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THE GAME TO PLAY: EXPANDING THE CO-OPETITION PROPOSAL

Eliezer Arantes da Costa, U. Estadual de Campinas, UNICAMP - elicosta@uol.com.br

Celso Pascoli Bottura, U. Estadual de Campinas, UNICAMP - bottura@dmcsi.fee.unicamp.br

João Maurício Gama Boaventura, São Paulo City U., UNIP - jm@boaventura.adm.br

Adalberto Américo Fischmann, São Paulo U., FEA/USP - aafischm@usp.br

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.

14712.doc 18

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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