Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly

Munich Personal RePEc Archive

Bertrand-Edgeworth games under oligopoly with a complete

characterization for the triopoly

De Francesco, Massimo A. and Salvadori, Neri

University of Siena, University of Pisa

30 September 2009

Online at https://mpra.ub.uni-muenchen.de/24087/

MPRA Paper No. 24087, posted 26 Jul 2010 02:07 UTC

Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly

Massimo A. De Francesco, Neri Salvadori University of Siena, University of Pisa

July 23, 2010

Abstract

The paper extends the analysis of price competition among capacity- constrained sellers beyond the cases of duopoly and symmetric oligopoly.

We first provide some general results for the oligopoly, highlighting features of a duopolistic mixed strategy equilibrium that generalize to oligopoly. Unlike in the duopoly, however, there can be infinitely many equilibria when the capacity of a subset of firms is so large that no strategic interaction among smaller firms exists. Then we focus on the triopoly, providing a complete characterization of the mixed strat- egy equilibrium of the Bertrand-Edgeworth game. The mixed-strategy region of the capacity space is partitioned according to key equilibrium features. We also prove the possibility of a disconnected support of an equilibrium strategy and show how gaps are then determined. Com- puting the mixed strategy equilibrium then becomes quite a simple task.

1 Introduction

The issue of price competition among capacity-constrained sellers has at- tracted considerable interest since Levitan and Shubik’s [13] modern reap- praisal of Bertrand and Edgeworth. Assume a given number of firms pro- ducing a homogeneous good at constant and identical unit variable cost up to some fixed capacity. Assume, also, a non-increasing and concave demand and that rationing takes place according to the surplus maximizing rule.

Then there are a few well-established facts about equilibrium of the price game. First, at any pure strategy equilibrium the firms earn competitive profit. However, a pure strategy equilibrium need not exist unless the ca- pacity of the largest firm is small enough compared to total capacity. When

a pure strategy equilibrium does not exist, existence of a mixed strategy equilibrium is guaranteed by Theorem 5 of [3] for discontinuous games.

Under the above assumptions on demand and cost, a mixed strategy equilibrium was characterized by Kreps and Scheinkman [12] for the duopoly within a two-stage capacity and price game. This model was subsequently extended to allow for non-concavity of demand (by Osborne and Pitchik, [15]) or differences in unit cost among the duopolists (by Deneckere and Kovenock, [9]). This led to the discovering of new phenomena, such as the possibility of the supports of the equilibrium strategies being disconnected and non-identical for the duopolists.

Yet there is still much to be learned about mixed strategy equilibria under oligopoly, even with constant and identical unit cost and concave demand, where a complete characterization of the mixed strategy equilib- rium is available only for some special cases. Vives [17], amongst others, analyzed the case of equal capacities among all firms. Within an analysis concerning horizontal merging of firms Davidson and Deneckere [4] pro- vided the complete analysis (apart for the fact that attention is restricted to equilibria in which strategies of equally-sized firms are symmetrical) of a Bertrand-Edgeworth game with linear demand, equally-sized small firms and one large firm with a capacity that is a multiple of small firm’s capac- ity.¹ More recently Hirata [11] provided an extensive analysis of triopoly with concave demand and efficient rationing: having highlighted the basic features of mixed strategy equilibria under triopoly, he was able to analyze how mergers between two firms would affect profitability in the different circumstances. Our analysis of the triopoly differs in scope from Hirata’s since we provide a complete characterization of mixed strategy equilibria in the triopoly: we reveal all qualitative features possibly arising in the tri- opoly, including the facts highlighted in [11].² In a still unpublished paper Ubeda [16] has compared discriminatory and uniform auctions and obtained a number of novel results on discriminatory auctions, a context equivalent

1Davidson and Deneckere [4] assumed a given number of equally-sized firms some of which merge. To see whether merger facilitates collusion in a repeated price game, they had to characterize equilibria of the static price game for the resulting special asymmetric oligopoly and hence mixed strategy equilibria when the new capacity configuration falls in the mixed strategy region of the capacity space. Our study shows that the equilibrium strategies of smaller firms may indeed be indeterminate (though each firm equilibrium payoff is the same at any equilibrium). Davidson and Deneckere avoided this problem by restricting their attention to equilibria that treat small firms symmetrically ([4], footnote 10, p. 123).

2Our own research and Hirata’s were conducted independently. (Results were made publicly available, in [7] and [10], respectively.)

to a Bertrand-Edgeworth game. Differences between our contribution and those of Hirata and Ubeda are further clarified below.

These references make it clear that the issue at hand is relevant in many respects, such as mergers (hence regulation), auctions, and price leader- ship.³ In contrast, a characterization of payoffs of all firms at a mixed strategy equilibrium of the price game does not seem to be needed to solve an oligopolistic two-stage capacity and price game, at least under Kreps and Scheinkman’s assumptions of convex cost of capacity, concavity of demand, and efficient rationing. In fact, it has recently been shown (see [2] and [14]) that the Cournot outcome then extends to oligopoly. This result basically derives from a fundamental property of mixed strategy equilibria, namely, the fact that the payoff of (any of) the largest firm is what is earned by the Stackelberg follower when rivals supply their capacity.⁴

As explained above, our ultimate goal was to deepen our understand- ing of mixed strategy equilibria under oligopoly and this paper provides a number of results in this connection. However, as soon as mixed strategy equilibria turned out to have different qualitative features depending upon the firms’ capacities, it occurred to us that a taxonomy was required in order to completely characterize such equilibria. This seemed hard to manage un- der general oligopoly and so we turned to the triopoly, to simplify the task and in the confidence of getting insights for subsequent generalizations to oligopoly. This research has led to several discoveries. Unlike in the duopoly, the equilibrium strategies need not have identical supports for all the firms:

the maximum and minimum of the supports need not be the same for all the firms⁵ and supports need not be connected (although their union is).

A further difference from the duopoly is that there can be infinitely many equilibria.

The paper is organized as follows. Section 2 contains definitions and the basic assumptions of the model along with a few basic results on equilibrium payoffs in oligopoly and a key Lemma. Section 3 is concerned with mixed

3The relevance of mixed strategy equilibria of price games for the analysis of mergers might also be viewed in a longer-run perspective, by allowing for capacity decisions by the merged firm and outsiders (on this, see Baik [1]). Characterizing mixed strategy equi- librium of the price game in a duopoly allows Deneckere and Kovenock [8] to endogenize price leadership by the dominant firm when the capacity vector lies in the mixed strategy region.

4Hence, at any capacity configuration giving rise to a mixed strategy equilibrium of the price subgame, the largest firm has not made a best capacity response: it would raise profit by reducing capacity. Having ruled out any such capacity configuration, the Cournot outcome follows straightforwardly.

5That minima may differ has also been recognized in [10] and [11].

strategy equilibria under oligopoly. Several features of a duopolistic mixed strategy equilibrium turn out to generalize to oligopoly: determination of the upper and lower bounds of the support of the equilibrium strategy of (any of) the largest firm; determination of the equilibrium payoff of the second-largest firm; the necessary symmetry of equilibrium strategies for equally sized firms (so long as the equilibrium is fully determined); the absence of atoms in equilibrium strategies, apart from the upper bound of the support of the largest firm, which it charges with positive probability when its capacity is strictly higher than for any other firm. Unlike in the duopoly, however, there can be infinitely many equilibria. Roughly speaking, this feature can arise when total capacity and the share of it held by a subset of firms are so large ⁶ that no strategic interaction exists among smaller firms: what is sold by any of them at some price only depends on prices set by firm(s) with larger capacities. In such a case, we show that there is a single equation constraining the equilibrium strategies of smaller firms.

Sections 4 to 6 are devoted to the triopoly. In Section 4 the region of the capacity space involving a mixed strategy equilibrium is partitioned into several subsets according to the features of the resulting equilibrium. This leads to a classification theorem which characterizes the firms’ payoffs and bounds the supports of the equilibrium strategies throughout the region of mixed strategy equilibria. Quite interestingly, there are circumstances where the smallest firm gets a higher payoff per unit of capacity than the larger ones’.⁷ Section 5 introduces the theoretical possibility of the support of the equilibrium strategy being disconnected for some firms. More specifically, we clarify when there is necessarily a gap in the support between the minimum and the maximum and how the gap is then determined. Having done this, we are able to complement our classification theorem with a uniqueness theorem: either the equilibrium is unique or not fully determined, and we identify the two complementary subsets of the region of mixed strategy equilibria where the former and the latter hold true, respectively.⁸ The event of a gap in some support is established in Section 6. Here we construct the mixed strategy equilibrium in the set where the supports of equilibrium strategies have the same bounds for all the firms. That set is, in turn, partitioned into two subsets according to the nature of the equilibrium: in

6In [11], as well as in the earlier version [7] of this paper, indeterminateness was only discovered for the case in which the largest firm’s capacity is higher than total demand.

7This fact was also discovered by [11]. Besides, we are able to compute that firm’s payoff, even in those circumstances.

8Uniqueness of the mixed strategy equilibrium of the price game with fixed capacities was proved, for the duopoly, by Osborne and Pitchik [15].

one, the supports are connected for all the firms; in the other, there is a gap in the support of the smallest firm. To show that gaps are a more general phenomenon, in Section 6 we also look elsewhere in the region of mixed strategy equilibria and provide an example with a gap in the support of the equilibrium strategy of the intermediate-size firm. Section 7 briefly concludes.

2 Preliminaries

There are n firms, 1,2, ..., n, producing a homogeneous good at the same constant unit cost (normalized to zero), up to capacity. The demand func- tionD(x) is defined forp>0, continuous, and decreasing and concave when positive. We define P(x) as the inverse function D⁻¹(x) for x ∈ [0, D(0)) and P(x) = 0 for x > D(0).⁹ Without loss of generality, we consider the subset of the capacity space (K₁, K₂, ..., K_n) where K₁ > K₂ > ... > K_n, and we defineK=K₁+...+K_n.

It is assumed throughout that any rationing is according to the efficient rule. Consequently, let Ω(p) be the set of firms charging pricep: the residual demand forthcoming to all firms in Ω(p) is maxn

0, D(p)−P

j:pj<pKj

o

= Y(p). If P

i∈Ω(p)K_i > Y(p), the residual demand forthcoming to any firm i∈ Ω(p) is a fraction α_i(Ω(p), Y(p)) of Y(p), namely, D_i(p₁, ..., p_n) = αi(Ω(p), Y(p))Y(p). Our analysis does not depend on the specific assump-

tion being made onα_i(Ω(p), Y(p)): for example, it is consistent with α_i(Ω(p), Y(p)) = K_i/P

r∈Ω(p)K_r as well as with the assumption that residual demand is shared evenly, apart from capacity constraints, among firms in Ω(p).¹⁰

At any given pure strategy profile, letp= max{p1, ..., pn}. Letp^c be the competitive price, that is,

p^c=

P(K) if D(0)>K 0 ifD(0)6K.

We now provide necessary and sufficient conditions for the existence of a pure strategy equilibrium and show that no pure-strategy equilibrium actually exists when the competitive price is not an equilibrium. These results are straightforward generalizations of similar results for the duopoly.

9A similar definition of functionP(x) can be found in Davidson and Deneckere [5].

10In this case,αi(Ω(p), Y(p)) = min{Ki/Y(p), β(p)}whereβ(p) is the solution inαof equationP

i∈Ω(p)min{Ki/Y(p), α}= 1. LetM ∈Ω(p) andKM > Ki (each i∈ Ω(p)).

ThenP

i∈Ω(p)min{Ki/Y(p), α}is increasing inαover the range [0, KM/Y(p)] and equal toP

i∈Ω(p)Ki/Y(p)>1 forα=KM/Y(p).

Proposition 1 (i) (p₁, ..., p_n) = (p^c, ..., p^c) is an equilibrium if and only if either

K−K₁ >D(0), if D(0)6K, (1) or

K16−p^c D^′(p)

p=p^c, if D(0)> K. (2) (ii) All firms earn the competitive profit at each pure strategy equilibrium and(p^c, ..., p^c) is the unique equilibrium if D(0)> K.

Proof. (i) If K >D(0), charging p^c = 0 is a best response of firm ito rivals chargingp^c if and only ifP

j6=iK_j >D(0). This holds for eachiif and only if P

j6=1Kj >D(0). If D(0) > K, charging p^c is the best response of firmito rivals chargingp^cif and only ifh

d[p(D(p)−P

j6=iK_j)]/dpi

p=p^c 60.

This holds for eachiif and only if K₁ 6−p^c[D^′(p)]_p=pc.

(ii) We must scrutinize strategy profiles such that p > p^c. Assume first D(p)−P

j:pj<pKj >0. If #Ω(p)>1, then at least some firm i∈Ω(p) has a residual demand lower than K_i and would raise profits by deviating to a price negligibly lower thanp, since output would jump up, from [D(p)− P

j:pj<pKj]αi(Ω(p), Y(p)) to minn

Ki, D(p−ǫ)−P

j:pj<pKj

o

. If #Ω(p)<

n, any firmj /∈Ω(p) is selling its entire capacity and therefore has not made a best response: it would still sell its capacity by raising the price, provided it remains lower thanp. Next assumeD(p)−P

j:pj<pKj 60. In order for any firm charging more than the lowest price p to have made a best response, it must be p = 0 and P

j:pj=0K_j > D(0) (the latter of course requiring thatK >D(0)): note that all firms are here earning the competitive profit (zero). But then, in order for each firm j charging p to have also made a best response, it must be P

s:ps=0,s6=jK_s >D(0).

Therefore, equilibria with p > p^c may only exist if inequalities (1) hold, the set of equilibria then being any strategy profile such thatP

s:ps=0,s6=jKs>

D(0) for each j such that p_j = 0; if inequalities (2) hold, then a unique equilibrium exists, in which all firms charge the competitive price p^c >

0; if neither (1) nor (2) holds, then no pure strategy equilibrium exists.

As a consequence, the existence of a pure strategy equilibrium depends upon the capacity of the largest firm to be sufficiently small compared to total capacity. In fact, either (1) or (2) holds if and only if K₁ 6 max{K−D(0),−p^c[D^′(p)]_p=pc}. It is assumed in the following that K₁ >

max{K−D(0),−p^c[D^′(p)]_p=pc}, so that we are in the region of mixed strat- egy equilibria.

We henceforth denote by (φ₁(p), ..., φ_n(p)) = (φ_i(p), φ_−i(p)) a profile of strategies at a mixed strategy equilibrium, where φ_i(p) = Pr(p_i < p) is the probability of firm i charging less than p. For the sake of brevity, we denote by Π^∗_i (rather than by Π^∗_i(φ_i(p), φ_−i(p)) firm i’s expected profit at equilibrium strategy profile (φ_i(p), φ_−i(p)), and by Π_i(p) firm i’s expected profit when it charges p with certainty and the rivals are playing the equi- librium profile of strategies φ_−i(p).¹¹ Further, S_i is the support of φ_i(p), and p⁽ⁱ⁾_M and p⁽ⁱ⁾m are the maximum and minimum of S_i, respectively. More specifically, we say that p ∈S_i when φ_i(·) is increasing at p, that is, when φi(p+h)> φi(p−h) for any 0< h < p, whereasp /∈Si ifφi(p+h) =φi(p−h) for someh >0.¹² Of course, anyφ_i(p) is non-decreasing and everywhere con- tinuous except at p^◦ such that Pr(p_i =p^◦) >0, where it is left-continuous (lim_p→p^{◦ −}φ_i(p) = φ_i(p^◦)), but not continuous. Let p_M = max_ip⁽ⁱ⁾_M and p_m = min_ip⁽ⁱ⁾m, M = {i:p⁽ⁱ⁾_M = p_M} and L ={i:p⁽ⁱ⁾m =p_m}. Moreover, if

#M < n, then we define pb_M = max_{i /∈M}p⁽ⁱ⁾_M. Similarly, if #L < n, then we definepb_m = min_{i /∈L}p⁽ⁱ⁾_m.

Obviously, Π^∗_i > Πi(p) everywhere and Π^∗_i = Πi(p) almost everywhere in S_i. Some more notation is needed to investigate further the properties of Π_i(p). Let N = {1, ..., n} be the set of firms, N_−i = N − {i}, and P(N₋i) ={ψ} be the power set of N₋i. Further, let

Z_i(p;φ_−i) :=p X

ψ∈P(N−i)

q_i,ψ Y

r∈ψ

φ_r Y

s∈N−i−ψ

(1−φ_s), (3) whereφi ∈[0,1] are real numbers andq_i,ψ = max{0,min{D(p)−P

r∈ψKr, Ki}}

is firm i’s output when any firm r ∈ ψ charges less than p and any firm s∈N_−i−ψcharges more thanp.¹³ FunctionZ_i(p;φ_−i) allows firmi’s payoff function Πi(p) to be decomposed into functions{p, φ₋i(p)}, so long as firm i’s rivals’ equilibrium strategiesφ_−i(p) are all continuous inp: namely, Π_i(p) = Z_i(p;φ_−i(p)). If instead Pr(p_j =p^◦)>0 for somej6=i, thenZ_i(p^◦;φ_−i(p^◦))>

Π_i(p^◦)>lim_p→p^◦₊Z_i(p;φ_−i(p)).¹⁴ Sometimes we factorize φ_j and (1−φ_j) in equation (3) to obtain

Z_i(p;φ_−i) =Z_i(p;φ_−i−j, φ_j) =φ_jZ_i(p;φ_−i−j,1) + (1−φ_j)Z_i(p;φ_−i−j,0).

11In principle the vector of equilibrium payoffs need not be unique if the equilibrium strategy profile is not so.

12Note thatφi(p) = 0 in a right neighborhood of zero.

13Note thatQ

r∈ψφr is the empty product, hence equal to 1, when ψ =∅; and it is similarlyQ

s∈N₋i−ψ(1−φs) = 1 whenψ=N−i.

14The exact value of Πi(p^◦) when Pr(pj=p^◦)>0 for somej6=idepends on function αi(Ω(p), Y(p)).

Z_i(p;φ_−i−j,1) andZ_i(p;φ_−i−j,0)) have a clear interpretation: ifφ_r =φ_r(p) (eachr 6=i, j), thenZ_i(p;φ_−i−j,1) andZ_i(p;φ_−i−j,0)) are firmi’s expected payoffs when chargingp, conditional onp_j < pandp_j > p, respectively. We establish some properties of functions Z_i(p;φ_−i) which will be useful later on.

Lemma 1 (i) Z_i(p;φ_−i) is continuous in p. For every p and every φ₋i there exists ǫ > 0 such that Zi(p;φ₋i) is concave in p in the intervals [p, p+ǫ] and [p−ǫ, p]: as a consequence, Z_i(p;φ_−i) is locally concave in p whenever it is differentiable in p. Wherever Z_i(p;φ_−i) is concave in p but not strictly so, there is a function h(φ₋i), 0 6 h(φ₋i) 6 1, such that Z_i(p;φ_−i) =h(φ_−i)pK_i.¹⁵

(ii) For given φ_−i and for any ψ ∈ P(N_−i), Z_i(p;φ_−i) is kinked at p=P(P

r∈ψKr)and locally convex there if Q

r∈ψφrQ

s∈N−i−ψ(1−φs)>0.

(iii) Z_i(p;φ_−i) is continuous and differentiable in φ_j (each j 6= i) and

∂Z_i/∂φ_j 6 0. More precisely, ∂Z_i/∂φ_j <0if and only if there exists some ψ∈ P(N₋i−j) such that¹⁶

Y

s∈ψ

φs

Y

t∈N−i−j−ψ

(1−φt)>0 (4)

and

0< D(p)−X

h∈ψ

K_h< K_i+K_j. (5) (iv) ∂Z_i/∂φ_j <0if and only if ∂Z_j/∂φ_i <0.

(v) If ∂Z_i/∂φ_j <0, then ∂Z_i/∂φ_r<0 for any r < j.

(vi) If ∂Z_i/∂φ_j = 0, then there is functionG(φ_−i−j)such thatZ_i(p;φ_−i) = G(φ_−i−j)pK_i and Z_j(p;φ_−j) =G(φ_−i−j)pK_j.

(vii) LetN˜ ={i∈N :∂Z_j/∂φ_i <0∀j∈N}and N˜˜ =N−N. Similarly,˜ φ˜ = {φ_i : i ∈ N˜} and φ˜˜ = {φ_i : i ∈ N˜˜}. Assume that N˜˜ is not empty.

Then, for eachr∈N˜,Z_r(p;φ_−r) =Q_r(p; ˜φ_−r)−pR_r( ˜φ_−r)P

s∈N˜˜ φ_sK_swhere Qr(p; ˜φ₋r) := pP

ψ∈P( ˜N−r)q_r,ψQ

t∈ψφtQ

v∈N˜−r−ψ(1−φv) and Rr( ˜φ₋r) :=

P

ψ∈P( ˜N−r),0<qr,ψ<Kr

Q

t∈ψφtQ

v∈N˜−r−ψ(1−φv).

(viii) Assume 0 < φ1 < 1, 0 < φs < 1 for some s ∈ N₋1, and P(P

i6=1K_i)> p > P(P

i∈ΦK_i) where Φ ={i∈N :φ_i>0}. Then:

(a) ∂Z₁/∂φ_i<0 and ∂Z_i/∂φ₁ <0 for any i∈N₋₁;

15Ifφ−i=φ−i(p^◦), thenh(φ−i) is the probability that the residual demand for firmi is not lower thanKi when firmichargesp^◦and the rivals are playingφ−i(p^◦).

16By slightly extending notation,N−i−j=N− {i, j}andP(N−i−j) is its power set.

(b) if p < P(P_r

h=1K_h) then ∂Z_r+1/∂φ_i < 0 and ∂Z_i/∂φ_r+1 < 0 for any i > r+ 1;

(c) if p>P(K₁), ∂Z_i/∂φ_j = 0 for any i∈N₋₁ and any j ∈N_−1−i. (ix) If K_i=K_j andφ_i 6φ_j, thenZ_i(p;φ_−i)6Z_j(p;φ_−j)andZ_i(p;φ_−i)<

Z_j(p;φ_−j) whenever φ_i< φ_j and ∂Z_i/∂φ_j <0.

(x) If K_i 6K_j and φ_i > φ_j = 0, then (K_j/K_i)Z_i(p;φ_−i)>Z_j(p;φ_−j).

Proof. (i) Zi(p;φ₋i) is a convex linear combination of functions which are concave in the intervals [p, p +ǫ] and [p −ǫ, p] for any p and suffi- ciently small ǫ. If Q

r∈ψφ_rQ

s∈N−i−ψ(1−φ_s) > 0 at some ψ such that q_i,ψ = D(p)−P

r∈ψK_r, then Z_i(p;φ_−i) is strictly concave; if not, then Q

r∈ψφ_rQ

s∈N−i−ψ(1−φ_s) > 0 only for ψ’s such that either q_i,ψ = K_i or q_i,ψ = 0.

(ii) At p=P(P

r∈ψK_r), the left derivative ofZ_i(p;φ_−i) with respect to pequals the right derivative pluspD^′(p)Q

r∈ψφ_rQ

s∈N−i−ψ(1−φ_s)<0.

(iii) DifferentiateZ_i(p;φ_−i) with respect toφ_j and rearrange to obtain

∂Z_i

∂φ_j =Z_i(p;φ_−i−j,1)−Z_i(p;φ_−i−j,0) =

=p X

ψ∈P(N−i−j)

(q_i,ψ∪{j}−q_i,ψ)Y

r∈ψ

φ_r Y

s∈N−i−j−ψ

(1−φ_s). (6) Then, ∂Z_i/∂φ_j 6 0 since q_i,ψ∪{j} −q_i,ψ 6 0. Clearly, ∂Z_i/∂φ_j < 0 if and only if there exists ψ ∈ P(N_−i−j) such that inequality (4) holds and q_i,ψ∪{j}−q_i,ψ <0, which leads to inequalities (5).

(iv) It follows from the symmetrical role of i and j in inequalities (4) and (5).

(v) Recall that, in order for∂Zi/∂φj <0 (∂Zi/∂φr<0), inequalities (4) and (5) must hold for someψ∈ P(N_−i−j) (resp., ψ^′ ∈ P(N_−i−r)). Suppose they hold for some ψ such that r /∈ ψ. For ψ^′ = ψ, inequalities (5) read 0< D(p)−P

h∈ψK_h < Ki+Kr, which hold too since the first inequality is unchanged andK_j 6K_r; inequality (4) holds ifφ_j <1. Suppose inequalities (4) and (5) hold for some ψ such that r ∈ ψ. For ψ^′ = ψ ∪ {j} − {r}, inequalities (5) read 0< D(p)−P

h∈ψ^′Kh < Ki+Kr, which hold too since the second inequality is unchanged andK_j 6K_r; inequality (4) holds ifφ_j >

0. Thus the claim is proved if φ_j ∈(0,1). Assume now that φ_j = 0. The claim is still proved if someφ’s for which inequalities (4) and (5) are satisfied do not includer. Assume the opposite, i.e., that allφ’s for which inequalities (4) and (5) are satisfied include r; then Z_i(p;φ_−i) = φ_rZ_i(p;φ_−i−r,1) and

∂Z_i/∂φ_r 6 0 only if Z_i(p;φ_−i) = 0. Assume now that φ_j = 1 and all

φ’s for which inequalities (4) and (5) are satisfied do not include r, then Z_i(p;φ_−i) = (1−φ_r)Z_i(p;φ_−i−r,0) and ∂Z_i/∂φ_r <0.

(vi) For eachψ⊆N_−i−jit is eitherq_i,ψ∪{j}=q_i,ψ = 0 orq_i,ψ∪{j}=q_i,ψ = K_i or Q

r∈ψφ_rQ

s∈N−i−j−ψ(1−φ_s) = 0. Hence in all positive addends of sum (3)q_i,ψ =K_i. Thus there is a functionG_i(φ_−i−j) such thatZ_i(p;φ_−i) = G_i(φ_−i−j)pK_i. Similarly, taking account of part (iv), we obtainZ_j(p;φ_−j) = Gj(φ₋i−j)pKj. Finally, Gi(φ₋i−j) < Gj(φ₋i−j) if and only ifq_i,ψ = 0 and q_j,ψ = K_j for some ψ ⊆ N_−i−j, i.e., K_j 6 D(p)−P

r∈ψK_r 6 0. This contradiction implies thatG_i(φ_−i−j) =G_j(φ_−i−j).

(vii) Let ψ ∈ P( ˜N₋r) and ψ^′ ∈ P( ˜N˜). It is easily checked that if q_r,ψ =K_r, then also q_r,ψ∪ψ^′ =K_r. Otherwise there are i, j ∈ ψ^′ such that

∂Z_i/∂φ_j <0 since q_i,ψ∪ψ^′_∪{r}−q_i,ψ∪ψ^′_∪{r}−{j} <0. Similarly, if 0 < q_r,ψ <

K_r, then q_r,ψ∪ψ^′ = q_r,ψ −P

s∈ψ^′K_s > 0. As a consequence, Z_r(p;φ_−r) = pP

ψ∈P( ˜N−r),qr,ψ=Kr

P

ψ^′∈P( ˜N˜)K_rQ

t∈ψφ_tQ

u∈ψ^′φ_uQ

v∈N−r−ψ−ψ^′(1−φ_v) + pP

ψ∈P( ˜N−r),0<qr,ψ<Kr

P

ψ^′∈P( ˜N)˜

hq_r,ψ−P

s∈ψ^′K_si Q

t∈ψφ_tQ

u∈ψ^′φ_uQ

v∈N−r−ψ−ψ^′(1− φ_v) =pP

ψ∈P( ˜N−r)q_r,ψQ

t∈ψφ_tQ

v∈N˜−r−ψ(1−φ_v)hP

ψ^′∈P( ˜N)˜

Q

u∈ψ^′φ_uQ

v∈N˜˜−ψ^′(1−φ_v)i

− pP

ψ∈P( ˜N−r),0<qr,ψ<Kr

Q

t∈ψφtQ

v∈N˜−r−ψ(1−φv)hP

ψ^′∈P( ˜N˜)

P

s∈ψ^′KsQ

u∈ψ^′φuQ

v∈N˜˜−ψ^′(1−φv)i

= Q_r(p; ˜φ_−r)−pR_r( ˜φ_−r)P

s∈N˜˜ φ_sK_shP

ψ^′∈P( ˜N˜)−P( ˜N˜−s)

Q

u∈ψ^′−{s}φ_uQ

v∈N˜˜−ψ^′(1−φ_v)i

= Q_r(p; ˜φ_−r)−pR_r( ˜φ_−r)P

s∈N˜˜ φ_sK_s. The first equality holds by definition.

The other equalities are obtained by rearranging the sum and by recognizing complementary events.

(viii.a)∂Z1(p)/∂φi<0 if at least one product on the right-hand side of (6) is strictly negative. This is certainly so forψ = Φ− {1, j}. (Note that if i ∈ Φ, 0 < q_1,ψ∪{i} < K₁ whereas if i /∈ Φ, 0 < q_1,ψ < K₁.) Part (iv) completes the proof.

(viii.b) Let Ψ₁ be the set of the subsets ψ of N_−(r+1)−i which satisfy inequality D(p) > P

h∈ψK_h. Ψ₁ is not empty since {1,2, ..., r} ∈ Ψ₁. Let Ψ2 be the set of the subsetsψ ofN_−(r+1)−i which satisfy inequalityD(p)<

P

h∈ψK_h+K_r+1+K_i. Ψ₂ is not empty since Φ− {r+ 1, i} ∈Ψ₂. Because of part (iii) the claim is proved if Ψ₁∩Ψ₂ 6= ∅. Assume contrariwise that Ψ1 ∩Ψ2 =∅. Then for any ψ ∈Ψ1, D(p)−P

h∈ψK_h > Kr+1+Ki > 0, while, for anyψ∈Ψ₂,D(p)−P

h∈ψK_h 60< K_r+1+K_i. Of course, there is some ψ_l ∈ Ψ₁ such that {1,2, ..., r} ⊆ ψ_l and ψ_l∪ {l} ∈ Ψ₂. Therefore Kl>D(p)−P

h∈ψlKh >Kr+1+Ki, which contradicts the fact that Kl6 K_r+1 and K_i>0. Statement (iv) completes the proof.

(viii.c) If 1 ∈/ ψ ⊆ N_−i−j, then q_i,ψ∪{j} −q_i,ψ = K_i −K_i = 0. If

1∈ψ⊆N_−i−j, thenq_i,ψ∪{j}−q_i,ψ = 0−0 = 0.

(ix) SinceK_i =K_j,Z_i(p;φ_−i−j, β) =Z_j(p;φ_−i−j, β). HenceZ_i(p;φ_−i)− Z_j(p;φ_−j) = (φ_j−φ_i)∂Z_i/∂φ_j.

(x) Sinceφ_i > φ_j = 0,Z_i(p;φ_−i) =Z_i(p;φ_−i−j,0), whereasZ_j(p;φ_−j)6

Z_j(p;φ_−i−j,0) because of part (iii). Thus it suffices to prove that (K_j/K_i)Z_i(p;φ_−i−j,0)>

Z_j(p;φ_−i−j,0). Note that for anyq_i,ψwith a positive coefficient inZ_i(p;φ_−i−j,0) there is a corresponding q_j,ψ with a positive coefficient in Z_j(p;φ_−i−j,0), based on the same ψ ∈ P(N_−i−j), and vice versa. The claim follows since (K_j/K_i)q_i,ψ >q_j,ψ for each ψ∈ P(N_−i−j).

Parts (iv)-(vii) of Lemma 1 allow a nice interpretation of the Jacobian matrix [∂Z_i/∂ψ_j]_i,j∈N. If # ˜N = s, ˜N = {1,2, ..., s}, because of part (v).

Submatrix [∂Z_i/∂ψ_j]

i,j∈N˜˜ is a zero (n−s)×(n−s) matrix, because of parts (iv), (v), and (vi). Submatrix [∂Zi/∂ψj]

i∈N,j˜˜ ∈N˜ is a negative (n−s)×s matrix whose rank is 1, because of parts (v) and (vi). Similarly, submatrix [∂Z_i/∂ψ_j]

i∈N ,j˜ ∈N˜˜ is a negative s×(n−s) matrix whose rank is 1, because of parts (v) and (vii). Finally, submatrix [∂Z_i/∂ψ_j]_i,j∈N˜ is an s×smatrix with all elements on the main diagonal nought and all off-diagonal elements negative, because of part (v).

3 Mixed strategy equilibria under oligopoly

In this section we establish a number of general properties of mixed strategy equilibria under oligopoly. Since [12] it has been known thatp_M = p⁽¹⁾_M = p⁽²⁾_M = arg maxp(D(p)−K₂) in a duopoly with K₁ > K₂; also, φ₁(p_M) <

φ₂(p_M) = 1 if K₁> K₂,while φ₁(p_M) =φ₂(p_M) = 1 ifK₁ =K₂. Therefore Π^∗_i = p_M(D(p_M)−K2) for any i such that Ki = K1. These results have recently been generalized to oligopoly, as summarized here.

Proposition 2 φ_i(p_M) = 1for anyisuch thatK_i < K₁,p_M = arg maxp(D(p)− P

j6=1K_j),p⁽ⁱ⁾_M =p_M for someisuch thatK_i=K₁, andΠ^∗_i = maxp(D(p)− P

j6=1K_j) for any i:K_i=K₁.

Proof. For a complete proof, see [2] and [6], and, more recently, [16], [14], and [11].

The following proposition establishes quite expected properties of mixed strategy equilibria.

Proposition 3 (i) For any i∈N, Π^∗_i = Π_i(p) for p in the interior of S_i. (ii) For any p^◦∈(p_m, p_M), p^◦ > P(P

i:p⁽ⁱ⁾m<p^◦K_i).

(iii) #L>2 and #M >2.

(iv) For any p^◦ ∈(p_m, p_M), #{i:p^◦ ∈S_i} 6= 1.

Proof. (i) Suppose contrariwise that Π^∗_i > Π_i(p^◦) for some p^◦ in the interior ofSi. This reveals thatp^◦ is not charged by i: it is Pr(pj =p^◦)>0 for somej 6=i and Z_i(p^◦;φ_−i(p^◦))>Π_i(p^◦)>lim_p→p^◦₊Z_i(p;φ_−i(p)). As a consequence, Π^∗_i >Π_i(p) on a right neighborhood ofp^◦: a contradiction.

(ii) Otherwise for i such that p⁽ⁱ⁾m < p^◦ it would be Π_i(p) = pK_i for p∈S_i∩[p_m, p^◦]: a contradiction.

(iii) Assume contrariwise that L={i}. Then, on a right neighborhood ofp_m, Π_i(p) =pmin{K_i, D(p)}, a non-constant function. A similar contra- diction would occur ifM ={1}.

(iv) If #{i : p^◦ ∈ Si} = 1, then φ₋i(p) are all constant for p close enough top^◦, and Π_i(p) = Π^∗_i cannot be positive there: by Lemma 1(i)-(ii),

∂Z_i(p;φ_−i)/∂p= 0 only if Z_i(p;φ_−i) = 0.

The following proposition about p_m and p_M generalizes well known re- sults concerning duopoly to oligopoly. Similar generalizations were also pro- vided by Ubeda [16] in a different context. In order to shorten notation, we henceforth denote lim_p→h+Π_i(p) and lim_p→h−Π_i(p) as Π_i(h⁺) and Π_i(h⁻), respectively, and lim_p→h+φ_i(p) as φ_i(h⁺).

Proposition 4 (i) p⁽ⁱ⁾m =p_m for any isuch that K_i =K₁.

(ii) p_m = max{bp,bbp} where pb= Π^∗₁/K₁ and bbp is the smallest solution of equation pD(p) = Π^∗₁; Π^∗₁ =pKb ₁ or Π^∗₁ =bbpD(bbp) according to whether pb>bbp or pb6bbp.

(iii) p_m> P(P

j∈LK_j).

(iv) p⁽ⁱ⁾_M =p_M for any isuch that K_i=K₁. Proof. (i) SinceD(pM)>P

j6=1Kj, ifp⁽ⁱ⁾m > pmfor somei6= 1 such that K_i = K₁, then a fortiori D(p) > P

j∈LK_j for p 6p_M: as a consequence, for anyj∈L,Π_j(p) is increasing for p∈[p_m, p⁽ⁱ⁾_m): a contradiction.

(ii) If p < max{bp,bbp}, then Π₁(p) 6 pmin{D(p), K₁} < Π^∗₁ = pKb ₁ = bbpD(bbp). Hence,pm>max{bp,bbp}. On the other hand, ifpm >max{bp,bbp}, then Π1(p⁻_m)>Π^∗₁. Indeed, ifp >b bbp, thenD(bp)> K1 so that it is eitherD(pm)>

K₁, hence Π₁(p⁻_m) = p_mK₁ > pKb ₁, or D(p_m) < K₁, hence Π₁(p⁻_m) =

p_mD(p_m) > bbpD(bbp) (since pD(p) is increasing for p ∈ [0, p_M]). If bbp > p,b thenD(bbp)< K₁ and hence Π₁(p⁻_m) =p_mD(p_m)>bbpD(bbp).

(iii) If #L = n and p_m 6 P(P

j∈LK_j), then each firm earns no more than its competitive profit, contrary to Proposition 2. If #L < n and p_m < P(P

j∈LK_j), then Π_j(p) is increasing over a right neighborhood of p_m, anyj∈L: an obvious contradiction. If #L < nandp_m=P(P

j∈LK_j), then Π^∗_i =p_mK_i even ifp_m were charged with positive probability by some j∈L− {i}. As a consequence,

Π^∗_i = Π_i(p) =p



D(p)− X

j∈L−{i}

K_j



 Y

j∈L−{i}

φ_j(p) +pK_i



1− Y

j∈L−{i}

φ_j(p)





=p[D(p)−D(pm)] Y

j∈L−{i}

φj(p) +pKi

in a neighborhood ofp_m.ThereforeQ

j∈L−{i}φ_j(p) = _p[D(p)^(pm−p)Ki

−D(pm)], which is decreasing in a right neighborhood ofp_msince lim_p→p+

mdQ

j∈L−{i}φ_j(p)/dp= [K_ip_mD^′′(p) + 2D^′(p)]/2p²_m[D^′(p)]²<0: an obvious contradiction.

(iv) Let K₂ = K₁, p⁽¹⁾_M =p_M and p⁽²⁾_M < p_M. Since φ₁(p) < φ₂(p) = 1 for p ∈ (p⁽²⁾_M, p_M), Π₁(p) < Π₂(p) there because of the following Lemma 2(a) and Lemma 1(ix). As a consequence, Π^∗₂ > Π2(p) > Π1(p) = Π^∗₁ for p∈(p⁽²⁾_M, p_M)∩S₁, contrary to the fact that Π^∗₁ = Π^∗₂ because of Proposition 2. Quite similarly, if (p⁽²⁾_M, p_M)∩S₁ =∅, i.e., Pr(p₁=p_M) = 1−φ₁(p⁽²⁾_M)>0, then Π^∗₂ >Π2(p⁻_M)>Π1(p_M) = Π^∗₁.

Lemma 2 If p∈(p_m, p_M), then

(a) [∂Z₁/∂φ_i]_{φ−1=φ−1(p)} < 0 and [∂Z_i/∂φ₁]_{φ−i=φ−i(p)} < 0 for any i ∈ N₋₁;

(b) if p < P(P_r

h=1K_h)then[∂Z_i/∂φ_j]_{φ−i=φ−i(p)}<0and [∂Z_j/∂φ_i]_{φ−j=φ−j(p)}<

0for any i6r+ 1and any j∈N_−i;

(c) if p > P(K₁), [∂Z_i/∂φ_j]_{φ−i=φ−i(p)} = 0 for any i ∈ N₋₁ and any j∈N_−1−i.

Proof. Proposition 2, Proposition 3(ii)-(iii), and Proposition 4(i) imply that for (φ_i, φ_−i) = (φ_i(p), φ_−i(p)) andp∈(p_m, p_M) the assumptions of part (viii) of Lemma 1 hold. Then the claim follows from Lemma 1(iv)-(v)&(viii) and from the fact that the demand function is not increasing.

Note that, since bp is decreasing in K₁, the event of bbp > pb arises at relatively large levels ofK1. Proposition 4(ii) has an immediate consequence:

Corollary 1. p_m >P(K₁) if and only ifbbp>p.b