage equilibrium release

times (STAGE 1 SOLUTION).

To derive the STAGE 2 SOLUTION, we first derive each movie’s best-response release time

function given the competing studio’s release time. The Nash equilibrium will then be the point of

intersection of the two movies’ best-response release time functions.

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Whether studio j will prefer to release its movie in period 1 or 2, given the competing studio’s

release time t3-j, 0, depends on the following quantity.

3 ,0 3 ,0 3 ,0, , 1, , 2, .j j j j j j j j jt S t S t (14)

Studio j’s release time is then determined by the following best-response function. 10

3 ,0*0 3 ,0

3 ,0

1 if , 0,,

2 if , 0.

j j j

j j j

j j j

tt

t

(15)

Suppose we define δj (θj, t3-j, 0) = Δj (θj, t3-j, 0) / θj1, then the best-response function in equation (15) can be

re-written as follows (since the sign of Δj and δj are the same).

3 ,0*0 3 ,0

3 ,0

1 if , 0,,

2 if , 0.

j j j

j j j

j j j

tt

t

(16)

Using equations (8) and (9), one can derive the following expressions for δj (θj, t3-j, 0).

31

1 2 1 1 2 1 3 1 1 2 2 2

1( ,1) ,j j j

j j jj j

q qkq

q q k q q k q k q q k q k

(17)

and

1

1 1 1 2 2

( , 2) 1 .j j jj j

j j

q k q

q k q k q q k

(18)

Noting that the RHS of equations (17) and (18) involve φj, but not θj, one can re-write δj (θj, t3-j, 0) as δj (φj,

t3-j, 0) and equation (16), which represents movie j’s best-response release time function can appropriately

be re-written as follows.

3 ,0*0 3 ,0

3 ,0

1 if , 0,,

2 if , 0.

j j j

j j j

j j j

tt

t

(19)

Further, equations (17) and (18) can be re-written as follows.

31

1 2 1 1 2 1 3 1 1 2 2 2

1( ,1) ,j j j

j j jj j

q qkq

q q k q q k q k q q k q k

(20)

and

1

1 1 1 2 2

( , 2) 1 .j j jj j

j j

q k q

q k q k q q k

(21)

10 The first argument for τ*j0 (.,.) must technically be θ instead of θj (see footnote 6). However, in our model, it turns out that it is sufficient to know θj in order to determine τ*j0 (.,.).

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From the above equations (20) and (21), it is clear that both δj (φj, 1) and δj (φj, 2) are linear and

increasing in φj. If the slope is steep enough (i.e., if φj < φjl), both δj (φj, 1) and δj (φj, 2) will be negative,

which implies that movie j will be released in period 2 (regardless of when the competing movie (3-j) is

released). Similarly, for a sliding-scale contract whose slope is shallow enough (i.e., if φj > φju), both δj (φj,

1) and δj (φj, 2) will be positive, which implies that movie j will prefer to release in period 1, regardless

of when movie (3-j) is going to release. The threshold values, φjl and φju, can be obtained using the

following equations.

, 2 0j jl (22)

,1 0j ju (23)

Lemma 1. The threshold values satisfy the ordering φjl < φju < 1.

(Proof of Lemma 1 is available in Appendix A).

Proposition 1: Second-Stage Nash Equilibrium for Movie Release Times

1. If the revenue-sharing contract θj has a slope φj > φju, early (i.e., period 1) release is a dominant

strategy for studio j, i.e., τ*j0 (θj, t3-j, 0) = 1 for any t3-j, 0.

2. If the revenue-sharing contract θj has a slope φj < φjl, late (i.e., period 2) release is a dominant

strategy for studio j, i.e., τ*j0 (θj, t3-j, 0) = 2 for any t3-j, 0.

3. If the revenue-sharing contract θj has a slope φjl < φj < φju, studio j prefers early (i.e., period 1)

release only if studio (3-j) releases late (i.e., in period 2), i.e., τ*j0 (θj, 1) = 2 and τ*j0 (θj, 2) = 1.

(Proof of Proposition 1 is available in Appendix A).

The first item in Proposition 1 says that revenue-sharing contracts that do not decline fast enough

(i.e., slightly declining, flat or increasing) induce studios to release their movies early. This is because

demand in period 1 is not much more important to the studio (from a profit standpoint) than demand in

period 2. The studio, therefore, aims to stimulate demand for its movie in both periods and induces as

many consumers to watch its movie as possible by releasing early and keeping the movie available in the

theater screen for two periods.

The second item in Proposition 1, in contrast to the first, says that revenue-sharing contracts that

decline fast enough (i.e., steep decline) induce studios to release their movies late. This is because

demand in period 2 is much less important to the studio (from a profit standpoint) than demand in period

1. The studio, therefore, releases its movie for one period only (period 2) and additionally saves itself the

need to compensate the theater, with shared revenues, for two periods. This motivation for the studio to

delay its movie release becomes even more pronounced when the outside good option in period 2 is

relatively unattractive compared to that in period 1 (i.e., period 1 is “low season” and period 2 is “high

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season”). In other words, fast declining revenue-sharing contracts make delayed release even more likely

for studios that have finished movie production well before the high season. Such studios wait for the

high season before they release their movie.

The third item in Proposition 1 says that revenue-sharing contracts that decline, but not fast

enough (i.e., shallow decline) produce an asymmetric equilibrium in movie release times, where one

movie releases early (in period 1) while the other movie releases late (in period 2). One studio finds it

viable to open its movie early in order to insulate itself from competition from the other studio in the first

period; the other studio also does not simultaneously open its movie early because it benefits from having

consumers who have watched the competing studio’s movie earlier to watch its movie in the later period

(instead of fighting head-on with the competing studio for those consumers) since consumers are assumed

not to watch the same movie twice. This asymmetric equilibrium becomes stronger when the two movies

have different levels of attractiveness, with the movie of higher attractiveness releasing first, and the

movie of lower attractiveness releasing later. This effect is consistent with the movie release timing

strategies derived by Krider and Weinberg (1998) for movies of differing opening strengths. However, in

our case, such an asymmetric equilibrium will not always hold, with both movies instead simultaneously

releasing early or late, when the revenue-sharing contracts are either almost flat or declining steeply,

respectively. In this sense, our model generalizes the insights in Krider and Weinberg (1998) by

endogenizing both revenue-sharing contracts and movie release times of studios. In fact, the analysis of

Krider and Weinberg (1998), which takes the revenue-sharing contracts as exogenous, cannot rationalize

the simultaneous delay in release of both competing movies under any circumstances. Our model allows

for this possibility, in addition to the other release strategies discussed in Krider and Weinberg (1998).

The analysis of Radas and Shugan (1998), which is predicated on a monopolistic movie scenario, argues

for the optimality of early release for movies with either strong predicted openings or strong word of

mouth effects.

In practice one rarely finds competing movies simultaneously releasing early (i.e., immediately

after production) during a season. Instead, movies often wait for a few months after production is

complete before releasing in theaters. Proposition 1 allows for such time lag between movie production

and release in two out three cases (with both movies delaying release in one case, and one of the movies

delaying release in the other case). Furthermore, both of these cases also correspond to the observed

reality that each studio’s revenue-sharing contract with a theater is declining over time (of course, the

sub-game-perfect equilibrium aspect of such a strategy remains to be proved, which we discuss next).

Therefore, there is natural face validity to our model in terms of describing institutional realities in the

movie industry.

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Proposition 2: Movie attractiveness and release timing

Suppose that the outside option is weakly more attractive in period 2 (which can be thought of as

representing high season, as discussed in Radas and Shugan (1998)): 1 2.k k Then, it is easier to

induce the weaker movie to postpone: 0jl for 1,2j , and 1 2u u when 1 2q q . Additionally,

it is harder to induce movies to postpone in low season: ju is decreasing with 2k , and if 1 2k k k ,

then ju is decreasing with k .

(Proof of Proposition 2 is available in Appendix A).

According to Proposition 1, inducing late release requires a sharing rule with slope below ju . Thus,

Proposition 2 shows that our model is consistent with Krider and Weinberg (1998) in that less attractive

movies are more likely to postpone their release, and thus enjoy a shorter display time. Proposition 2 also

suggests that movies completed in low season will likely release in low season, while movies completed

in high season will likely postpone their release, especially if they are weak. We provide empirical

support for both implications in section 5.

Claim 1: Existence of Equilibrium with Sliding-Scale Contracts

There always exists a pure-strategy sub-game-perfect equilibrium, as defined in Definition 1, for our

game-theoretic model. While multiple equilibria are possible, there is always an equilibrium where both

studios have sliding-scale revenue-sharing contracts, i.e., φ*1 < 1 and φ*2 < 1.

(Proof of Claim 1 is available in Appendix A).

Claim 1 establishes that the sliding-scale revenue-sharing contract is an equilibrium choice (albeit not

unique) of a studio, regardless of the quality of its own movie, or that of its competing studio’s movie,

and whether the production and release periods are of low- or high-season.

Claim 2: Conditions for Lack of Existence of Equilibrium with Flat-Scale or Increasing-Scale Contracts

Equilibria involving flat-scale or increasing-scale revenue-sharing contracts are not possible when: (1+r)

δj (1, 1) – Cj1 < 0, (1+r) δj (1, 2) – Cj1 < 0, and δj (0, 2) < 0 for both movies (j=1,2).


Claim 2 establishes that the sliding-scale revenue-sharing contract is the only equilibrium choice of a

studio under some conditions.

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Claim 3: Conditions for Channel Profits Generated By a Movie Being Maximal

The channel (i.e., two studios plus theater) profits are maximized in equilibrium as long as either

concession profits are sufficiently high, or δj (0, 2) ≤ 0 and first period display costs Cj1 are sufficiently

high. Furthermore, when these conditions for maximum channel profit are satisfied, there exists an

equilibrium with sliding-scale revenue-sharing contracts that achieves this maximum.


Claim 3 establishes that the sliding-scale revenue-sharing contract could serve as a channel coordination

mechanism, by aligning the interests of the studio and the theater, when it comes to maximizing the

channel profits generated by a movie. It highlights the importance of the theater’s concession profits and

display costs in achieving such channel coordination. Even though the theater has no bargaining power in

the channel, the theater’s concession profits, through its influence on the theater’s participation constraint,

ends up influencing the equilibrium revenue-sharing contracts and, therefore, equilibrium release times of

the studios.

To summarize, two types of sliding-scale revenue-sharing contracts – (1) steep and (2) shallow -- lead to

improved profit outcomes for the studios and the theater (“win-win”) compared to the base case of flat

contracting, for the following reasons. Under the steep sliding-scale revenue-sharing contract, the

theater’s display costs greatly decrease, which increases the ability of both studios to demand a much

higher share of the box-office revenues during the release period. Under the shallow sliding-scale

revenue-sharing contract, one studio finds it viable to open its movie early in order to insulate itself from

competition from the other studio in one period; the other studio also does not simultaneously open its

movie early because it benefits from having consumers who have watched the competing studio’s movie

earlier to watch its movie in the later period (instead of fighting head-on with the competing studio for

those consumers); each studio’s share of box-office revenues during the release period goes down to

counter the theater’s increased display costs from screening in both periods. Therefore, a sliding-scale

contract arises as an equilibrium that not only improves the studio’s profits (compared to a flat-scale

contract), but also is incentive compatible with the release times / lengths of the movies that improve the

theater’s profits (by decreasing the theater’s display costs).

EMPIRICAL EVIDENCE

In order to ascertain the validity of our proposed theoretical model, we test some implications of

our theoretical model (presented in Proposition 2) using empirical data. For this purpose, we use

empirical data pertaining to an identified sample of 10 duopoly markets, each involving two competing

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movies of similar genre that finished production at roughly the same time (mimicking the assumptions

behind our theoretical model). We summarize the model implications, as well as the supporting empirical

evidence, below.

1. Model Implication: When the two competing movies are of dissimilar attractiveness, the higher

attractiveness movie would release in the first period, while the lower attractiveness movie would

release in the second period.

Empirical Support: We operationalize movie attractiveness using five proxy variables: (i) average

IMDB viewer rating for the movie, (ii) number of IMDB viewer votes for the movie, (iii) number of

external reviews for the movie, (iv) movie production budget, and (v) gross box-office revenues for

the movie. Using one proxy variable at a time, we compute the difference in movie attractiveness

between the two movies, referred to as ATTRACTDIFF. We operationalize the time delay in movie

release using the observed time lag between movie completion date and movie release date. We

compute the difference in this release time lag between the two movies, referred to as RELEASEDIF.

The correlation between ATTRACTDIF and RELEASEDIF turns out to be -0.77, -0.57, -0.52, -0.58

and -0.49, respectively, for the five proxy variables for ATTRACTDIF discussed above. Since these

correlations are all negative and large, they are directionally consistent with the model implication

that the more attractive movie is released sooner while the less attractive movie is released later.

2. Model Implication: When the two competing movies are of dissimilar attractiveness, the higher

attractiveness movie is displayed in both periods, while the lower attractiveness movie is displayed in

one period (i.e., the second period) only.

Empirical Support: We operationalize the difference in movie attractiveness between the two movies,

referred to as ATTRACTDIFF, in the same manner as in point 1. We operationalize the run length of

a movie using the number of days for which the movie showed positive box office receipts. We

compute the difference in the run length between the two movies, referred to as RUNDIF. The

correlation between ATTRACTDIF and RUNDIF turns out to be 0.73, 0.61, 0.51, 0.44 and 0.63,

respectively, for the five proxy variables for ATTRACTDIF discussed above. Since these correlations

are all positive and large, they are directionally consistent with the model implication that the more

attractive movie runs longer while the less attractive movie runs shorter.

3. Model Implication: Given that the first period is regular season, movies are more likely to delay their

release for the second period if the second period is high season (i.e., Summer, Christmas) than

otherwise.

Empirical Support: We separate the 10 duopoly movie markets in to two types: (i) those in which the

movies are released during the regular season, (ii) those in which the movies are released during high

season, referred to as REGSEASON and HISEASON, respectively. We then compute the average

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time lag between movie completion date and movie release date for movies within each type. These

averages turn out to be 108 days and 122 days for REGSEASON and HISEASON, respectively. This

is consistent with the model implication that there will be a time delay in movie release when a movie

is completed prior to the high season.

CONCLUSIONS

Our paper provides a theoretical explanation for the prevalence of the sliding-scale revenue

sharing contracts in the motion picture industry, and our analysis indicates that sliding scale contracts can

arise in equilibrium as a mechanism that can induce the optimal product release schedule that can (under

certain realistic conditions) achieve channel coordination even in the presence of competition. Using a

two-period game-theoretic model that involves two studios and a theater (which is assumed to have

sufficient screen capacity to display both movies in both periods), as well as utility-maximizing

consumers who choose between the available movies and the outside good in each period, we show that

under a flat-scale or increasing-scale contract, it is optimal for both studios to release their movies in

period 1. Under a sliding-scale contract, however, one of two sets of outcomes is optimal: (1) Steep

Sliding Scale: Both studios release their movies in period 2, or (2) Shallow Sliding Scale: One studio

releases its movie in period 1, while the other does in period 2. Both of these cases lead to improved profit

outcomes for the studios and the theater (“win-win”) compared to the base case of flat (or increasing)

contracting, for the following reasons. Under case (1), the theater’s display costs greatly decrease, which

increases the ability of both studios to demand a much higher share of the box-office revenues during the

release period. Under case (2), one studio finds it viable to open its movie early in order to insulate itself

from competition from the other studio (and this decreased competition also benefits the second studio

that releases its movie later); each studio’s share of box-office revenues during the release period goes

down to counter the theater’s increased display costs from screening in both periods. These findings may

have implications for other industries where it may be important to induce the producers to postpone the

product release, and suggest that sliding-scale contracts may achieve this goal.

We show that our key results are robust to the relaxation of various modeling assumptions (see

Appendix B). If consumers maximize the sum of their utilities over both periods, or have a positive

discount rate (non-myopic consumers), we show that the sliding-scale contracts still arise in equilibrium

and flat or increasing contracts do not (Proposition 1, and Claims 1 and 2 hold). Allowing for endogenous

closing times and longer time horizons, we find that the sliding-scale contracts arise in equilibrium

(Proposition 1 and Claim 1 hold), although the conditions under which there are no flat or increasing-

scale contracts need to be modified to incorporate the endogenous closing times (Claim 4 replaces Claim

2). Finally, we show that allowing for theater competition (i.e., more than one theater) would optimally

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induce the studio to offer the movie to only one of the competing theaters (a finding consistent with

business practice in the industry), and all of the Propositions and Claims would still hold.

We also find empirical support for some of the key implications of our theoretical model.

Specifically, our model predicts that (1) the more attractive movie is released sooner while the less

attractive movie is released later; (2) the higher attractiveness movie is displayed in both periods, while

the lower attractiveness movie is displayed in one period (i.e., the second period) only; (3) if the first

period is regular season, movies are more likely to delay their release for the second period if the second

period is high season than otherwise. We provide empirical support for all three theoretical predictions.

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Eliashberg, J., Elberse, A., Leenders, M. A. A. M. (2006). The Motion Picture Industry: Critical Issues in

Practice, Current Research, and New Research Directions. Marketing Science, 25, 6, 638-661.

Eliashberg, J., Sawhney, M. S. (1994). Modeling Goes to Hollywood: Predicting Individual Differences in

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Filson, D., Switzer, D., Besocke, P. (2005). At The Movies: The Economics Of Exhibition Contracts.

Economic Inquiry, 43, 2, 354-369.

Friedberg, A. A. (1992). The Theatrical Exhibitor. in Squire, J. E. (Ed.) The Movie Business Book, Simon &

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APPENDIX A: PROOFS

Proof of Lemma 1. From (20) and (21), it follows that 0)1,1( j and (1,2) 0j when 1j :

0))()((

)))((1

))(()1,1(

222113

231323

121

2

3

221

1

13121

2

121

kqkqqkq

qkqkqq

kqq

kq

q

kqq

k

kqkqq

q

kqq

q

jj

jjjj

j

j

j

jjj

1 1 2 2

(1, 2) 1 0j jj

j

q q

q k q q k

Because both ( ,1)j j and ( , 2)j j increase in j and are positive when 1j , we can infer that the

values of j that set these functions to zero are less than 1: , 1j l and , 1j u . Setting (21) equal to

zero to solve for ,j l and substituting the value into (8), we obtain

2 63 1 1 2 2 3 2 3 3 1

,1 3 1 2 1 2 1 1 2 2

( )( )( )( ,1) 0

( )( )( )( )j j j j j j

j j lj j

q q k q q k q k q q q k

k q k q k q q k q q k

Thus, it must also hold that , ,j l j u . Q.E.D.

Proof of Proposition 1. All three proposition statements follow from Lemma 1 applied to the best-

response strategies *, 3 ,( , )j o j j ot

given by (19). Specifically, for ,j j u , we have ( ,1) 0j j and

( , 2) 0j j , and therefore *, 3 ,( , ) 1j o j j ot for any ojt ,3 . Similarly, for ,j j l , we have

( ,1) 0j j and ( , 2) 0j j ; and, therefore, *, 3 ,( , ) 2j o j j ot for any ojt ,3 . Finally, for

, ,j l j j u , we have ( ,1) 0j j and ( , 2) 0j j , and therefore *, ( ,1) 2j o j and

*, ( , 2) 1j o j . Q.E.D.

Proof of Proposition 2. Suppose 1 2k k . From (15), we have 0jl . From (14), if 1 2q q , then we

also have

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2

21 2 21 2

12 1 2 1 1 1 2 2 1 2 2

1 2 21 2

2 1 1 2 1 2 1 2 1 2 2

1 2 21 2

1 1 1 1 2 2 1 1 1 2 2

2

1

1

u

u

q q q kk k

q k q k k q k q q q k

q q kk k

q k k q q k k q q q k

q q kk k

q k k q q k k q q q k

where the inequality is obtained by observing that 1 2k k and thus replacing 2q with a smaller 1q

decreases both parts of the expression. The above expressions for ju are also decreasing with 2k , and

when 1 2k k k , they are also decreasing with k

Proof of Claim 1. To prove the claim, we are going to construct an equilibrium with declining-scale

sharing rules for both studios. We start by specifying release strategies and sharing rules (both of which,

naturally, depend on model parameters), and then we show that the specified release strategies and

sharing rules form an equilibrium. Because our goal is to specify equilibrium strategies, we find it

convenient to use starred variable names for these strategies from the moment they are defined, even

though the proof that they form an equilibrium is offered a few lines below.

The way we specify release strategies and sharing rules will depend on the signs of the following two

expressions:

1(1 ) (1,1)j jr C (24)

1(1 ) (1,2)j jr C (25)

To interpret the above two expressions, note first that, although the function 3 ,( , )j j j ot was defined as

3 ,0 3 ,0 1( ,{1, }) ( ,{2, }) /j j j j j j jS t S t , it can be equivalently defined as

3 ,0 3 ,0{1, },{1, } {1, },{2, }j j j j j jS t S t because 1j in the denominator cancels out with the 1j

in the numerator. Thus, 3 ,0(1, )j jt represents the change in the studio’s revenue obtained when the

movie is released in period 1 instead of period 2 when the studio receives 100% of the revenues in both

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periods; or equivalently, the change in total box office revenue generated by movie j . Thus, expressions

(24) and (25) represent the change in the total channel revenue generated by movie j , which consists of

the change in the box office revenue, the corresponding change in the concession stand receipts, and the

change in display costs (which is the display cost 1jC for movie j in period 1). Expression (24) assumes

that the competing movie j3 is released in period 1, while expression (25) assumes that movie j3

is released in period 2.

The remainder of the proof consists of the following three parts. In part (I), we specify sharing

rules and release strategies for all possible parameter combinations, which we group into three cases; in

part (II) we prove that the specified rules and strategies are feasible, and in part (III) we prove that they

form an equilibrium.

(I) Sharing rule and release strategy specification.

(a) If either of the following two conditions hold

(a.1) expression (24) is nonnegative for both 1j and 2j (neither movie can generate higher

profits by postponing), or

(a.2) expression (24) is negative for either one or both movies, and, for the movie (or movies) j for

which (24) is negative, we have (0,1) 0j (the movies that generate higher profits by postponing

cannot find a sharing rule that will make late release incentive-compatible);

then we can specify the release strategies and sharing rules as follows. For

1 2*1 *

(1 ) (1;{1,1}) (2;{1,1})

(1;{1,1}) (2;{1,1})

dj j j j j

jj j j

r D D T C C

D D

; (26)

and pick * ( ,1)j ju that is sufficiently high to ensure that *1j given by (26) is below 1

(feasibility is shown below in part (II) of the proof). Next, let *0 ( ) 1j jt if 0),1( jj and

*0 ( ) 2j jt if 0),1( jj . Note that, according to Proposition 1, the above specification of the

sharing rule implies that * * * *10 20 2( ) ( ) 1jt t .

(b) If either of the following two conditions hold

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(b.1) expression (24) is negative for movie j but not for the movie j3 , and 3 (0,1) 0j

(studio j can generate higher profit by postponing and finds it incentive-compatible to postpone,

while studio j3 cannot generate higher profit by postponing), or

(b.2) expression (24) is negative for both movies, (0,1) 0j , and (i) 3 (0,1) 0j , or (ii)

2 (0,1) 0 and (25) is nonnegative for movie j3 , or (iii) 2 (0,1) 0 , (25) is negative for both

movies, and 3 (0,2) 0j (both movies could generate higher channel profit by postponing but

only movie j postpones because, for the competitor, postponing at the same time is either not

incentive-compatible or generates lower channel profit);

then we can specify the release strategies and sharing rules as follows. Let *3 1j be given by (26),

pick the slope *3 3( ,1)j jl that is sufficiently high to ensure that *

3 j given by (26) is below 1

(feasibility is shown below in part (II)), let

0 3 ,0 2*1

0 3 ,0

(1 ) (2;{ 2, 1})

(2;{ 2, 1})

dj j j j j

jj j j

r D t t T C

D t t

; (27)

let *j take an arbitrary value in (0, )ju , and let *

, ( ) 1j o jt if (1, ) 0j j , *, ( ) 2j o jt if

(1, ) 0j j , *3 , 3( ) 1j o jt if 3 3(2, ) 0j j , *

3 , 3( ) 2j o jt if 3 3(2, ) 0j j , Note

that, with this specification, we have * *, ( ) 2j o jt and * *

3 , 3( ) 1j o jt .

(c) If both (24) and (25) are negative for both movies, and we also have 1(0,2) 0 and

2 (0,2) 0 (both movies can generate higher channel profits and find it incentive-compatible to

postpone) we can specify the release strategies and sharing rules as follows. Let

2 2*1

2

(1 ) (2;{2, 2})

(2;{2, 2})

dj j

j

r D T C

D

; (28)

pick * (0, )j jl (feasibility is shown in part (II)). and let *, ( ) 1j o jt if 0),2( jj and

*, ( ) 2j o jt if 0),2( jj . Note that, with this specification, we have * * * *

1, 1 2, 2( ) ( ) 2o ot t .

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(II) Feasibility. To show feasibility of the release strategies and sharing rules defined in (a) – (c) we

need to prove that, in all of the cases, *1 [0,1]j and * 0j , 1, 2j . Because we aim to construct an

equilibrium with declining scale sharing rules, we also need to show that * 1j , 1, 2j . For each of

the three cases we next show that *1 1j and *0 1j . Then, we show that we also have *

1 0j in

all three cases.

In case (a), (26) can be rearranged as:

1 2*1 * *

(1;{1,1}) (2;{1,1}) (1;{1,1}) (2;{1,1})

(1;{1,1}) (2;{1,1}) (1;{1,1}) (2;{1,1})

dj j j j j j j

j

j j j j j j

D D r D D T C C

D D D D

In the above, when * 1j , the first fraction equals to 1, while the second fraction has a negative

numerator due to assumption A2 and a positive denominator. Therefore, the right hand side of the above

expression is strictly smaller than 1 at * 1j . Because the right hand side of the above expression is

continuous in *j , we can find * ( ,1)j ju that is sufficiently close to 1 so that, when used in (26),

produces *1 1j (interval ( ,1)ju is nonempty because, from Proposition 1, 1, uj ).

In case (b), *3 j

and *3 1j are obtained using the same expressions as in case (a), and are, therefore

feasible. To show that *j and *

1 are feasible, we rearrange (27) as

, 3 , 2*1

, 3 ,

(2;{ 2, 1})1 ,

(2;{ 2, 1})

dj j o j o j j

jj j o j o

rD t t T C

D t t

which is less than 1 due to assumption A2. We can pick a feasible * (0, )j ju as long as the interval

(0, )ju is nonempty, which can be shown as follows. By definition, ( ,1) 0j ju , and ( ,1)j j is

increasing in j . Additionally, in case (b), we have (0,1) 0j . Thus, 0ju .

Finally, in case (c), similar logic shows that *1 1j and *0 1j .

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To show feasibility, it remains to verify that *1 0j in all three cases. In all of the three expressions,

(26), (27) and (28), used to define *1j we have a positive denominator. Thus, we need to show that the

numerator is also positive. Assumption A3 states that

3 ,00 3 ,0 0( ) max ({ , 3})

j

dj j j

tT t t t

Set 0t in the first term of the right hand side in the above inequality equal to * * * * *10 1 20 2{ ( ), ( )}ot t t , as

specified in each of the three cases. Let jt 3̂ denote the release time that maximizes the second term:

3 3 ,0 0ˆ arg max ({ , 3})j j jt t t . Substituting for profit from (13), we can rewrite the above as

* * *1 0 1 10 2 0 2 20

3 3 ,0 0 3 , 3 ,0

* *0 0

1 , 1 ,

ˆ ˆ1 ,{ , 3}

1 ,

dj t t

t t

j j j j t jt

j jt jt

T r D t t C I t t r D t t C I t t

r D t t t C I t t

r D t t C I t t

The second inequality holds because, the profit generated by movie j3 when shown alone cannot be

below the profit generated by movie j3 when it is shown along with the competing movie j . The

above implies that *1 0j in all three cases.

(III). No incentives to deviate. We next show that * * *0 10 1 20 2( ) ( ( ), ( ))t t t and the declining scale

sharing rules ),( *2

*1

* , defined above, form an equilibrium. By construction, the theater’s

participation constraint is satisfied ((26), (27), and (28) are obtained from the theater’s participation

constraint). Additionally, studio j finds offering sharing rule *j optimal: studio j cannot gain from

deviation because the theater will not accept a different sharing rule that leaves the theater with less profit,

and the total profit generated by movie j cannot be increased further, keeping the strategy of studio

j3 fixed. To see why this is the case, note that, the change in profit from deviation can be evaluated

using (24) or (25) for movies 1 and 2 respectively. By construction, this change is negative in cases (a.1),

(b.1), (b.2)(ii), and (c). In the remaining cases, if the change is positive for movie j , the alternative

opening time is not incentive-compatible. Consider, for example, case (a.2) where 2 (0,1) 0 and we

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have *20 1t . Although deviation to 20 2t would increase the profit generated by movie 2, positivity of

2 (0,1) implies that 0ju , which, according to Proposition 1, in turn implies that there does not exist

a feasible sharing rule that is compatible with movie 2 opening in period 2. Thus, release strategies

*0 ( )t , together with declining scale sharing rules * form an equilibrium. Q.E.D.

Proof of Claim 2. The result follows from the proof of Claim 1. The conditions of Claim 2 fall under

case (c) in the proof of Claim 1. In this case, the equilibrium release time for both movies is period 2

(otherwise, either studio would have an incentive and ability to deviate). Q.E.D.

Proof of Claim 3. Let * * *0 10 20( , )t t t be the set of opening times that maximizes 0( )t . To prove the

claim, we will analyze under what conditions there exists an equilibrium where the release time strategies

*0 ( )j jt and sharing rules * * * *

1( , )j j j j satisfy: * * *0 0( )j j jt t . We will consider the following three

cases: (a) * *10 20 1t t , (b) * *

0 3 ,01; 2j jt t , and (c) * *10 20 2t t . In case (b), without loss of

generality, we will set 1j to ease exposition.

(a) Suppose * *10 20 1t t . In this case, if there is a channel-coordinating equilibrium, it has the

following properties. For 2,1j , the equilibrium slopes satisfy * ( ,1)j ju ; from theater’s

participation constraint, *1j is given by (26). Because feasibility of these values is proven in Claim 1, we

only need to verify that neither studio wishes to deviate from the specified strategies. We show below

that neither studio wishes to deviate if r is sufficiently high.

(b) Suppose next that * *0 3 ,01; 2j jt t (note that this case does not happen if concession profits are

sufficiently high). If 2 (0,1) 0 , then channel coordination fails because it is impossible to find a

sharing rule that is incentive-compatible with *20 2t . Suppose next that 2 (0,1) 0 (which holds if

(0,1) 0j ). Then, in the channel coordinating equilibrium, we have: *1 1,l , *

2 2(0, )u , *1 is

given by (26), and *2 is given by (27). As shown in the proof of Claim 1, these values are positive and

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bounded by 1, and therefore feasible. Thus, it remains to verify that neither studio wishes to deviate from

the specified strategies. We show below that neither studio wishes to deviate if r is sufficiently high.

(c) Finally, suppose that * *10 20 2t t (note that this case does not happen if concession profits are

sufficiently high). If (0,2) 0j for at least one of the movies, then channel coordination fails because

it is impossible to find a sharing rule that is incentive-compatible with *0 2jt . Suppose next that

(0,2) 0j for both movies. If there is a channel coordinating equilibrium, then * (0, )j jl and *1j

is given by (28) for 2,1j . As in Claim 1, these values are positive and bounded by 1, and therefore

feasible. Thus, it remains to verify that neither studio wishes to deviate from the specified strategies. We

show below that neither studio wishes to deviate if r is sufficiently high.

In all three cases: (a), (b), and (c), studio j wishes to deviate if:

* * *0 3 ,0 0

* * *0 3 0 0

0 (1 ) ( ,{3 , }) (3 )

(1 ) ( ,{ , }) ( )

dj j j jt j j

t

dj j j jt j j

t

r D t t t C I t t T

r D t t t C I t t T

, (29)

where *0t maximizes 0( )t . The definition of *

0t also implies that:

* * * *0 3 ,0 ,0 3 ,0

* * *0 3 ,0 0

* * *3 0 3 ,0 3 , 3 ,0

* * *0 3 ,0 0

0 ({ , }) ({3 , )

(1 ) ( ,{ , }) ( )

(1 ) ( ,{ , }) ( )

(1 ) ( ,{3 , }) (3 )

j j j j

j j j jt jt

j j j j t jt

j j j jt jt

t t t t

r D t t t C I t t

r D t t t C I t t

r D t t t C I t t

* * *3 0 3 ,0 3 , 3 0(1 ) ( ,{3 , }) ( ))j j j j t j

t

r D t t t C I t t

Using the above, we can rewrite (29) as:

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* * * *0 3 ,0 0 3 ,0

* *1 0 1 0

* * * *3 0 3 ,0 3 0 3 ,0

0 (1 ) ( ,{3 , }) ( ,{ , })

( 1) ( ( 2)

(1 ) ( ,{ , }) ( ,{3 , })

j j j j j jt

j j j j

j j j j j jt

r D t t t D t t t

C I t C I t

r D t t t D t t t

(30)

When *0 1jt , the first inequality is violated for high enough r , while the second inequality is violated

for high enough 1jC . Similarly, when *0 2jt , the second inequality is violated for high enough r ,

while the first inequality is violated for high enough 1jC . Studio j wishes to deviate as long as at least

one of the inequalities is violated.

To complete the proof, note that, in all cases, if there is a channel coordinating equilibrium, then there

is a channel-coordinating equilibrium where all studios have declining scale sharing rules. Q.E.D.

APPENDIX B: EXTENSIONS

NON-MYOPIC CONSUMERS

Suppose consumers discount future period utility by , 10 . This is a generalization of our base

model, which assumes 0 . In an extreme case with no discounting ( 1 ), each consumer i makes

consumption choices for both periods that maximize the sum of the utilities in the two periods. When

0 , equations (1) and (2), central to our derivations, correctly describe movie demand only in period

2 (as given by (2)). Nevertheless, Proposition 1 and Claims 1 and 2 still hold, as we show next.

For any , a consumer who finds it optimal to watch movie j if it is released in period 2, would still

find it optimal to watch movie j if it is released in period 1.1 Because the reverse is not true with a

positive probability, releasing movie j in period 1 results in a strictly larger total demand for movie j

than releasing the movie in period 2:

0 3 ,0 0 3 ,0 0 3 ,0(1,{ 1, }) (2,{ 1, }) (2,{ 2, }j j j j j j j j jD t t D t t D t t .

If the sharing rule of studio j 1 1( , )j j j j is with 1j , then, from the above inequality, studio j

1 This can be formally shown by contraposition. Suppose that, when movie j is released in period 1, the consumer

prefers to purchase either the outside option or the competing movie in each of the two periods. Because this

consumption choice is also available when movie j is released in period 2, thus consumer must prefer this choice

to any alternative that involves watching movie j . This contradicts the assumption that consumer chooses to

what movie j if it is released in period 2.

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finds it optimal to release its movie in period 1. Thus, as stated in Proposition 1, there exist 1jl and

1ju such that, studio j ’s best response to 3 ,0 2jt is to release in period 2 if and only if j jl ;

and similarly, studio j ’s best response to 3 ,0 1jt is to release in period 2 if and only if j ju

(formally, *3 ,0( ; ) 2j j jt if and only if either j jl and 3 ,0 2jt or j ju and 3 ,0 1jt ).

Unlike for the main model, for this extension we do not obtain, in general, that jl ju . However, this

inequality is not used in the proofs of Claims 1 and 2.

In this extension, Claims 1 and 2 still hold (their proofs go through) because the proofs rely on the shape

of the demand functions only through the results of Proposition 1. Since the relevant results of

Proposition 1 hold in this extension, Claims 1 and 2 also hold.

LONGER HORIZONS AND ENDOGENOUS CLOSING TIMES

The assumption that the world in our model exists only for two periods may appear as a limitation.

However, the model readily extends to longer time horizons by observing that we can interpret the second

period in our model as the present value of all the subsequent periods.2 With additional periods, however,

our assumption that all movies are displayed until the final period (exogenous closing times) becomes less

reasonable. Thus, we next consider an extension where closing times are instead determined

endogenously. This extension is particularly interesting because, with endogenous closing times,

releasing the movie early does not guarantee a longer display time, and postponing the release is no

longer equivalent to a shorter display time. Thus, in this version, it is possible that, absent incentives

from sharing rules, a studio would prefer a late release while the theater prefers an early release (if, for

example, both demand and display costs are high in period 2). As we show in this section, our main

results (Proposition 1 and Claim 1) are robust to this extension. While the statement of Claim 2 requires

some modification to take closing times into account, the basic message remains the same: while

increasing sharing rules are not necessary to support an equilibrium (according to Claim 1), there are

parameter values for which all equilibria have a decreasing scale sharing rule for either one or both of the

studios.

In practice, theaters typically have an option of closing a movie once it starts generating less than a

pre-specified amount of weekly revenue. Thus, it is reasonable to assume that the theater chooses closing

times for all movies to maximize its profit. Let 0jc jt t be the closing time for movie j that opens at

time 0jt (so if 0jc jt t t , movie j is displayed in period t only, while if 0jc jt t , then we have

2 An extension to continuous time is considerably less tractable, but produces similar qualitative results.

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02; 1jc jt t , and therefore movie j is displayed in both periods). Endogeneity of closing times does

not impact the movie demand in period 1: if the movie opens in period 2, period 1 demand is zero, as

before; if the movie opens in period 1, period 1 demand is a function of the contemporaneous

competition, and does not depend on whether the movie is closed in the same or later period. Movie

demand in period 2, on the other hand, depends on the closing time for movies released in period 1 (if

released and closed in period 1, demand in period 2 is zero; if released in period 1 but closed in period 2,

demand in period 2 is determined as before, by the strength of contemporaneous competition). Thus, the

expressions for demand for movie j in periods 1 and 2 in this version can be stated as follows.

00

1 10 2 20 1

( 1)(1; )

( 1) ( 1)j j

j

q I tD t

q I t q I t k

(31)

0 1 0 2 01 1 2 2 2

3 02

( 2)(2; , ) 1 (1; ) (1; )

( 2) ( 2)

( 2)(1; )

( 2)

j jcj c

c c

j jcj

j jc

q I tD t t D t D t

q I t q I t k

q I tD t

q I t k

(32)

The profit 0( , , )j c jS t t of studio j is

0 0 0 0 0( , , ) (1) 1; ( ) (3 ) 2; ,j c j j j j j j j cS t t D t I t t t D t t , (33)

and the profit ),,( jcoj ttT of the theater from movie j is

0 1 0

0 0 2

( , , ) (1 (1) ) 1; ( 1)

(1 (3 ) ) 2; , ( 2)

j c j j j o j j

j j j c j jc

T t t r D t C I t

t r D t t C I t

.

After the theater and studios agree on sharing rules, and studios choose their release times, the theater can

choose closing times.

Introduction of closing time merits some additional discussion. For movies released in period 2, the

closing time is automatically set to period 2 because this is the last period in the model. Thus, the choice

of closing time is interesting only for movies released in period 1. Because the theater is assumed to

control the closing time, we can conclude that movie j that opens in period 1 is still displayed in period 2

if and only if

0 1 0 3 , 2

0 1

0 (1 (1) ) (1; ) (1 (2) ) 2; ,{2, }

(1 (1) ) (1; )

j j j j j j c j

j j j

r D t C r D t t C

r D t C

,

which can be expressed as:

2 0 3 ,(1 (2) ) 2; ,{2, }j j j j cC r D t t . (34)

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Time consistency of sharing rules imposes restrictions on equilibrium sharing rules when closing

times are endogenous, as we discuss next. In particular, if a movie released in period 1 generates

sufficient revenue in period 2 to cover the display costs, then time consistency requires the equilibrium

sharing rule *j to leave the theater with a sufficient portion of the revenue to justify late closing.

Formally, if

2 0 3 ,(1 ) 2; ,{2, }j j j cC r D t t , (35)

then (34) should hold for the equilibrium strategies *j , *

0t , and *,3 cjt . Otherwise, the theater and the

studio would find it beneficial to renegotiate to a new period 2 sharing rule that would give the theater a

sufficient portion of revenue to induce late closing, and would thus leave the studio with the remaining

(positive) portion of the revenue.

Moreover, time consistency on off-equilibrium paths requires that we allow the studio to renegotiate

its equilibrium sharing rule if the competing studio switches to a different opening time. This is an

equilibrium refinement that is not crucial for our results, as all of the analysis would go through under the

assumption that sharing rules cannot change on off-equilibrium paths. However, making a specific

assumption simplifies the discussion, and letting studios renegotiate as described above is particularly

attractive because it reflects the practice of adjusting sharing rules on the weekly basis and insuring that

the theater receives a minimum payment (sometimes referred to as house nut) that covers the operating

expenses.

Given the above discussion, we define the equilibrium in this version as follows.

Definition 3. (Equilibrium with Endogenous Closing Times)

An equilibrium is a set of period 1 sharing rules, ))1(),1(()1( *2

*1

* , period 2 sharing rule strategies

* * *0 1 1 0 2 2 0(2, ) ( (2; (1), ), (2; (1), ))t t t θ θ θ , release strategies * * *

0 1,0 1 2,0 2( ) { ( ), ( )}t t t , and closing

strategies * * *0 1, 0 2 0( , ) { ( , (2)), ( , (2))}c c ct t t t t t that satisfy

D3.1. (Incentive Compatibility of Release Times) For any set of sharing rules ),( 21 ,

release strategies * * *0 1,0 1 2,0 2( ) { ( ), ( )}t t t form a Nash equilibrium: for *

0 0 ( )j j jt t ,

* * * * * *0 0 , 3 ,0 3 ,0 3 ,0 3

ˆ ˆ ˆ ˆ( ( ), ( ( ), ), ) ({ , ( )}, ({ , ( )}, ), )j jc j j j o j j jc j j j jS t t t S t t t t t

where 1, 2,j

}ˆ,ˆ{ˆ21 , *

0ˆ ( (1),θ (2; (1), ))j j j j t ,

D3.2. (Theater’s Participation Constraint) For 1, 2,j

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* * *0( , , ) d

j c jT t t T

D3.3. (Optimality of Sharing Rules) Studio 2,1j cannot benefit by deviating to a different sharing

rule: for any *j j that satisfies the theater’s participation constraint,

* * * * * * * * *0 0 0 0( ( ), ( ( ), ), ) ( ( ), ( ( ), ), )j c j j c jS t t t S t t t

Where ),( *3

**jj ,

* * * * *

0( (1), (2; (1), ))j j j j jt θ , and ),( *3 jj .

D3.4 (Incentive Compatibility of Closing Times). For any set of release times 0 1,0 2,0( , )t t t and

sharing rules ),( 21 , closing strategy *0t ( , )c t maximizes the theater’s profit from movie

j :

* * *0 0 3 , 0 0 3 , 0( ,{t ( , ), t ( , )}, ) ( ,{ , t ( , )}, )j jc j c j j jc j c jT t t t T t t t .

D3.5 (Time consistency of sharing rules) For any set of period 1 sharing rules ))1(),1(()1( 21

and release times 0 1,0 2,0( , )t t t , period 2 sharing rule strategies

* * *0 1 1 0 2 2 0(2, ) ( (2; (1), ), (2; (1), ))t t t θ θ θ form Nash equilibrium:

* * *0 0 0 0

* * *0 0 3 , 0 3 3 0

( , ( , (2; (1), )), ( (1), (2; (1), )))

({ ,{ ( , (2)), ( , (2; (1), ))},( (1), (2)))

j c j j j

j jc j j c j j j j

S t t t t t

S t t t t t

θ θ

t θ

Subject to the theater’s participation constraint:

* * *0 0 0 0

* * *0 0 3 , 0 3 3 0

( , ( ,θ (2; (1), )), ( (1),θ (2; (1), )))

( ,{ ( , (2)), ( , (2; (1), ))}, ( (1), (2)))

j c j j j j j

j jc j j c j j j j

T t t t t t

T t t t t t t

θ.

Time consistency requires that, if movie j is released in period 1 and generates sufficient period 2

demand, ((35) holds in equilibrium), then the equilibrium sharing rule in period 2 ensures that movie j is

closed in period 2. Thus, the equilibrium sharing rule satisfies the inequality (34), which can be restated

as: * * * max * *1 3 ,0 3 ,(2) ( , )j j j j j j ct t , where

2max * *

3 ,0 3 , * *3 ,0 3 ,

( , ) 12;{1, },{2, }

jj j j c

j j j c

Ct t r

D t t

.

Otherwise, the equilibrium places no restrictions on the value of )2(*j . Thus, the equilibrium closing

time for movie j released in period 1 is period 2 if (35) holds in equilibrium, and period 1 otherwise.

The above analysis also suggests that, when closing times are endogenous, Proposition 1 holds for

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sharing rules that satisfy max1 3 ,0 3 ,( , )j j j j c jt t because they induce late closing (assumed in the

main model version) for movies open in period 1. Because time consistency implies that the equilibrium

sharing rules satisfy this inequality when the second period demand is large enough, and the slope of the

sharing rule is not relevant when the second period demand is below display costs, Claim 1 still holds in

this version, as we show next.

Let 3 ,0 3 ,( , )j j j ct t be defined as follows:

* *3 ,0 3 ,0 3 ,0 3 ,0 3 ,0

*3 ,0 3 ,0

( ,1) (1;{1, }, τ ({1, })) (2;{1, }, τ ({1, }))

(2;{1, }, τ ({2, }))

j j j j c j j j c j

j j c j

t D t t D t t

D t t

,

where optimal closing time * * *0 1 1,0 2,0 2 1,0 2,0τ ( ) (τ ({ , }), τ ({ , }))c c ct t t t t can be described as follows:

0

0 3 ,0 3 ,

1,0 2,0 2*

(2,2) if, for 1,2, either 2,

or 1 and (23) holds for movie given and 2 ;

(1,2) if 1, (23) is violated for movie 1 given and 2, anτ ( )

j

j j j c

cc o

j t

t j t t

t t tt

2,0

1,0 1

1,0 2,0 3 ,0 3 ,

d either 1

or (23) holds for movie 2 given 1;

(1,1) if, 1 and (23) is violated for both movies given 1;

(2,1) otherwise.

c

j j c

t

t t

t t t t

Note that this definition of 3 ,0 3 ,( , )j j j ct t coincides with the definition of 3 ,0(1, )j jt in the main

model if *3 ,0τ ({1, }) 2jc jt . Substituting this new definition of 3 ,0 3 ,( , )j j j ct t for 3 ,0(1, )j jt , and

using equilibrium closing times given by * * *1 1,0 2,0 2 1,0 2,0τ ( ) (τ ({ , }), τ ({ , }))c o c ct t t t t , the rest of the proof of

Claim 1 applies (with the appropriate modifications for demand, as given by (31) and (32), and thus the

statement of Claim 1 holds.

The statement of Claim 2 does not directly carry over to this version of the model. According to

Claim 2, studios that find late release optimal choose declining scale sharing rules. In this version,

because closing time is endogenous, late release may also be incentive compatible with flat or increasing

sharing rules when early release means early closing because period 2 demand does not cover display

costs. In this case, the shape of the sharing rule is not relevant, and the opening time is determined only

based on the relative size of the demand in period 1 and period 2. On the other hand, because time

consistency imposes an upper limit on the equilibrium sharing rule in period 2, in this version, declining

scale sharing rules may appear in all equilibria even if late release is not attractive. Thus, we can modify

the statement of Claim 2 as follows.

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Claim 4. There is no equilibrium where studio j chooses a flat or increasing scale sharing rule if

0 2(1 ) (2; , )j c jr D t t C for any 0t and ct (i.e. (35) holds for movie j given any opening and

closing times) and either

(a) 1(1 ) (1,1) 0j jr C and (0,1) 0j , or

(b) 1 2

ˆ ˆ(2;{1,1}, ) (2;{1,1}, )

ˆ ˆ(1 ) (2;{1,1}, ) (1 ) (2;{1,1}, )j c j c

dj c j j j c j

D t D t

r D t T C r D t C

,

where 3 ,ˆ { 2, 1}c jc j ct t t .

Moreover, if the above holds for both movies, and, in (a), we also have 1(0, 2) 0 and 2 (0, 2) 0 ,

then there is no equilibrium where either of the two studios chooses a flat or increasing scale sharing rule.

In the above statement, the additional condition that (35) holds reduces the parameter space, but the

additional case (b) increases the parameter space where declining scale sharing rules strictly dominate flat

or increasing sharing rules (as compared to the main model).

THEATER COMPETITION

Our choice of model setup is motivated by the existing practice where studios negotiate contract terms for

each movie separately with each individual movie theater (even if the theater belongs to a chain; see

Vogel (2004)). In the interest of reducing competition between exhibitors, movie releases are often

scheduled so that geographically close exhibitors do not display the same movies at the same time (see

Vogel (2004)). This practice is consistent with the implications obtained by extending our model to

multiple theaters, as we discuss next.

Competition between theaters can be introduced into our model as follows. Suppose there are two

exhibitors, 2,1e that are perfect substitutes from the moviegoers’ point of view: when a given movie is

displayed in both theaters simultaneously, each theater receives one half of the demand for the movie.

Consistent with practice, the equilibrium in this version of the game has each movie j released only in

one of the two theaters. This is a standard result in the channel literature when retailers are perfect

substitutes (see, for example, Shaffer (1991), Martimort (1996)). The intuition for this result in our model

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can be presented as follows. Let )(, tej be the sharing rule between studio j and theater e in period t .

For studio j , it is rational to display its movie in both theaters 2,1e in period t if and only if

)()( 2,1, tt jj . However, if )()( 3,, tt ejej and movie j is displayed in both theaters in period t ,

both theater 1 and studio j can benefit by renegotiating and choosing a new sharing rule to be

)()(ˆ1,1, tt jj where 0 is small. This new sharing rule induces studio j to release its movie

only in theater 1, thereby almost doubling the profit of theater 1, and increasing the profit of studio j by

2 . Thus, in equilibrium, movie j cannot be displayed in two theaters simultaneously, implying that our

results presented in section 4 are robust to theater competition.

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