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THE GAME TO PLAY: EXPANDING THE CO-OPETITION PROPOSAL
Eliezer Arantes da Costa, U. Estadual de Campinas, UNICAMP - [email protected]
Celso Pascoli Bottura, U. Estadual de Campinas, UNICAMP - [email protected]
João Maurício Gama Boaventura, São Paulo City U., UNIP - [email protected]
Adalberto Américo Fischmann, São Paulo U., FEA/USP - [email protected]
ABSTRACT
Game theory has grounded many concepts in business strategy, but an essential problem arises to
apply this theory in business: what is the game to play? Brandenburger & Nalebuff (1995)
emphasized the need of choosing the right game and also that one may need to play more than
one game at the same time, thinking about both cooperative and competitive ways, concept they
named co-opetition. In academic papers of business strategy, cooperation and competition have
been frequently a subject of study, in particular when involving game theory, like for Lado, Boyd
and Hanlon (1997). Specifically for helping the strategist choosing the right game to play, we
understand that other dimensions than competition and cooperation should be considered,
specially the power ratio between players, and in this sense, this paper expands the original idea
of co-opetition providing a broader view for choosing a game. A ‘map’ for this choice is
presented, the Strategic Games Matrix – SGM, informing what games are recommended for each
particular situation represented in the cells of this matrix. Nine different situations, result of a
combination of different degrees of competition/cooperation and power ratio of players, are
addressed by the SGM. The nine different situations required six different games to be solved,
four of them could be solved using classical games: Minimax, Pareto, Nash and Stackelberg, and
the other two situations needed new games to be defined and solved: the Paternalistic-Solidary
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and the Dominant-Marginal, these last two games, although not yet solved, are here
characterized and mathematically described.
1 – INTRODUCTION
Business strategy is a broad subject and employs different concepts, like the ones from
the game theory, to deal with its conflict of interest problems. For better distinguishing the game
theory strategy perspective form others, an overview of different concepts of strategy is
presented below.
After bounding business strategy grounded in game theory, the problem issue of this
study can be better understood: what is the right game to play? This question was previously
discussed by Brandenburger & Nalebuff (1995, 1996) whom proposed the co-copetition concept.
We expand their discussion including basically two questions:
(i) What is the corresponding ‘name of the game’ a manager would want or have to play
in terms of the classical game theory concepts for the different business situations she or
he faces? This question has a pragmatic purpose: managers need first to identify what is
the name of the game they want or have to play to then implement a model for daily
business works. In this sense, this study presents a tool, the Strategic Games Matrix -
SGM, to help managers to identify the games to be played.
(ii) Competition and cooperation are the only parameters managers need to consider
before choosing a game to play? The answer to this question is no, and another relevant
parameter, expanding the co-opetition original proposal, should be introduced: the
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player’s power ratio. The present study explains that the power of one player when
compared to the power of other players also changes the type of the game to play.
Considering the wide variety of conflict of interests situations in the real business
world, we understand that another dimension than competition and cooperation should be
considered, specifically the power ratio between players, for helping the strategist choice of the
right game to play. In this sense, this paper expands the original idea of co-opetition providing a
broader view for choosing a game.
A ‘mapping tool’ for this choice is presented, the Strategic Games Matrix – SGM,
indicating what games are recommended for each particular situation represented in the cells of
this matrix. Nine different situations, result of a combination of different degrees of
competition/cooperation and power ratio of players, are addressed by the SGM.
The nine different situations required six different games to be solved, four of them could
be solved using classical games: Minimax, Pareto, Nash and Stackelberg, and the other two
situations needed new games to be defined and solved: the Paternalistic-Solidary and the
Dominant-Marginal. These last two games, although not yet solved, are here characterized and
mathematically described.
2 - THE DIFFERENT KIND OF STRATEGY CONTENT CONCEPTS
Before focusing the discussion in the kind of strategy approached in this study, based in
the theory of games, a broad view of the several concepts of strategy, and where this approach is
located, is useful.
Through the analysis of how distinct authors define strategy, an interesting overview of
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the different concepts of strategy can be made. An analysis of the strategy content is presented
bellow, grouping, by similarity, the different ways the investigated authors define strategy. Hax
& Majluf (1996: 1) understand that, for analysis purpose, the strategy content should be
separated from the strategy process; these authors (1996: 14) also identified nine dimensions in
the strategy concept: (1) determine long term objectives and resource allocation priorities, (2)
select the businesses the organization is in, (3) attempt to achieve sustainable advantage, (4)
identify the different levels of management, (5) be an integrative pattern of decisions, (6) define
the contributions to the stakeholders, (7) state the strategic intent, (8) aim developing and
nurturing the core competencies, and (9) be a means for investing in resources to develop a
sustainable competitive advantage. Hofer & Schendel (1978: 25) also studied the strategy
concept, and pointed out that the main strategy components are: (1) scope, (2) resource
deployments, (3) competitive advantages, and (4) synergy.
To classify authors in different categories, the assumption used in this paper is: authors
emphasize their strategy concept when defining it. Hence, from authors’ research, is possible to
recognize four categories for the strategy content. The types, named as follows, are neither
incompatible between each other nor exclusive:
1- Strategy as means to attain goals
2- Strategy oriented to competitive advantage
3- Strategy focused in core competencies
4- Strategy based in the interaction with opponents
Some authors made definitions using more than one category and in these cases they
were classified in all identified categories. One example is Quinn, in Mintzberg & Quinn (1992:
5), whom stated that a well formulated strategy helps to marshal and allocate an organization’s
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resources into a unique and viable posture based on its relative internal competencies and
shortcomings, anticipated changes in the environment, and contingent moves by intelligent
opponents. Others who used more than one type were Fahey & Randall (1998: 23). They
explained that strategy must confront and resolve three sets of elemental choices: scope,
competitive differentiation, and goals.
This study focus the last listed type, “Strategy based in the interaction with opponents”.
This kind of strategy is grounded in the theory of games and is explained in Section 3.
2.1 - STRATEGY AS MEANS TO ATTAIN GOALS
There is a group of authors that perceives strategy as a way to attain goals. Such as, in a
causal relation, where the cause is a way to using the organization resources and the effects are
goals achievement. Chandler (1962: 13), for example, states that strategy can be defined as
determination of the basic long-term goals and objectives of an enterprise, and the adoption of
courses of action and the allocation of resources necessary for carrying out these goals.
Christensen et al. (1978: 125) have a similar view of strategy and define it as the pattern of
decisions in a company that (1) shapes and reveals its objectives, purposes, or goals, (2) produces
the principal policies and plans for achieving these goals, and (3) defines the business the
company intends to be. Other authors with the same approach are Learned et al. (1965), Andrews
(1971), Ackoff (1973), Rhenman (1973), Rumelt (1974), Drucker (1977), Lorange & Vancil
(1977), Hofer & Schendel (1978), Miles & Snow (1978), Steiner (1979), Henderson (1979),
Fahey & Randall (1998) and Johnson & Scholes (1989).
2.2 - STRATEGY ORIENTED TO COMPETITVE ADVANTAGE
The literature shows a group of definitions linking strategy to competitive advantage. The
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basic assumption is that the strategy should be oriented to achieve and maintain a competitive
advantage. Porter introduced this new concept in 1985 and after that several other authors
adopted it. Hax & Majluf (1991: 6) agree that connecting strategy to competitive advantage was
a Porter’s innovation and also adopted it defining strategy as a way to achieve and maintain the
competitive advantage.
Henderson (1991) in Montgomery & Porter (1991: 5) changed his earlier definition of
strategy to follow Porter’s concept and redefined strategy as a deliberate search for a plan of
action that will develop a business’s competitive advantage and compound it. Even Andrews
(1971: 28), who initially defined strategy as the means to achieve goals, renewed its definition.
Years after 1971, Andrews (1987: 14) reformulated its definition including the competitive
advantage concept. Pfeffer (1998) and Fahey & Randal (1998) also defined strategy in this
category.
2.3 - STRATEGY FOCUSED IN CORE COMPETENCIES
Another group of researchers emphasized the core competencies issue, proposing that the
competitive advantage or the advantage in the interaction within opponents would be the result
of the core competencies of the organization. This concept, or something close to it, was initially
proposed by Hofer & Schendel (1978: 27) when they stated that the distinctive competencies
were the most important part of the strategy. But, for sure, were Hamel & Prahalad (1994: 23)
who made this idea widely known and pointed out the need of a strategic architecture that
provides a blueprint for building the competencies needed to dominate future markets.
Werther & Kerr (1995: 14) also adopted this idea and explained that the absence of
competencies blocked the market leadership in different organizations. Other authors who used
the same concept for strategy formulation were Andrews (1987) and Quinn (1992).
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2.4 - STRATEGY BASED IN THE INTERACTION WITH OPPONENTS
There is also a fourth group of definitions linking strategy and the interactions with
opponents. These authors gave priority to the actions and reactions between the organization and
its opponents. In other words, they believe that it is not possible to conceive a strategy without
considering the actions and reactions of the opponents.
The route of this concept leads to Von Neumann and Morgenstern who in 1944 published
the book The Theory of Games and Economic Behavior. Von Neumann and Morgenstern (1947:
79) defined strategy as a complex plan which specifies what choices the player will make in
every possible situation, for every possible actual information which the player may possesses at
that moment, in conformity with the pattern of information which rules the game provided for
him or her.
For Schelling (1960: 3) the term strategy focus the interdependence of the decisions
involving competitors and their expectation regarding the behavior of each other. This concept is
also used by Dixit & Nalebuff (1991: ix) who stated that strategic thinking is the art of outdoing
an adversary, knowing that the adversary is trying to do the same to you. Other authors follow
this idea, such as Simon (1947), Newman (1950), and Brandenburger & Nalebuff (1996).
3 – STRATEGY ANALYSIS BASED IN GAME THEORY
This present work is grounded in the fourth strategy content concept approach – strategy
based in the interaction with opponents – cited in 2.4. We choose this approach for being that
one that, as we see, more adapts to the competitive strategies analysis, considering that the
competitors are always present in any business and their actions and decisions can interact with
the company’s decisions and results.
3.1 –STRATEGY ANALYSES BASED IN GAME THEORY APPLIED TO
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BUSINESS MANAGEMENT
Some attempts in applying game theory concepts and results to business environments
were reported by authors. Dixit describes the competitive movements of the companies, as
moves in a board game, and make considerations regarding the possible reactions by opponents
upon experiencing the adversary’s moves (Dixit & Nalebuff, 1991, Dixit & Skeath, 1999). Porter
mentions the application of the classical theories of equilibrium strategies from game theory to
interpret situations of strategic confrontation and the choices of moves by each competitor
(Porter, 1980).
Smit & Ankun describe the application of game theory to decision making in investment
strategies under competitive conditions (Smit & Ankun, 1993).
Oster develops the concepts of competitive rivalry, applying game theory concepts to
analysis and interpretation for the rationales of company strategic decisions. She defines and
applies the ‘payoff matrices’, ‘dominant strategy’, ‘credible threats’, ‘threat points’ and ‘strategic
core’ concepts to the business world to model the business behavior between various companies,
discussing the feasibility of the various possibilities of partial or total coalitions (Oster, 1994).
Ghemawat mentions various situations of competition between companies for which
game theory may be very useful in the analysis and decision between the various competitive
strategies available (Ghemawat, 1999). He warns, however that very frequently the decision
makers do not follow the ‘paths of rationality’, since frequently businesspersons carry out their
strategic choices based more on psychological, political or diplomatic reasons, such as, for
instance, the need for justifying past decisions, due to a selective perception of reality, due to
unjustified hostility or, simply, due to ‘intuitive hunches’, and mentions some known cases from
the literature to illustrate these phenomena.
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3.2 - CLASSIC GAME THEORY FUNDAMENTALS
First of all is useful to clarify that the meaning of some technical terms used in this paper
is the same as used in the game theory context:
A game can be conceptualized as a mathematical model representation of a
problem of conflict among two or more autonomous agents or decision-makers or controllers,
called the players. They are supposed to act on a same system, intending, each one of them, to
optimize some particular criteria, as measurable outputs of that system. In typical classical game
problems, some type of conflict of interests is present, considering that the optimum result for
one player can imply in a loss for another player.
The aim of the theorists in a game problem is two find a ‘solution’ for the game problem,
in the sense of finding a point, called the ‘equilibrium point’, in a manner that each player
understands having achieved the optimum result that could be obtained in this particular
condition of the game. The saddle-point is a particular case of equilibrium point typically for
zero-sum games.
The term strategy used in game theory is slightly different of the ones used in the
business environment. For the game theorists, (Basar & Oster, 1999: 5), the term strategy is
equivalent to the ‘decision rules’ that should be taken by all players involved in the game to
achieve or to reach an equilibrium point of the game.
Başar & Olsder (1999), classical authors in game theory, describe cooperative and
competitive game theory classic situations, and indicate forms of obtaining ‘equilibrium
strategies’ for several types of games. Among these typical game situations, for the development
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of this paper, the following have been chosen:
• Zero-sum games, where the saddle-point equilibrium strategy is applicable;
• Non-cooperative variable-sum games, where the Nash equilibrium strategy is applicable;
• Cooperative variable-sum games, where the Pareto equilibrium strategy is applicable;
• Hierarchical games, where the stronger player carries out its strategic move and announces
it to the remaining players, to which the Stackelberg equilibrium strategy is applicable:
leader strategy for the stronger player, and follower strategy for the weaker player.
3.3. EQUILIBRIUM POINT AND EQUILIBRIUM STRATEGIES CONCEPTS
An equilibrium strategy for a game can be understood as a ‘solution’ to the game’s
problem. It leads towards the decisions that must be taken by the players considering the
objective functions that express the conflicting interests of the players involved.
The application of one or another equilibrium strategy depends, among others, on the
game’s structure, the number of participants, the attitude of cooperation or of rivalry between the
participants, the information structure available to each player and the existence or not of any
‘privileged’ player in condition to impose its own strategy upon the remaining players.
The formal concepts of equilibrium point and of equilibrium strategy, using the
terminology presented (1), are here described: For simplicity sake, let us assume a simple game
(1) The generic notation used here for the formalization of discrete time dynamic non-cooperative games, is
summarized below:
Pi, with i=1, .., N, is the designation of the ith player; k=0, 1, ..., K is the index that defines each of the K+1 stages
of the game; xk is the game state vector at stage k; uk is the decisions vector taken by player Pi at stage k;
),...,,,...,( 1
111 uuxxJz N
kKii += is the game objective function for player Pi; ),...,,( 1
1 uuxfx N
kkkkk =+ is the game
transition equation from stage k to stage k+1; )(ηγ i
k
i
kku = is the game’s strategic function, where η i
k is the
information set available to player Pi at stage k.
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with only two players P1 and P2, whose decision-control variables, at stage k, are designated by 1
ku and 2
ku , and whose strategic functions are 1(...)
kγ and
2(...)
kγ , respectively.
An equilibrium point of the game is defined as a set of decisions1 2( ),k kû û such that, for
them, each player believes to have found the optimum possible for its objective function,
respecting the game’s limitations and acting according to its own player posture and its own
power ratio assumption, as in SGM cells, soon to be presented.
The equilibrium strategy for a game, if it exists, is a set of rules 1 2( (...), (...))k k
γ γ that leads
to a game equilibrium point. In this sense, an equilibrium strategy is a ‘solution’ to the game’s
problem. It leads towards the decisions that must be taken by the players considering each
distinct objective function, expressing the conflicting interests of the players involved.
4. THE STRATEGIC GAMES MATRIX (SGM)
The Strategic Games Matrix (SGM) (2), described in this Section, intends to support the
managers in diverse conflict management situations, helping them to identify the right game to
play. The SGM can be useful to alert each manager, explicitly, in each conflict of interests
situation that, before deciding: “How should I compete?”, she or he should first decide: “What
game should I play now?”.
To construct the SGM, two dimensions of the game are described in the following topics.
4.1. THE RELEVANT DIMENSIONS FOR GAME ANALYSIS
(2) A preliminary version of the concepts and structure used in this matrix was presented in Costa & Bottura (2004).
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There are various relevant dimensions that could be used for a conceptual mapping of the
attitude or behavior a player may assume in explicit situations of conflict of interests. For our
conceptual model, we opted in favor of two dimensions, suggested by some distinguishing
characteristics present in the classical games equilibrium strategies mentioned in topic 3.2.
In the application of this concepts to the business world, a player (3) may be seen as a
decision-maker which, individually or as a team, considering the risks and opportunities
involved, makes ones decisions and implements them, even aware that such decisions imply in
risks due to the results of the actions influencing and being influenced, positively or negatively,
by independent and unpredictable decisions by others decision makers, with others interests at
stake.
Studying the different conflicts of interests situations between players, we choose two
strong conceptual conditioners that characterize and distinguish the classic games and strategies
mentioned in topic 3.2: these are the player typical attitude, or behavior, upon facing its
competitors, and the power ratio balance between the player being analyzed and the opponent
with whom one understands there are conflicts of interests. These two conceptual conditions
became the two axes for the Strategic Games Matrix.
4.1.1. THE HORIZONTAL GSM AXIS: PLAYERS POSTURE ASSUMPTIONS
A basic issue that involves situations of conflicts of interests between players is the
manner by which a particular player faces ones opponent. For sake of simplicity, we adopt, for
(3) For sake of this paper, we will use the word player to represent an entity – indistinctly, it may be a company, a
decision-maker, a person, or a group of them – which has an objective function, or criterion, that it intends to
optimize, aware that the results for it depend on actions or decisions made by it and by other players, over which it
has not direct control.
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this type of posture, three mutually exclusive typical types that we break down into three distinct
statements:
(1) “If possible, I would like to destroy my competitor; if not, I would like to weaken it as
much as possible, so that it can no longer generate any threat to me in the future”, what we
entitle as a ‘rival posture’ (war attitude);
(2) “My competitor exists and is right there, but there are opportunities for all. Although I
recognize that there will always be conflicts of interests between us, I will act so as to
obtain and maintain my vital space for survival and growth”, what we entitle as a
‘individualistic posture’ (competitive attitude); and
(3) “I must survive, so must my competitor. Hence it must be possible to find some
coordinated form of action, in order to find a conciliatory solution that is the best for the
whole”, what we entitle as an ‘associative posture’ (cooperative attitude);
For the purpose of this analysis, one must recognize that these three possibilities truly
exist, and that they profound the condition for the analysis of the postures, behaviors, and actions
that are taken in each interests conflict situation, in each type of game. The three grades – or
levels – of player postures assumptions characterized above are named here as rival (warrior
attitude), individualistic (competitive attitude) and associative (cooperative attitude),
respectively, and they are illustrated in Figure 1, which shows typical situations where these
assumptions are applicable, results desired by the players, ethical assumptions and typical mottos
for each.
In this paper, one does not question if this choice is a subjective issue, or if it is an
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objective matter that can be explained by a basically economic motivation for company survival.
For the purpose of this conceptual model, we note that they are present in the conflicts of
interests’ situations, without adding, so far, any moral judgment of value to the choice made.
Predatory competition Loyal competition
Alliances, consortiums and
partnerships
Typical situations
“All are against me!”
“Each one takes care of themselves, and may the best win”
Typical mottos
Rival Individualistic AssociativeCompetitive
Postures
Eliminate or reduce competitors Win and survive The best possible for
the wholeDesired results
“All is valid for survival”
“Win, yes, but with dignity”
“We are all in the same boat”
Ethical assumptions
Predatory competition Loyal competition
Alliances, consortiums and
partnerships
Typical situations
“All are against me!”
“Each one takes care of themselves, and may the best win”
“One for all and
all for one”Typical mottos
Rival Individualistic AssociativeCompetitive
Postures
Eliminate or reduce competitors Win and survive
the wholeDesired results
“All is valid for survival”
“Win, yes, but with dignity”
“We are all in the same boat”
Ethical assumptions
Figure 1 – Players Posture Assumptions
Predatory competition Loyal competition
Alliances, consortiums and
partnerships
Typical situations
“All are against me!”
“Each one takes care of themselves, and may the best win”
Typical mottos
Rival Individualistic AssociativeCompetitive
Postures
Eliminate or reduce competitors Win and survive The best possible for
the wholeDesired results
“All is valid for survival”
“Win, yes, but with dignity”
“We are all in the same boat”
Ethical assumptions
Predatory competition Loyal competition
Alliances, consortiums and
partnerships
Typical situations
“All are against me!”
“Each one takes care of themselves, and may the best win”
“One for all and
all for one”Typical mottos
Rival Individualistic AssociativeCompetitive
Postures
Eliminate or reduce competitors Win and survive
the wholeDesired results
“All is valid for survival”
“Win, yes, but with dignity”
“We are all in the same boat”
Ethical assumptions
Figure 1 – Players Posture Assumptions
4.1.2. VERTICAL AXIS: PLAYERS POWER RATIO ASSUMPTIONS
In a second dimension, one seeks to represent the power ratio assumptions that a specific
player adopts with regards to its main opponents. In favor of simplicity, three typical power
ratios are characterized by the following statements:
(1) “I am the strongest and I have conditions to impose my interests upon my opponents”,
what we entitle as a ‘hegemonic power ratio’;
(2) “I am similar; I and my direct main opponents have equivalent strengths”, what we
entitle as a ‘balanced power ratio’; and
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(3) “I am too weak. I cannot make my opponents act according to my interests. If possible,
I prefer to wait for their decisions, before taking mine”, what we entitle as a ‘weak power
ratio’.
The power ratio assumptions described above are illustrated in Figure 2, which shows typical
situations where these assumptions are applicable, results desired by the players, ethical
assumptions and typical mottos for each.
For the purposes of these analyses, we will not go into the objective issue of the true power
balance between the player and its direct opponent. One needs only to recognize that these three
assumptions are present in situations of conflicting interests, without adding, so far, any
connotation involving moral judgment of value associated to the position taken by the player in
that specific confrontation.
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Figure 2 – Players Power Ratio Assumptions
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Starter, or terminal Free market Monopoly, control or regulation
Typical situations
“I am too small!” “I am one of them” “I am the strongest”
Typical mottos
Weak Balanced HegemonicPower
balance assumptions
Survive WinMaintain the
sovereign position
Desired results
“All is valid for survival”
“Win, yes, but following the rules”
“I make the rules and gain from
them”
Ethical assumptions
Figure 2 – Players Power Ratio Assumptions
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4.2. THE STRATEGIC GAMES MATRIX (SGM) STRUCTURE
Upon combining the three players’ posture assumptions with the three players’ power
ratio assumptions, a matrix with nine cells representing nine typical strategic games, the
Strategic Games Matrix results, as in Figure 3. For each cell a strategic position name is
indicated. Each one of the nine cells in the resulting SGM represents a situation of a typical
strategic game, and, to five of them, correspond classical games of game theory, as mentioned in
topic 3.2.
Despotic Leader Paternalistic
Predatory Competitive Cooperative
Follower Solidary /SolitaryMarginal
Rival
Wea
kB
alan
ced
Heg
emon
ic
Despotic Leader Paternalistic
Predatory Competitive Cooperative
Follower Solidary /SolitaryMarginal
Rival
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Player Postures Assumptions
Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant Leader Paternalistic
Predatory Competitive Cooperative
Follower SolidaryMarginal
Rival Individualistic
Wea
kB
alan
ced
Heg
emon
ic
Figure 3 – Strategic Games Matrix (SGM)
Despotic Leader Paternalistic
Predatory Competitive Cooperative
Follower Solidary /SolitaryMarginal
Rival
Wea
kB
alan
ced
Heg
emon
ic
Despotic Leader Paternalistic
Predatory Competitive Cooperative
Follower Solidary /SolitaryMarginal
Rival
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Player Postures Assumptions
Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant Leader Paternalistic
Predatory Competitive Cooperative
Follower SolidaryMarginal
Rival Individualistic
Wea
kB
alan
ced
Heg
emon
ic
Figure 3 – Strategic Games Matrix (SGM)
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On the other hand, four specific game situations, corresponding to four SGM vertices
cells, there is not a correspondent classical game. So, the SGM clarifies there are two new games
to be added to the classical ones to fill all the cells of the GSM; the new games are described in
section 5.4.
The nine GSM cells are respectively entitled: Dominant, Leader, Paternalistic, Predatory,
Competitive, Cooperative, Marginal, Follower, and Solidary, as shown in Figure 3. These names
better represent mnemonically each one of the typical conflict of interests players may face.
5 - MAPPING CLASSICAL GAMES AND THEIR EQUILIBRIUM STRATEGIES
TO THE SGM
Four classical games can be explained by the five central SGM cells, representing five
typical game situations. These classical games are named: Cooperative Game, Competitive
Game, Predatory Game and Leader/Follower Game and on the SGM are illustrated in Figure 4,
and are described in the following topics.
In this Section the classic game theory equilibrium structures mentioned in topic 3.2
corresponding to the five central cells of the SGM, as shown in Fig. 4, are briefly discussed:
5.1. NASH EQUILIBRIUM STRATEGIES – COMPETITIVE GAMES
The strategic position at the center-center SGM cell, named here as Competitive Game,
describes situations of ‘perfect competition’, or ‘free market’, with many suppliers, where none
of them is capable of dominating the remainders. In the non-cooperative variable-sum games,
where a player decides to assume a competitive strategic position, it seeks to optimize its
objective function ignoring what the other players are doing or intending to do.
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Leader:Leader:Stackelberg Stackelberg EquilibriumEquilibrium
Predatory:Predatory:SaddleSaddle--pointpointEquilibriumEquilibrium
Competitive:Competitive:Nash Nash
EquilibriumEquilibrium
Cooperative:Cooperative:ParetoPareto
EquilibriumEquilibrium
Follower:Follower:StackelbergStackelberg
EquilibriumEquilibrium
Rival Individualistic
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Players Posture AssumptionsAssociative
Wea
kB
alan
ced
Heg
emon
ic
Despotic
Solidary / Solitary
Marginal
Leader:Stackelberg Stackelberg EquilibriumEquilibrium
Predatory:Saddle-pointEquilibrium
Competitive:Nash Nash
EquilibriumParetoPareto
EquilibriumEquilibrium
Follower:StackelbergStackelbergEquilibrium
Rival Individualistic Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant
Solidary Marginal
Figure 4 – The Four Classical Games in SGM
Paternalistic
Cooperative:
Leader:Leader:Stackelberg Stackelberg EquilibriumEquilibrium
Predatory:Predatory:SaddleSaddle--pointpointEquilibriumEquilibrium
Competitive:Competitive:Nash Nash
EquilibriumEquilibrium
Cooperative:Cooperative:ParetoPareto
EquilibriumEquilibrium
Follower:Follower:StackelbergStackelberg
EquilibriumEquilibrium
Rival Individualistic
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Players Posture AssumptionsAssociative
Wea
kB
alan
ced
Heg
emon
ic
Despotic
Solidary / Solitary
Marginal
Leader:Stackelberg Stackelberg EquilibriumEquilibrium
Predatory:Saddle-pointEquilibrium
Competitive:Nash Nash
EquilibriumParetoPareto
EquilibriumEquilibrium
Follower:StackelbergStackelbergEquilibrium
Rival Individualistic Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant
Solidary Marginal
Figure 4 – The Four Classical Games in SGM
Paternalistic
Cooperative:
If this solution exists, it is characterized by the situation where none of the players is able
to improve its result by changing only its own decision-control variables. Such set of decisions is
called the Nash equilibrium point, defined below:
A Nash equilibrium point 1* ( ,. . . , ,. . . , )i N Uû û û û= ∈ , if it exists, for a non-cooperative
game, with K=1, and variable sum, with N players, is defined if, for all i iu U∈ , i N∈ , it obeys
simultaneously the N following objective function inequalities:
1 11 1( ,..., ,..., ) ( ,..., ,..., )i N i Nû û û u û ûJ J≤ , ... ,
1 1( ,..., ,..., ) ( ,..., ,..., )i N i Ni iû û û û u ûJ J≤ , ... ,
1 1( ,..., ,..., ) ( ,..., ,..., )i N i NN Nû û û û û uJ J≤ .
14712.doc 19
5.2. PARETO EQUILIBRIUM STRATEGY – COOPERATIVE GAMES
For variable-sum games – at the right-center SGM cell – the cooperation among players
may lead to results – for all of them – that are better than those they would obtain if each one
tried to optimize its objective function without an a priori knowledge of other’s decisions. These
are the cooperative games: when players decide to share information on the respective
constraints and conditions, alternative actions and objective functions, it is possible for them to
find a point of equilibrium, called ‘Pareto optimum’, which is ‘the best’ possible for all players.
This point, if it exists, is characterized by the fact that none of the players can improve its
result without, with its action, harming the other’s results. These are the so called ‘win-win
games’. The cooperative game environment implies, however, that there must be an explicit or
implied agreement between the players for them do not exacerbate their individual interests in
detriment of others. This type of game therefore requires good-faith and loyalty among all
participants.
For a variable-sum cooperative game (K=1) with N players, the point
* 1( ,. . . , ,. . . , )i Nû û û û U= ∈ is defined as a Pareto optimum if there is no other point
1( ,. . . , ,. . . , )i Nu u u u U= ∈ such that ( ) ( )i i
ii ûuJ J≤ , Ni∀ ∈ .
This condition requires that ( ) ( )ii
i i ûJ u J≤ , Ni∀ ∈ , only if ( ) ( )ii
i i ûJ u J= , i N∀ ∈ , with a
strict inequality for at least one i N∈ .
5.3. MINIMAX EQUILIBRIUM STRATEGIES – RETALIATORY GAMES
This strategic positioning applies to ‘lose-win type’ and ‘lose-lose type’ games – at the
left-center SGM cell, where the players assume, explicit or implicitly, that a gain for one implies
in losses to the remainder, characterizing a retaliatory competitive position. To formalize this
14712.doc 20
strategic position, the zero-sum game concept is used: a zero-sum game is defined as a game for
which
1
1 1 1 1( ) ( ( ,..., , ,..., ,..., )) 0i N
i i K k Ki N i N
J x x u u uz +∈ ∈
≡=∑ ∑
For a zero-sum game (4), a solution, if it exists, for which each player acts towards what it
understands as the most favorable to optimize its objective function, considering all the
possibilities the others could do, is called a saddle-point. This point has the peculiar
characteristic that any deviation from it, by any of the players, makes its result worsen in relation
to its objective function.
Applying this concept to a game with only two players (N=2), a saddle-point is defined
as being a pair of decisions,1 2( , )û û , that satisfy the following inequality:
1 2 1 2 1 2
1 1 1( , ) ( , ) ( , )J û u J û û J u û≤ ≤ for every 1 1u U∈ and 2 2u U∈ .
Generalizing this concept for N players, a strategic decision Uu ii ∈ˆ by each player Pi is
defined as a saddle-point equilibrium solution if, for every admissible set
1{ ,..., ,..., }i N Uu u u ∈ , the following relation is valid:
1 1 1 1
1 1 1 1 1 1,..., , ,..., ,..., , ,...,
( ,..., , , ,..., ) ( ,..., )max maxi i N N
i ii i N i i Nu u u u u u u u
iûJ u u u u J u u− +
− + − +≤
A strategy that leads to a saddle-point is the Minimax strategy. Note that the saddle-point
calculation for player Pi depends exclusively on the objective function for Pi. This occurs
because, on this strategy, player Pi do not take into account the objective functions for the other
players, since one cannot rely on their good-faith or their ‘rationality’. This strategy applies also
to real situations where a player Pi can imagine that another player may have non-rational or
(4) In reality, it is sufficient that the result of this operation is a constant, instead of zero.
14712.doc 21
erratic or unpredictable behaviors, or even malicious, i.e., that an adversary may make moves to
‘damage’ Pi’s objectives, even if, doing so, its own interests could be damaged.
5.4. STACKELBERG EQUILIBRIUM STRATEGIES – LEADER / FOLLOWER
GAMES
These strategies apply to hierarchical games where there is a strongest player, the leader,
and another weaker player, the follower (Cruz Jr, 1978), both with individualistic posture, are
called Stackelberg strategies. They correspond to two opposed positions: center-upper and
center-lower SGM cells.
Consider a simplified hierarchical game between a player M, called leader, and a player
P, called follower, with strategic decisions λ and u , and objective functions ( , )R uλ and
( , )J uλ , associated to players M and P, respectively. Let us suppose also that, by the structure
and rules of the game, player M selects first its strategic decision λ and, then, player P selects
its strategic decision u , knowing beforehand the M’s decision.
The pair ( , ) ( , )u L Uλ ∈ , if it exists, defines a Stackelberg equilibrium point for which:
(a) There is a transformation : UT L → such that, for any given Lλ∈ ,
( , ) ( , )J J uTλ λ λ≤ for every u U∈ , and
(b) There is a Lλ∈ such that ),()ˆ,ˆ( λλλλ TRTR ≤ for every Lλ∈ , where λ̂ˆ Tu = .
Note that, to obtain a Stackelberg equilibrium point, it is necessary that the follower be a
rational decision-maker, always making optimal decisions under its limitation. For this game
structure, one can determine a pair of Stackelberg strategies – for the leader and for the follower.
It is typically applied to situations of conflict of interests between a very strong player
14712.doc 22
and another very weak, both with individualistic concurrent assumptions.
6. THE PROPOSED GAMES: GSM LIMIT CASE SITUATIONS
The four situations described in the SGM corner cells for which we did not find a
classical game to explain can describe other games, and we name them limit-case situations.
As these limit-case situations may really occur in the business world for instance, one
relevant contribution of this paper is to explain these games and formally describe their main
characteristics.
The Figure 5 illustrates these two proposed games that can explain the four missing situations
on the previous SGM.
Figure 5 – Typical strategic games for the two limit-case situations
Leader:Stackelberg
game
Retaliatory:Minimax
Cooperative:Pareto
Follower:Stackelberg
Rival Individualistic
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Players Posture Postures
Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant
Solidary / Solitary Marginal
PaternalisticLeader:
game
Retaliatory:Minimax
Cooperative:Pareto
Follower:
Dominant
SolidaryMarginal
Paternalistic
NashNashCompetitive:
Figure 5 – Typical strategic games for the two limit-case situations
Leader:Stackelberg
game
Retaliatory:Minimax
Cooperative:Pareto
Follower:Stackelberg
Rival Individualistic
Play
ers P
ower
Rat
io A
ssum
ptio
ns
Players Posture Postures
Associative
Wea
kB
alan
ced
Heg
emon
ic
Dominant
Solidary / Solitary Marginal
PaternalisticLeader:
game
Retaliatory:Minimax
Cooperative:Pareto
Follower:
Dominant
SolidaryMarginal
Paternalistic
NashNashCompetitive:
Further developments to this work should be to mathematically model these games, and
to provide their respective equilibrium strategies, where possible.
14712.doc 23
6.1. PATERNALISTIC-SOLIDARY GAMES
These new limit-case games are characterized by having two opposed antagonist players,
in different and opposed situations cells in the SGM, both with cooperative posture, as follows:
(a) The paternalistic strategic position, at the upper-right SGM cell: It occurs in games where a
more powerful player, by its own decision, shapes its own actions and those of the remaining
weaker players in the game, seeking preservation and development of its economic-business
system as a whole. These games are similar to the situation of a family father, supposed to have
complete authority over the small children: he does all he comprehends to be necessary to
promote the development, growth and harmony within his family, in a paternalistic way.
A paternalistic equilibrium point limit-case game can be found as follows: Let 0 iα≤ ≤ 1 be a
relative importance weight for the player Pi such that 1
N
iiα
=
∑ = 1, and let 1
(...)N
ii iz Jα
=∑= be a
multi-criteria objective function, encompassing all the objective functions of all the N players,
the new function to be minimized.
A paternalistic equilibrium point for this limit-case game can be found as a solution to a multi-
criteria optimization problem where the new objective function is a linear combination of all the
objective functions for all players. Otherwise, the Paternalistic player should take in account, on
its decision, the ‘risk’ of an undesirable Solidary player decision for the solitary strategy, quitting
the game.
(b) Solidary strategic positioning: In opposition to the paternalistic position described above is
the Solidary position – at the right-lower SGM cell –, that represents the situation of a player, in
a game, in a weaker, however associative posture which, without the power to impose its
interests upon the others, seeks to follow the rules established by the ‘ruling power’, looking for
14712.doc 24
some individual advantage. Otherwise it prefers to quit the game. This is how a member behaves
in relation to its cooperative organization: it simply needs to decide whether it should join the
‘collective’ and obtain some advantage of it or, alternatively, it should rather act on its own. A
Solidary equilibrium solution for this limit-case game can be treated as a simple decision tree
problem with only two branches, representing the alternative decisions: ‘join the collective’, or
‘work alone’.
6.2. DOMINANT – MARGINAL GAMES
These new limit-case games are characterized as having two opposed antagonists players,
in different and opposed strategic positioning cells in the SGM, both with rival postures, as
follows:
(a) A Dominant strategic position – at the left-upper SGM cell – characterizes the game where
one player has all strength and has the intention to destroy the smaller competitors. Its attitude
may be of intimidation, blackmail, price war, for instance, to try to bankrupt the small ones. It
may pressure its clients not to purchase from the small ones. A Dominant equilibrium point
limit-case for this game can be obtained through the solution of a mono-criterion optimization
problem in which the player in Dominant position ignores all the objective functions of its
‘small’ opponents and simply minimizes its own objective function. The player at a Dominant
position could treat the possible actions of ‘small’ competitors simply as random noises.
(b) Marginal strategic positioning: Countering the Dominant position at the game described
above, is the marginal strategic position – at the left-lower SGM cell –, where a weaker however
courageous and rival posture player in the game does everything it understands as necessary to
survive, trying, as much as possible, to obtain some advantages upon causing losses to the major
game dominator. A marginal equilibrium point limit-case for this game can be obtained through
14712.doc 25
the solution of an optimization problem in which the Marginal position player tries to maximize
the main and stronger competitor’s objective function with the purpose of infringing upon it the
maximum possible damage.
7. USING THE SGM TO CHOOSE THE RIGHT GAME TO PLAY
The problem of deciding to which of the games a manager or a decision-maker should
choose to play was already treated by many authors, for example, Brandenburger & Nalebuff
(1995).
7.1. HOW TO DECIDE WHICH GAME TO PLAY
In the processes of strategic planning, some anomalies generally observed in the making
of cooperative and competitive strategies can be avoided by a complete understanding and
adequate use of the Strategic Games Matrix (SGM) concepts.
Through the identification of the malfunctions and inconsistencies caused by the
inadequate choices of the strategic positioning, and, in consequence, of the choice of the game to
be played, it is possible to clarify the analyses and interpretations of real risky situations, losses,
or absence of entrepreneurial success.
The concepts derived from the use of the SGM should be incorporated to the
methodologies of formulation of strategies and to the management training programs for the
companies.
This way, decision-makers and managers could be much more prepared, adding to their
repertory of management tools the following specific skills:
a) Being able to recognize that each conflict of interests situation is ‘unique’ and that
there is no standard solution that can be applied to all of them equally;
b) Knowing how to evaluate the situation in detail and the correlation of forces in
14712.doc 26
particular and selecting the best applicable power balance to assume in each case;
c) Being able to examine the real situation, and deciding if he or she can trust in the
good-faith and the fidelity of the opponents, choosing the adequate attitude in each case; and
d) Being able to identify, in the SGM, the game to be played and choosing the most
adequate strategy.
7.2 – IMPLICATIONS OF NOT CHOOSING THE RIGHT GAME
The basic decisions and actions, in each conflict of interest confrontation, can be
classified in two main categories:
(a) according to the adequate choice of the game to be played, and
(b) according to the correct way to play the game.
The combination of those two options leads us to the following four consequences:
(i) If both choices (a) and (b) are correct, we have an indication of Success in business.
(ii) If the choice (a) is correct and (b) is incorrect, we have an indication of Learning. It
means that, although the way of playing is wrong, with the use of partial and
intermediate results it is possible to learn the correct way of playing the game, and
gradually moving towards the quadrant of Success.
(iii) If the choice (a) is incorrect and (b) is correct, we have and indication of Frustration,
because the player understands she or he played correctly, but the results, however,
were unsatisfactory. The frustration with the results can lead the player to try another
way of playing the same game, moving towards the failure quadrant, instead of
questioning if he or she would be playing the right game, and finally,
(iv) If the choice (a) is wrong and so is choice (b), we have certainly an indication of
Failure, with few or no chance of recovery.
14712.doc 27
The message this matrix tries to transmit to the managers and decision-makers is the
following: all effort must be put mainly in the discussion of which is the right –or the possible–
game to play, even when it demands the participation of the highest command of the company.
The consequences of an incorrect choice of the game to be played are much worse in the long
term, than the wrong decision about the way to play the game, for in this case there is at least the
opportunity of learning.
Unfortunately many companies invest time and resources in teaching their people how to
play the game correctly, instead of teaching them to decide on the correct game to play.
8. CONCLUSIONS AND COMMENTS
In this study the co-opetition proposal has been expanded by the utilization of the
Strategic Games Matrix, here discussed in detail.
Through the application of game theory concepts to the business strategy context,
inspired on Brandenburger & Nalebuff (1995) concern on the right game to play, the SGM points
games to play according not only to cooperative-competitive attitudes but also to competitors
power ratio, offering a wider perspective to strategic games to be played.
Furthermore the two new games here discussed and elaborated –the GSM limit-case
situations– amplify the repertoire of games one should examine, in a structured form. We
understand that both the business strategy and the game theory fields mutually benefited from the
types of propositions and discussions here made through SGM.
14712.doc 28
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