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Research Article
Market Share Delegation in a Bertrand Duopoly:
Synchronisation and Multistability
Luciano Fanti,1Luca Gori,2Cristiana Mammana,3and Elisabetta Michetti3
1Department of Economics and Management, University of Pisa, Via Cosimo Ridol 10, 56124 Pisa, Italy
2Department of Law, University of Genoa, Via Balbi 30/19, 16126 Genoa, Italy
3Department of Economics and Law, University of Macerata, Via Crescimbeni 20, 62100 Macerata, Italy
Correspondence should be addressed to Elisabetta Michetti; elisabetta.michetti@unimc.it
Received October ; Revised March ; Accepted March
Academic Editor: Jinde Cao
Copyright © Luciano Fanti et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation.
e dynamics of the economy is characterised by a dierentiable two-dimensional discrete time system. e paper stresses the
importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their
substitutability. is means that when products are complements, players may coordinate their behaviour even if initial conditions
are dierent. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very
dierent dynamic patterns.
1. Introduction
Strategic delegation is a relevant topic in both oligopoly the-
ory and industrial organisation, and several papers have con-
tributed to clarify questions related to the dierences between
the behaviour of prot-maximising rms and managerial
rms (e.g., [–]). In the former kind of rms, ownership and
management coincide, and consequently the main aim they
pursue is prot maximisation. In the latter, ownership and
management are separate and managers may be driven by
incentive schemes that only partially take into account prot
and the other objectives of the rms, such as output, revenues,
relative performance evaluation, and market share [,–].
In addition to the above-mentioned theoretical papers, there
also exist some empirical works that stress the importance of
market share delegation contracts in actual economies [,].
e present paper studies a nonlinear duopoly game with
price competition and market share delegation and extends
the study carried out by Fanti et al. []tothecaseofcomple-
mentary or independent products. To this end, by following
an established literature led by Bischi et al. [], we assume
that players have limited information and analyse how
a managerial incentive scheme based on market share aects
thelocalandglobaldynamicsofatwo-dimensionaldiscrete
time system. e paper stresses the dierences with the
analysis carried out by Fanti et al. []onthesubstitutability
between products in the case with managerial rms and
market share contracts and compares the results achieved.
e rest of the paper is organised as follows. Section
describes the model. Section shows some preliminary
global properties of the two-dimensional dynamic system
(feasible set). Section studies the xed points of the system,
the invariant sets, and local stability. Section is concerned
with multistability and shows that synchronisation may arise
when managers receive the same bonus. It also stresses the
dierences with Fanti et al. [] and takes into account the
asymmetric case in which bonuses are not equally weighted
inthemanagers’objectivefunction.Section outlines the
conclusions.
2. The Model
Consider a duopoly game with price competition, horizon-
tal dierentiation, and market share delegation contracts
Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 394810, 13 pages
http://dx.doi.org/10.1155/2015/394810
Discrete Dynamics in Nature and Society
(see [] for details). Market demands of goods 1and 2are,
respectively, given by
1=1−1−1−2
1−2,
2=1−2−1−1
1−2,()
where ∈ (−1,0] is the degree of dierentiation of
(complementary) products, while ≥0and ≥0are
quantity and price per unit of good of rm (=1,2).
Both the rms have the same marginal cost 0≤<1and
hire a manager, who receives a bonus based on market share
/(+)(,=1,2, =), where +is total supply. e
objective function of manager is
=Π+
+, ,=1,2, =, ()
where Π=(
−)are prots and >0is the
(constant) delegation variable of player . Hence, by using (),
() becomes
=1−−1−
1−2−+1+
2−−,
,=1,2, =, ()
from which we get the following marginal bonus:
=1−2−1−+
1−2−(1+)1−
(1−)2−−2,
,=1,2, =.
()
We now assume a discrete time (∈Z+) dynamic setting,
where each player has limited information, as in Bischi et al.
[], and uses the following behavioural rule to set the price
for the subsequent period:
,+1=,+,,,,
, ,=1,2,∈Z+,()
where >0.Wewanttodescribethequalitativeand
quantitative long-term price dynamics when products are
complementary or independent, that is, ∈ (−1,0],and
underline the similarities and dierences with the case of
substitutability investigated in Fanti et al. [].
Assume =1,+1,=1,,=2,+1,and=2,.By
using () and (), the two-dimensional discrete time dynamic
system is as follows:
:
=,
=1+1−2−1−+
1−2
−1(1+)1−
(1−)2−−2
=,
=1+1−2−(1−)+
1−2
−2(1+)(1−)
(1−)2−−2.
()
3. The Feasible Set
It is of importance to observe that system () is economically
meaningful only whether, at any time , the two state variables
and are not negative; that is, they belong to ,whereis
theconvexpolygonwithvertices(0,0),(0,1−),(1,1),and
(1−,0).
Let ((0),(0)),=0,1,2..., denote the th iterate
of system for a given initial condition ((0),(0)) ∈ .
en, the sequence ={((),())}∞
=0is called trajectory.A
trajectory is said to be feasible for if ((),())∈for
all ∈N;otherwise,itisunfeasible.eset⊆whose
points generate feasible trajectories is called feasible set.A
pointbelongingtothefeasiblesetiscalledfeasible point.
efeasiblesetofsystemis depicted in white in Figures
(a) and (b) fortwodierentparameterconstellations,while
the unfeasible points belonging to are depicted in grey. e
followingevidencecanbeimmediatelyobserved:similarly
to the substitutability case, (i) set is nonempty such that
⊂and (ii) set may have a simple structure (as in
Figure (a)) or a complex structure (as in Figure (b)). e
rst evidence can be easily demonstrated by considering that
the origin is a feasible point and that there exists a >0small
enough such that (1−,1−)is not a feasible point.
With regard to thestudy of the structure of the feasible set,
anumericalprocedurebasedonthestudyoftheproperties
of the critical curves can be used (see, e.g., [–]). By taking
into account the results proved in Fanti et al. [], it is easy to
verify that is of 4−2−0type since can be subdivided
into regions whose points have 4,2,or0preimages, and the
boundaries of such regions are characterised by the existence
of at least two coincident preimages.
Wedenotethecriticalcurveofrank-1by LC (it represents
the locus of points that have two or more coincident preim-
ages) and the curve of merging preimages by LC−1;thatis,the
set of points (,)∈such that |(,)|=0,where
Discrete Dynamics in Nature and Society
0
011−d
1
1−d
y
x
(a)
1−d
1−d
1
1
y
00
x
(b)
0
01
1
1−d
1−d
y
x
LCb
−1
LCa
−1
(c)
0
1−d
1−d0
1
1
Z2
Z4
Z0
y
x
LCb
LCa
(d)
F : (a), (b). e feasible set ⊂is depicted in white; the gray points are initial conditions producing unfeasible trajectories. (a)
=1.5,=−0.1,=0.5,and1=2=0.2.(b)=1.5,=−0.2,=0.5,and1=2=0.2. (c) Critical curves of rank-, LC−1,forsystem
and the parameter values as in (b). (d) Critical curves of rank-, LC =(LC−1), for the same parameter values as in panel (c). ese curves
separate the plane into the regions 4,2,and0, whose points have a dierent number of preimages.
,=1+1−4−1−+
1−2−1(1+)1−2+−
(1−)2−−3
1−2−1(1+)−
(1−)2−−3
1−2−2(1+)−
(1−)2−−31+
1−4−(1−)+
1−2−2(1+)(1−)2−+
(1−)2−−3,
()
is the Jacobian matrix of system (see Figures (c) and (d)).
e study of the structure of set is of interest from
both economic and mathematical perspectives, since the
long-term evolution of the economic system becomes path-
dependent, and a thorough knowledge of the properties of
becomes crucial in order to predict the system’s feasibility.
We now x all parameter values but .en,asisshown
in Figures (a) and (b),if=−0.15,sethas a simple
structure(connectedset),whilewhen=−0.2,setconsists
of innitely many nonconnected sets. is is due to the fact
that the LCcurve moves upwards as parameter decreases
and consequently a threshold value −0.1551does exist
such that a contact between a critical curve and the boundary
ofthefeasiblesetoccurs.iscontactbifurcationcauses
the change of from a connected set to a nonconnected
set. In fact, a portion of the unfeasible set enters in a
region characterised by a high number of preimages so
that new components of the unfeasible set suddenly appear
aer the contact (in Figure (c), the feasible set is depicted
immediately aer the contact bifurcation creating grey holes
inside the white region). e complexity of the structure of
the feasible set increases if decreases and the grey area
Discrete Dynamics in Nature and Society
1−d
1−d
1
1
y
0
0
x
(a)
1−d
1−d
1
1
y
0
0
x
(b)
1−d
1
1−d1
y
0
0
x
(c)
1−d
1
y
0
x
1−d
10
(d)
F : Parameter values: =1.5,=0.5,1=0.4,and2=0.2.(a)has a simple structure (=−0.15). (b) has a complex
structure (=−0.2). (c) Set aer the contact bifurcation (=−0.16): gray holes are depicted. (d) Immediately before the nal bifurcation
(=−0.216) almost all trajectories are unfeasible.
increases too (i.e., the set of initial conditions generating
unfeasibletrajectories);however,itcanbealsoobservedthat
if decreases further, a nal bifurcation occurs. In fact, as
crosses a value −0.218,almostalltrajectoriesbecome
unfeasible (see Figure (d)).
Notice that in the simulations presented in Figure we
have assumed 1=
2.However,asimilarbehaviourholds
also when 1=
2: in this case, the contact bifurcation
between the critical set and the boundary of the feasible set
occurs at two points that are symmetric with respect to the
main diagonal, so that the resulting feasible set is symmetric
too (see Figures (a) and (b)). e previous arguments show
that the bifurcations concerning the structure of the feasible
set is strictly related to the value of the two key parameters
and , which represent the degree of horizontal product
dierentiation and the level of market share bonuses. e
following results can be proved.
Proposition 1. Let be the dynamic system given by (6).
(i) If →−1+,then={(0,0)}.
(ii) If →+∞,=1,2,then={(0,0)}.
Proof. (i) If →−1+,then→={(,)∈R2+:+≤
2,≥0,≥0}. Observe rst that it must be +<2
for being well dened and that (0,0)∈, and hence we
consider initial conditions such that (0)+(0)<2and at
least one component of ((0),(0))is strictly positive. Taking
into account system (),itcanbeobservedthat
(1)+(1)=(0)+(0)+,(0),(0)()
and that ∀((0),(0))∈ : (0)+(0) <2and (0)+
(0) =0;if→−1+,then(,(0),(0))diverges; that is,
((0),(0))produces an unfeasible trajectory.
(ii) is statement can be proved simply considering the
limits lim1→+∞(1)and lim2→+∞(1)for any given initial
point ((0),(0))∈,(0)+(0) =0.
According to Proposition ,ifis high enough or
products tend to be complements (is low enough), the
feasible set is consists of only the origin. is result conrms
the one obtained in Fanti et al. [] for the substitutability
case; that is, economic meaningful dynamics are produced
only when the degree of complementarity or substitutability
Discrete Dynamics in Nature and Society
between products is not too high. More precisely, in the
case under investigation, economically meaningful long-
term dynamics can be produced only for ∈−(0)and ∈
+(0)(=1,2), thus conrming the numerical experiments
previously presented.
By taking into account the above-mentioned arguments,
in what follows we will focus on the study of the dynamics
produced by by assuming that is suciently small and
is suciently high.
4. Fixed Points, Invariant Sets, and
Local Stability
Let be the dynamic system given by () and consider a
feasible initial condition. We now recall the results proved in
Fanti et al. [] concerning xed points and other invariant
sets of for ∈(0,1). It can be veried that they still hold
also when ∈(−1,0].Weherepresentasketchoftheproof
and we refer to Fanti et al. []forafurtherdiscussion.
Remark 2. Let be given by ().en,onehasthefollowing.
(i) ={(,):0≤≤1−,=0}and ={(,):
0≤≤1−,=0}are invariant sets. e dynamics
of on such sets are governed by =(,0)and
=(0,)and they can be complex; in any case,
and are repellor.
(ii) If 1=
2=,thenalso={(,)∈R2+:=
,∈[0,1)}is an invariant set. e dynamics of on
such a set are governed by ()=[1+(/(1−))
(((1−)(1−)−+)/(1+)−((1+)/4(1−)))]
and they can be complex; furthermore, canbean
attracting set.
(iii) e origin 0=(0,0)is a xed point for all parameter
values; it can be a stable node, an unstable node, or
a saddle point. Up to two more xed points on
and may be owned. ey are given by 1=(0,0)
and 2=(0,0)and can be unstable nodes or saddle
points.
(iv) If 1=2=,thenadmits a unique interior xed
point ∗=(
∗,∗)for all <4(1−+)/
(1+)2= ,where∗= 1 −((1−)+
(1−)2+(2−)(1+)2)/2(2−).∗can be a
stablenode,anunstablenode,orasaddlepoint.
Proof. Consider the system given by ().
(i) (,0)=(,0)and (0,)=(0,);thatis,and
are invariant, and consequently the dynamics of on
such lines are governed by the two one-dimensional
maps =(,0)and =(0,).Asboth
and are unimodal for suitable parameter values (as
proved in []),complexdynamicscanbeproduced.
and are repellor as the eigenvalue of (,0)and
(0,)associated with the direction orthogonal to
each semiaxis is greater than one for all parameter
values.
(ii) Assume 1=
2=;then,(,) =(,),and
hence is invariant and the dynamics of system on
are governed by ()=(,).Giventheproper-
ties of ()studied in Fanti et al. [], nonmonoton-
icity can occur for suitable parameter values and
complex dynamics may emerge. Finally, canbean
attracting set as the eigenvalue of (,)associated
with the eigenvector orthogonal to the diagonal may
be less than one in modulus (see again Fanti et al.
[]).
(iii) Since (0,0)=(0,0),0is a xed point; furthermore,
while considering (0,0),itcanbeobservedthat0
canbeastablenode,anunstablenode,orasaddle
point. By solving =and =and following
Fanti et al. [], it can be easily veried that up to two
positive solutions exist, namely, 1and 2;suchxed
points cannot be stable nodes as and are repellor.
(iv) Assume 1=2=; then, by solving equation ()=
it can be veried that it admits a unique positive
feasible solution ∗i <(see Fanti et al. []for
more details); as each eigenvalue of (∗,∗)can be
greater or less than one in modulus, then ∗canbea
stablenode,anunstablenode,orasaddlepoint.
e question of the existence of an interior xed point
in the general case 1=2(i.e., with dierent market share
bonuses) cannot be addressed analytically. However, itwill be
discussed later in the paper by using numerical techniques.
By taking into account parts (i) and (iii) in Remark ,inwhat
follows we will focus on the study of the dynamics produced
by for a feasible initial condition belonging to the interior
of , thus focusing on economically meaningful initial states.
5. Synchronisation and Multistability
In order to study the evolution of system when products are
independent or complementary and compare this case with
that of substitutable goods, we concentrate on the particular
case of identical market share bonuses; that is, 1=2=.As
aconsequence,systemin () takes the symmetric form
given by
:
=,
=1+1−2−1−+
1−2
− (1+)1−
(1−)2−−2
=,
=1+1−2−(1−)+
1−2
− (1+)(1−)
(1−)2−−2. ()
Discrete Dynamics in Nature and Society
Since map is symmetric, that is, it remains the same
when the players are exchanged, then either an invariant set
of the map is symmetric with respect to or its symmetric set
is invariant. By considering part (iv) in Remark ,thelocal
stability analysis of the unique interior xed point ∗can
be carried out by considering the Jacobian matrix associated
with system given by
,=1+1−4−1−+
1−2−(1+)1−2+−
(1−)2−−3
1−2− (1+)−
(1−)2−−3
1−2− (1+)−
(1−)2−−31+
1−4−(1−)+
1−2−(1+)(1−)2−+
(1−)2−−3.
()
Let
1()=1+
1−2
⋅4(4−)(1−)3+4(−3)(1−)2−(1+)2
4(1−)2,
2()=
1−2. ()
en, the Jacobian matrix evaluated at a point on the diagonal
is of the kind (,)=1()2()
2()1(), ()
so that the eigenvalues of (,)arebothrealandgivenby
()=1()+2(),
⊥()=1()−2(),()
while the corresponding eigenvectors are, respectively, given
by V=(1,1)and V⊥=(1,−1).
e eigenvalues evaluated at the xed point ∗are,
respectively, ∗=1∗+2∗,
⊥∗=1∗−2∗. ()
us, ∗can be attracting for suitable values of parameters
such that both (∗)and ⊥(∗)belong to the set (−1,1).
Dierent from the case in which products are substitutes,
the following Proposition can easily be veried.
Proposition 3. If =0,then(∗)=⊥(∗);if∈
(−1,0),then(∗)<⊥(∗).
Proof. If =0,then2() = 0∀,andconsequently
(∗)=⊥(∗);if∈(−1,0),then2()<0∀>0,
and hence (∗)<⊥(∗).
From Proposition it follows that if =0,then
the interior xed point is a stable or an unstable node.
Furthermore, conditions (∗)>1or ⊥(∗)<−1are
sucient for ∗to be an unstable node.
e following condition for the local stability of ∗
(which holds if products are substitutes) applies also to the
case ∈(−1,0]and it can be recalled below (see Fanti et al.
[] for the proof ).
Proposition 4. Let system be given by (9).ena>0does
exist such that ∗is locally asymptotically stable ∀∈(−,),
given the other parameter values.
By taking into account Propositions and ,asu-
cient condition for ∗to be locally stable for =0is
→4(1+)−.Inthiscase,giventhegeometricproperties
of map ()described in Fanti et al. [], the initial conditions
belonging to converge to ∗with independent products.
In addition, the trajectories starting from initial conditions
close to it, that is, ((0),(0)) ∈ (∗,),with(0) =
(0),alsoconvergeto∗.isbehaviouroccursaslong
as ∈
−(0),while,asdecreases, the xed point loses
its stability rstly along the diagonal, thus giving rise to a
dierent scenario compared to that presented in Fanti et al.
[] for the substitutability case.
Denition 5. Afeasibletrajectory= {(),()}∞
=0,
((0),(0))∈, is called synchronised trajectory. A feasible
trajectory starting from ((0),(0))∈−,thatis,with
(0) =(0),issynchronisedif|()−()|→0as →+∞.
With regards to the dynamics of synchronised trajecto-
ries, we rst recall the result proved in Proposition .
An attractor located on the invariant set exists for
if products are not too complementary (is not too
low) and the market share bonus is not too large (see
Figures (a) and (c)). is fact can also be conrmed by
considering the following properties of map ():(0)=0,
lim→1−()=−∞,(0)>1∀∈(0,),()<0∀∈
[0,1). As a consequence, there exists a unique−∈(0,1)such
that (−)=0and a unique maximum point ∈(0,−)
such that ()isthemaximumvalueofin [0,1).is
implies that is unimodal in [0,−]. By considering that
()increases when decreases and ()→+∞if →
−1,thenadoes exist such that ()>−if −1<<
(see Figure (b)). On the other hand, by considering the role
of parameter , a cascade of period doubling bifurcations is
observed when decreases (see Figure (c)).
Discrete Dynamics in Nature and Society
d
x
b = 0.8
b = 2.5
0.8
0
0
−0.35
d
x
1
0
0
−0.25
(a)
01
1
0
x
𝜙(x)
d = −0.2
Δ
d = −0.4
(b)
x
x
d = −0.05
d = −0.2
10.625
6.8698
0
0
0
0
1
1
b
b
(c)
F : Parameter values: =0.5,=1.5. (a) One-dimensional bifurcation diagrams with respect to for two xed values. (b) Map
is plotted for dierent values and =1. (c) One-dimensional bifurcation diagrams with respect to (0<<)for two xed values.
By taking into account the previous results and looking
at the one-dimensional bifurcation diagrams in Figures (a)
and (c), it can be observed that synchronised trajectories
convergetotheuniqueinteriorxedpointifis close to (the
manager bonus is close to its upper limit). In addition, similar
to what occurs with the logistic map, cycles can emerge due
to period doubling bifurcation of when decreases (i.e., the
degree of complementaritybetween products increases). is
evidence is also conrmed when products are substitutes,
thus proving that complexity in synchronised trajectories
arises when moving away from the hypothesis of independent
products.
Let us consider now a duopoly with identical players that
startfromdierentfeasibleinitialconditionsandlet⊆be
an attracting set of .If=∗for =0, then by considering
() and Proposition the following proposition holds.
Proposition 6. Let ∗be an attracting xed point of for =
0.en,∗is an attracting xed point of and ∃−<0such
that ∗is an attracting xed point of ∀∈(−,0)∩(−1,0).
At =−,thexedpoint∗loses stability along the diagonal.
Proof. Let =0and assume that ∗is an attracting xed
point of ;thatis,(∗)=(∗)∈(−1,1).Since(∗)=
⊥(∗),then∗is an attracting xed point of .Asboth
(∗)and ⊥(∗)are continuous with respect to ∗and
,then∃−(0)such that |(∗)|<1and |⊥(∗)|< 1
∀∈(−,0)∩(−1,0),and∗is an attracting xed point of .
Finally, since (∗)<⊥(∗)∀∈(−1,0)and they are
both increasing with respect to ,thenascrosses −the
eigenvalue (∗)must cross −1;thatis,∗loses its stability
along the diagonal.
According to Proposition , if synchronised trajectories
converge to ∗for =0, then trajectories starting from
feasible initial conditions close to it, with (0) = (0),
aresynchronisedinthelongtermaslongas∈
−(0).
If decreases further, the xed point loses stability rstly
Discrete Dynamics in Nature and Society
along the diagonal. is contrasts with the result obtained
in Fanti et al. [] where products are substitutes. In fact, if
passes from zero to positive values, the xed point rst
loses transverse stability and consequently the trajectories are
not synchronised. erefore, synchronisation in this model
is strictly related to the assumption of complementarity
between products.
If consists of a -cycle, then, similarly to what happens
for the xed point, several numerical computations show
that if is a -cycle for =0, then the -cycle loses
stability rstly along the diagonal when decreases, so
that synchronisation may occur. Dierent from the case of
substitutability, this evidence conrms that when products
are complements, players may coordinate their behaviour
towards a situation in which prices are equal (and the market
is equally shared). In our analysis it is also stressed that the
emergence of synchronisation with negative values of also
conrms the result obtained in Fanti et al. []withprot-
maximising rms.
Consider now a more complex situation; that is, is a
chaotic attractor on . In order to study its transverse stability,
itispossibletousetheprocedureproposedinBischietal.[],
Bischi and Gardini [], and Bignami and Agliari []. Recall
that the transverse Lyapunov exponent is dened as follows:
Λ⊥=lim
→∞1
=0ln ⊥,()
where 0∈and is a generic trajectory generated by .
If 0belongs to a generic aperiodic trajectory embedded
within the chaotic set ,thenΛ⊥is the natural transverse
Lyapunov expone nt Λ⊥,wherenatural indicates that the
exponent is computed for a typical trajectory taken in
the chaotic attractor . Since innitely many cycles (all
unstable along the diagonal) are embedded inside the chaotic
attractor , a spectrum of transverse Lyapunov exponents
can be dened and the natural transverse Lyapunov exponent
represents a sort of weighted balance between the transversely
repelling and transversely attracting cycles. If all cycles
embedded in are transversely stable, that is Λmax
⊥ <0,then
is asymptotically stable in the Lyapunov sense for the two-
dimensional map . Nevertheless, it may occur that some
cycles embedded in the chaotic set become transversely
unstable; that is, Λmax
⊥ >0, while Λ⊥ <0.Insucha
case, is not stable in the Lyapunov sense but it is a stable
attractor in the Milnor sense. If a Milnor attractor of
exists, then some transversely repelling trajectories can be
embedded in a chaotic set which is attracting only on average.
In addition, such transversely repelling trajectories can be
reinjected toward so that their behaviour is characterised
by some bursts far from the diagonal, before synchronization
or convergence towards a dierent attractor. is situation is
called on-o intermittency.
In order to investigate the existence of a Milnor attractor
, we numerically estimate the natural transverse Lyapunov
exponent Λ⊥, represented with respect to in Figure (a)
for a xed negative value of .Itispossibletoobservethatit
can take negative values. As an example, we consider =2.07
at which Λ⊥<0while Λmax
⊥ >0and the one-dimensional
map exhibits a 4-piece chaotic attractor. is an attractor
of system belonging to the diagonal (see Figure (b)), but
a trajectory starting from an initial condition that does not
belong to the diagonal has a long transient before converging
to (see Figure (c)). In fact, by considering the dierence
()−()for any we can observe that the transient part
of the trajectory is characterised by several bursts away from
. e typical on-o intermittency phenomenon occurs. e
whole trajectory starting from (0)=0.1and (0)=0.2is
shown in Figure (d).
e study of the geometrical properties of the critical
lines may be used to estimate the maximum amplitude of
the bursts by obtaining the boundary of a compact trapping
region of the phase plane in which the on-o intermittency
phenomena are conned. Following Mira et al. [], we obtain
the boundary of the absorbing area in Figure (e) for the case
presented in Figure (d). Observe that such a region contains
the whole trajectory presented in Figure (d).However,not
all trajectories are synchronised as also admits a coexisting
attractor, that is, a 2-period cycle whose basin is depicted in
orange in Figure (f).
If admits an attractor ⊂and there exist
feasible trajectories starting from interior points that are not
synchronised, then the question of multistability has to be
considered. In fact, several attractors may coexist (each of
which with its own basin of attraction) so that the selected
long-term state becomes path dependent, as in the situation
shown in Figure (f). In this case, the structure of the basins
of dierent attractors becomes crucial to predict the long-
term outcome of the economic system. In Figure (a),the
unique Nash equilibrium is locally stable, as the market share
bonusisclosetoitsupperlimit.Itisalsoglobally stable,in
the sense that it attracts all feasible trajectories taken into
the interior of the feasible set (that represents economic
meaningful initial conditions).
If we compute /,thenitispossibletoobservethat
increases as decreases, and →+∞as →−1
+.
As a consequence, in order to obtain the convergence to the
unique Nash equilibrium, must be set at higher levels as the
degree of horizontal product dierentiation decreases (i.e.,
products tend to be more complementary). Furthermore, if
products are complements, then crosses −1as decreases
and a ip bifurcation occurs along the invariant set ;thatis,
a -period cycle appears close to the xed point and it is stable
along the diagonal. By taking into account the result proved
in Proposition ,whenthisbifurcationoccurs,⊥is still
smaller than 1in modulus and consequently (immediately
aer the rst ip bifurcation) the 2-period cycle attracts all
interior feasible trajectories. However, if is still decreased,
a period doubling bifurcation cascade occurs along the
diagonal, so that more complex bounded attractors (such as
periodic cycles) may exist on around the unstable Nash
equilibrium. As a consequence, the long-term synchronised
dynamics may be characterised by bounded periodic (or
even aperiodic) oscillations around the Nash equilibrium.
If the attractor is transversely unstable, the situation may
become very complicated, as it is shown in Figure (b) where
a2-period cycle attracts all synchronised trajectories, while
two attractors coexist out of the diagonal: a 2-period cycle
Discrete Dynamics in Nature and Society
2 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.1
0.4
0.3
0.2
0.1
0
−0.1
−0.2
b
(a)
01.2
1.2
0
y
x
(b)
0 8000
−0.5
0
0.5
t
x(t) − y(t)
(c)
1.2
0
y
0 1.2
x
(d)
y
x
0 1.2
0
1.2
(e)
y
x
01
0
1
(f)
F : Parameter values: =1.5,=−0.2,and=0.5. (a) e natural transverse Lyapunov exponent with respect to parameter .(b)
Four-piece chaotic attractor of system belonging to the diagonal for =2.07. (c) Bursts away from the diagonal before synchronization
for =2.07,(0)=0.1,and(0)=0.2. (d) e whole trajectory starting from initial condition as in (c) and converging to the attractor in
(b). (e) e minimal absorbing area in which on-o intermittency phenomenon occurs for the same parameters as in (c). (f) e attractor
coexists with an attracting -period cycle for =2.07.
Discrete Dynamics in Nature and Society
0 1.2
0
1.2
E∗
b
y
x
(a)
0 1.2
0
1.2
y
x
(b)
0 0.8
0
0.8
y
x
(c)
01
0
1
x
y
(d)
F : Parameter values: =1.5and =0.5. (a) e white region represents the basin of ∗for =5.1and =−0.2. (b) A -period
cycle together with a 2-period cycle coexists with the attractor belonging to the diagonal (a two-period cycle) for =2.4and =−0.2.
(c) For =2.3,a4-piece quasi-periodic attractor has been created, =−0.2. (d) A particular scenario is presented for =0.2inthecaseof
independent products =0.
whosebasinisdepictedinorangeanda4-period cycle whose
basin is depicted in yellow; note that the periodic points are
in symmetric position with respect to the diagonal. As the
parameter decreases, a further ip bifurcation occurs and
a4-period cycle is created on the diagonal (see Figure (c));
furthermore, the 4-period cycle existing out of the diagonal
becomes a stable focus and then undergoes a Neimark-Sacker
bifurcation at which it becomes a 4-cyclic attractor formed
by a 4-piece quasi-periodic attractor (green basin) coexisting
with a 2-period cycle (orange basin), while synchronisation
is avoided. is is how the situation presented in Figure (f)
is approached: a nal bifurcation that causes the transition to
more complex basin boundaries occurs. Consequently to the
nal state sensitivity it is impossible to predict the long-term
outcomes of the economy.
Finally, we consider the case in which products are
independent from each other and each manager behaves as
amonopolist(=0). When a ip bifurcation along the
diagonal creates a -period cycle, it can be observed that
a-period cycle is simultaneously created out of the diagonal
as the eigenvalues of cycles embedded into the diagonal are
identical. As a consequence, any period doubling bifurcation
along is associated with a period doubling bifurcation
orthogonal to . A similar phenomenon of multistability
is presented in Bischi and Kopel [].iscaseisshown
in Figure (d): the green points represent initial conditions
converging to a 4-period cycle on the diagonal while the
yellow points represent initial conditions converging to a
4-period cycle out of the diagonal. is scenario occurs
when the 2-period cycle along the diagonal undergoes the
second period doubling bifurcation: two stable 4-period
cycles are created, one along the diagonal (red points) and
one with periodic points symmetric to it (black points); these
two stable cycles coexist with the 2-period cycle previously
created out of the diagonal (white points). Observe that an
economy starting far away from the diagonal may become
synchronised, as the basin of attraction on the diagonal
is comprised of several nonconnected sets. is result is
Discrete Dynamics in Nature and Society
1.2
01.2
0
y
x
1.2
01.2
0
y
x
E∗
E∗
Δb = −3
Δb = 3
(a)
1
00
y
x
1
(b)
1
01
0
y
x
(c)
1
01
0
y
x
(d)
F : (a) Parameter values: =1.5,=0.5,=5.1,and=−0.2. e Nash equilibrium for positive and negative values of .(b)If
=−0.3, the Nash equilibrium is unstable and a 2-period cycle is globally stable. (c) A complex attractor (black points) coexists with a
2-period cycle (white points) if =1.5,=0.5,=2.07,=−0.2,and=0.15. (d) Coexisting attractors if =0.2,=0while =0.01.
Discrete Dynamics in Nature and Society
relevant from an economic point of view. In fact, it implies
coordination even though the manager hired in each rm
behaves as a monopolist in his own market.
5.1. e Asymmetric Case. Wenowconsiderthecaseinwhich
managers’ bonuses are evaluated dierently; that is, 1=2.
Obviously, in this case isnolongerinvariant(i.e.,iftherms
start from the same initial feasible condition ((0),(0))∈
,theywillbehavedierentlyinthelongterm),sothat
synchronisation cannot occur. However, similar to the case
in which products are substitutes, multistability still emerges.
Assume 1=and 2=+,where∈(−,+∞).
With regard to the existence of the Nash equilibrium, we
recall that Proposition states a necessary condition; that is,
parameter shouldnotbetooclosetoitsextremevalue−1
and, in addition, and should not be too high. A Nash
equilibrium, if it exists, is given by a point (∗,∗)∈such
that (∗,∗)=(∗,∗)=0.Asaconsequence,aninterior
xed point can be obtained by considering the intersection
points of the two curves (,) = 0and (,) = 0in
the phase plane. Of course, if these curves intersect in a point
∗=(∗,∗)∈,thenitisaNashequilibriumfor.
By considering the analytical properties of and ,
numerical simulations allow us to conclude that the main
results of Fanti et al. []areconrmedalsowhenproducts
are complements or independent. ese results are collected
in the following list.
(i) If the Nash equilibrium exists, then it is unique such
that the equilibrium price is higher for the variety
associated with a lower market share bonus (see
Figure (a)).
(ii)IftheNashequilibriumislocallystableinthe
symmetric case, then it is also locally stable in the
asymmetric case if and only if the perturbation on
is small enough (i.e., isclosetozero).
(iii) In the case of heterogeneity, the Nash equilibrium
loses stability via a ip bifurcation at which it becomes
asaddlepointandastable2-period cycle appears
close to ∗(see Figure (b)).
(iv) Synchronised trajectories do not emerge and syn-
chronisation cannot occur, while multistability still
emerges (compare Figures (c) to (f)).
To better describe point (iv) above, we recall that a
situation in which synchronisation may occur is depicted
in Figure (f):thesymmetricsystemadmits a complex
attractor on the diagonal that coexists with a 2-period cycle
out of the diagonal. If we consider a slight dierence between
weights attached to market share bonuses, that is, is small
enough, we obtain the situation depicted in Figure (c):the
attractor on the diagonal disappears while a complex attractor
coexists with the 2-period cycle previously found.
Similarly, observe that, with independent products and
homogeneous managers, three coexisting attractors are
owned (see Figure (d)). is situation drastically changes
if = 0.01. In fact, as shown in Figure (d),asmall
perturbation on causes the disappearance of the attracting
4-period cycle symmetric to the diagonal, while a 4-period
cycle close to the diagonal persists together with the 2-period
cycle existing out of the diagonal. Obviously, due to the het-
erogeneity between the weights , the shape of the bound-
aries of the coexisting attractors is no longer symmetric with
respect to the diagonal.
Although the managers’ behaviours are no longer coor-
dinated, it is interesting to stress that, dierent from the
case of substitutability between products, the structure of
the basin of attraction seems to become simpler than under
homogeneous delegation contracts.
6. Conclusions
is paper has studied the mathematical properties of a non-
linear duopoly game with price competition and market share
delegation contracts. e main aim was to extend the analysis
carried out by Fanti et al. [] to the case in which products are
complementary or independent. e most important result
is that the interaction between the degree of complemen-
tarity and the delegation variable (which aects managerial
bonuses) may produce synchronisation in the long term. is
result does not emerge when products are substitutes.
From an economic point of view, synchronisation is
relevant because it implies coordination between players.
en, in a model with managerial rms and market share
delegation contracts, coordination can (resp., cannot) hold
when products are complements (resp., substitutes). In addi-
tion,wehavealsoshownthatmultipleattractorsmayexistso
that initial conditions matter.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
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