Rivalry between Strategic Alliances
Anming Zhang
Sauder School of Business, University of British Columbia
Yimin Zhang
China Europe International Business School (CEIBS), and Department of Economics and
Finance, City University of Hong Kong
December 2003
Revised: April 2005
Abstract: Rivalry between strategic alliances is investigated in a model where each alliance
member maximizes its own profit and some share of its partner’s profit. A complementary
alliance confers a strategic advantage by allowing the partners to credibly commit to greater
output, owing to both within-alliance complementarities and cross-alliance substitutabilities.
Although rivalry between different alliances can sometimes lead to a Prisoners’ Dilemma for
firms, it tends to improve economic welfare. On the other hand, an alliance that arises due purely
to the threat of entry may reduce welfare.
Keywords: Competing strategic alliances; Partial alliance; Supermodularity; International
airline alliances
JEL classification: L2; L4; L9
Correspondence: Anming Zhang, Sauder School of Business, University of British Columbia,
2053 Main Mall, Vancouver, B.C. V6T 1Z2, Canada. Phone: (604) 822-8420; fax: (604)
822-9574; e-mail: [email protected].
1. Introduction
Strategic alliances represent an important form of cooperation between two or more business
entities. A strategic alliance might be viewed as a lesser form of a merger. It is not a merger per
se, since alliance partners remain separate business entities and retain their decision-making
autonomy. While merger activities have slowed down significantly since 2000, strategic
alliances are increasingly, and widely, used by firms. They are particularly prevalent in
network-oriented industries such as airline, shipping, telecommunications, multimodal
transportation and logistics industries. Consider the airline industry. The four largest global
alliances – namely, Star Alliance, OneWorld, Wings, and SkyTeam – accounted for 57% of the
world’s revenue passenger kilometres in 2002 (Airline Business, November 2003). 1 Like
alliances in logistics and other transportation industries, an airline alliance is a multi-product
network, with each of its products corresponding to travel (either by people or by cargo) in a
particular city-pair market. The extent of alliance will affect not only the partners but also the
nature of interaction between competing alliances in various markets. The strategic alliance
between, for instance, United and Lufthansa – the two major partners of Star Alliance – will thus
have a major impact on the alliance decision by American and British Airways in OneWorld,
which is competing with Star Alliance in the U.S., European and North Atlantic markets.
In this paper, we investigate the issue of strategic alliances and alliance rivalry – where each
alliance member maximizes its own profit and some share of its partner’s profit – in the context
of both within-alliance and cross-alliance interactions. We focus on the strategic motives of an
alliance and on how a strategic advantage is conferred. Our second objective is to investigate
whether strategic alliances should be viewed as causes for anticompetitive concerns. Strategic
1
The merger of Air France and KLM was recently approved by EU regulators. As a consequence, KLM is expected
to leave Wings and join SkyTeam, whose major partners are Air France and Delta. KLM’s current alliance partner,
Northwest, is also expected to join SkyTeam, with Continental (Northwest’s long-time domestic alliance partner)
joining as well (Brueckner and Pels, 2005). If the Wings and SkyTeam alliances are consolidated into a single
mega-alliance, the three largest global alliances would each have about 20% of the world’s passenger kilometres.
1
alliances allow firms to expand their networks, take advantage of product complementarities,
realize economies of scale and scope, and improve product quality and customer service. 2
Despite these potential benefits to firms and/or consumers, antitrust authorities often view
strategic alliances with suspicion. In the airline example, most international routes have only a
few competitors, so an alliance between any two significant competitors may considerably
reduce the degree of competition on certain routes. Here an important consideration is whether
government policy should encourage or restrict domestic carriers from expanding their networks
through foreign alliances, while taking account of any potential anticompetitive effects. More
generally, since alliances are already an important component in each of the transportation
modes, 3 further consolidation may raise competition concerns that will depend on the barriers to
entry and/or expansion.
We examine a complementary alliance in which two firms link up their complementary
products in the context of competing strategic alliances. We find that such an alliance confers a
strategic advantage by allowing the partners to credibly commit to greater output levels, owing
to both within-alliance complementarities and cross-alliance substitutabilities. Even if an
alliance creates a negative direct effect on profit, it might be pursued, either because it is a
dominant strategy in alliance rivalry or because it would deter entry. Although rivalry between
2
For example, airline alliances enable partners to better coordinate their flight schedules to minimize travelers’ wait
time between connecting flights. The partners also benefit from sharing information, research and experiences in
different market segments and from linking their existing networks; thereby, reducing the need for developing new
hubs: a major cost of entry and particularly so for international aviation due to bilateral treaties and foreign
ownership restrictions (Baker and Pratt, 1989; Pitelis and Schnell, 2002).
3
The three largest strategic alliances in the air cargo sector, namely, New Global Cargo, SkyTeam Cargo, and
OneWorld, accounted for 49% of international freight tonne kilometers in 2002. In water transportation, the five
shipping alliances – namely, Maersk/Sea-Land, Grand Alliance, New World Alliance, The K-Line/COSCO Group,
and United Alliance – accounted for 61% of international freight tonne kilometers in 2000. The same trend has
recently appeared in the U.S. domestic airline industry. Over the past few years, virtually all of the U.S. hub-spoke
carriers have entered into broad codesharing partnerships, including the United/US Airways alliance that began in
January 2003, and the three-way alliance between Northwest, Continental and Delta initiated in June 2003.
2
different alliances can sometimes lead to a Prisoners’ Dilemma for firms, it tends to improve
economic welfare because it would, owing to the strategic effect, result in greater output levels
than would be found in the absence of the rivalry. On the other hand, an alliance can arise due
purely to the threat of entry; such an alliance may reduce welfare.
Although there is not a formidable body of theoretical analysis on competition between
alliance networks, 4 there have been some relatively recent important empirical contributions to
the literature. Brueckner (2001) provides a theoretical analysis of an international airline alliance
with a monopoly pair of carriers. In his model, the alliance partners provide complementary
products when their respective domestic spoke-to-hub routes are used in combination by
connecting passengers. In the common international hub-to-hub route, however, the partners’
outputs are substitutes. Brueckner and Whalen (2000) analyze duopoly-pair alliances on a
similar network with the international hub-to-hub route suppressed. Both papers obtained
interesting results under the assumptions of linear demand functions, linear marginal costs, and
symmetric route structures. In particular, Brueckner points out that the cooperative pricing of
trips by alliance partners would increase traffic in the connecting market since portions of a
connecting trip are complements. 5 We extend this literature by analyzing generic strategic
alliances and developing the welfare results in a more general setting. We also explicitly
consider rivalry between alliance networks where potential partners decide not only on whether
to form an alliance, but also on the extent of cooperation if they were to form an alliance.
2. The Model
4
This is in sharp contrast to the vast theoretical literature on single-firm, single-product oligopoly and on mergers.
See, e.g. Farrell and Shapiro (1990a), Cabral (2003) and Pesendorfer (2005).
5
Oum, et al. (2000) obtained similar results based on restrictive demand and cost specifications and restrictive
strategy sets. Empirical investigation of international airline alliances includes USDOT (1994), USGAO (1995),
Oum, et al. (1996, 2000) and Brueckner and Whalen (2000). In particular, Oum, et al. and Brueckner and Whalen
provide empirical support for the pro-competitive potential of complementary airline alliances. Chen and Ross
(2000) explore possible entry-deterrence effects of strategic alliances involving the sharing of production capacity.
3
We consider a four-firm, four-product model that is likely the simplest structure in which
strategic alliance rivalry can be addressed. The four firms, denoted 1, 2, 3 and 4, produce goods
1, 2, 3 and 4, respectively. The inverse demand function is written as:
p i = p i (q1 , q 2 , q3 , q 4 )
(1)
with
p12 (Q) > 0,
p12 (Q) > 0,
p 43 (Q) > 0,
p34 (Q) > 0 .
(2)
In (2), Q ≡ (q1 , q 2 , q3 , q 4 ) denotes the output vector and the subscripts denote partial derivatives,
e.g. p 12 (Q) ≡ ∂p 1 (Q ) / ∂q 2 . The first two inequalities show demand complementarity between
goods 1 and 2, whereas the other two inequalities show demand complementarity between 3 and
4. Using ci (q i ) to denote total cost, the profit of each firm can be written as:
π i (Q) = qi p i (Q) − ci (qi ) ,
i = 1,2,3,4 .
(3)
Condition (2) is then equivalent to the following condition:
π 21 (Q ) > 0,
π 12 (Q) > 0,
π 43 (Q ) > 0,
π 34 (Q) > 0
(4)
since, e.g. π 21 (Q) ≡ ∂π 1 (Q) / ∂q 2 = q1 p 12 by (3). Inequalities (4) say that an increase in the output
of firm 2 (firm 4) will increase the profit of firm 1 (firm 3), and vice versa. Further, we assume:
π 121 (Q) > 0,
π 343 (Q) > 0,
π 212 (Q) > 0,
π 434 (Q) > 0
(5)
that is, changes in the output of firm 2 (firm 4) raise the marginal profit of firm 1 (firm 3), and
vice versa. Condition (5) indicates that goods 1 and 2 (3 and 4) are “strategic complements”
(Bulow, et al., 1985). Observe that the fact the goods are complements is conducive to their
1
1
strategic complementarities. For example, since π 12
and p 12 > 0 by (2) or (4), it
= p 12 + q1 p12
1
1
follows that π 12
> 0 if p12
≥ 0 . Thus condition (5) will hold, given (2) or (4), if the demand
functions are linear. Nonetheless, (5) can hold for more general demand specifications.
We shall for simplicity assume that firms 1 and 2 always form a (potential) alliance pair,
with 3 and 4 forming the other pair. The situation is depicted in Figure 1, where the former is
4
referred to as the North alliance, while the latter the South alliance. For illustration, “North” may
be Star Alliance, in which United and Lufthansa provide complementary products since their
respective domestic routes are used in combination by international passengers, while “South”
may be OneWorld with American and British Airways providing respective domestic services.
Alternatively, one may consider firms 1 and 3 producing (differentiated) peanut butter, whereas
2 and 4 producing (differentiated) jelly. Thus, firms 1 and 2 form one peanut butter/jelly alliance
pair, while 3 and 4 form the other pair. Since within each firm pair the products are (strategic)
complements, such alliance may be called a “complementary” alliance.6
1
2
North Alliance
3
4
South Alliance
Figure 1. Basic Model Structure
We examine complementary alliances in a model where each alliance member maximizes
its own profit and, to a certain extent, its partner’s profit. The decision problems faced by firms 1
and 2 may be expressed as:
6
Observe, however, that although the complementarity conditions (4) and (5) are here driven by demand
complementarities, they can also arise through complementarities on the cost side – e.g. there are economies of
scope in the partners’ joint production process. More generally, therefore, we can refer to strategic alliances that
satisfy (4) and (5) as complementary alliances. We discuss the issue further in the concluding remarks.
5
Max π 1 + απ 2 ≡ Max π 12 (Q;α )
q1
q1
(6)
Max π 2 + απ 1 ≡ Max π 21 (Q;α ).
q2
q2
That is, in making its optimal quantity decision a firm incorporates, apart from its profit, a
fraction α of its partner’s profit, with 0 ≤ α ≤ 1 . One natural interpretation for this formulation
is cross shareholding in equity alliances. A significant proportion of strategic alliances in the
airline industry are equity alliances in which one airline buys a share of stock in its partners. 7 An
equity alliance tends to yield greater firm values, measured in stock returns, than other types of
strategic alliances. 8 Although alliances of this type do not necessarily change production
possibilities at any partner firm, they do alter the incentives of the alliance partners. 9 In
particular, when α = 0 , each firm owns zero share of the other firm so it maximizes its own
profit. Partial alliance occurs when 0 < α < 1 , in which two firms own the same fraction of each
other’s shares. Finally, a “full” alliance occurs when α = 1 , where each firm pays full attention
to its partner’s profit and so the firms maximize the joint profit. Similarly, the decision problems
faced by firms 3 and 4 are expressed as:
Max π 3 + βπ 4 ≡ Max π 34 (Q; β )
q3
q3
Max π 4 + βπ 3 ≡ Max π 43 (Q; β )
q4
7
(7)
q4
Most recent international examples include the Air France/KLM alliance, the Cathay Pacific/Air China alliance,
and the proposed Qantas/Air New Zealand alliance.
8
Oum, et al. (2000) find, in the airline industry, that conditional on a broad scope of cooperation between partners,
equity alliances had a greater abnormal return on the day of alliance announcements than non-equity alliances. More
generally, it has been argued that equity participation can reduce transaction cost and uncertainty, and can also
lesson partner perceptions of opportunistic behavior (e.g. Pisano, 1989; Parkhe, 1993).
9
Farrell and Shapiro (1990b) have made a similar point in the context of joint ownership of assets in Cournot
oligopoly. In a different context, Mandy (2001) examines “vertical control” that a vertically integrated upstream
monopolist exercises over its subsidiary and its implications for pricing and economic welfare. Vertical control in
his model means the extent to which the parent company can align the objective of its downstream affiliates with the
objective of the overall firm. This is not dissimilar from the problem that we examine in which one alliance partner
attempts to maximize its profit and some fraction of its partner’s profit.
6
where the extent of alliance or partial ownership is captured by parameter β , 0 ≤ β ≤ 1 . 10
In addition to the degree of cross shareholding, parameters α and β may be interpreted as
the degree of cooperation between potential partners. When α = 0 , firms 1 and 2 act
independently. As α increases, the firms strengthen their cooperation, with α = 1
corresponding to full integration in which the firms act as if they were a single decision-making
unit in their joint-profit maximization. The discussion holds similarly for β .
Our goal is to explore the implications of conditions (4) and (5) for alliance strategies and
economic welfare in an environment of competing alliances. This environment is relevant for
studying strategic alliances in a number of network-based markets. For example, an international
passenger making an interline trip can usually choose between several airline pairs, either allied
or non-allied. For the allied pairs, the competing choices include Star Alliance, OneWorld and
SkyTeam. In our model, the competing aspect is reflected by the following assumption:
π 1j + π 2j < 0,
π i3 + π i4 < 0,
j = 3,4 ;
i = 1,2
(8)
that is, the outputs of the two alliances, North and South, are substitutes. Furthermore, following
the standard practice in models of quantity competition (e.g. Tirole, 1988) we assume that an
alliance’s marginal profit declines when the output of the rival alliance rises:
π 112j < 0, π 221j < 0, j = 3,4 ;
π 334i < 0, π 443i < 0, i = 1,2
(9)
implying the outputs of the two alliances are “strategic substitutes” (Bulow, et al., 1985).
In addition to output decisions, the firms within each of these competing pairs together need
to decide on whether to form an alliance and the extent of alliance α or β . Since a strategic
alliance (e.g. equity alliance) brings the participants into what is essentially a medium- to
long-term contractual relationship, the alliance structure, once decided upon, cannot easily be
10
In (6), (7) we assume a Cournot game in the product-market competition. Brander and Zhang (1990, 1993), for
example, find some empirical evidence that rivalry between duopoly airlines is consistent with Cournot behavior.
7
altered in a major way. 11 The alliance relationship is a strategic decision, which might
reasonably be regarded as given at the time of output competition. This suggests a two-stage
alliance game: In stage 1 each firm pair decides on its alliance structure, α or β ; then in stage 2
the four firms choose their output levels. 12
3. Effects of Alliance
We now examine the subgame perfect equilibrium of the alliance game. In the second-stage
competition, the four firms choose their quantities to maximize profits, taking the alliance
structure (α , β ) as given. Assuming the second-order conditions π iiK < 0 hold, for K = 12, 21,
34, 43, then the Cournot equilibrium is characterized by the first-order conditions,
π 112 (Q; α ) = π 11 + απ 12 = 0
(10)
π 221 (Q; α ) = π 22 + απ 21 = 0
(11)
π 334 (Q; β ) = π 33 + βπ 34 = 0
(12)
π 443 (Q; β ) = π 44 + βπ 43 = 0 .
(13)
Regularity conditions are imposed so that the equilibrium exists and is stable.
In the absence of the South alliance, this Cournot game of firms 1 and 2 is supermodular:
Roughly speaking, a game is called supermodular if the increment in a player’s payoff due to an
increase in its strategy variable is increasing in the strategy variable of any of the other players.
Thus, supermodularity may be viewed as the mathematical characterization of the notion of
strategic complementarity (e.g. Topkis, 1998; Vives, 2000). With competing alliances, however,
our four-firm Cournot game is no longer supermodular, because the outputs of the North and
11
Parkhe (1991) defines a strategic alliance as a “relatively enduring interfirm cooperative arrangement, involving
flows and linkages that utilize resources and/or governance structures from autonomous organizations, for the joint
accomplishment of individual goals.” A strategic alliance is thus different from an ordinary alliance in that the
partners here make a more serious commitment to cooperation. As a result,
α ’s or β ’s are chosen, since equity participation is
presumably a unilateral decision made by each firm. Extending the analysis to individual α ’s and β ’s would add
12
We do not address here the issue of how firm-specific
mathematical complexity; nevertheless, our basic results would continue to hold in that case.
8
South alliances are strategic substitutes. Observe, nevertheless, that the game is supermodular in
34
3
12
12
1
= π 34
+ απ 344 > 0 by (5), and π 13
<0,
(q1 , q 2 ,−q3 ,−q 4 ) , because π 12
= π 12
+ απ 122 > 0 and π 34
π 1412 < 0 , π 3134 < 0 and π 3234 < 0 by (9). Similar inequalities hold for π 21 and π 43 , so the game is
supermodular in (−q1 ,− q 2 , q3 , q 4 ) .
Thus the strategic complementarity within an alliance, together with the strategic
substitutability across the alliances, make the second-stage game supermodular in
(q1 , q 2 ,−q3 ,−q 4 ) and (−q1 ,− q 2 , q3 , q 4 ) . Given this result, the effects of alliance variables α and
β on the equilibrium quantities, denoted q i (α , β ) , can be derived in Proposition 1. These
comparative static effects are central to our subsequent analysis.
Proposition 1. Under complementary alliances,
qαi > 0, q βj > 0, qαj < 0, q βi < 0,
i = 1,2 ;
j = 3,4
(14)
that is, the strengthening of one of the alliances increases the output of both member firms’
products and decreases that of their rival firms.
Proof: Since the game is supermodular in (q1 , q 2 ,−q3 ,−q 4 ) , Topkis (1998, Theorem 4.2.2)
indicates that qαi (α , β ) > 0 for i = 1,2 and qαj (α , β ) < 0 for j = 3,4 if the game has increasing
differences in α . The latter holds because, by (4),
π 112α = π 12 > 0,
π 221α = π 21 > 0,
π 334α = 0,
π 443α = 0
i.e., π iKα ≥ 0 , for K = 12, 21, 34, 43 and i = 1, 2, 3, 4. The other part of the Proposition is shown
in a similar way.
Q.E.D.
Alliance structures therefore influence subsequent output competition, which in turn affects
firms’ overall profitability. Taking the second-stage equilibrium output into account, firm i’s
profit in the first stage, denoted φ i , is given by:
φ i (α , β ) ≡ π i (Q(α , β )) , i = 1,2 ;
φ j (α , β ) ≡ π j (Q(α , β )) , j = 3,4 .
(15)
The alliance equilibrium arises when each firm pair chooses its profit-maximizing alliance
9
structure, taking the alliance structure of the other pair as given at the equilibrium value.
Proposition 2. At the alliance equilibrium both firm pairs form a full alliance.
Proof: From (15), it follows that π α12 + π α21 = φα1 + φα2 + (π 1 + π 2 ) + α (π α1 + π α1 ) . Substituting
π α1 + π α2 = φα1 + φα2 and recollecting the terms yields: π α12 + π α21 = (1 + α )(φα1 + φα2 ) + (π 1 + π 2 ) .
Solving for
φα1 + φα2 and using the expressions for π α12 and π α21 , we then obtain:
φα1 + φα2 = (1 − α )(π 12 qα1 + π 21 qα2 ) + [(π 31 + π 32 )qα3 + (π 41 + π 42 )qα4 ]
(16)
where π 12 , π 21 are positive by (4) and π 1j + π 2j < 0 by (8), j = 3,4 . Thus, by Proposition 1,
∂ (φ 1 + φ 2 ) / ∂α > 0 . That ∂ (φ 3 + φ 4 ) / ∂β > 0 can be similarly shown.
Q.E.D.
Proposition 2 shows that a complementary alliance can be both an offensive strategy and a
defensive strategy in an alliance rivalry. It improves the alliance’s own profit, given the alliance
structure of the rival firms; it defends the allied firms when the rival firms intensify their
cooperation. Under the specified conditions a full alliance is, in effect, each firm pair’s dominant
strategy. Note furthermore that the effect of a change in α (or, β ) on profit can be split into two
parts: i) a direct effect of the shift in the alliance’s profit function, and ii) an indirect effect of the
shift in the marginal profits, which in turn changes the equilibrium. The direct effect may be
illustrated by considering a monopoly pair of firms (say, 1 and 2) who are unconcerned with
entry. In this case, the bracketed term on the right-hand side of (16) vanishes. The remaining
term, (1 − α )(π 12 qα1 + π 21 qα2 ) , then represents the direct effect of alliance on the pair’s profit.
This direct effect is positive so long as α < 1 . The positive sign arises because of the
monotonicity properties of the supermodular game between firms 1 and 2. As indicated above,
supermodularity captures the notion of complementarity: in our case, complementarity between
q1 and q 2 means that the marginal returns to one variable increases in the level of the other
variable. Therefore, the term captures the internalization of the familiar “complementarity
externality:” the alliance allows partners to internalize the positive effect of an increase in one
10
firm’s output on its partner’s profit. This internalization of demand externalities is similar to the
mitigation of the “double marginalization” problem in vertical integration (e.g. Tirole, 1988).
Brueckner (2001, 2003) and Brueckner and Whalen (2000) have a clear demonstration of the
problem for an international airline alliance, where double marginalization occurs because each
carrier of an international interline itinerary tries to maximize the profit from its own segment
independently from the other carrier. Consequently, independent carriers typically charge
segment fares higher than a single decision maker controlling prices over the joint itinerary
would. In other words, cooperative pricing of an alliance enables carriers to internalize part of
the double-marginalization externality associated with joint pricing on interline tickets. In our
context, both firms use quantities as their strategy variable and we show the result in a general
setting using recent developments in supermodular games.
The monopoly-pair case serves as a useful base for comparison with the duopoly-pair case.
Whilst the term (1 − α )(π 12 qα1 + π 21 qα2 ) captures a “direct” advantage of complementary alliances,
the “indirect” effect is unique to competing alliances. This effect is represented by the term
(π 31 + π 32 )qα3 + (π 41 + π 42 )qα4 in (16). Since this effect works by indirectly influencing the
behavior of the rival firms (which in turn improves own profits), it may be referred to as the
“strategic effect” of alliance. Observe that this indirect, strategic effect augments the direct
effect – recall that conditions (4) and (5) tend to reinforce each other – that is, parametric shifts
dα will shift both the total and marginal profits in the same direction. Our analysis therefore
suggests that the rivalry between multiple alliances may, owing to the strategic effect, result in a
higher degree of cross shareholding or alliance than would be had in the absence of rivalry, such
as in the monopoly-pair case. 13
13
More specifically, let (α , β ) denote the equilibrium duopoly-pair alliance structure and (α , β ) the
d
d
m
m
optimal monopoly-pair alliance structure. Then, other things being equal, the strategic effect tends to produce
αd ≥αm, β d ≥ β m
when the alliance yields large negative synergies for the participants. The negative
synergies of alliance may arise due to partners’ opportunistic behavior or to their incompatible structures in financial
11
The above discussion is useful in explaining Proposition 1. A basic insight of Proposition 1
is that to obtain the monotone comparative static results on the Cournot equilibrium, the
complementarity conditions (supermodularity and increasing differences) are the key ingredient.
For illustration, consider the creation of an alliance between one peanut butter firm and one jelly
firm. Since an increase in the output of peanut butter will increase the demand for jelly, the
peanut butter firm will produce more after being allied with the jelly firm. In other words, this
alliance will lead these two firms to internalize demand externalities. With the rising marginal
profits for both firms, their reaction functions will shift so as to produce more output, holding the
output choice of their competitors constant. Since the outputs of competing alliances are
strategic substitutes, the other two firms will react by reducing their output. 14
Although Proposition 2 has established that full alliance will be the dominant strategy for
two firms offering complements, it is not always true, shown in Proposition 3 below, that
alliance partners are better off if they both form full alliance than if they both choose partial
alliance or remain independent.
Proposition 3. Rivalry between different alliances can result in a Prisoners’ Dilemma for firms.
The proof of Proposition 3 is given in the Appendix. To explain the intuition behind the
result, it is useful to first examine the effect of an alliance on the rival’s profit. From (15) and (7),
φα3 + φα4 = [π 134 qα1 + π 234 qα2 + π 334 qα3 + π 434 qα4 − β (π 14 qα1 + π 24 qα2 + π 34 qα3 + π 44 qα4 )]
[
]
+ π 143 qα1 + π 243 qα2 + π 343 qα3 + π 443 qα4 − β (π 13 qα1 + π 23 qα2 + π 33 qα3 + π 43 qα4 ) .
Since π 334 = 0 by (12) and π 443 = 0 by (13), rearranging then yields:
status, asset composition, labor contracts and management style (e.g. Parkhe, 1993). Further, these negative effects
can be fixed effects in the sense that they are independent of the level of output. Since the fixed effects are additive
constraints and do not affect most of the results, they are assumed away from the analysis presented in the paper.
14
This is classic behavior in quantity competition: Here, forming an alliance allows the allied firms to be more
aggressive. We add to the analysis by looking at the issue when there are four firms interacting with each other,
rather than the usual two firms interacting with each other.
12
φα3 + φα4 = [(π 13 + π 14 )qα1 + (π 23 + π 24 )qα2 ] + [(1 − β )π 34 qα3 + (1 − β )π 43 qα4 ] .
(17)
Noticing π 13 + π 14 < 0 , π 23 + π 24 < 0 by (8) and π 43 > 0 , π 34 > 0 by (4), so by Proposition 1 the
four terms in (17) are all negative. That φ β1 + φ β2 < 0 can be similarly shown, leading to:
Proposition 4. The strengthening of an alliance by partners reduces the profit of their rival
firms.
Now consider the possibility of an alliance between a peanut butter firm and a jelly firm.
This alliance will make the two firms increase their output; as a consequence, the competing
peanut butter and jelly firms will reduce theirs (Proposition 1). The end result of increased
production of the allied firms and reduced production of the rival firms will, according to (17),
reduce the profit of the rival firms (Proposition 4). This negative pecuniary externality on the
rival firms is not taken into consideration by the alliance firms when they decide on the degree of
alliance between them. When both pairs choose full alliance, the strategic gains tend to offset
each other while industry output tends to rise beyond the levels produced under partial alliances
or independence. If the resulting prices are sufficiently low, then full alliances reduce overall
industry profits relative to partial alliance or independence, giving rise to a Prisoners’ Dilemma.
Next, we investigate the effect of alliances on price and total surplus. For this purpose, we
assume that products 1 and 3 (similarly, 2 and 4) are perfect substitutes; thus there are two
markets: a (say) peanut butter market consisting of output q1 + q3 with price p1 , and a jelly
market consisting output q 2 + q 4 with price p 2 . The vector of changes in total equilibrium
output in each market can be denoted as:
∂q~
∂ ⎡ q1 + q 3 ⎤ ∂ ⎡ q1 ⎤ ∂ ⎡q 3 ⎤
≡
⎢
⎥=
⎢ ⎥+
⎢ ⎥.
∂α ∂α ⎣q 2 + q 4 ⎦ ∂α ⎣q 2 ⎦ ∂α ⎣q 4 ⎦
(18)
We have the following result (the proof is given in the Appendix):
Proposition 5. Assuming homogeneous products 1 and 3 (2 and 4), a complementary alliance
13
increases total output, and hence reduces price, in at least one market. In the case of symmetric
firms where four firms face symmetric demands both within a market and across the markets and
have the same costs, the alliance increases total output and reduces price in both markets.
Thus consumers in at least one market are better off (in terms of lower price) following a
complementary alliance, and are better off in both markets if firms are reasonably symmetric. To
examine the effect on total surplus, we consider a partial equilibrium framework in which
consumer demand is derived from a utility function that can be approximated by the form
u (Q) + z , where z is expenditure on a competitively supplied numeraire good. The welfare
4
function is then given by W = u (Q ) − ∑ ci ( qi ) . Differentiating W with respect to α and using
i =1
∂u / ∂qi = p i , we obtain:
4
Wα = ∑ ( p i − ci' )qαi .
(19)
i =1
The signs of the mark-up terms, p i − ci' , are positive by the first-order conditions, while the qαi
terms are either positive or negative depending on i. As a result, the sign of Wα is in general
ambiguous, depending on the magnitude of two positive ( p i − ci' )qαi terms ( i = 1,2 ) relative to
that of two negative ( p j − c 'j )qαj terms ( j = 3,4 ). However, Wα > 0 for symmetric firms:
Proposition 6. Assuming homogeneous products and symmetric firms, a complementary
alliance improves social welfare.
Although the rivalry between complementary alliances will likely improve economic
welfare, a complementary alliance might be used to deter entry; as such, it can reduce welfare.
Specifically, suppose now that a pair of firms, say, 1 and 2, have an exogenously given
opportunity to choose its alliance structure prior to the entry (and alliance) decision of firms 3
and 4, and that there exists a sunk entry cost. The entry cost can include market-specific
irrecoverable research, advertising and promotional expenditures, as well as investments in
setting up the network and other initial operations. Then Proposition 4 indicates that there exists
14
a range of entry costs, such that rival entry will be pre-empted if and only if the incumbents form
an alliance. An incumbent pair might then ally if it is better off with alliance and no entry than
with no alliance and entry. Indeed, numerical examples show that the threat of entry alone can
result in a complementary alliance, in the sense that forming an alliance is not profitable (owing,
for example, to negative synergies) in the absence of entry. 15
4. Concluding Remarks
We have examined alliance incentives for competing strategic alliances. A complementary
alliance confers a strategic advantage by allowing alliance partners to credibly commit to greater
output, owing to both within-alliance complementarities and cross-alliance substitutabilities.
The strategic effect of a complementary alliance tends to augment its positive direct effects on
profit that arises from the mitigation of the double-marginalization problem. Even if an alliance
creates other negative synergies, it might be pursued, either because it is a dominant strategy or
because it would deter entry. Although sometimes it may be profit dissipating, rivalry between
complementary alliances tends to enhance economic welfare, because the strategic effect results
in a higher degree of alliance, and hence greater output levels, than would be the case in the
absence of such rivalry. 16 We also found that a complementary alliance can arise purely for the
purpose of entry deterrence; such an alliance might reduce total surplus since it creates negative
synergies for incumbents and it reduces market competition. In these cases, the socially optimal
level of alliance would balance the social costs with the social benefits of internalizing demand
externalities. Partial alliance (or limited cross shareholding/control) may arise because of
15
This result has a strong tie to Dixit’s (1980) insight on the role of capital investment in entry deterrence.
Essentially, incumbent firms that choose to fully ally with each other are trading off higher fixed costs for higher
total and marginal profit effects arising from complementarity externalities. The higher marginal profit credibly
commits the alliance to produce more, thereby reducing the profitability of entry.
16
It is worth noting that these results are quite close to those of Brander and Spencer (1983) and Spencer and
Brander (1983). In particular, firms seek commitment devices to increase output in Cournot rivalry, but this, while
raising welfare, can make firms worse off in equilibrium.
15
external restrictions imposed by regulators on the interfirm relationship.
We have concentrated our analysis on complementary alliances which, apart from the
aspect of entry deterrence, generally lead to improved welfare. In contrast, horizontal alliances,
which are formed by firms who produce substitutes, tend to be anti-competitive. Indeed, we can
show that a horizontal alliance may reduce competition not only in the market where prior
competition between the partners takes place, but also in other markets of the alliance network –
and hence may be restricted by regulatory agencies. Furthermore, a real-world alliance is likely
to have both complementary and horizontal elements; consequently, such a “hybrid” alliance
may have both pro- and anti-competitive effects. For example, an international airline alliance is
likely to reduce competition in the hub-to-hub markets in which both partners operate and
usually dominate, while increasing competition in the connecting markets to which the partners
provide cheaper airfares. The welfare implications are generally less definitive for hybrid
alliances than for the other two alliance types. Existing work by Brueckner and others focuses on
specific structures, and a more complete treatment of hybrid alliances in a general setting of
competing alliances remains a subject of future research.
In this article, the results of complementary alliances are driven by the demand
complementarities underlying (4) and (5). However, (4) and (5) can also arise from cost
complementarities in producing multiple outputs. Our results will continue to hold in this case.
Cost interaction brings in another important aspect of strategic alliances: alliance partners
usually coordinate the use of inputs. It would be interesting to examine the impact of strategic
alliances on partners’ productivity and their competitiveness in product-market competition.
The framework of this paper may be useful in addressing strategic network rivalry in other
situations. One potential area of application is in supply chain management and logistics. Supply
chain management typically lies between full integration (when one firm manages the entire
material and information flow) and independent operation of each channel member. Therefore,
coordination to reduce costs and risks between the various players in the chain is key to its
16
effective management. Often the result is a mutually beneficial logistics partnership or strategic
alliance. It would be interesting to analyze the rivalry of one supply chain against another supply
chain, in a way similar to our analysis of the rivalry between different strategic alliances.
Acknowledgement: We are very grateful to two anonymous referees and especially the editor
(Simon Anderson) whose comments have led to a significant improvement. Earlier versions of
this paper have been presented at University of British Columbia, University of Calgary,
American Economic Association’s Transportation and Public Utility Group Conference, Kobe
University, and the 7th Air Transport Research Society Conference. We thank the seminar
participants and Xuan Zhao for helpful comments. Financial support from the Social Science
and Humanities Research Council of Canada (SSHRC) is gratefully acknowledged.
17
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19
Appendix
Proof of Proposition 3: Consider the auxiliary function, φ 1 (α , α ) + φ 2 (α , α ) , which is, by (15),
φ 1 + φ 2 = π 12 (Q(α , α );α ) − απ 2 (Q(α , α )) + π 21 (Q(α , α );α ) − απ 1 (Q(α , α )) .
(A1)
Totally differentiating (A1) with respect to α and applying (10) and (11) yields:
d (φ 1 + φ 2 )
dq1
dq 2
dq 3
dq 4
= (1 − α )π 12
+ (1 − α )π 21
+ (π 31 + π 32 )
+ (π 41 + π 42 )
dα
dα
dα
dα
dα
(A2)
where dq i (α , α ) / dα represents total differentiation, i = 1,2,3,4 . The signs for the terms in front
of dq i / dα terms are fixed (i.e. either positive or negative). It remains to see whether we can
sign dq i / dα . Totally differentiating the first-order conditions (10)-(13) with respect to α after
replacing β with α , solving for dQ / dα and substituting, we obtain:
[
]
dQ
12
21
34
43 d ∏
= −( I − R ) −1 diag π 11
, π 22
, π 33
, π 44
dα
dα
[
(A3)
]
where dQ / dα is a column vector, and d ∏ / dα = π 12 , π 21 , π 34 , π 43 is a column vector and its
four elements are, by (4), all positive. In (A3), I is the identity matrix and
⎡ 0
⎢ 21
R
R ≡ ⎢ 21
⎢ R3134
⎢ 43
⎢⎣ R41
R1212
R1312
0
R3234
R2321
0
R4243
R4343
R1412 ⎤
⎥
R2421 ⎥
R3434 ⎥
⎥
0 ⎥⎦
with RijK ≡ −(π iiK ) −1 π ijK , for i ≠ j . Note, e.g. Rij12 is the derivative of firm 1’s reaction function.
By (5), (9) and the second-order condition, matrix R has the following sign pattern:
⎡0
⎢+
R=⎢
⎢−
⎢
⎣−
+ −
0 −
− 0
− +
−⎤
− ⎥⎥
.
+⎥
⎥
0⎦
(A4)
We now determine the sign of [I – R]-1. The stability of the Cournot-Nash equilibrium implies
that the magnitude of the eigenvalues of R must be less than unity (Zhang and Zhang, 1996).
Hence, by Neumann Lemma, [I – R]-1 exists and [I − R ] = I + R + R 2 + L + R n + L . Hence,
−1
by (A4), [I – R]-1 must have the following signs:
20
⎡+
⎢+
−1
[ I − R] = ⎢
⎢−
⎢
⎣−
+
+
−
−
−
−
+
+
−⎤
− ⎥⎥
.
+⎥
⎥
+⎦
(A5)
Given the sign pattern of [I – R]-1, each element of dQ / dα will be the difference of two positive
numbers and, hence, will in general have an ambiguous sign. It then follows from (A2) that the
sign of d (φ 1 + φ 2 ) / dα is ambiguous, and in the cases where (such numerical examples exist):
φ 1 (1,1) + φ 2 (1,1) < φ 1 (α , α ) + φ 2 (α , α ) , φ 3 (1,1) + φ 4 (1,1) < φ 3 (α , α ) + φ 4 (α , α )
for 0 ≤ α < 1 , a Prisoners’ Dilemma will rise.
Q.E.D.
Proof of Proposition 5: Differentiating (10)-(13) with respect to α yields:
Φ
∂Q (α , β ) ∂ ∏
+
=0
∂α
∂α
(A6)
where
12
⎡π 11
⎡ q1 (α , β ) ⎤
π 1212 π 1312 π 1412 ⎤
⎢ 21
⎥
⎢ 2
⎥
π
π 2221 π 2321 π 2421 ⎥
⎢q (α , β )⎥ ,
,
Q
α
β
(
,
)
≡
Φ ≡ ⎢ 21
34
⎢π 31
⎢ q 3 (α , β ) ⎥
π 3234 π 3334 π 3434 ⎥
⎢ 43
⎥
⎢ 4
⎥
43
43
43
⎣⎢q (α , β )⎦⎥
⎣⎢π 41 π 42 π 43 π 44 ⎥⎦
⎡π 12 ⎤
⎢ ⎥
∂ ∏ ⎢π 21 ⎥
.
≡
∂α ⎢ 0 ⎥
⎢ ⎥
⎣⎢ 0 ⎦⎥
Partition matrix Φ into:
⎡Φ
Φ = ⎢ 11
⎣Φ 21
Φ 12 ⎤
Φ 22 ⎥⎦
where Φ ij is a 2-by-2 matrix. Solving the second equation in (A6) and substituting then yields:
∂
∂α
1
⎡q 3 ⎤
34 ∂ ⎡ q ⎤
⎢ 4 ⎥ = R12
⎢ ⎥,
∂α ⎣q 2 ⎦
⎣q ⎦
where R1234 ≡ −Φ −221 Φ 21 . By the stability condition the norm of matrix R1234 must be less than unity
(Zhang and Zhang, 1996). If follows that either q α3 < q α1 or q α4 < q α2 , and by Proposition 1,
∂q~ / ∂α must have at least one positive element. In the case of symmetric firms, dq 1 = dq 2 and
dq 3 = dq 4 , thereby implying ∂q~ / ∂α > 0 and hence prices fall as α increases.
21
Q.E.D.