Comments are welcome.
Stable Market Structure and Sufficient Conditions for Horizontal Mergers By Jingang Zhao* January 24, 2001 Department of Economics Iowa State University 260 Heady Hall Ames, Iowa 50011-1070
[email protected] Fax: (515) 294-0221 Tel: (515) 294-5245 Abstract: Consider the four possible mergers in a market with three firms: 12, 13, 23, and 123. Which one, if any, will take place? Although the question seems fairly simple, it has not yet been fully answered. The existing literature provides two necessary conditions for a merger: profitability and a non-empty core (i.e., a merger will not be formed if it is unprofitable or if its core is empty). This paper shows that a stable partition of the firms, which is immune to breakups and new mergers, is sufficient for mergers in the partition to take place. The paper provides a tractable condition for checking stability. By applying the condition, the paper completely characterizes stability in all three firm linear markets, and it partially characterizes stability in n-firm markets. JEL Classification Number: C62, C71, C72, D43, L10
Keywords: Core, Cournot/Nash equilibrium, hybrid solution, merger, stable market structure
*
I would like to thank Beth Allen, Harvey Lapan, Herbert Scarf, Donald Smythe, Rajiv Vohra, and
seminar participants at the Fall 2000 Midwest Economic Theory and Trade Meeting at U. Minnesota for their encouragements and valuable comments. An excel program for computing the optimal and stable partitions in linear markets with n = 3 (it has been used to compute Examples 1-6) is available from the author upon readers’ requests. All errors, of course, are my own.
1. Introduction Consider the four possible mergers in a market with three firms: 12, 13, 23, and 123. Which one, if any, will take place? Although the question looks fairly simple, one needs to go beyond both Nash equilibrium and the core and use the concept of a hybrid equilibrium (Zhao, 1992; see Allen (2000) for the significance of hybrid games) to find an answer. The existing literature provides two necessary conditions for a horizontal merger: profitability (i.e., observed mergers must be profitable), and a non-empty core (i.e., observed mergers must have a non-empty core). In contrast, this paper provides a sufficient condition: mergers will take place if they form a stable market structure -- or, in other words, a stable partition of the firms. Roughly speaking, a partition is stable if it is immune to breakups and to new mergers. This rules out any incentives, for any coalitions, including those consisting of individual firms, to form a new partition1. In order to exhaust all possible deviations, this paper first converts an oligopoly market to a partition function game (Thrall and Lucas, 1963), and it establishes a tractable necessary and sufficient condition for the stability of the grand coalition (i.e., monopoly) and a general partition. The paper then uses the condition to characterize stable partitions based on cost and demand parameters in linear markets with n firms, and it completely characterizes stability in three firm linear markets. This paper extends work on stable partitions and cartel stability by a number of previous authors. Since Thrall and Lucas (1963), scholars have approached the stable partition problem by representing stable partitions as the equilibria of new, derived games. Shenoy (1979) represents a stable partition (in coalitional and partition function games) as the core of a derived abstract game; Hart and Kurz (1983) use partition values to represent stable partitions as strong equilibria of a partition formation game due to von Neumann and Morgenstern (1944; Section 26 on pages 243-45). More recently, Bloch (1996) and Yi (1997) use fixed distribution rules to represent stable partitions as Nash equilibria of sequential games; Ray and Vohra (1999) tackle the same problem using stationary subgame perfect equilibria of a bargaining game without any fixed distribution rules; Zhao (1996)
1
Such internal stability differs from the external stability or comparative statics of Hahn (1962) and
Kohlberg and Mertens (1986). See van Damme (1987) for survey.
2
and Ray and Vohra (1997) study stable partitions in normal form games2. Study of the cartel stability problem began with a discussion by Postlewaite and Roberts (1977) and the paper by d'Aspremont, Jacquemin, Gabszewicz and Weymark (1983), followed by those of Donsimoni, Economides and Polemarchakis (1985), and Donsimoni (1986). These studies evaluate the incentives to breakup and enlarge a cartel by one firm, assuming the cartel is a price leader and fringe firms are price takers. Recently, Ellison (1994), and Griffin and Xiong (1997) examine the cartel stability empirically, and Shaffer (1995) extends the model to Cournot fringe firms3. As readers will see, this paper pushes the earlier works to the limit by exhausting all possible breakups and enlargements. What is missing from the literature is a tractable condition that can be used to check the stability of all partitions. As a result, most of the earlier studies covered only a subset of the possible deviations, and the resulting equilibrium partitions are often only partially stable4. This paper helps ameliorate the problem by providing a computationally tractable (necessary and sufficient) condition for stable partitions. One advantage of the condition is that it allows one to characterize a stable partition in terms of the economic forces behind the model. Consider, for example, a linear market with three firms. Assume firms 1, 2 and 3 are, respectively, the large, middle and small firms, and assume monopoly merger is 2
See Greenberg and Weber (1993), Perry and Reny (1994), Ferreira (1999), Banerjee, Konishi and
Sonmez (2000), Konishi and Ray (2000), and the Manresa Conference Program (May 1999) organized by Salvador Barbera and Matt Jackson for recent works related to stable partition. Other related works (see Kurz (1988), Greenberg (1994) for survey) include: (1) games with cooperation structure (Myerson, 1977; Aumann and Myerson, 1988; Dutta, Nouweland and Tijs, 1995; Qin, 1996); (2) coalition formation in networks (Jackson and Wolinsky, 1996; Dutta and Mutuswami, 1997), in trade (Riezman, 1985; Tesfatsion, 1995; Bagwell and Staiger, 1997) and in business games (Brandenburger and Stuart, 1996). 3
See also Rajan (1989), Kamien and Zang (1990), Bloch (1995), Horn and Persson (1996),
Gowrisankaran (1999), Nocke (1999), Prokop (1999), Fridolfsson and Stennek (2000), Gowrisankaran and Holmes (2000) on merger formation and cartel stability. 4
See the interesting results of Bloch (1996, Proposition 5.1) and Ray and Vohra (1999, Theorem
3.9). In symmetric linear markets, the two models result in an identical partition with one proper coalition and singletons, given by a minimum coalition size k ≥ (3+2n- 5+4n)/2. More interestingly, the condition coincides with merger profitability in two other studies: [(n+1)2-(m+1)(n-m+1)2] >0 in Salant, Switzer and Reynolds (1983; (3’)) for mergers of size (m+1) in symmetric market, and [(n+1)2-k(n-k+2)2] > 0 in Smythe, Zhao and Heubeck (2000; Proposition 1) for mergers of size k in asymmetric market.
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impossible (due to high merging costs associated with regulations). In this situation, it can be shown that the merger 23 will never be formed (i.e., the partition {1; 23} is always unstable) if the cost difference between 2 and 1 is sufficiently large, and it will be formed if the cost difference is sufficiently small (see Section 5 for complete characterizations). By exhausting all possible deviations, this study captures a key feature of the recent historical changes in market structure. Like the turn of the twentieth century, the recent changes are characterized by waves of horizontal mega mergers; unlike the turn of the twentieth century, recent changes are also accompanied by voluntary breakups5. Following Zhao (1992), such hybrid phenomena, involving simultaneous mergers and breakups, are represented as movements from one partition to another. Such generality implies a wide range of applications in economics. For example, it will be useful to check the stability of the finest partition in ALL previous studies of Cournot equilibria with three or more firms. Because it covers all deviations, stable partition theory provides an answer to the fundamental question “what is the solution for a game”: If players’ rationality is restricted to non-cooperative behavior -- or, in other words, the finest partition (the coarsest partition, a general partition ∆), then the solution for a game is its Nash equilibrium (its core, its hybrid equilibrium for ∆). If players’ rationality is free of the above restrictions, then the only solution for a game is the stable partition (with its choices and allocations). The paper is organized as follows. Sections 2 and 3 define the problem and the concept of stable partitions; Section 4 establishes key conditions for a stable partition in games with n players and in markets with n firms; Section 5 characterizes stability in three firm linear markets; Section 6 concludes, and the Appendix provides all the proofs. 2. The Problem This section reviews two necessary conditions for a merger and a new core method (Zhao, 2001) for proving the main results.
An oligopoly market with n firms for a
homogeneous good is equivalent to a normal form TU (transferable utility) game:
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For example, the largest twenty mergers in history are all created during the three year period
November 1997-2000. Zhao (1998) has a list of these largest mergers. High profile breakups includes AT&T (98), AT&T (00), and Pepsi’s spinning off its tri-restaurants business.
4
(1)
Γ = {N, Zi, π i},
where N = {1, 2, ..., n } is the set of firms; Zi = [0, zi], i∈ N, is i's production set, zi is its capacity (0 < zi < ∞); π i(x) = p(Σ xj) xi – Ci(xi), x = (x1, ..., xn) ∈ Z = Πi∈NZi, is its profit function, Ci(xi), xi∈Zi, its cost function, and p(Σ xj) the inverse demand function. A market structure is a partition of the firms or “the number and size distribution of mergers”, which includes as a polar case the finest partition ∆0 = {(1), ..., (n)} (i.e., Bain’s definition as the number and size distribution of firms, 1959). Let ∆ = {S1, ..., SJ} be a general partition (i.e., ∪Sj = N and Si∩Sj = ∅ for all i ≠ j), then the mergers S1, ..., SJ are a change from ∆0 to ∆, and a merger S with k firms (k = |S| < n) is a change from ∆0 to ∆S = {S, (i1), ..., (in-k)}. The following assumption, A0, is used throughout the paper. A0 (Assumption 0):
(i) p(⋅) is decreasing, and each π i(x) is continuous in x and is
quasi-concave in xi; (ii) at any equilibrium, the optimal supply by any coalition of firms is an interior solution; (iii) for each merger S ⊆ N, its capacity and cost function are: (2)
zS = Σ j∈Szj, and
(3)
CS(q) = Min {Σ j∈SCj(xj) | q = Σ j∈Sxj ≤ zS, xj ≥ 0, j∈S}. The above capacity makes the core of a merger non-trivial, and it adds a new
dimension to merger study. The cost function (3) assumes “no synergy” (see Farrell and Shapiro, 1990) (i.e., S could use the most efficient technology up to zS = Σj∈Szj)6. ~ ~ ~S(∆) | S∈∆} and π Let ~ x(∆) = {x (∆) = {π S(∆) | S∈∆} be post-merger supply and the
~;θ} unique7 post-merger profits for ∆. The merger contract for each S ∈∆ is a pair {x S S
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No capacity is considered in Salant, Switzer and Reynolds (1983), Perry and Porter (1985), and
Farrell and Shapiro (1990). Without capacity, v(S) defined in (4) is zero, therefore the core would be trivial. One may assume stronger synergies (like Perry and Porter, 1985) or no “weak synergy” (i.e., replacing "Σ j∈Sxj ≤ zS" in (3) by "xj≤ zj for all j∈S"), these merit separate future studies. 7
See Zhang and Zhang (1996) for uniqueness conditions.
If there are multiple equilibria, the
πS = πS(xS(x-S), x-S) contracts can be changed as a "best response" (xS(x-S); θS(x-S)): (i) 0≤ Σ j∈Sxj(x-S) ≤ zS; (ii) ~ πS. See Zhao (1996) for more discussions. ≥ πS(yS; x-S) for all 0≤ Σ j∈Syj ≤ zS; (iii) θi ≥ 0, Σi∈Sθi = ~
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~ with θS ≥0 and Σ j∈Sθj = π S(∆), specifying its members, production, and profit allocation.
This covers all mergers including full mergers and virtual mergers like airline alliances. Let the guaranteed profit for a merger S (S ⊆ N) be given by (4)
v(S) = Max Min Σi∈Sπi(xS, y-S) = Min MaxΣi∈Sπi(xS, y-S), x y y x S
-S
-S
S
where (xS, y-S) is a vector w∈ Rn such that wi = xi if i∈S, = yi if i∉S; the Min is taken over Z-S = ∏j∉SZj; and the Max over {xS∈RS+ | Σ j∈Sxj ≤ zS}. In general, one would have vα(S) = Max Min Σi∈Sπi(xS, y-S) < vβ(S) = Min MaxΣi∈Sπi(xS, y-S), x
S
y
y
-S
-S
x
S
indicating that the β-core is a proper subset of the α-core (Aumann, 1959). As shown in Zhao (1999), there is no need to make the α- and β-distinction in oligopoly study, and one can simply use the term core (i.e., α-core = β-core always holds in an oligopoly market). π and Definition 1: A profit vector θ ∈ Rn+ is in the core of the market (1) if Σ θi = −
Σi∈Sθi ≥ v(S) for all S ≠ N, where v(S) is given by (4) and −π is the monopoly profit. The core selection problem is a separate issue, it could be resolved by values or bargaining8. The core of the monopoly merger is the same as the core of the market (1). Similarly, the core of a merger S ∈∆ is the core of the following strategic TU game9: (5)
ΓS(~x-S) = {S, Zi, πi(xS, ~x-S)},
where πi(xS, ~ x-S) = p(Σ j∈Sxj+Σ j∉S~ x j )xi – Ci(xi), i∈S, are parameterized by ~ x-S. Definition 2 below defines the equilibrium at a market structure ∆ = {S1, ..., SJ}.
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The core is “basic to the theory”, because “other solutions gain some support if it can be shown that
they are in some way related to it” (Maschler, Peleg and Shapley, 1979, p. 305). Its selection is solved by values like nucleolus (Schmeidler, 1969), Shapely value (1953), and τ-value (Tijs, 1981), and by solutions in the bargaining problem (d, X), where di = v(i), all i, and X = C(Γ) is the core in Γ = {N, v(.)}. 9
~ ~ ,~ ~ Precisely, θS is in the core of (5) if Σj∈Sθj = πS = πS(x S x-S) and Σj∈Tθj ≥ v(T, x-S) for all T≠S, where
x-S) = [p(Σ j∈Txj +Σ j∈S/Tyj +Σ j∉S~ xj )Σ j∈Txj – CT(Σ j∈Txj)]. v(T, ~ x-S) = Max Min πT(xT, yS/T, ~ xT yS/T
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Definition 2: The market equilibrium at ∆ = {S1, ..., SJ} is a list of merger contracts ~, θ) = (x ~(∆),θ(∆)) = {(x ~ ,θ )|S∈∆} such that each θ is in the core of Γ (~ (x S S S S x -S) given by (5). Therefore, an equilibrium is reached if each S∈∆ maximizes its joint profit and splits it within its core. Such a hybrid equilibrium (Zhao, 1992) includes the core and Cournot equilibrium as two polar cases (when ∆ = ∆m or ∆0)10. Lemma 1 below reviews the known core results, where a linear market11 is a (2n+1)-vector (a, c, z) ∈R2n+1 ++ , with a>0 as the intercept of inverse demand, c = (c1, ..., cn) ∈ Rn++ as marginal cost (c1≤ ... ≤ cn), and z ∈Rn++ as capacity (i.e., p = a-Σ xi, Ci(xi) = cixi, 0 ≤ xi ≤ zi). As shown in Example 1 below, the equilibrium for ∆ captures both internal cooperation within each of its mergers and Nash behavior across the mergers. ~ Example 1: The allocation for ∆ = {1; 23} is: θ1 = π 1(∆), θ2(t) = v2+td23, and θ3(t) = ~ v3+(1-t)d23, where vi and π S(∆) are pre- and post-merger profits, t∈[0, 1] is firm 2’s share of ~ the merger gains d23 = π 23(∆)-(v2+v3). For (a; c; z) = (6; 0.5, 1.05, 2.46; 3, 3, 3) and t = 0.1, ~ ~ (v1, v2, v3) = (4.01, 2.11, 0.002); π 1(∆) = 4.067, π23(∆) = 2.152, d23 = 0.04; and θ(0.1) =
(4.067, 2.114, 0.038). As shown in Example 6, ∆ with θ(0.1) will not be formed, ∆ with ~ θ(0.2) = (4.067, 2.118, 0.034) will be formed (note for ∆3 = {12; 3}, π 12(∆3) = 6.184).
Lemma 1: Suppose parts (i) and (iii) of A0 hold. (I) If the market (1) is linear, then its core and the core of (5) are both non-empty and both have a non-empty relative interior (Zhao, 1998); (II) the core of (1) is non-empty if all π i(x) are concave; (III) the core of (5) is non-empty if π i(xS, x-S), i∈S, are all concave in xS (Zhao, 1999). 10
If the core is empty, one can use weaker concepts like kernel (Davis and Maschler, 1965) and
bargaining set (Aumann and Maschler, 1964; Zhou, 1994). This has been extended to hybrid solution with a distribution rule (HSDR; Zhao, 1996, 1999). Let D = {core, equal share, … } be a finite set of distribution ~ ,θ ) (S∈∆) such rules, then a DR (distribution rule) for ∆ is a map from ∆ to D. A HSDR for ∆ is a list (x S S that each S∈∆ maximizes its joint profit and θS is a split according to DR(S). 11
In a linear market, a firm's profit is π i(x) = p(x) xi – ci xi = (a-Σ xj)xi–cixi, part (iii) of A0 is
simplified as: cS = Min{cj | j∈S}, CS(q) = cS q, 0 ≤ q ≤ zS = Σ j∈Szj.
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Part (II) is an extension of Scarf’s α-core theorem (1971) to TU β-core. Lemma 2 ~ below lists two necessary conditions for a merger, where x^, πj(x^), ~ x (∆), and π (∆) are pre-
_ and post-merger supply and profits, v(S) is given by (4), and x is monopoly supply. Lemma 2: (I) Suppose a monopoly merger (N; − x ; θ) is observed in (1). Then θ satisfies: (i) θj > πj(x^) for all j; and (ii) Σj∈Sθj ≥ v(S) for all S⊆N. (II) Suppose a list of ~ ; θ ), S∈∆ = {S , ..., S }, are observed in (1). Then for each S∈∆, θS satisfies: mergers (x S S 1 J πS(∆) and θj > πj(x^) for all j∈S; and (ii) θS is in the core of (5). (i) Σj∈Sθj = ~ As shown in Example 2, the two necessary conditions are independent. The lemma is useful in understanding the recent three-way A.P.A. merger (August 1999), which passes Alcoa as the leader in Aluminum (it has sales revenues of $21.6 billion per year). Its shares are split as (44%, 29%, 27%) among Canadian Alcan, French Pechiney and Swiss Algroup. By the lemma, such a split must be in its core. The main results in this paper are established by the following minimum noblocking payoff method (MNBP; Zhao, 2001), which has been used to prove Lemma 1. Definition 3: The MNBP of a coalitional TU game Γ = {N, v} is given by (6)
ìï Min Σxi MNBP(Γ) = í n ïî subject to x ∈ R+; Σi∈Sxi ≥ v(S) for all S≠N. In other words, the MNBP is the grand coalition’s minimum payoff below which
the core is empty ( i.e., core ≠ ∅ ⇔ v(N) ≥ MNBP). Although characterizing MNBP is a demanding task, the logic behind this new method is fairly intuitive: the core is non-empty if and only if the grand coalition’s payoff is sufficiently large. In addition to such intuition, the MNBP method has the advantage of revealing the core’s relative interior12. 12
The linear programming (6) is fundamentally different from the following one for balancedness:
ì Min Σi∈Nxi í n î subject to x ∈ R+; Σi∈Sxi ≥v(S) for all S ⊆ N; and Σi∈Nxi = v(N); though they look similar. The above is a degenerate problem (i.e., its objective function is a constant and its feasible set is empty when the core is empty), while (6) is non-degenerate and it always has a solution.
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Example 2: For n = 3 and (a; c; z) = (6; 0.5, 0.5, 0.5; 2, 2, 2) ∈ R7++, the two necessary conditions for monopoly both hold: v(N) = 7.56 > MNBP = 4.59, v(N) > Σ π i = 5.67. However, if monopoly has a merging cost MCM = 2, then profitability condition fails and the core condition still holds: MNBP = 4.59 < v(N)-MCM = 5.56 < Σ π i = 5.67. 3. The Stable Market Structure Let Π denote the set of all partitions of N. For each ∆ ∈Π, let its unique post~ ~ (∆) = {π merger profits be π S(∆) | S∈∆}. This translates the oligopoly market (1) into the
following partition function game (Thrall and Lucas, 1963): (7)
ΓPF = {N, φ},
~ where for each B ∈ Π, the game specifies a joint profit φ(B, S) = π S(B) for each S∈ B.
~ , θ ) | T∈∆,}, consider the deviation by a S∉∆. Let Given ∆ and its equilibrium {(x T T (8)
Π(S) = {B ∈Π | B = {S, T1, ..., Tm}}
denote the set of partitions of which S is a member. S has incentives to move to a partition ~ B ∈ Π(S) if its profits at B exceed its current joint profits (i.e., if φ(B, S) = π S(B) > Σj∈Sθj).
Therefore, "whether S ∉ ∆ will deviate from ∆" depends on which B ∈Π(S) will be formed. Because S has no control over the partition of N/S, its action depends crucially on its belief. We examine three such beliefs: cautious belief, complete breakup belief, and loyal relation belief, which lead to three notions of stability. Though it is customary to say “a partition ∆ is stable,” the precise definition is “its merger contracts are stable” or “a partition ∆ with a profit allocation θ is stable.” Under the cautious belief, the deviating S is prepared for the worst partition: it could receive the least (among Π(S)) at the new structure. Precisely, the worst partition (9)
Bα(∆,S) ≡ Bα(S) = {S, T1, T2, ..., Tmα}
is the solution of Min {φ(B, S) | B ∈Π(S)}, which is a constant for all ∆ S. ~, θ) = (x ~(∆), θ(∆)) be an equilibrium at ∆ in the market (1). The Definition 4: Let (x
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partition ∆ with the allocation θ is stable in the sense of cautious belief (i.e., it is α-stable13) if Σj∈Sθj ≥ φ(Bα(S), S) for all S⊆N and S∉∆, where φ is given by (7). To put it differently, mergers in ∆ will be formed under the cautious belief if there is no new merger S whose joint profits at all B ∈Π(S) exceed its current profits. Under the complete breakup belief, a coalition would breakup into singletons if it has any deviator to join S, and it remains unchanged otherwise. Precisely, this is given by (10)
Bγ(∆,S) = {S, T1, T2, ..., Tmγ},
where for i = 1, …, mγ, Ti = Sj for some j with Sj∩S= ∅, = (ij)⊆ Sj for some j with Sj∩S≠ ∅. ~, θ) = (x ~(∆), θ(∆)) be an equilibrium at ∆ in the market (1). The Definition 5: Let (x partition ∆ with the allocation θ is stable in the sense of complete breakup (i.e., it is γstable) if Σj∈Sθj ≥ φ(Bγ(∆,S), S) for all S⊆N and S∉∆, where φ is given by (7). Under the loyal relation belief, if a coalition Sj ∈ ∆ has any deviator to join S, then its remaining firms remain loyal to each other and stick together as a new coalition. Precisely, the new partition under the loyal relation belief is given by (11)
Bδ(∆,S) = {S, T1, T2, ..., Tmδ},
where for i = 1, …, mδ, Ti = Sj / S = {i | i∈ Sj and i∉ S} for some j. Note Ti = Sj for some j with S∩Sj = ∅, and Ti ⊂ Sj for some j with S∩Sj ≠ ∅. ~, θ) = (x ~(∆), θ(∆)) be an equilibrium at ∆ in the market (1). The Definition 6: Let (x partition ∆ with the allocation θ is stable in the sense of loyal relation (i.e., it is δ-stable) if
Σj∈Sθj ≥ φ(Bδ(∆,S), S) for all S⊆N and S∉∆, where φ is given by (7). To see the difference between the complete breakup and loyal relation beliefs, let n = 5, S = (1,2). For ∆ = {1, (2,3,4,5)}, Bγ(∆,S) = {(1,2), 3, 4, 5}, Bδ(∆,S) = {(1,2), (3, 4, 5)}. It is useful to see what an unstable partition means. A partition ∆ with θ is α-unstable (by the cautious belief) if there is a merger S (∉∆) having higher joint profits at all B∈ Π(S). Similarly, it is γ- (δ-) unstable by the complete breakup (loyal relation) belief if there is an S 13
We follow Aumann's tradition (1959) of labeling stable concepts by Greek letters. See Thrall and
Lucas (1963), Hart and Kurz (1983), and Zhao (1996) for more discussions.
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(∉∆) having higher joint profits at Bγ(∆,S) (at Bδ(∆,S)). The relation among these three notations of stabilities is summarized in the following Remark 1. Remark 1: (i) Let ∆ be a market structure and θ be its profit allocation in the market (1). If (∆, θ) is δ-stable, then (∆, θ) is γ-stable; if (∆, θ) is γ-stable, then (∆, θ) is αstable. (ii) If (∆, θ) is α-stable, then θ is in the core of the market (1). By the remark, loyal relation belief is stronger than complete breakup belief, which is stronger than cautious belief. The α-stability of any ∆ refines the core of (1) (i.e., φ(Bα(S),S) > v(S)), because a deviating S under α-stability is prepared for the worst partition in Π(S); while in the core (= α-core = β-core), it is prepared for the worst x-S. Alternatively, the above differences in the stability of ∆ can be given below14: Xδ ⊂ Xγ ⊂ Xα⊂ Cβ = Cα ⊂ XE, which is also illustrated in Example 3 in Section 4, where XE, Cα, Cβ are the efficient, αand β-core allocations for ∆m, and Xα, Xγ, and Xδ are the α-, γ-, δ- stable allocations for ∆. 4. Stable Market Structure in n Firm Markets Without merging costs, monopoly profit is the highest among all partitions, each ∆ ≠ ∆m is therefore unstable. We will focus on ∆m, the only candidate for a stable partition without merging costs. For a deviating S ≠ N, let Bα(∆m,S), Bγ(∆m,S) and Bδ(∆m,S) be given in (9)-(11). Under A0, a merger raises non-members’ profits, so S receives the worst joint profits (among Π(S)) when the outsiders are singletons. This leads to Bα(∆m,S) = Bγ(∆m,S) ≡ Bα(S) = {S, (i1), ..., (in-k)}, and Bδ(∆m,S) = {S, N/S}, with |S| = k < n, and N/S = {i | i∉S}. One therefore only needs to check its α- and δstabilities. The minimum no blocking profits against the α- and δ-deviations are given by 14
This is an extension of Hart and Kurz’s result (1983). For ∆m in the market (1), the relation
becomes: Xδ ⊂ Xγ = Xα ⊂ Cβ = Cα ⊂ XE (i.e., one now has Xγ = Xα); for ∆m in a general normal form game, it becomes: {Xδ ∪ Xγ} ⊂ Xα ⊂ Cβ ⊂ Cα ⊂ XE (i.e., one no longer has Cβ = Cα and Xδ ⊂ Xγ).
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(12)
MNBPα = {Min Σxi | x∈Rn+ ; Σi∈S xi ≥ φ(Bα(S), S) for all S ≠ N},
(13)
MNBPδ = {Min Σi∈Nxi | x∈Rn+ ; Σi∈S xi ≥ φ(Bδ(∆m,S), S) for all S ≠ N}. In Theorem 1, the statement “the monopoly ∆m is stable” is meant “there exists an
allocation of the monopoly profits θ such that ∆m with θ is stable.” Theorem 1: Suppose A0 hold in (1), let v(N), MNBPα, and MNBPδ be given by (4), (12) and (13). (i) The monopoly merger ∆m is stable in the sense of cautious belief ⇔ v(N) ≥ MNBPα; (ii) ∆m is stable in the sense of loyal relation ⇔ v(N) ≥ MNBPδ. In other words, monopoly will be formed under the cautious (loyal relation) belief if monopoly profits exceed the minimum no blocking profits MNBPα (MNBPδ). The above sufficient conditions for ∆m will not hold if the monopoly incurs merging costs15. Let MC(∆0, ∆) denote the merging costs for the mergers in ∆ (i.e., the costs of moving ∆0 to ∆), then the monopoly merging costs can be denoted by MCM ≡ MC(∆, ∆m) for all ∆ ≠ ∆m. Remark 2: Given the market (1), suppose (i) A0 hold; (ii) MCM > 0; (iii) for all ∆', ∆’’ ≠ ∆m, MC(∆', ∆’’) = 0; (iv) the total profits at any ∆ ≠ ∆m is less than (v(N)-MCM). Then, (i) ∆m is stable in the sense of cautious belief ⇔ (v(N)-MCM) ≥ MNBPα; (ii) ∆m is stable in the sense of loyal relation belief ⇔ (v(N)-MCM) ≥ MNBPδ. Example 3: For (a; c; z) = (6; 0.5, 0.5, 0.5; 2, 2, 2) ∈ R7++ (as in Example 2), MNBPδ = 10.08 > v(N) = 7.56 > MNBPα = MNBPγ = 5.67, so monopoly is α- and γ-stable but not δ-stable. However, with MCM >1.89, monopoly becomes α-unstable (i.e., v(N)MCM < 5.67 = MNBPα). By MNBPcore = 4.59 < MNBPα = 5.67 < MNBPδ = 10.08, δstability is stronger than α-stability, which is stronger than the core. Another advantage of the MNBP method is the following result on the sensitivity or the comparative statics of a stable monopoly merger16. 15
Such costs could be significant. For example, the largest known merging cost is the $1.8 billion
break-up fee (paid to American Home for dropping its agreement with Warner-Lambert) in the $90 billion Pfizer/Warner-Lambert deal (February 2000). 16
Precisely, a stable ∆m in (a, c, z) = t ∈R2n+1 ++ remains stable against small shocks if there exists ε > 0
12
Remark 3: Suppose A0 hold and there is no merging cost. An α- (δ-) stable ∆m remains as a stable partition against small shocks in the market if and only if v(N) > MNBPα (v(N) > MNBPδ). In other words, an α- (δ-) stable monopoly merger is subject to breakdown for some shocks in the market if and only if v(N) = MNBPα (v(N) = MNBPδ). These two remarks on merging costs and sensitivity can be similarly extended to Theorems 2-6, we therefore will not repeat these extensions in the rest of this paper. As shown in Example 3, the α- and γ-stabilities of ∆m = {(1, ..., n)} are identical. However, for a partition ∆ ≠ ∆m, its α- and γ-stabilities will generally be different. Given ∆ ≠ ∆m, assume its total profits exceed the net monopoly profits: Σ θj > v(N)-MCM (this rules out deviation to monopoly). For any S ∉ ∆, the new partitions for S, Bα(∆,S), Bγ(∆,S), and Bδ(∆,S) are given in (9)-(11). Let (14)
Yα(∆) = { x∈Rn+ | Σi∈S xi ≥ φ(Bα(∆,S), S) for all S ≠ N and S ∉ ∆}, Yγ(∆) = { x∈Rn+ | Σi∈S xi ≥ φ(Bγ(∆,S), S) for all S ≠ N and S ∉ ∆}, Yδ(∆) = { x∈Rn+ | Σi∈S xi ≥ φ(Bδ(∆,S), S) for all S ≠ N and S ∉ ∆},
denote the sets of allocations immune to the α-, γ-, and δ-deviations from ∆, and let (15)
MNBPα(∆) = {Min Σxi | x ∈ Yα(∆)},
(16)
MNBPγ(∆) = {Min Σxi | x ∈ Yγ(∆)}, and
(17)
MNBPδ(∆) = {Min Σxi | x ∈ Yδ(∆)}
denote respectively the MNBP against the α-, γ- and δ-deviations from ∆. Theorem 2: Consider a partition ∆ ≠ ∆m with a profit allocation θ ∈ Rn+ in (1). Assume: (i) A0 holds; (ii) MCM ≡ MC(∆, ∆m) > 0, and MC(∆', ∆) = 0 for all ∆, ∆’ ≠ ∆m; (iii) Σθj > (v(N)-MCM). Then the following two claims hold: (I) ∆ with θ is α-unstable (γ-, δ- unstable) if Σθj < MNBPα(∆) (Σθj < MNBPγ(∆), Σθj < MNBPδ(∆)); (II) ∆ with θ is αstable (γ-, δ- stable) if and only if θ ∈ Yα(∆) (θ ∈ Yγ(∆), θ ∈ Yδ(∆)). such that ∆m is stable for all t’ ∈ Bε(t), where for t∈ RK, ||t||2 = Σt2i , and Bε(t) = {y∈ RK | ||t–y|| < ε}.
13
Theorems 1 and 2 provide a computational solution to stable partition problems in oligopoly markets. To check the stability of ∆m, one simply solves the linear programming problem (12) or (13) and compare the MNBPα or MNBPδ with v(N). The stability of ∆ ≠ ∆m is interesting only if monopoly is ruled out by high merging costs. It can be checked out by first computing MNBP(∆) and then evaluating the related inequalities. Theorems 1 and 2 also hold in partition function and normal form TU games, except that the α- and γ-stabilities of ∆m in Theorem 1 could be different (i.e., the α- and γstabilities of ∆m are equivalent only in (1)). They can be alternatively presented using the balancedness of a coalitional TU game17, but this does not produce any additional benefits. Note that θ ∉ Yα(∆) and Σθj ≥ MNBPα(∆) could both hold (i.e., Σθj ≥ MNBPα(∆) is only necessary for α-stability). Because the allocation θ is a key determinant for the stability of ∆ ≠ ∆m, each ∆≠ ∆m needs to be evaluated separately so as to determine the stable ranges of θ. Hence, checking the stability of (∆, θ) (∆ ≠ ∆m) is more involved, and this provides a rich source of future studies. The rest of this section uses Theorem 1 to characterize monopoly stability in two classes of liner markets with n firms. (i.e., cj ≡ c1 for all j). Lemma 3: Given a symmetric market (a, c, z) ∈R2n+1 ++ Suppose parts (ii)-(iii) of A0 holds. Then (12) and (13) are: (18)
if 2 ≤ n ≤ 5 ì n(a-c1)2 / (n+1)2 MNBPα = MNBPα(a, c, z) = í î n(a-c1)2 / [9(n-1)] if 6 ≤ n;
(19)
MNBPδ = n(a-c1)2 / 9. Theorem 3: Given a symmetric market (a, c, z) ∈R2n+1 ++ . Suppose parts (ii) and (iii)
of A0 holds. (i) The monopoly merger ∆m is α-stable for all n; (ii) ∆m is δ-stable for n = 2; and (iii) ∆m is δ-unstable for all n ≥ 3. Therefore, monopoly in a symmetric linear market is unstable in the sense of loyal relation (except in duopoly market), and it is always stable in the sense of cautious belief and in the sense of complete breakup belief. 17
For example, ∆m is α-stable ⇔ the game Γα is balanced ⇔ C(Γα) ≠ ∅, where Γα= {N, vα} is
14
Now consider an asymmetric market (a, c, z) ∈R2n+1 ++ with cj ≡ c2 ≥ c1 for j ≥ 3. Let (20)
ε = ε2 = (c2-c1)/(a-c1)
be the relative saving on marginal costs by firm 2 (and all j ≥ 3), and define n0 and ω0 by 9 − 4, 1+4ε2
(21)
n0 =
(22)
ω0 = (4n-9)/(8n-6). Note that 0< ε 0 ⇔ ε3 0 ⇔ ε3 > θ4; and (III) d23 > 0 ⇔ ε3 > θ6. By θ4< θ6 < θ0 and the lemma, d23 > 0 implies d13 > 0 (see Figure 1). A larger ε3 represents larger cost savings by S = 13 or 23. Similarly, large ε2 or large θ2 = (15ε2 -1) represents large cost savings by S = 12. Therefore, Lemma 5 shows that a merger is profitable if and only if its cost savings are sufficiently large.
16
The next lemma computes MNBP. Readers uninterested in technical details may prefer to skip (28)-(36) and go directly to (37). Let ρi, vi and yi be given by (28)
2-2ε2+5ε22, ρ1 = (5-11ε2)/11, ρ2 = -1+27ε2- 4 -3ε2+42ε22,
ρ0 = -1+
19+27ε2+4 17-125ε2+218ε22 19+27ε2-4 17-125ε2+218ε22 ρ3 = , ρ4 = , 89 89 ρ5 = (29)
125-3 89 ≈ 0..22; 436
v1 = π1 = (a-c1)2(1+ε2+ε3)2/16, v2 = π2 = (a-c1)2(1-3ε2+ε3)2/16, v3 = π3 = (a-c1)2(1-3ε3+ε2)2/16;
(30)
v12 = (a-c1)2(1+ε3)2/9, v13 = (a-c1)2(1+ε2)2/9, v23 = (a-c1)2(1-2ε2)2/9, v3δ = (a-c1)2(1-2ε3)2/9, and v123 = v(N) = (a-c1)2/4;
(31)
y1 = (v12 +v13 -v23)/2, y2 = (v12 +v23 –v13)/2, y3 = (v13 +v23 –v12)/2. Lemma 6: Given (a, c, z) ∈R7++, let εi, θi, ρi, vi and yi be given by (25)-(31). Under
parts (ii) and (iii) of A0, the MNBPδ and MNBPα as defined in (12) and (13) are: (32)
ìïv13 + v23 + v3δ MNBPδ = í δ îï v12 + v3
(33)
For ε2 ≤ 14,
(34)
For 14 ≤ ε2 ≤ 11,
if ε3> ρ0;
ì v1+v2+v3 if ε3 < θ4 MNBPα = í if ε3 ≥ θ4; îv2+v13
1
1
if ε3 ≤ ρ0
1
MNBPα = v2+v13;
ì ïv +v = í v +v ïîv +v
v2+v13
(35)
For ε2 ≥
1 11, ε3 ≤θ6,
MNBPα
17
1
16
if 11 ≤ ε2 ≤ 77 16
5
16
5
2
13
if ε3≤ ρ1; 77≤ ε2≤ 22
3
12
if ε3> ρ1; 77≤ ε2≤ 22
3
12
if ε2 ≥ 22;
5
(36)
1
For ε2≥ 11, ε3>θ6,
MNBPα
ì ï = í ï î
ìïv2+v13 if ε3≤ ρ2 16 íy1+y2+y3 if ρ2 v(N) = 7.56), so monopoly is δ-unstable. By ε2 = 0.09 < 7/38 = 0.18 and ε3< ω2 = 0.14, ∆* = ∆0. With c2 =1.9 and c3 = 2 or (a; c) = (6; 0.5, 1.9, 2), cost savings are large, monopoly becomes δ-stable (i.e., ε3 = 0.27 > ω1 = 0.17; or ε2 = 0.26 > 1/6 = 0.17; or MNBPδ = 6.79 < 7.56). By ε2 = 0.26 > 5/22 = 0.23, monopoly is also optimal. As illustrated in Example 5, in Figure 2 and in part (a) of Figure 3, monopoly is optimal if ε2 ≥ 5/22, and δ-stable if ε2 ≥1/6. Therefore, there is no need for anti-trust if cost savings are large (ε2 ≥ 5/22), as monopoly is both stable and optimal. In general, however, a stable monopoly will not be optimal, and anti-trust regulation would be desirable. Now consider the stability of ∆1 = {1; 23}, ∆2 = {13; 2}, and ∆3 = {12; 3}. Although the α-, γ- and δ-stabilities of ∆ ≠ ∆m are generally different, they are identical with n = 3. Hence, the feasible sets in (14) become Y(∆1), Y(∆2) and Y(∆3). Assume that S = 12, 13, and 23 are all profitable (i.e., d12 , d13, d23 >0, see Lemma 5). Let (41)
~ 2 2 v1δ = π 1(∆1) = (a-c1) (1+ε2) /9 = v13; ~ 2 2 v2δ = π 2(∆2) = (a-c1) (1-2ε2) /9 = v23; and ~ 2 2 v3δ = π 3(∆3) = (a-c1) (1-2ε3) /9
be the single firm’s profits at ∆1, ∆2 and ∆3. Let the equilibrium profit allocation θ = θ(t) at ∆1, ∆2 and ∆3 be respectively denoted as: (42.1)
For ∆1, θ1 = v1δ = v13, θ2 = v2 + t d23, and θ3 = v3 + (1-t) d23;
(42.2)
For ∆2, θ1 = v1 + t d13, θ2 = v2δ = v23, and θ3 = v3 + (1-t) d13;
(42.3)
For ∆3, θ1 = v1 + t d12, θ2 = v2 + (1-t)d12, and θ3 = v3δ ,
where the v’s and d’s are given in (27), (29), (30), and (41), and t ∈ [0, 1] is the efficient member’s share of a merger’s gain. Let µi(εj, t) be given by: (43)
µ1(ε2, t) =
-14 -54ε2+36t(1+3ε2)+8 7+14ε2+88ε22+t(34-244ε2+352ε22)+9t2(1-4ε2+4ε22) , 2(7+90t)
19
(44)
-14+18ε2+36t(1+ε2)+8 7-28ε2+37ε22+t(34-256ε2+430ε22)+9t2(1+2ε2+ε22) µ2(ε2, t) = , 2(7+90t)
(45)
µ3(ε3, t) = µ2(ε3, t). Theorem 6: Given (a, c, z) ∈R7++, suppose (i) A0 holds; (ii) MCM ≡ MC(∆, ∆m) >
0, and MC(∆', ∆) = 0 for all ∆, ∆’ ≠ ∆m; and (iii) d12 , d13, d23 >0. Let εi, θ(t), and µi(ε2, t) be given by (25) and (42)-(45). (I) For ∆1 = {1; 23} and θ(t), assume Σθj > (v(N)-MCM). (a) For ε2 ≤ 1/11, ∆1 with θ(t) is stable for all t; (b) for 1/11 < ε2 < 113/316, ∆1 with θ(t) is stable ⇔ ε3 ≤ µ1(ε2, t); (c) for 113/316 ≤ ε2 ≤ 1/2, ∆1 with θ(t) is unstable for all t. (II) For ∆2 = {13; 2} and θ(t), assume Σθj>(v(N)-MCM), let e2(t) = (2t-9)/[14(2t-3)]. (a) For ε2 ≤ 1/11, ∆2 with θ(t) is stable for all t; (b) for 1/11 < ε2 ≤ e2(t), ∆2 with θ(t) is stable ⇔ ε3 ≤ µ2(ε2, t); (c) for e2(t) < ε2 ≤ 1/2, ∆2 with θ(t) is unstable for all t. (III) For ∆3 = {12; 3} and θ(t), assume Σθj > (v(N)-MCM). (a) For ε3 ≤ 3/14, ∆3 with θ(t) is stable for all t; (b) for 3/14 < ε3 ≤ 1/2, ∆3 with θ(t) is stable ⇔ ε2 ≤ µ3(ε3, t). (Figures 4 and 5 about here) Parts (I) and (II) are illustrated in Figure 4. As illustrated in Figure 5, part (III) can also be characterized by a function of ε2 and t, but this is more involved18. Theorems 5 and 6 imply a negative relationship between the stable number of mergers and the degree of cost savings: (1) an intermediate partition is stable (i.e., the stable number of firms is two) if cost savings are sufficiently small; (2) monopoly is stable (i.e., the stable number of firms is one) if cost savings are sufficiently large. 18
The two solutions of ε2 = µ3(ε3, t) are: µ5(ε2, t) ≥ µ50(ε2, t), which are the right and left halves of *
µ3(ε3, t). As shown in Figure 5, they meet at the minimum point of µ3(ε3, t): ε2(t) = min {µ3(ε3, t) | ε3} *
*
*
*
achieved at ε3(t) ( ε2(0) = 0.179, ε3(0) = 0.293). Now, part (III) becomes: (i) for ε2≤ ε2(t), θ(t) is stable; (ii) for *
*
*
*
ε2> ε2(t), ε3≤ ε3(t), θ(t) is stable⇔ ε3≤ µ50(ε2, t); (iii) for ε2> ε2(t), ε3> ε3(t), θ(t) is stable⇔ ε3 ≥ µ5(ε2, t).
20
It is clear now that the share t, representing cooperation within a merger, is a key determinant for stability. This is precisely characterized in the next remark. Let (46)
27ε2 4 µ10(ε2) = µ1(ε2, 0) = −1− 7 + 7
7+14ε2+88ε22 , and
11 27ε2 4 µ11(ε2) = µ1(ε2, 1) = 97+ 97 + 97
50-266ε2+476ε22 .
be the two polar cases of µ1(ε2, t) in part (I). As shown in part (b) of Figure 3, µ10(ε2) ≤ µ1(ε2, t) ≤ µ11(ε2). By inverting µ1(ε2, t) = ε3, one has: (47)
1 7ε32-97ε22 +54ε2ε3+14ε3+22ε2-9 t1(ε2, ε3) = −2 45ε 2+13ε 2 -54ε ε -18ε +14ε +1. 3 2 2 3 3 2 Remark 5: The stability result of part (I.b) can be alternatively characterized by the
share t: for 1/11 < ε2 < 113/316, ∆1 with θ(t) is stable ⇔ t ≥ t1(ε2, ε3). Parts (II) and (III) can be similarly given by the share t. As shown in Example 6 below, t1(ε2, ε3) is defined locally in the relevant domain. Example 6: For (a; c; z) = (6; 0.5, 1.05, 2.46; 3, 3, 3)∈ R7++(as in Example 1), ε2 = 0.1, ε3 = 0.356, µ10(0.1) = 0.3550, and µ11(0.1) = 0.3607.
By ε3 = 0.356 > µ10(0.1) =
µ1(0.1, 0), ∆1 with θ(0) = (4.067, 2.11, 0.042) is unstable. In this case, stability requires t ≥ t1(0.1, 0.356) = 0.12 = 12%. Hence, ∆1 with θ(0.1) = (4.067, 2.114, 0.038) is unstable; and ∆1 with θ(0.2) = (4.067, 2.118, 0.034) is stable. If c3 rises so ε3 rises to 0.358 and 0.359, the critical share will be increased to t1(0.1, 0.358) = 45% and t1(0.1, 0.359) = 68%. Finally, Lemma 5 and Theorem 6 imply a characterization for the orginal Cournot structure ∆0 = {1; 2; 3} to be stable. Note the three stabilities for ∆0 are all identical. Remark 6: ∆0 = {1; 2; 3} is stable if and only if the following (48) holds: (48)
θ2 < ε3 ≤ θ4, and MCM > [v(N)-(v1 + v2 + v3)],
where εi, θi, and vi are respectively given by (25), (26), (29) and (30), and MCM is the monopoly merging cost.
6. Concluding Remarks
21
This study has illustrated the fact that Nash behavior across mergers (or alliances) and internal cooperation within each merger both are key determinants of a stable partition. Because a Nash equilibrium and the core are only the two end points along the entire spectrum of possible hybrid equilibria, one needs to use the concept of a hybrid equilibrium in order to understand stable market structures and stable partitions. Using the MNBP method, this paper has computationally solved the stable partition problem in oligopoly markets (as well as in partition function and normal form TU games). To check the stability of a monopoly partition, one simply computes the MNBP and compares it with the monopoly profits v(N). Because the stability of an intermediate partition is defined for a specific profit allocation, the MNBP only provides a necessary condition for such stability. It will be useful to check the range of stable allocations for each intermediate partition, and this will provide a rich source of future studies. Theorems 1 and 2 allow one to characterize a stable partition in terms of the underlying cost and demand parameters. By characterizing stability in three firm linear markets and in a large class of asymmetric linear markets, this study suggests that there exists a negative relationship between the number of stable mergers and the degree of potential cost savings: a monopoly merger is stable if cost savings are sufficiently large, and some intermediate partition is stable if cost savings are sufficiently low and if monopoly is ruled out by high merging costs. Such a relationship seems to hold in more general models with multiple goods or exits and entries (or synergy or non-linear demand or asymmetric information or dynamics), and it will obviously be very useful (and challenging) to work out these ambitious extensions in future studies. The generality of a stable partition implies a wide range of useful applications. In addition to checking the stability of Cournot equilibria in previous studies, it will be useful to extend the characterizations of stability in linear markets from cases where there are three firms to cases where there are four to ten firms. These new characterizations will tell us how market power and cost differences affect structural stability in a much wider range
22
of real-world markets. For example, the new characterizations for market with six firms will shed lights in studying the stability of the {United-Delta, NW-Continental, Ame.AirUS Air} alliance structure in the six major carriers’ domestic market. Because it exhausts all deviations, the stable partition theory presented in this paper provides an answer to the question “what is the solution for a game”: Nash equilibrium (the core, hybrid equilibrium for a partition ∆) is the solution for a game if players’ rationality is limited to strategic behavior (cooperative behavior, the hybrid behavior with ∆); and stable partition is the solution if players’ rationality is free of the above restrictions. APPENDIX Proof of Theorem 1: Suppose v(N) ≥ MNBPα. Let x solve (12), and let θ be given by θi = xi + (v(N)−MNBPα)/n, all i. By the definition of (12), Σj∈Sθj ≥ Σj∈Sxj ≥ φ(Bγ(∆m,S), S) for all S≠N. Therefore, ∆m with the profit allocation θ is α-stale when v(N) ≥ MNBPα. If v(N) < MNBPα, no allocation of v(N) could satisfy all coalitions, therefore the monopoly merger with any θ will not be α-stale. The proof for part (ii) is similar.
Q.E.D
Proof of Remarks 2 and 3: The proof of Remark 2 is similar to that of Theorem 1. The proof of Remark 3 follows from Zhao (2001).
Q.E.D
Proof of Theorem 2: The proof is similar to that of Theorem 1.
Q.E.D
Proof of Lemma 3: The proof completes in six steps. Step 1. Determine φ(Bα(S), S) and φ(Bδ(∆m,S), S). For each 1≤ k ≤ (n-1), let S(k) = {T⊆ N
| |T| = k } denote the set of mergers with precisely k members.
By A0 and by
symmetry, for any S∈ S(k) deviating from ∆m, its α- and γ-profits are equal to the Cournot profits with (n-k+1) firms, and its δ-profits are equal to the Cournot profits with two firms: (A1)
(a-c1)2 φ(Bα(S), S) = φ(Bγ(∆m,S), S) = vα(k) = (n-k+2)2,
(A2)
(a-c1)2 φ(Bδ(∆m,S), S) = vδ(k) ≡ 9 .
23
Step 2. Determine MVα(k) and MVδ(k). For each k = 1, ..., n-1, let (A3)
MVα(k) = {Min Σi∈Nxi | x∈Rn+ ; Σi∈S xi ≥ vα(k), for all S ∈ S(k)}, and MVδ(k) = {Min Σi∈Nxi | x∈Rn+ ; Σi∈S xi ≥ vδ(k), for all S ∈ S(k)}
n denote the minimum profits to satisfy all coalitions in S(k). There are (k) constraints, n n–1 therefore vα(k) and vδ(k) appears (k) times. Each xi appears (k–1) times on the left-hand n n–1 n n–1 side. Summing up, we have (k–1) Σxi ≥ (k) vα(k), and (k–1) Σxi ≥ (k) vδ(k), or Σxi ≥ nvα(k)/k, and Σxi ≥ n vδ(k)/k. Using (A1) and (A2), we obtain (A4)
n(a-c1)2 n(a-c1)2 MVα(k) = k(n-k+2)2, and MVδ(k) = 9k ,
and the minimum solutions are xi = (a-c1)2/[k(n-k+2)2], all i; and yi = (a-c1)2/(9k), all i. Note that all constraints are binding at the minimum solutions. Step 3. Determine the maximum of MVα(k) and MVδ(k). By MVδ(k)’ < 0, (A5)
Max { MVδ(k) | 1≤ k≤ (n-1) } = MVδ(1) =
n(a-c1)2 9 .
By MVα(k)’ ≤ 0 if k ≤ (n+2)/3, and > 0 if k > (n+2)/3, we have Max { MVα(k) | 1≤ k≤ (n-1) }= Max {MVα(1), MVα(n-1)}. Define n(a-c1)2 n(a-c1)2 f(n) = MVα(1)-MVα(n-1) = (n+1)2 - 9(n-1) . The sign of f(n) is the same as g(n) = 9(n-1) - (n+1)2. Note g(n) is ∩-shaped and has two roots: n = 2, 5. Therefore, MVα(1) ≥ MVα(n-1) if 2 ≤ n≤5, and MVα(1) < MVα(n-1) if n ≥ 6. This leads to (A6)
ìn(a-c1)2/(n+1)2 if 2≤ n≤ 5 Max {MVα(k)|1≤ k≤(n-1)} = í în(a-c1)2/[9(n-1)] if 6 ≤ n. Step 4. Enlarge the feasible sets for MVα(k) and MVδ(k). Let the feasible sets in
(A3) be denoted by:
24
FRα(k) = {x∈ Rn+ | Σi∈S xi ≥ vα(k), for all S ∈ S(k)}, and
(A7)
FRδ(k) = {x∈ Rn+ | Σi∈S xi ≥ vδ(k), for all S ∈ S(k)}. Now, for each k = 1, ..., n-1, define FRα(k)* = {x∈ Rn+ | Σ xi ≥ MVα(k)}, and
(A8)
FRδ(k)* = {x∈ Rn+ | Σ xi ≥ MVδ(k)}. It follows from Step 2 that FRα(k) ⊆ FRα(k)* and FRδ(k) ⊆ FRδ(k)*
(A9) hold for all k.
Step 5. Enlarge the feasible regions for MNBPα and MNBPδ. The feasible regions for MNBPα and MNBPδ as given in (12) and (13) are: (A10)
FRα = {x∈ Rn+ | Σi∈S xi ≥ φ(Bα(S), S) for all S ≠ N } n-1
= {x∈ Rn+ | Σi∈S xi ≥ vα(k), for all S∈S(k), k = 1, ..., n-1}= ∩i=1 FRα(k); (A11)
FRδ = {x∈ Rn+ | Σi∈S xi ≥ φ(Bδ(∆m,S), S) for all S ≠ N} n-1
= {x∈ Rn+ | Σi∈S xi ≥ vδ(k), for all S∈S(k), k = 1, ..., n-1}= ∩i=1 FRδ(k). n-1
n-1
Now consider FR*α = ∩i=1 FRα(k)* and FR*δ = ∩i=1 FRδ(k)*. By the maximality in Step 3 and by (A8), one has (A12)
n-1
FR*α = ∩i=1 FRα(k)* = {x∈ Rn+ | Σ xi ≥ Max {MVα(1), MVα(n-1)}} ìï{x∈ Rn | Σ x ≥ MVα(1)} if 2≤ n≤ 5 i + = í ïî{x∈ Rn+ | Σ xi ≥ MVα(n-1)} if 6 ≤ n;
(A13)
n-1
FR*δ = ∩i=1 FRδ(k)* = {x∈ Rn+ | Σ xi ≥ MVδ(1)}.
It follows from (A9)-(A13), the following relations hold: (A14)
n-1
n-1
FRα ⊆ FR*α = ∩i=1 FRα(k)*, and FRδ ⊆ FR*δ = ∩i=1 FRδ(k)*. Step 6. Determine MNBPα and MNBPδ. Observe that the minimum value of 25
{Min Σi∈Nxi| x∈ FR*δ } is equal to (A15)
{Min Σi∈N
xi| x∈ FR*δ } = MVδ(1) =
n(a-c1)2 9 .
By the maximality in Step 3, the symmetric minimum solution (i.e., xi = (a-c1)2/9, all i) is included in all FRδ(k) and therefore also in FRδ. Since a global min must be a local min when it is locally feasible, it follows from (A14) and (A15) that n(a-c1)2 MNBPδ = {Min Σ xi | x∈ FRδ } = {Min Σxi| x∈ FR*δ } = 9 . Using similar arguments, one obtains MNBPα = {Min Σ xi | x∈ FRα } = {Min Σxi| x∈ FR*α } if 2 ≤ n ≤ 5 ì n(a-c1)2 / (n+1)2 =í î n(a-c1)2 / [9(n-1)] if 6 ≤ n. This completes the proof of Lemma 3.
Q.E.D
Proof of Theorem 3: By Theorem 1, Lemma 3, and v(N) = (a-c1)2/4, part (i) follows from ¼ > n/(n+1)2 and ¼ > n/[9(n-1)]. By v(N)- MNBPδ = (9-4n)(a-c1)2/36, v(N)>MNBPδ if n ≤ 2, and v(N) 0 (i.e., f(k) is ∪-shaped), Max f(k) = Max {f(1), f(n-1)}. Therefore, (A29)
Max { vαII (k)-(k-1)y* | 1≤ k≤ n-1} = (a-c1)2 Max {f(1), f(n-1)}.
Define d(n) = f(1)-f(n-1) =
[1+(n-1)ε]2+(n-2)(1-2ε)2 [1+ε]2 - 9 . (n+1)2
One can check that d(n)’’ < 0 (so d is ∩-shaped), and d(n) = 0 has two roots: (A30)
n0 = - 4 +
9 , and n1 = 2. 1+4ε2
(Figure 6 about here) By n0 ≥ 2 ⇔ ε ≤ 1/8 (see part (a) of Figure 6), one has: (A31)
ìf(1) if ε≤ 1/8 and n≤ n0 Max {f(1), f(n-1)} = í îf(n-1) if ε≤1/8 and n0 0 for all 2≤ n ≤ 5. Case 3. 6≤ n. d is given by d = d3(n) = (a-c1
1 (1+ε)2 (1-2ε)2 4 - 9 - 9(n-1) },
)2{
By d3(n)’ > 0, d3(n) is increasing. By d3(6) > 0, d > 0 for all 6≤ n. This proves part (i). Now we prove parts (ii) and (iii). d = v(N)−MNBPδ is given by (4n-3)(a-c1)2 1 1 (1+ε)2+(n-1)(1-2ε)2 4n-9 d = d4(n) = (a-c1)2{4 } = ( 2-ε) (ε -8n-6 ). 9 9 By ε ≤ 1/2, d > 0 if and only if ε ≥ ω0 = (4n-9)/(8n-6).
Q.E.D
The following expressions are used in the proofs for Lemmas 5 and 6 and Theorems ~ ~ ~ 5 and 6, where π 12(∆3) = v12, π13(∆2) = v13, and π23(∆1) = v23 (see (30)).
(B1)
vα1 = π1 = (a-c1)2(1+ε2+ε3)2/16, vα2 = π2 = (a-c1)2(1-3ε2+ε3)2/16, vα3 = π3 = (a-c1)2(1-3ε3+ε2)2/16; v1δ = (a-c1)2(1+ε2)2/9, v2δ = (a-c1)2(1-2ε2)2/9, v3δ = (a-c1)2(1-2ε3)2/9; α = v δ = (a-c )2(1+ε )2/9, v = v α = v δ = vδ = (a-c )2(1+ε )2/9, v12 = v12 13 1 3 1 2 12 13 13 1 α = v δ = vδ = (a-c )2(1-2ε )2/9. v23 = v23 1 2 23 2
~ Proof of Lemma 5: Consider first S = 12. Let d12 = π 12(∆3) – ( π1+π2). (B1) leads to
31
d12(ε3) = (a-c1)2 (1+ε3-3ε2)(15ε2-1-ε3) /72. By A0, (1+ε3-3ε2) > 0. Hence, d12 > 0 ⇔ ε3 < θ2, which is given by (B2)
θ2 = 15ε2-1. ~ Now consider S = 13. Let d13 = π 13(∆2) – ( π1+π3). (B1) leads to
d13(ε3) = (a-c1)2 (1+ε2-3ε3)(15ε3-1-ε2)/72. By (1+ε2-3ε3) > 0, d13 > 0 ⇔ ε3 > θ4, which is given by (B3)
1+ε2 θ4 = 15 . ~ Finally, consider S = 23. Let d23 = π 23(∆1) – ( π2+π3). By (B1), one has
d23(ε3) =
5(a-c1)2 1+ε2 1+13ε2 5(a-c1)2 ( ε )(ε (θ0-ε3)(ε3-θ6). 3 3 8 3 15 ) = 8
By (θ0-ε3) > 0, d23 > 0 ⇔ ε3 > θ6, where θ0 and θ6 are given by (B4)
θ0 =
1+ε2 1+13ε2 3 ; θ6 = 15 .
This completes the proof for Lemma 5.
Q.E.D
The following relations (see Figure 1) are useful in proving Lemma 6. (B5)
ε3 ≥ θ2 if ε2 ≤ 1/14; ε3 < θ2 = 15ε2-1 if ε2 > 1/11;
(B6)
1+ε2 ε3 > θ4 = 15 if ε2 > 1/14;
(B7)
θ4 < θ6 ≤ θ0; ε2 ≤ ε3 ≤ θ0; and ε2 ≤θ6.
Proof of Lemma 6: We first compute MNBPδ. There are six constraints: x1 ≥ v1δ, x2 ≥ v2δ, x3 ≥ v3δ ; x1+ x2 ≥ v12, x1+x3 ≥ v13, x2+x3 ≥ v23. By v1δ = v12, v2δ = v23, the problem becomes: (B8)
MNBPδ = Min { x1+x2 +x3| x1 ≥ v1δ, x2 ≥ v2δ, x3 ≥ v3δ ; x1+ x2 ≥ v12},
of which the minimum value is equal to (B9)
v3δ + Max {v1δ+ v2δ, v12}.
Let d(ε3) = v1δ + v2δ- v12=
(a-c1)2 2 2 2 9 [(1+ε2) +(1-2ε2) -(1+ε3) ].
32
By d’’< 0, d is ∩-shaped. d(ε3) = 0 has two roots: µ1 < 0 < µ2, where µ2 is given by µ2 = ρ0 = -1+
2-2ε2+5ε22
Hence, Max {v1δ+ v2δ, v12} = v1δ+ v2δ if ε3 ≤ ρ0, and = v12 if ε3> ρ0. By (B1), (B8) and (B9), one gets (32). We only provide an outline for proving (33)-(36), because complete proofs like those for (32) would make the paper too long. Figure 1 illustrates all the sub-cases. 1
Case 1. ε2 ∈ [0, 14]. By Lemma 5, d12 ≤ 0, so only five constraints are left: x1 ≥ v1 = vα1 , x2 ≥ v2 = vα2 , x3 ≥ v3 = vα3 ; x1+x3 ≥ v13, x2+x3 ≥ v23. Let h1 = v13–v1, h2 = v23–v2, one has d(ε2,ε3) = max{h1, h2}= h1, and v3 ≥ d(ε2,ε3) ⇔ ε3 ≤ θ4. By MNBPα = v1 + v2 + max {v3, d(ε2, ε3)}, MNBPα = v1 + v2 + v3, if ε3 < θ4, and MNBPα = v2 + v13 if ε3 ≥ θ4. This proves (33). 1
4
Case 2. ε2 ∈ [14, 53]. One has θ4 ≤ ε2 ≤θ2 < θ6. If ε3 ≥ θ2, then d12 ≤ 0. By Case 1, MNBPα = v2 + v13. If ε3 < θ2< θ6, then d23 ≤ 0. So the constraint x2+x3 ≥ v23 can be removed. Using similar steps as in Case 1, one can show MNBPα = v2 + v13. 4
Case 3. ε2 ≥ 53, and ε3 ≤ θ6. By d23 ≤ 0, x2+x3 ≥ v23 is removed. Similar to Case 2, and by d13 > 0, one can show MNBPα = v2 + v13 if ε3 ≤ ρ1, and = v3+v12 if ε3> ρ1. One can 4
16
5
also show that 53 ≤ ε2 ≤ 77 implies ε3≤ ρ1, and ε2 ≥ 22 implies ε3> ρ1. 4
4
1
Case 4. ε2 ≥ 53, and ε3 ≥ θ2 > θ6. This can only occur for ε2 ∈ [53, 11]. By Case 1, d12 ≤ 0, and ε3 ≥ θ4, MNBPα = v2 + v13. By Cases 2-3, one gets (34) and (35). 4
Case 5. ε2 ≥ 53, and θ2 ≥ ε3 > θ6. One has d12 > 0, d13 > 0, d23 > 0. Note at most one of x1 ≥ v1, x2 ≥ v2, x3 ≥ v3 can be binding. First solving each of the three cases: Case 5.1, x1 = v1; Case 5.2, x2 = v2; Case 5.3, x3 = v3. Now solve Case 5.4, x1>v1, x2>v2, x3>v3. In case 5.4, one must have x1+x2= v12, x1+x3= v13, and x2+x3 = v23. Solving these equations, one gets y1, y2, y3. By checking yi > vi, and using Cases 5.1-5.3, one can get (36).
Q.E.D
Proof of Theorem 5: Part (i). For each of the values of MNBPα, one can show v(N) > MNBPδ. Now consider part (ii). If ε3 ≤ ρ0, d = v(N)−MNBPδ is given by 33
1 (1+ε2)2+(1-2ε2)2 +(1-2ε3)2 d(ε3) = (a-c1)2 [4 ]. 9 Note that d’’< 0, and d(ε3) = 0 has two roots µ1 < µ2, where µ2 is outside the feasible range of ε3, and the first root µ1 is given by 1 µ1= ω1 = 2-
1+8ε2-20ε22 . 4
One can show ω1≤ ρ0. Therefore, (B10)
d > 0 ⇔ ω1≤ ε3 ≤ ρ0.
If ε3 > ρ0, d is given by (a-c1)2 1 (1+ε3)2+(1-2ε3)2 d(ε3) = (a-c1)2 [4 ] = 9 36 (1+10ε3) (1-2ε3) >0. By (B10), one gets d > 0 ⇔ ω1 ≤ ε3, which completes the proof of part (ii).
Q.E.D
Proof of Remark 4: Since ∆1 = {1; 23} and ∆2 = {13; 2} have identical welfare, ∆2 can be ignored. First, evaluate six cases below. Case 1. ∆0 →∆3. Let d1(ε3) = W3 - W0 be the welfare change. Using d1(ε3)’’ < 0 (i.e., d1 is ∩-shaped) and the two roots of d1 = 0, one can show: d1(ε3) ≥ 0 ⇔ ε3 ≤ λ1 = (-7+69ε2)/31, ε313/44, and ε3 > λ1 if ε2 < 7/38; Case 2. ∆0 →∆1. Let d2(ε3) = W2 - W0. Similar to Case 1, one can show: d2(ε3) ≥ 0 ⇔ ε3 ≥λ2, and ε3>λ2 if ε2> 7/38, where λ2 is given by (B11)
ω2 = λ2 = ω2 = (7+31ε2)/69; Case 3. ∆3 →∆1. Let d3(ε3) = W1 - W3. Similar as before, one can show: d3(ε3) ≥ 0
⇔ ε3 ≤ λ3 = (-ε2 + 8/11), ε3≤ θ0 < λ3 if ε2 < 13/44, and ε3≥ε2>λ3 if ε2>4/11; Case 4. ∆1 →∆m. Let d4(ε2) = Wm – W1, then d4(ε2) ≥ 0 ⇔ ε2 ≥ λ4 = 5/22; Case 5. ∆0 →∆m. Let d5(ε3) = Wm – W0. Similar as before, one can show: d5(ε3) ≥ 0 ⇔ λ5 ≤ε3 ≤ λ6, d5 < 0 if ε2 < λ7 = 5/14- 23/28 ≈ 0.19, and d5 > 0 if ε2> 5/22, where λ5 and λ6 are given by (B12)
λ5 = (5+9ε2-2 -11+80ε2-112ε22)/23, λ6 = (5+9ε2+2 -11+80ε2-112ε22)/23; Case 6. ∆3 →∆m. Let d6(ε3) = Wm - W3, d6(ε3) ≥ 0 ⇔ ε3 ≥ 5/22. Second, graphing cases 1-6 on [0, 0.5] for ε2 (similar to Figure 1). Third, picking up
34
the maximal welfare W*, and one can get: W* = Wm if ε2> 5/22; = W1 if 7/38< ε2≤ 5/22; = W1 if ε2≤ 7/38 and ε3 ≥ ω2; = W0 if ε2≤ 7/38 and ε3 < ω2.
Q.E.D
Proof of Theorem 6: Part (I) Consider the stability of ∆1 = {1; 23} with y given by y 1 = v1δ = v13, y 2 = v2+ td23, and y3 = v3+(1-t)d23. By d23 > 0 (i.e., ε3 > θ6) and the definition of y, y ∈Y(∆1) = { y | y1≥ v1δ = v13, y2 ≥ v2, y3 ≥ v3, y1+ y2≥ v12, y1+ y3≥ v13, y2+ y3≥ v23} = Yα(∆1) = Yδ(∆1) is equivalent to (B13)
d(ε3) = v13 +v2 + t d23-v12≥ 0.
Note d’’ < 0 (i.e., d is ∩-shaped), and d(ε3) = 0 has two roots: (B14) µ1(ε2,t) =
-14 -54ε2+36t(1+3ε2)+8 7+14ε2+88ε22+t(34-244ε2+352ε22)+9t2(1-4ε2+4ε22) , 2(7+90t)
µ10(ε2,t) =
-14 -54ε2+36t(1+3ε2)-8 7+14ε2+88ε22+t(34-244ε2+352ε22)+9t2(1-4ε2+4ε22) . 2(7+90t)
It can be checked that the following three claims hold: µ10(ε2,t) < θ6; (B15)
θ0 ≤ µ1(ε2,t) ⇔ ε2≤ 1/11; and θ6 ≤ µ1(ε2,t) ⇔ ε2≤ 113/316.
Therefore, by (B15), by the ∩-shape of d(ε3), and by θ6 ≤ ε3≤ θ0, one has
(B16)
d(ε3) =
ìï > 0 í ≥ 0 ⇔ ε3 ≤ µ1(ε2; t) ïî < 0
if ε2 ≤ 1/11 if 1/11 < ε2 < 113/316 if ε2 ≥ 113/316;
which leads to part (I). One can double check the above results by evaluating the special cases of t = 1 and t = 0 separately. The results obtained for t = 0 and 1 are the same as those by replacing the above t by 0 and 1 respectively. The proofs for parts (II) and (III) are similar, it is interesting to see that the formula for part (III) is the same as that of part (II), after switching ε2 and ε3.
Q.E.D
Proof of Remark 5: The discussions before the remark serve as its proof.
Q.E.D
Proof of Remark 6: The discussions before the remark serve as its proof.
Q.E.D
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40
p(x) = (a-Σxj), Ci(xi) = ci xi; ε2 = (c2-c1)/(a-c1), ε3 = (c3-c1)/(a-c1); θ0 = (1+ε2)/ 3, θ2 = 15ε2-1, θ4 = (1+ε2)/ 15, θ6 = (1+13ε2)/ 15; ρ1 = (5-11ε2)/11, ρ5 = (125-3×891/2)/436 ≈ 0..22 θ6 θ0
θ4
θ2
θ4
θ6 θ0
ε2
ρ1
ε2 θ2
ε2 θ2
θ0 ρ1
θ6
θ4
θ0
θ4 ρ5 ≈.22
ε2 ρ1 θ2 θ4
ρ1
θ6 θ0
ε2 ρ1 1/14 ≈.07
0
θ4 θ6
4/53 ≈.08
d12 > 0 ⇔ ε3 < θ2
d12 < 0
5/22 ≈.23
d12 > 0
d13 > 0 ⇔ε3 > θ4
d13 > 0 d23 > 0 ⇔ ε3 > θ6 Figure 1. The profitability for n = 3 and the relations among its parameters.
ε3
δ- s tab le
0 .2 5
δ- s ta b le
δ- u n s ta b le
ω 1 ( ε2 )
450
1 /2 = .5
0 .1 0
0
ε2
1 /6 ≈ .1 7
Fi g u r e 2 . T h e δ -s t a b ilit y o f m o n o p o ly p a rt it io n ∆ m . N o t e t h a t t h e fe a s ib le r e g i o n is t h e a r e a a b o ve t h e 4 5 0 lin e .
41
ε2 θ2
θ2
16/77 ≈.21
1/11 ≈.09
ρ1
θ6 θ0
1/2 = .5
ε2
ε3
ε3
Pa rt (a ) ∆* = ∆m
0. 5
Pa rt (b)
0. 5
∆* = ∆1 or ∆ 2 0.36
∆* = ∆1 or ∆2
ω2 (ε2 )
µ11(ε2 ) = µ1 (ε2 ,1)
0.18
0.10 µ1 (ε2 ,0.5)
∆* = ∆ 0
7/38 ≈ .18
0
µ10(ε2 ) = µ1 (ε2 ,0)
ε2
45 0 5/22 ≈ .23
ε2
45 0
0 1/11 ≈ .09
1/2 = .5
1/2 = .5
Figure 3. (a) The o ptimal pa rtitions for n = 3, the fe asible regi on is the are a ab o ve the 4 5 0 line; (b ) the critical val ue o f ε 3 fo r the stability of a n interm ediat e pa rtition with n = 3 dec re ases as the share t decre ases.
ε3
0 .5
P a rt (i ) θ 0 (ε 2 )
µ 1 ( ε2 ,0) 0 .3 8 0 .3 6
0 .5
0 .3 6
∆ 1 is sta b le 0 .2 14
θ 6 (ε 2 ) 1 /1 5
450
0
1 /1 1 ≈ .0 9
0 .5 1 1 3 /3 1 6 ≈ .3 6
θ 0 (ε 2 )
µ 2 ( ε2 ,0)
∆ 1 is u ns ta b le
1 /1 5
P a r t (i i)
ε3
ε2
0
∆ 2 is u ns ta b le ∆ 2 is sta b le
θ 4 (ε 2 )
450 1 /1 1 ≈ .0 9
0 .2 1 4 = e 2 ( 0)
Fi g u r e 4 . (i ) S t a b il it y o f ∆ 1 , fe a s ib le re g io n is M a x { ε 2 , θ 6 }≤ ε 3 ≤ θ 0 ; (i i ) s t a b ilit y o f ∆ 2 , fe a s ib le re g io n is M a x {ε 2 , θ 4 }≤ ε 3 ≤ θ 0 . In b o t h c as es , t is s e t a t 0 .
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0 .5
ε2
ε3
θ 2 (ε 2 )
0 .5
µ 5 ( ε 2 ,0 )
θ 0 (ε 2 ) s ta b le
s ta b le .29 3
u n s ta b le
3 /14 ≈.21 4
s ta b le
µ 5 0 ( ε 2 ,0 )
450 0 .5
0
1 /1 1
.1 7 9
ε2
.2 1 4
Fi g u r e 5 . T h e S t a b i lit y o f ∆ 3 , fe a s ib le re g io n is ε 2 ≤ ε 3 ≤ M i n {θ 0 ,θ 2 }, t is s e t a t 0 , a n d µ 3 ( ε 3 ,0 ) is re p re s e n t e d b y µ 5 (ε 2 ,0 ) a n d µ 5 0 (ε 2 , 0 ) .
d(n)
ε > 1 /8
d(n)
P a r t (a )
ε ≤ 1 /8 n
0
2
n
n0
n0
2
d 1 ( n)
P a r t (b ) 0 n1
n 2
n3
F i g u r e 6 . (a ) T h e t w o ro o t s o f d ( n ); (b ) t h e t h r e e r o o t s o f d 1 (n ).
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