BUSINESS STRATEGY FORMULATION MODELING
VIA HIERARCHICAL DYNAMIC GAME
Celso Pascoli Bottura; Eliezer Arantes da Costa
Laboratório de Controle e Sistemas Inteligentes – LCSI
Faculdade de Engenharia Elétrica e de Computação – FEEC
Universidade Estadual de Campinas – UNICAMP
<bottura@dmcsi.fee.unicamp.br>; <eliezer@futuretrends.com.br>
Campinas, São Paulo, Brazil
Abstract – Stochastic infinite dynamic games can be used as a conceptual basis for the development of strategic business
game models for cooperative and competitive environments and under turbulent conditions. This study presents an
analytical framework to develop a challenging and motivating tool for the improvement of executive strategic
management capability, using discrete time stochastic strategic hierarchical dynamic games. The dynamic game is
organized in three hierarchical levels: at the upper one lies the coordinating module, representing the market actions, at
the intermediate level are the competing companies, and at the lower level those responsible for each company’s
management units. Each company’s team has a computational model to simulate the company’s dynamic behavior. Some
classic game theory equilibrium strategies are presented and applied to the proposed strategic game. Stackelberg, Nash
and Pareto equilibrium strategies, duly combined in the proposed hierarchical structure, allow the attainment of
theoretical lower-bound solutions for didactic games.
Keywords – Business strategic games – hierarchical dynamic games – strategic management training
Introduction
Several efforts have been made to model and to solve problems of optimization for large-scale systems with hierarchical
structures, as much for tactical as for strategic use, in productive systems [1, 2] and in decision support systems [3]. The
formalization of non-cooperative dynamic game theory in [4] serves as the basis for a series of interesting applications,
among them the treatment of an optimization problem utilizing distributed and parallel processing in hierarchical
structures [5]. Non-cooperative dynamic game theory is a field that can be explored in educational applications for
development and training of executives, which is the purpose of this study. Concepts about company strategy have been
consolidated in [6], reviewed and expanded in [7], and applied to manufacturing for instance, in [8]. A tutorial
consolidation of the concepts of strategic management is presented in [9].
More recently, an effort toward strategic planning was developed in [10], in which a strategic business game is
employed to generate turbulent scenarios. Such research creates a business game using dynamic simulation resources,
applying principles originally proposed in [11], revised and expanded to apply in the business arena in [12]. The work
presented here, expanding the application of [10], employs competitive and turbulent scenarios, in which companies are
simulated vying for markets and clients and, at the same time, disputing limited resources and supplies. A stochastic
strategic hierarchical dynamic game for application in executive training programs is proposed here as a model for a
competitive and turbulent business environment
1. Executive capability development
Experiments in the development of human resources
and specialized studies about adult education – also
called ‘andragogy’, by some authors – show that
traditional teaching methods, based on presentations,
lectures, readings and seminars, lack something as they
do not motivate, nor involve participants on their real
day-to-day problems. In contrast, playful activities such
as business games and firm simulations in cooperative
or competitive conditions and under turbulent
environments are methods that, if well applied, can
stimulate emulation, enthusiasm, and motivation –
essential elements for efficient learning.
Some modern pedagogical applications for the
development and improvement of companies’ future
managers and decision-makers have used business
games, mainly implemented as computer models for
companies’ operational programming in short-term time
frames, usually based on spreadsheets concepts.
It is a fact that a vast amount of literature on this
subject [6, 8] – even some of the most recent – [7, 9],
treat the concepts of strategic management from a more
qualitative than quantitative point of view.
Nevertheless, some efforts have been made to quantify
strategic actions and their long-term consequences
through mathematical models [10, 12].
Company modeling using optimization and
simulation concepts and methods may reasonably
represent these business environments, at least for
educational purposes. Concepts and resources for
modeling and simulation of continuous systems can be
found in [11, 12]. For the present work, important
contributions are in [4, 5, 13, 14, 15, 16, 17].
y
Business games were greatly developed mainly from the
1950s on as an adaptation, for the company’s
environment, of war games concepts, methodology and
results that had long been used in the military field.
Experiments have shown that participants of business
games end up enthusiastically motivated to increase their
capabilities, including abilities to use more quantitative
tools. Well-elaborated games are able to integrate
segmented knowledge and concepts from various fields
or disciplines, thus enabling participants to form a global
vision of a company and its market as a whole.
The team decision-making interactive process fosters
interpersonal relationships; it provides incentive for the
search for objective data, for the ability to seek out
consensus and for solutions by negotiation. Leadership,
clear and assertive communication, and giving and
receiving feedback capabilities are also highlighted.
Otherwise, competition among distinct teams encourages
the ability to make decisions under pressure, uncertainty,
conflicting situation and tension, seeking consensual
solutions internally, while under intense external
competition.
3. Non-cooperative deterministic game
In order to implement a conceptual platform for business
games, we begin at the concepts and formulations of
dynamic games theory. With a sufficiently broad
meaning for this purpose, we can define a noncooperative deterministic dynamic game [NDDG] with
several participants and multiple stages as a systems
optimization problem with multiple decentralized and
autonomous decisions.
Thus, from the point of view of systems control
theory, a [NDDG] can be associated with a particular
problem of optimal control with multiple controllers. In
this type of game, each of the N participants – or players
– receiving information progressively disclosed by the
structure of the game, makes a sequence of decisions,
stage by stage, attempting to optimize one’s objective
function (1) – while obeying the game constraints.
For a formal presentation of the optimization problem
introduced above, we adopt the following notation,
derived from the terminology of systems theory [4].
For the basic model, we call:
x
k
the vector that represents the state of the game at
stage k;
(1 )
We always make use of the concept that the players must minimize
their objective function.
k
an observation of the state
x
k
for the player Pi at
stage k;
η ki the information available to the player Pi at stage k;
u
2. Educational uses of business games
i
i
k
the action decided by the player Pi at stage k.
So, we can define an infinite non-cooperative
deterministic dynamic game [NDDG] as a structured
set of logical-mathematical conceptual elements, whose
relationships are graphically represented in Figure 1.
In fact, the above-referred definition assumes some
simplification hypotheses: (a) the duration of the game
is fixed; (b) time is assumed to be discrete, by stages;
and (c) all equations, functions and variables are
admitted to be deterministic. Restrictions (a) and (b) of
the [NDDG] do not limit its applicability to strategic
business games for educational purposes, but the item
(c) will be relaxed in Section 4 below.
4. Non-cooperative stochastic game
The [NDDG] described above does not adequately
represents a business environment due to the failure to
encompass random possibilities in the process, which
are common in the business world, full of surprises
brought about by both external and internal factors.
An extension of the [NDDG] concept to deal with
these stochastic aspects can be accomplished with small
changes in some of the formal elements described
above. A simple way to understand and to treat the
randomness of the game is to add into the model a new
‘player’, the (N+1)th, called nature, S, whose decisions
and actions influence the state of the game evolution
and the objective function value calculation.
Nature’s actions can be assumed to obey a
probabilities law established a priori and thus, the state
transition equation in [NDDG] is replaced by a function
of a conditional state probability distribution, given the
previous actions of the players and the previous values
of the state [4].
Keeping these considerations in mind, we can
define an infinite non-cooperative stochastic dynamic
game [NSDG] discrete in time, with N players, with a
pre-defined fixed duration, as a structured set of
logical-mathematical conceptual elements, whose
relationships can also be illustrated graphically in
Figure 1, by merely introducing one more player, S,
the nature, and replacing the deterministic character by
a stochastic one for some variables and functions.
From the point of view of systems control theory,
a [NSDG] non-cooperative stochastic game can be
associated with a particular problem of optimal
stochastic control with multiple controllers [1, 4].
Considering the sufficient generality of the
[NSDG] game, it is adopted as the conceptual reference
platform for the strategic business games treated in this
work.
U
i
k
: u k ∈U k
i
U
Γ
Γ :γ
i
i
k
k
∈ Γk
i
u
i
k
η
Informat ion Space
Ν :η
i
i
k
k
Ν
i
=
γ (η
i
i
k
k
u
η
∈ Νk
i
k
u
⊂ { y ,...,
1
k
u
i
i
i
1
y ,u
k
(z
zi* = imin
∈N
f
1
x
1
k
k +1
k
..., u k ,..., u k )
x
k
,...,
y
N
i
y
k
=
i
h (x
k
k
x
⇐
k
x
Observat ion Funct ion
,..., u1 ,..., u k −1}
yes
K +1 ?
no
k −1
i
=
x
N
i
)=
i
z
( xk , u k ,
i
k
u
:
N
k −1
xk +1 =
Delay
i
i
X
St ate Transit ion
i
)
k
k
∈
X
i
η
k
k
Informat ion i St ruct ure
k
x
i
Decision / Control
k
:
X
i
St rat egy Space
End of Game: *
The Winner is i
for which
*
St ate Space
Control Space
K +1
k +1
k +1
x
k
Object ive Funct ion
z :
J ( x ,..., x
u ,..., u ,..., u )
i
z
i
=
K +1
1
i
1
i
N
1
k
K
,
Init ializat ion:
Init ial st at e is given
k
)
x
k
Y
z
i
k
⇐
x
1
i
k
Observat ion Set
Y
i
k
:
y
i
k
∈Y k
i
Figure 1 – Sche mat ic representation of a non-cooperative dynamic ga me – [NDDG] / [NSDG]
5. Strategic reference game
Applying the concepts and formulation of [NSDG] to
model business games for educational purposes
requires several simplifications and specifications
without loss of the generality and applicability intended
for the given purpose. We will work with a special
class of games, which we call strategic reference game
[SRG], defined as a particular case of [NSDG], which
is, simultaneously, strategic, stationary, equitable, and
closed-loop, encompassing the following concepts:
(a) A strategic game can be defined as a [NSDG] for
which the objective function to be minimized is an
exclusive function of the final state of the game, x K +1 ,
that is,
z
i
=
J (x
i
(i )
K +1
) ; i ∈ N ; ( 2)
(b) A stationary game can
be defined
as a i[NSDG] for
(i )
i
which the functions f k (...) , hk (...) and γ k (...) and the
space Ν k can be rewritten as
i
and
f
(i )
(...) ,
i
h (...)
and
γ
i
(...)
Ν , respectively, discarding, thus, the subscript k ;
i
(c) An equitable game can be defined as a [NSDG] for
which the superscript i, which differentiate the
characteristics of (ithe
players,
cani be discarded. Thus,
)
i
the functions f k (...) , hk (...) , γ k (...) , J i (...) and the
space
J
(...)
Ν
can be rewritten as
and
Ν , respectively;
i
k
f
k
(...) ,
h (...) , γ
k
k
(...) ,
k
(d) A closed-loop
game can be defined as a [NSDG]
i
for which η k = { x1 ,..., xk} , for every k ∈ K , that is, the
players have access to all information about the past
states of the game.
6. Games with hierarchical structure
Proceeding the work with [SRG], we further specify its
structure, in order to obtain a more useful formal model
for the intended purpose.
6.1. Hierarchical game in two levels
A non-cooperative hierarchical game in two levels is
modeled through a process of forming a group of
subsystems, each one representing a competing
company, with its own operative rules for state
transition, information, decision, and objective
function. Each company, here represented by a
subsystem [CS]i , vies in the market for raw materials,
specialized production man power, managerial
resources, financial resources, technology, and other
supplies. On the other hand, they also compete in the
market for clients’ preferences.
The market, in the broader sense, also interferes on
the game, acting on prices and quantities transacted by
the N competitors, their clients and providers. A
method of representing the market actions is to
introduce into the game a new player, the (N+2) th,
which we will call ‘the market’, symbolized by Q,
which has the duty of seeking out the balance between
the aggregate demand and the aggregate offer for both
supplies and products.
The [SRG], organized as described above, is
defined as a hierarchical game in two levels, [HG2].
The formulation of this concept can be obtained
through a convenient partition and segmentation
process of the [SRG] game, resulting in two types of
subsystems described below:
I. Company subsystems [CS]i – The mathematical
formulation of a company subsystem [CS]i, for
k ∈ K and i ∈ N can be written as follows:
(a)
i
x
k +1
The problem of optimal stochastic control with a strategic
objective function such as defined here can be seen as a specific
application of predictive control [19]. The goal is to optimize the
desired ‘final result’. That is, the ‘closer’ the players are able to place
their companies to the target final state, the better their game
performance is graded.
f ( x ,u , λ ) ,
i
i
i
i
k
k
k
k
x
given the initial state
i
1
, for
each subsystem [CS]i , and
(b)
(2 )
=
z =J
i
y = h ( x ) , η k ⊂ { yk , uk −1} ,
i
i
i
k
k
k
i
K +1
i
i
i
u = γ (η )
i
i
i
k
k
k
and
i
( xK +1) .
II. Market Coordinator Subsystem [MCS] – The
mathematical formulation of the market coordinator
subsystem [MCS], k ∈ K and i ∈ N , similarly to the
those prices.
[CS]i subsystem (3) can be written, respectively, as:
Coordination by prices is what mostly resembles
the situation prevalent in a free competitive market,
with multiple suppliers and multiple buyers, without
any of them regulating or controlling the market prices
and/or quantities.
(a)
m
=
k +1
g (m , λ ,..., λ
1
k
k
k
v = ρ (m ) , µ
δ = R (m ) .
(b)
k
k
k
k
k
k
,..., λ k , u k ,..., u k ,..., u k ) , and
k
i
N
1
= {vk , λ k −1} ,
i
N
λ = β (µ )
i
i
i
k
k
k
and
k
The [CS]i modules communicate only with the
market coordinator subsystem, [MCS], which informs
to each one of them, at the beginning of each new
i
stage, its decision parameter, λ k . The [CS]i, in turn,
i
informs the [MCS] about their decisions uk .
6.2. Hierarchical game in three levels
By analogy with the treatment given to the [SRG]
game, we can additionally expand the dynamic
hierarchical game [HG2], applying a further
segmentation process to each company subsystem
[CS]i: each of the competing i companies is assumed to
consist of G Managerial Units, [MU]ij, where
j ∈ {1,2,..., G} , introducing G new players for each
company.
These managerial units, [MU]ij, represent the main
functional or managerial areas of the company. In this
sense, each [MU]ij has its own state transition equation,
information structure, strategies, and decisions, and a
specific objective function to be minimized. However,
leaders of managerial units should seek out consensus
with their peers to reach the optimal objective function
for their company as a whole.
Therefore, by analogy with the segmentation
presented at Section 6.1, the model for [HG2] can be
expanded to obtain [HG3] wherein the coordination of
a second level is achieved by a new module called
[CSC]i, representing the coordination of all the [MU]ij,
by the company’s chief executives.
7. Coordination of hierarchical game
Resuming the problem of multi-level coordination, we
choose, for simplicity sake, the [HG2] as a basic
reference. Two classic ways to treat the matter of multilevel coordination have been proposed: coordination by
quotas and coordination by prices.
We will only describe the latter, which is the one
utilized in this study. (4) Within this option, player Q
i
establishes market prices, λ k , as much for supplies as
8.
Equilibrium strategies
To formulate a dynamic hierarchical game, such as the
game [HG2], or even better, as [HG3], the best
strategies for the player ‘market’, Q, for the company
directors, Pi, for the company coordination the [CSC]i
and for those in charge of the Managerial Units [MU]ij
should be those ones that optimize their respective
objective functions.
In strategic hierarchical dynamic games, the
purpose is to find a sequence of decisions that brings
the system to the final desired state– or as close as
possible to it. The classic ways of solving this problem
would be to treat it as an optimal control problem,
solvable, in theory, by Pontryagin’s Minimum
Principle, by the Calculus of Variations or by
Dynamic Programming [4, 13, 18, 19, 20], depending
on the case. These theoretical methods, however, are
not adopted in this study, as they do not make use of
the specific knowledge of the problem and its topology,
on the way we are implementing and exploring, with
multiple heuristic human decision-makers.
From a careful analysis of possible alternative
strategies for each configuration of the relationship
between players in [HG3] results the following
conclusions:
(i) The competitive relationship among players Pi can
be treated as a typical Nash equilibrium strategy, (5)
since they act independently from each other and are
prevented from sharing information and from
cooperating among each other for making coordinated
decisions in order to optimize together their objective
functions;
(ii) The relationship among those responsible for the
Managerial Units [MU]ij of the same company i can be
characterized as a game to which the Pareto
equilibrium strategy (6) applies, since it is a game of
(5 )
A Nash equilibrium point,
The symbols
m, g , λ , v, ρ , µ , β , δ , R
used for the [MCS]
subsystem are defined accordingly by analogy to [CS]i.
(4 )
In coordination by quotas, the market establishes quantities of
supplies and of products for each player. The players offer prices for
supplies and products they wish to buy and to sell.
1
i
N
for a non-
cooperative game (K=1) of variable sum, with N players, may be
i
i
defined if, for every u ∈U , i ∈N , the following N inequalities are
simultaneously true:
1
i
J (u ,..., u ,..., u
N
1
1
fori final products, and players Pi make their decisions
uk about the quantities they want to buy and sell at
(3 )
*
u = (u ,...,u ,...,u ) ∈U
i
J (u ,..., u ,..., u
N
i
1
i
J (u ,..., u ,..., u
N
)≤
1
i
J (u ,..., u ,..., u
N
1
1
) , ...,
) ≤ J i (u ,..., u ,..., u ) , ...,
N
)≤
i
1
N
i
J (u ,..., u ,..., u
N
N
).
(6 )
A Pareto optimum in a variable sum cooperative game (K=1),
with N players, if it exists, can be defined as the point
*
1
i
N
for which there does not exist any other point
= ( ,..., ,..., ) ∈
u u u u U
u = ( u ,..., u ,..., u
1
i
J (u ) ≤ J (u ) ,
i
i
i
i
N
) ∈U
∀i ∈ N ,
that
meaning
satisfies
that
the
J (u
i
inequalities
i
)≤
J (u
i
i
),
variable sum, in which there should be cooperation
among the managers in charge in order to optimize the
objective function for their company as a whole;
(iii) The relationship between the player Q,
coordinator, representing the market, and each player Pi
can be interpreted as a typical Stackelberg equilibrium
strategy (7), in which the market is the leader and each
player Pi acts as a follower;
(iv) The relationship between the internal coordinator
of each company, called the [CSC]i, and each [MU]ij
can also be considered as a typical Stackelberg
equilibrium strategy, since the strategic decisions of
the coordinator are assumed to be known a priori by
each unit manager.
Despite these considerations, the direct application
of the conditions mentioned above – the Stackelberg,
Nash and Pareto equilibrium strategies – to the game
in question ends up handicapped for practical reasons
because, with the exception of Q, the other players are
human beings making heuristic decisions.
Nonetheless, these conclusions are certainly useful
to the conceptual organization of the games,
considering that: (a) they enable the creation of formal
models for business games, which expand our
understanding of alternative strategies available to the
players; (b) they enable the creation of a correct
taxonomy for business games within the concepts of
classic game strategies; and (c) they enable, under some
specific conditions, the generation of algorithms for the
calculation of a lower-bound theoretical solution, (8)
which could serve as the lower limit to the heuristic
solutions human players can produce.
Figure 2 illustrates the application of the concepts
presented here, for a hierarchical game in three levels,
indicating applicable classic equilibrium strategies for
each case.
1st Level
Market
Coordinator
Stackelberg
Strategy
2nd Level
Co mpany 1
...
Co mpany i
...
Co mpany N
Stackelberg
Strategy
Nash Strategy
3rd Level
Managerial
Unit i,1
...
Managerial
Unit i,j
...
Managerial
Unit i,G
Pareto Strategy
Figure 2 – Classic strategies applicable to a
hierarch ical ga me on three leve ls - [HG3]
9. Implementing business games
A business game in three hierarchical levels, such as
[HG3], can be computationally implemented through
the interconnection of the following modules. (9)
9.1. Implementing subsystems [CS]i
The computational implementation of a subsystem
[CS]i is achieved in such a way that the player Pi can
‘simulate’ as many times as he/she deems necessary,
within the allowed time limit, within each stage, the
dynamic behavior of its company, within the planning
horizon span.
Among the computational environments that
generate dynamic simulation tools available on the
market, the software Vensim® DSS32 5.1 (10) can be
adopted, utilizing the concepts of level variables (11) and
of flow variables, (12) duly inter-related. The software
Simulink® (13) can also be used.
9.2. Implementing subsystems [MU]ij
∀i ∈ N , only if
J (u ) = J (u
i
i
i
inequality written for at least one
i∈N
i
),
∀i ∈ N , with the
.
(7)
Let a hierarchical game between player M, called the leader, and a
player P, called the follower, be with strategic decisions λ and u ,
J (λ , u ) , respectively,
in which, the player M, opts first for his decision λ and, following
and objective function
R (λ , u )
that, P opts for his decision
u
and
, knowing
λ
beforehand. A
Stackelberg’s equilibrium point, if it exists, is defined as equal to
( , u ) ∈ ( L ,U ) for which:
λ
T : L →U such that, for any
J (λ , T λ ) ≤ J (λ , u ) for every
The computational implementation of the [MU]ij
modules uses the same methodology described above,
with dynamic models in Vensim or Simulink. In order
to implement a strategic cooperation among those in
charge of the [MUs]ij, they are encouraged to share
information and heuristically seek the optimization of
the objective function of the company as a whole.
The players – the executives of each company –
make their decisions in a cooperative way, with the
purpose of minimizing a multi-criteria global function,
which should take into account the ‘strategic health of
the company’, (14) assuring the state trajectory of their
a) There is a transformational relation
fixed
λ ∈L
,
u ∈U
b) There is
for every
(8 )
λ ∈L
λ ∈L
such that
, where
R (λ ,T λ ) ≤ R (λ ,T λ )
u =T λ
.
We call the lower-bound solution that set of decisions for a
strategic game, which optimizes all the objective function for all the
players at all levels.
(9 )
The [HG2] game can be treated as a particular case of the [HG3].
Ventana® Simulation Environment, of Ventana Systems, Inc.
(11)
The level variables represent the quantities that can be modeled as
results of an accumulation process. For example: raw materials in
stock, orders backlog, employees level, etc.
(12)
The flow variables represent quantities that can vary over time as
functions of decisions made by managers, clients, providers,
competitors, etc.
(13)
Simulink® is a module of MATLAB 6.0, of MathWorks, Inc.
(14)
A successful implementation of a solution for this problem [5] is
the one that adopts a linear quadratic objective function, associated
(10)
companies is maintained within a hyper-tube feasibility
region. (15)
9.3. Implementing modules [CSM]
and [CSC]i
The algorithm for the subsystem coordination [CSM]
module generates the prices for supplies and products
that are informed to the players – executives of
competing companies. For the aggregate supply, the
algorithm forces the prices to rise whenever the
aggregate demand has a tendency to overtake its supply
availability; and it forces prices to fall, in the opposite
situation. For the aggregate supply of products to the
market, the behavior of the coordinator is the reverse.
The [CSM] module also generates randomly,
according to specific probability distributions,
the
i
numeric sample values for the variables θ k ∈ Θ and
ϕ ∈Ψ
i
k
influencing
the
game,
thus
creating
probabilistic scenarios with graded rigor levels, both
externally and internally.
The module that coordinates all the [MU]ij, called
[CSC]i , implemented at the level of the [SSE]i, follows
the same principles and mechanisms of the [CSM]
module, with appropriate adaptations.
10. Conclusions
Some comment and conclusions can be summarized as
follows:
• The classic equilibrium strategies from game theory
are applicable to adequately represent the optimal
cooperative and/or competitive players’ strategic
decisions in hierarchical business games;
• Game theory concepts and models can be used for
computation of a solution to be used as a lower-bound
for the heuristic solution human beings can produce;
• Complex competitive and turbulent business
environments can be adequately modeled as stochastic
hierarchical strategic games;
• Dynamic non-cooperative game theory can be used as
a conceptual and formal platform for the development
of strategic business games for educational purposes;
• Business games constitute a support decision tool for
good strategic decision formulation via heuristic
teamwork and game theory equilibrium strategies
concepts utilization.
• The educational experiment under implementation
using the model here presented is sufficient to show that
the framework and modeling proposed can be
efficiently used as a didactic tool for the education and
of decision makers and for fostering their capability for
planning and strategic management, as intended.
to the ‘distance’ – within a given metric – between the values
obtained by the players and the established target-value.
(15)
We call feasibility hyper-tube that region of the space X consisting
of all the possible state trajectories of the system, x ik ∈ X , obeying
the rules of the game.
Bibliografical References
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