The present note first provides an alternative formulation of the Cancian, Bills and Bergström (1995)-problem which discards the non-existence difficulty and consequently allows to consider some extensions of the TV-newscast scheduling game.The extension we consider consists in assuming that viewers’preferences between the competing channels do not depend only on the timing of their broadcast, but also on some other characteristics, like the content of the show or the identity of the newscaster. Then we identify a sufficient condition on the dispersion of these preferences over the viewers’population guaranteeing the existence of a unique Nash equilibrium. It turns out that, at this equilibrium, In an interesting contribution, Cancian, Bills and Bergström (1995) (henceforth CBB) have exhibited some technical difficulties which may occur when TV-broadcasters compete in program scheduling. These authors consider the problem of scheduling evening television newscasts: two TV-networks hav

Didier Laussel

Jean Gabszewicz

Jean J. Gabszewicz

Nathalie Sonnac

Université Catholique De Louvain

CiteSeerX

2008/44
The TV News Scheduling Game When the
Newscaster\'s Face Matters.
Jean  GABSZEWICZ
Didier  LAUSSEL
Nathalie  SONNACThe TV news scheduling game when
the newscaster's face matters
Jean J. Gabszewicz
CORE, Universit e catholique de Louvain
Didier Laussel
Universit e de la M editerrann ee
Nathalie Sonnac
Universit e Paris II
May 2008Abstract
The present note rst provides an alternative formulation of the Cancian, Bills
and Bergstr om (1995)-problem which discards the non-existence diculty and
consequently allows to consider some extensions of the TV-newscast scheduling
game.The extension we consider consists in assuming that viewers'preferences
between the competing channels do not depend only on the timing of their
broadcast, but also on some other characteristics , like the content of the show
or the identity of the newscaster. Then we identify a sucient condition on
the dispersion of these preferences over the viewers'population guaranteeing the
existence of a unique Nash equilibrium. It turns out that, at this equilibrium,
both networks broadcast their news at the same instant.1 Introduction
In an interesting contribution, Cancian, Bills and Bergstr om (1995) (hence-
forth CBB) have exhibited some technical diculties which may occur when
TV-broadcasters compete in program scheduling. These authors consider the
problem of scheduling evening television newscasts : two TV-networks have
each to decide non cooperatively at what time to broadcast their show so as to
maximise audience size. Assuming that viewers want to watch the news as soon
as they get home from work, the authors show that the resulting game has no
pure strategy Nash equilibrium1. The present note rst provides an alternative
formulation of the CBB-problem which discards the non-existence diculty and
consequently allows to consider some extensions of the TV-newscast scheduling
game.The extension we consider consists in assuming that viewers'preferences
between the competing channels do not depend only on the timing of their
broadcast, but also on some other characteristics , like the content of the show
or the identity of the newscaster. Then we identify a sucient condition on
the dispersion of these preferences over the viewers'population guaranteeing the
existence of a unique Nash equilibrium. It turns out that, at this equilibrium,
both networks broadcast their news at the same instant.
2 The CBB-problem
Consider a time interval [Tmin;Tmax] corresponding to some period in the evening.
At each instant t in this interval there is a TV-watcher, which we denote by
t; coming back at home and willing to attend the TV-channel which is the
rst to broadcast its news after his arrival. There are two TV-channels i;
i = 1;2, and each one of them has to decide non cooperatively at which in-
stant ti in [Tmin;Tmax] to broadcast its news. For simplicity, we take the interval
[Tmin;Tmax] to be the [0;1] interval. Each network aims at maximising audience.
1The existence of a unique mixed strategy Nash equilibrium for the CBB program schedul-
ing game has been proved in Gabszewicz, Laussel and Lebreton (2007)
1The only dierence between the two channels is their time of news' diusion.
This problem can be formalised in terms of a normal form game, -we call it the
TV newscast scheduling game- , with the channels as players, the time interval
[0;1] as strategies, and audiences Ai as payos, namely
Ai(t1;t2) = ti
when ti < tj;
Ai(t1;t2) = ti   tj
when tj < ti; and
Ai(t1;t2) =
ti
2
when tj = ti;ti;tj 2 [0;1];i = 1;2: It is easy to show that this game has no
Nash equilibrium in pure strategies (see Cancian, Bills and Bergstrom, 1995).
The non-existence diculty obtained within the original CBB problem cru-
cially hinges on their assumption concerning the behaviour of TV-watchers with
respect to their "ideal" hours of broadcasting time.Here we propose an alterna-
tive version of the problem which allows to discard this diculty. This alterna-
tive version is directly inspired from the median voter problem in voting theory.
To introduce this version, we start by supposing that the whole potential au-
dience is already present at home in the beginning of the time interval [0;1].
Furthermore, we assume now that TV-watchers dier among them because, for
some unspecied reasons, some of them prefer to watch the news in the begin-
ning of the evening while others on the contrary prefer to delay this event. More
precisely,we suppose that TV-watchers are uniformly ranked in the interval [0;1]
by order of their ideal time for attending news. We alsoassume that the farther
the time a channel broadcasts its news from this ideal instant, the lower the
utility of the viewer for watching that channel. Finally, all TV-watchers are
assumed to view the news within one channel. Given these assumptions and
assuming t1  t2, the payos of the program scheduling game now write as2
A1(t1;t2) =
t1 + t2
2
(1)
2In the reverse case, A1(t1;t2) becomes A2(t1;t2) and vice-versa.
2A2(t1;t2) = 1  
t1 + t2
2
when t1 < t2; and
A1(t1;t2) = A2(t1;t2) =
1
2
(2)
when t1 = t2: It is easy to show that the above version of the CBB-game has
a unique pure strategy Nash equilibrium, namely, the pair (t
1;t
2) =
 1
2; 1
2

at
which both channels obtain half of the total audience.3. Thus, this alternative
formulation of the program scheduling game, with payo functions (1) and (2)
above, allows to go round the non-existence technical diculty encountered in
the Cancian and al. initial proposal.
3 A variant of the TV-news scheduling game
Generally, TV-viewers'preferences about channels'news not only depend on the
timing of their broadcast, but also on other elements like the news'content, the
identity of the presentator, or the brand image of the channel. These character-
istics, combined with viewers'preferences about broadcasting hours, determine
which channel a viewer will nally select. In order to extend the analysis to
such extra-characteristics, let us start by assuming that all TV-watchers pre-
fer to watch channel 1 to channel 2 whenever they broadcast their news at the
same instant of time. This assumption is easily introduced into the model by
supposing that viewer t incurs a utility loss equal to jt   j when he watches
channel 1 broadcasting its news at time ; and equal to jt    + j; > 0; when
watching channel 2 broadcasting at : The number  thus measures the increase
in utility a TV-viewer obtains when watching his preferred channel when both
channels broadcast their news simultaneously. It is easy to see that, due to
the uniform preferences of viewers for the extra-characteristics, whatever the
strategy t2 selected by its opponent, channel 1 can evict channel 2 by selecting
3The proof goes as for the median voter theorem in the case of a unform distribution of
voters.
3t1 = t2: Consequently there exists no equilibrium with both channels sharing
the total audience.
More signicant is a situation where the preferences of viewers with respect
to the extra-characteristics of the channels are dispersed over the population.
For instance, it seems reasonable to think that viewers' preferences for the TV-
newscasters are heterogeneous : not only some may prefer the newscaster of
channel 1 to the one of channel 2 while the reverse is true for others, but their
preferences may also vary in intensity. To introduce formally this heterogeneity
into the model, we suppose that, at each point of time t; there is a set of viewers
who are identical in terms of their preferences for the broadcasting time of the
channels : all of them consider the instant t as their ideal broadcasting instant.
This set is identied as the viewers of type t: However we now suppose that
the number ; measuring the increase (or decrease) of utility that TV-viewers
of type t obtain when watching channel 1 when both channels broadcast their
news simultaneously, is no longer constant over the type, but varies over the set
of viewers of type t: More precisely, we suppose that a viewer of type t incurs a
utility loss (or a utility gain, when  is negative) equal to (t   )2 +  when he
watches channel 1 broadcasting its news at time ; and equal to (t   )
2 while
watching channel 2 when it broadcasts at time ; with  uniformly distributed on
some interval [min;max] with density equal to 1
max min
4: Figure 1 represents
the set of TV-viewers on the rectangle [0;1]  [min;max]:
insertfigure1
For each type t and each pair of strategies (t1; t2);t1 < t2; dene by  (t;(t1;t2))
the TV-viewer who is indierent between watching news at channel 1 and chan-
nel 2, namely,  (t;(t1;t2)) solves
(t   t1)2 +  (t;(t1;t2)) = (t   t2)
2
4Notice that  (t) < 0 implies that type t prefers channel 2 to channel 1 when they broadcast
their news at the same time.
4or
 (t;(t1;t2)) = (t1   t2)(t1 + t2   2t): (3)
The existence of  (t;(t1;t2)) requires that  (t;(t1;t2)) 2 [min;max]; a prop-
erty which is not a priori guaranteed for any pair of strategies (t1; t2): This
would require that, whatever the pair of strategies (t1; t2) , there exists, for
each type t of TV-watchers, some of them who prefer to watch channel 1's
news, while some others of the same type prefer to watch those of channel 2.
The next lemma identies a sucient condition for this property to be satised,
in terms of the dispersion of viewers'preferences on the extra-characteristics of
the channels.
Lemma 1 [ 1;1]  [min;max] )  (t;(t1;t2)) 2 ]min;max[:
Proof:
The proof follows immediately from the denition of  (t;(t1;t2)) (see (3.1)),
noting that the inequality
(t1   t2)(t1 + t2   2t)  max
must be satised for the largest value of (t1   t2)(t1 + t2   2t); which implies
max  1; and the inequality
min  (t1   t2)(t1 + t2   2t)
must be satised for the smallest value of (t1   t2)(t1 + t2   2t), which implies
min   1: Q.E.D.
It follows from the above lemma that all type-t viewers whose  -value
exceeds  (t;(t1;t2)) watch news at channel 1 while the remaining ones ( <
 (t;(t1;t2))) watch channel 2. Given a pair of strategies (t1;t2); the audience
A1 (t1;t2;t) of channel 1, among TV-viewers of type t; is thus equal to the length
of the interval [ (t;(t1;t2));max]; multiplied by the density 1
max min; or
A1 (t1;t2;t) =
max    (t;(t1;t2))
max   min
:
5Furthermore, the total audience A1 (t1;t2) of channel 1 obtains as
A1 (t1;t2) =
Z 1
0
A1 (t1;t2;t)dt (4)
=
Z 1
0
max   (t1   t2)(t1 + t2   2t)
max   min
dt
=
1
max   min
[max   (t1   t2)(t1 + t2   1)]:
Similarly, we obtain the total audience A2 (t1;t2) of channel 2 as
A2 (t1;t2) =
1
max   min
[(t1   t2)(t1 + t2   1)   min]: (5)
The functions Ai (t1;t2); i = 1;2; are the payo functions of the program
scheduling game. As shown in proposition 1, the sucient condition identied
in lemma 1 is also sucient to guarantee the existence of a unique equilibrium
for the program scheduling game.
Proposition 1 If [ 1;1]  [min;max]; there exists a unique Nash equilibrium
in pure strategies, namely (t
1;t
2) =
 1
2; 1
2

; furthermore, this equilibrium is in
dominant strategies.
Proof : Using the rst order necessary condition, which is also sucient, we
get from (3.3)
@A1
@t1
=
1
max   min
(1   2t1) = 0 , t1 = t
1 =
1
2
:
Furthermore, this condition is independent of the value of t2 so that 1
2 is the
unique best reply against any strategy t2 of channel 2. A similar reasoning,
using (3.4), applies to show that t
2 = 1
2 is also the unique best reply for channel
2 against any strategy of its opponent. Q.E.D.
3.1 Bibliography
References
[1] Cancian,M., A.Bills and T. Bergstr om (1995). Hotelling location problem
with directional constraints : an application to television news scheduling.
The Journal of Industrial Economics. 43, 121-124.
6[2] Gabszewicz, J. D. Laussel and M. Lebreton (2007). The mixed strategy
Nash equilibrium of the television news scheduling game. forthcoming, The
Journal of Industrial Economics.
7

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