issn 0732-2399 eissn 1526-548X 06 2505 0538 doi 10.1287/mksc.1050.0183 ©2006 INFORMS
Commentary
A Comment on “Is Having More Channels Really
Better? A Model of Competition Among Commercial
Television Broadcasters”
Shan-Yu Chou
Department of Business Administration, National Taiwan University, 1 Section 4, Roosevelt Road, Taipei 106, Taiwan, Republic of China, chousy@ccms.ntu.edu.tw
Chi-Cheng Wu
Department of Business Management, National Sun Yat-Sen University, 70 Lianhai Road, Kaohsiung 804, Taiwan, Republic of China, ericwu@bm.nsysu.edu.tw
T
his paper shows that the analysis of Liu et al. (2004) contains a substantive error—the asserted pure-strategy Nash equilibrium leading to their Theorems 1 and 2 is really not an equilibrium. We show that in their model, either pure-strategy Nash equilibria do not exist or, unlike their asserted main result, when a pure-strategy equilibrium exists, increasing the number of commercial television broadcasters does not result in lower-quality programs. Possible modifications of Liu et al.’s model that may help restore the desired result are discussed.Key words: imperfect competition; game theory; market structure; media
History: This paper was received February 3, 2005, and was with the authors 3 months for 2 revisions.
1. Introduction
Liu, Putler, and Weinberg (2004; hereforth LPW) pro-vided an interesting analysis of equilibrium program choices made by imperfectly competitive commercial television broadcasters. In LPW’s static model, view-ers have different ideal points for program type, but they all prefer high-quality to low-quality programs. Taking viewers’ preferences as given, two competing broadcasters simultaneously choose program types and quality levels to maximize profits—broadcaster’s revenue from advertising slots minus the cost of producing the TV program. LPW focused on pure-strategy Nash equilibria and showed that in equilib-rium, either the duopolists become local monopolists or the quality of the equilibrium programs is strictly lower than the quality level wanted by a monopo-list; and, in either case, the duopolists always choose different program types. They also considered a two-period model in which viewers’ lead-in effect can be taken into account.
This paper intends to point out a substantive error in LPW’s analysis—the asserted pure-strategy Nash equilibrium leading to their Theorems 1 and 2 is really not an equilibrium. Our main theorem will show that in LPW’s static model, a pure-strategy Nash equilib-rium exists if and only if in equilibequilib-rium the duopolists become local monopolists, and thus the assertion that
duopolists may provide lower-quality programs than a monopolist will be disproved.
That LPW failed to deliver their desired result is rather unfortunate, and we contend that it was their model rather than the equilibrium concept (i.e., Nash equilibrium) that should be responsible for this fail-ure. In general, a monopolist’s incentives to provide quality may differ from those of duopolists for two reasons. First, quality provision is costly, and com-pared to duopolists, a monopolist can better capture the social benefits resulting from its private efforts of quality provision; after all, the monopolist has no rivals by definition. This suggests that increasing the number of broadcasters may reduce program qual-ity, an effect that LPW intended to show. However, it is also likely that the duopolists may provide more quality than their monopolistic counterpart, because the buyers can credibly threaten to switch from one firm to the other in the duopoly case but not in the monopoly case. For example, if viewers share the same ideal point for the program type, then the duopolistic broadcasters must engage in a Bertand-type quality competition, leading to an equilibrium quality level higher than the quality level a monop-olist would choose. Even if viewers have heteroge-neous ideal points, the same conclusion may follow if viewers care very little about program type rela-tive to their concern about quality level. We believe
that LPW’s model places too much weight on view-ers’ concerns about program type, which rules out the possibility that program quality may be higher in the duopoly case than in the monopoly case.
Because LPW chose to focus on the above first effect, and because the desired result did not occur in equilibrium, one would ask how LPW’s model might be modified to ensure that the equilibrium pro-gram quality is lower in the duopoly case than in the monopoly case. We offer several possible solu-tions in the conclusion section, including the recogni-tion of rare resources that may be required to produce a stylish program, the sequentiality of the decisions about program type and program quality, the explicit modeling of how program choice transforms into advertising revenue, and the threat of retaliation in an infinitely repeated game.
The remainder of this paper is organized as follows. In §2, we review LPW’s static model and their two theorems. In §3, we prove our main theorem, which, together with LPW’s results, gives a complete charac-terization of the pure-strategy Nash equilibria for the LPW model. Possible modifications of LPW’s model that may help restore LPW’s desired result are dis-cussed in §4.
2. The LPW Model and Main Results
LPW considered two TV broadcasters, A and B, fac-ing a continuum of viewers in a model à la Hotellfac-ing (1929). Each broadcaster, i, can choose a program
vi di, where di ∈ 0 1 and vi∈ 0 + stand for
the program type and program quality, respectively. A viewer, k, is identified by his ideal point, xk, for
program type, and the collection of all viewers is rep-resented by the distribution of xk, which is uniform
on the unit interval 0 1. Viewer k receives zero util-ity if he chooses not to watch any program, and he receives utility
ui
k= vi− xk− di
if he watches broadcaster i’s program vi di. Viewers
seek to maximize utilities, and broadcasters seek to maximize profits. Broadcaster i’s profit given its pro-gram vi di is
i= qi− cvi2
where qiis the population of viewers watching
broad-caster i’s program, and c > 0 is a cost parameter. Because a viewer can choose not to watch any pro-gram and a broadcaster i can always offer vi= 0, ui
k
and i are both nonnegative in equilibrium.
LPW first showed that for a monopolist, the opti-mal program is vm dm = 1 2 1 2 if 0 < c ≤ 2 1 c d otherwise
where 1/c ≤ d ≤ 1−1/c. The corresponding monopoly profits in the above two cases are m= 1/c and m=
1 − c/4, respectively.
LPW then considered the duopoly market. The game proceeds as follows. Broadcasters A and B first choose their programs vA dA and vB dB
simul-taneously. Upon seeing both vA dA and vB dB,
viewers decide which program to watch, or watch neither program. Then, A and B are realized, and
the game ends. A pure-strategy Nash equilibrium for this game is a pair of programs vA dA vB dB such
that for i j ∈ A B, j = i, given broadcaster j’s pro-gram vj dj, vi di maximizes i.
LPW showed that in the duopoly market, no pure-strategy Nash equilibria can exist if 0 < c < 8/3; and if c ≥ 4, then in equilibrium, the two broadcasters become local monopolists with vA= vB= A= B=
1/c.
In the remaining case, where 8/3 ≤ c < 4, LPW showed that in equilibrium, vA = vB = 1/4, and
dA dB equals either 1/4 3/4 or 3/4 1/4,
result-ing in the profits A = B= 1/2 − c1/42. Because
vA = vB= 1/4 < vm when 8/3 ≤ c < 4, having more
competitors reduces equilibrium program quality in this case. Moreover, with dA= dB, the equilibrium
pro-gram types chosen by the two broadcasters exhibit the counterprogramming property. Based on these obser-vations, LPW reported their two main theorems:
Theorem LPW-1. In a duopoly market, television
broad-casters tend to differentiate from each other and adopt a “counterprogramming” strategy.
Theorem LPW-2. Having more competitors can result
in each competitor offering lower-quality programs com-pared to a situation with fewer competitors.
Our main theorem in the next section will disprove Theorem LPW-2 and show that LPW’s analysis for the case 8/3 ≤ c < 4 contains a substantive error.
3. Main Theorem
Our main theorem (Theorem 1) below will show that in LPW’s static model, no pure-strategy Nash equi-librium can exist if 8/3 ≤ c < 4. Thus, combined with LPW’s results for the cases 0 < c < 8/3 and c ≥ 4, our theorem will imply that an equilibrium exists if and only if c ≥ 4, in which the two broadcasters must become local monopolists in equilibrium. Hence, in
the LPW model, increasing the number of competitors need not result in lower program quality.1
Lemma 1. Suppose that 8/3 < c < 4. In any
equilib-rium, A B< 1/c and qA qB> 0.
Proof. We first show that A B≤ 1/c. Observe
that for broadcaster i = A B,
i= qi− cvi2≤ 2vi− cvi2≤1
c = m
where all three equalities hold if and only if vi= 1/c
and qi= 2vi= 2/c. Thus, if i= 1/c, then
qi= 2 1 c > 2 1 4 ⇒ qj<1 2< 2 c ⇒ j< 1 c
However, expecting vi di, broadcaster j could have
chosen vj dj = vi+ di and captured all
broad-caster i’s viewers, which would have generated a profit greater than or equal to
qi−cvi+2= i−c2vi+2 =1
c−c2vi+2 > j
where the last inequality holds for sufficiently small
> 0 because j< 1/c. Hence, vj dj does not
maxi-mize j given vi di, a contradiction to the
assump-tion that vA dA vB dB is an equilibrium.
Now suppose that in equilibrium, qi = 0, so that
i = −cvi2, implying that vi = 0 = i. Expecting
vi di, broadcaster j becomes a monopolist and j=
1/c, which is a contradiction to the above first asser-tion.
Lemma 2. Suppose that 8/3 < c < 4. In any
equilib-rium, the program types chosen by the two broadcasters must differ, i.e., di= dj.
Proof. Suppose instead that di= dj= d. First,
sup-pose that vi = vj = v. This implies that qi = qj =
q and i= j = . Expecting vi di, broadcaster j
could have chosen vj dj = v+ d and captured all
broadcaster i’s viewers, which would have generated for broadcaster j a profit greater than or equal to
2q − cv + 2= + q − c2v + 2 > = j
where the last inequality holds for sufficiently small
> 0 because q > 0 (Lemma 1). This implies that vj dj does not maximize j given vi di, a
contra-diction.
Now suppose that vi> vj. This implies that qj= 0,
a contradiction to Lemma 1.
Because of Lemma 2, if 8/3 < c < 4, then di= dj in
equilibrium, and we assume dA< dB without loss of
generality.
1We thank an anonymous referee for recommending the following
proof to us, which is much more concise than our original proof.
Lemma 3. Suppose that 8/3 < c < 4. In any
equilib-rium, with dA< dB, dA= vA and 1 − dB= vB.
Proof. By symmetry, we will only prove that dA=
vA. Recall that by definition of a Nash equilibrium,
vA dA must maximize A given vB dB. We show
that this cannot hold if either dA> vA or dA< vA.
(1) First, suppose that dA> vA. In this case,
view-ers with ideal points xk∈ 0 dA− vA do not watch
any program. (Otherwise, some viewers may choose to watch broadcaster B’s program, but then qA = 0,
which contradicts Lemma 1.) There are two possible cases: either (i) dA + vA ≥ dB− vB or (ii) dA + vA <
dB− vB. Note that in case (i), there is a (nonnegative)
population of viewers xk with uAk ≥ 0 and uBk≥ 0 (and
only some of these viewers choose to watch broad-caster A’s program). By contrast, in case (ii), all view-ers xk with uAk ≥ 0 must have uBk< 0.
• Consider (i). We claim that in case (i), given
vB dB, vA dA − yields a higher profit than
vA dA for broadcaster A, where dA − vA > > 0.
Note that replacing vA dA by vA dA− involves
moving broadcaster A’s program type from dA to
dA − (away from dB by a small distance of >
0). Two effects result from this program change: it would attract a population of new viewers with ideal points on the left of dA, but it would also cause
a loss of some viewers with ideal points on the right of dA. However, the population of the lost viewers
never exceeds !
• Next, consider (ii). In this case, qA= 2vA, and
by Lemma 1, A = 2vA− cvA2< 1/c. This implies
that broadcaster A is a local monopolist in equilib-rium, and yet its program quality is too low relative to the monopoly optimal level (vA< 1/c). Thus,
broad-caster A could have benefited from replacing vA dA
by vA+ dA, where 1/c − vA > > 0, which
con-tradicts the assumption that vA dA maximizes A
given vB dB.
(2) Now suppose that dA< vA. We first claim that
vB< vA+ dB− dA. Note that if vB> vA+ dB− dA,
then every viewer strictly prefers broadcaster B’s pro-gram to broadcaster A’s propro-gram, and hence qA= 0, a
contradiction to Lemma 1. On the other hand, if vB=
vA+ dB− dA, then viewers with ideal points on the
left of dA feel indifferent about the two programs, but
viewers with ideal points on the right of dA strictly
prefer broadcaster B’s program to broadcaster A’s pro-gram, and hence qA= dA/2. However, broadcaster A
could have chosen vA+ dA and obtained all the
viewers with ideal points on the left of dA, which
would have yielded a profit greater than or equal to
A+qA
2 − c2vA + 2 > A
if > 0 is sufficiently small. This is another contradic-tion. Now, given that vB< vA+ dB− dA, we claim
that vA dA is dominated by vA dA+ , where 0 <
< vA − vB+ dB− dA: moving broadcaster A’s
pro-gram type from dA to dA+ (moving toward dB by
a distance of ) does not cause any loss of viewers with ideal points on the left of dA, but it allows
broad-caster A to gain more viewers with ideal points on the right of dA. This proves that vA dA does not
maxi-mize A given vB dB, a contradiction.
We thus conclude that in equilibrium, dA= vA and,
by symmetry, 1 − dB= vB.
Lemma 4. Suppose that 8/3 < c < 4. In any
equilib-rium, vA≤ 1/c.
Proof. Recall that the marginal revenue of vi is 2
in the monopoly case,
qi
vi =
2vi
vi = 2
By Lemma 3, dA= vA, and thus the marginal revenue
of vAis strictly less than 2 in the duopoly case: raising
vA to vA+ , say, does not gain a population of new
viewers by more than because there are no view-ers with ideal points on the left of xk= 0! Because the marginal cost function of vA, which is 2cvA, remains
the same in both the monopoly case and the duopoly case, the duopoly equilibrium choice vA can never
exceed the monopolist’s optimal choice 1/c. Note that LPW’s asserted equilibrium satisfies all four lemmas derived above, and, hence, it does look like an equilibrium. Moreover, Lemma 4 shows that the equilibrium program quality in the duopoly case can never exceed the quality level chosen by a monop-olist, and, hence, if there exists one equilibrium, then Theorem LPW-2 will be valid. Unfortunately, our main theorem below shows that no equilibrium can actually exist if 8/3 < c < 4.
Theorem 1. No pure-strategy Nash equilibrium can
exist in the static model of LPW if 8/3 < c < 4.
Proof. Suppose instead that there is a pure-strat-egy Nash equilibrium vA dA vB dB with dA< dB.
By Lemma 1, we have B< 1/c. By Lemmas 3 and 4,
we have vA= dA≤ 1/c. However, in this case,
broad-caster B could have replaced vB dB by vB dB =
1/c + 1/c to capture all broadcaster A’s viewers
and to obtain a profit 1 c− c 2 c + 2 > B
where the inequality holds if > 0 is small enough. This proves that vB dB does not maximize
broad-caster B’s profit given vA dA, a contradiction.
LPW used a numerical example to demonstrate their asserted equilibrium. In that example, c = 27, and LPW asserted that vA dA = 1/4 1/4 and
vB dB = 1/4 3/4 formed a Nash equilibrium.
Following Theorem 1, however, vA dA vB dB =
1/4 1/4 1/4 3/4 is actually not a Nash
equilib-rium because expecting broadcaster B’s strategy
vB dB = 1/4 3/4, broadcaster A could have
cho-sen vA
dA
= 1/27 + 10−100 1 − 1/27 and obtained
a profit strictly greater than broadcaster A’s profit in LPW’s asserted equilibrium, which is 1/2 − 271/42.
4. Concluding Remarks
In this section, we propose several modifications to LPW’s static model, which may help restore Theo-rem LPW-2. First, in the LPW model, a duopolist can always mimic its rival’s program style without much difficulty, which implies that a duopolist has nearly the same capability as a monopolist of inter-nalizing the surplus generated by its private effort of quality provision. This explains why low-quality pro-grams cannot be sustained in equilibrium, or equiva-lently, no equilibrium can exist (Lemma 4 shows that whenever an equilibrium exists, then the duopolists must offer low-quality programs). In reality, however, producing TV programs with distinguished styles (or “program types,” in the terminology of LPW) usually requires special and scarce resources. TV celebrities like Larry King, Connie Chung, and David Letterman, for example, represent key resources in the produc-tion of stylish TV programs, and one broadcaster can-not access these scarce resources if the latter are under contracts with another broadcaster. The scarcity of these resources suggests that mimicking the program type of a rival may be more difficult than assumed in LPW’s model. Hence, a model that recognizes the effects of scarce resources is more likely to admit an equilibrium.
Second, LPW assumed that broadcasters’ choices of program type and quality level cannot affect r. This assumption may not be so innocuous as it may seem at first. Gal-Or and Gal-Or (2005) considered how a monopolistic broadcaster facing two compet-itive advertisers should optimally design the sched-ule R charged to each advertiser as a function of the type and the number of commercials that the tiser wishes to air (where R corresponds to the adver-tising revenue rq in the LPW model). An important implication of Gal-Or and Gal-Or’s analysis is that a high-quality TV program may allow only a few com-mercial breaks, which imposes a more stringent upper bound on the capacity of advertising time slots than a low-quality program.2 More precisely, a program
with a large v may produce a large q, but it may reduce r at the same time! By considering the relation-ships between the capacity constraint of time slots, 2We thank Steve Shugan for directing our attention to the two
articles by Gal-Or and Gal-Or (2005) and Steenkamp et al. (2005), which have enriched our discussions in this section.
the program quality, and the feasible number of com-mercial breaks, a new model may work better in dis-couraging broadcasters from mimicking each other’s program types: mimicking the rival’s program type can be profitable only if it is accompanied by a qual-ity level higher than the rival’s, but that would imply very few commercial breaks, resulting in lower adver-tising revenue. Thus, an equilibrium is more likely to exist after r is explicitly modeled and endogenously derived.
Third, the timing of events in LPW’s model mat-ters also. LPW assumed that a broadcaster can choose program type d and program quality v at the same time. In reality, it is not unusual that in producing a program, a broadcaster must first choose d, but v can-not be determined until production starts. This may happen if, for example, there is uncertainty about the cost parameter, c, at the time that d is chosen. In this sequential version of LPW’s model, a Bertrand-type quality competition may occur in subgames where
dA and dB are close to each other, and rationally
expecting this unpleasant consequence, the broadcast-ers would rather make dA and dB sufficiently
differ-entiated, which in turn implies an equilibrium quality level lower than the monopoly quality level. The idea that competitors try to differentiate in an early stage to avoid aggressive competition in a later stage is not new; it has also appeared in the analysis of Gal-Or and Gal-Or (2005), among others.
Fourth, if the interactions between the broadcast-ers last indefinitely, and if the broadcastbroadcast-ers care enough about their future profit streams, then the well-known folk theorem (see, for example, Chap-ter 5 in Fudenberg and Tirole 1991) ensures that the duopolists can attain collusive profits with their pro-gram choices approximating those of a monopolis-tic broadcaster capable of offering two programs to viewers at one time. If such a monopolistic broad-caster existed, would it offer two different programs to viewers, and would the quality levels of the offered programs be lower than vm? The answer is
appar-ently positive: the monopolist should offer two differ-ent programs instead of one, and for each program
the monopolist, expecting fewer viewers than when it can only offer one program, should choose a quality level strictly lower than vm.3
Finally, we remark on the solution concept of Nash equilibrium. It is often proposed that dynamic equi-librium analysis can be replaced by a conjectural vari-ation in the static model, in which firms choose not to compete aggressively because they “conjecture” that retaliation may follow if they take actions to hurt their rivals. Such a conjectural variation is incompat-ible with the concept of Nash equilibrium. In a static model, by definition, the firms can act only once, and any conjecture that involves the rivals’ retaliation is incorrect. To incorporate the rivals’ retaliation into the analysis, one does not have to abandon Nash equi-librium; explicitly formulating the long-term inter-actions between strategic broadcasters in a dynamic model and finding the (subgame perfect) Nash equi-librium, in our view, is more promising than the above conjectural variation approach.
References
Fudenberg, D., J. Tirole. 1991. Game Theory. MIT Press, Cambridge, MA.
Gal-Or, E., M. Gal-Or. 2005. Customized advertising via a common media distributor. Marketing Sci. 24(2) 241–253.
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Steenkamp, J.-B. E. M., V. R. Nijs, D. M. Hanssens, M. G. Dekimpe. 2005. Competitive reactions to advertising and promotion attacks. Marketing Sci. 24(1) 35–54.
3Although we advocate the merits of a dynamic model, we do not
contend that static models are never adequate in correctly capturing the rational program choices made by commercial television broad-casters. In fact, the recent findings of Steenkamp et al. (2005) show that the net outcome of most promotion and advertising attacks in their sample is not influenced by the defender’s reaction. Put differ-ently, the ultimate competitive impact of most advertising and pro-motion attacks is due primarily to the nature of consumer response, not to the vigilance of competitors.
