Independent Research Project in Applied Mathematics

2007-11-1

Perspectives on Team Dynamics: Meta Learning and Systems Intelligence

Helsinki University of Technology

Department of Engineering Physics and Mathematics

Systems Analysis Laboratory

Jukka Luoma 61320J

2007-11-1

2007-11-1

Abstract

Losada (1999) observed management teams develop their annual strategic plans in a lab

designed for studying team behavior. Based on these findings he developed a dynamical

model of team interaction and introduced the concept of Meta Learning (ML) which

represents the ability of a team to avoid undesirable attractors. This paper analyzes the

dynamic model in more detail and discusses the relationship between Meta Learning and

the new concept of Systems Intelligence (SI) introduced by Saarinen and Hämäläinen

(2004). In our view, the ML ability of a team clearly represents a SI competence.

Losada’s mathematical model predicts interesting dynamic phenomena in team

interaction. However, our analysis shows how the model also produces strange and

previously unreported behavior under certain conditions. Thus, the predictive validity of

the model also becomes problematic. It remains unclear whether the model behavior can

be said to be in satisfactory accordance with the observations of team interaction.

Key words: team interaction, meta learning, systems intelligence, chaotic dynamics,

Lorenz attractor

2007-11-1

Contents

Introduction.........................................................................................................................1

Meta Learning and Systems Intelligence ............................................................................1

Background of the Model ...................................................................................................2

The Model...........................................................................................................................3

Analysis of the Model.........................................................................................................6

Model Validation.............................................................................................................6

Analysis of the Model Behavior ......................................................................................7

Qualitative Behavior of the Model ..................................................................................7

Qualitative Behavior of the Model: three performance categories..................................9

Analysis of the Model Predictions.................................................................................11

Discussion.........................................................................................................................15

References.........................................................................................................................16

1

Introduction

The perspective taken in Systems Intelligence1 (SI) (Saarinen and Hämäläinen 2004) is

the acknowledgement that we are always a part of systems involving interaction and

feedback. The study of SI is concerned with the behavioral intelligence of human agents

in systemic settings. SI looks for efficient ways for an individual to change her own

behavior in order to influence the behavior of a system in different environments.

We see the concept of a system as a useful one in understanding human action in

dynamic settings. Today, many organizational studies describe organizational phenomena

by drawing analogies to concepts which originate in the mathematical modeling of

dynamical systems (e.g. Senge 1990; Senge et al. 1994; Stacey 1995, 2001; Axelrod and

Cohen 1999; Morel and Ramanujam 1999; Keskinen et al. 2003). Examples of related

concepts include attractors, chaos, bifurcations, and equilibrium and non-equilibrium

dynamics. Since SI is interested in human behavioral intelligence in organization

dynamics these concepts are of interest to SI research as well. Mathematical modeling of

human systems provides a convenient way of testing and analyzing the consequences of

different types of interventions. In particular, modeling human interaction allows us to

analyze how micro-level phenomena can influence macro-level outcomes.

The literature on the modeling of social interaction is extensive. For example, the early

work of Simon (1952) illustrated how mathematical modeling could be used in

“clarifying of concepts” of a theory, and in “derivation of new propositions”. Without

attempting to provide a comprehensive list of references, only some recent studies, which

in our view have connections to SI research, are referred to. Collins and Hanneman

(1998) suggested a common framework for modeling the theory of interaction rituals (IR)

by Collins (1981, 2004). Young (2001) introduced a dynamical model of conformity

based on Blume’s (1993) work on strategic interaction. Gintis et al. (2005) used game

theoretical models and experimental game theory to study decision making in social

interaction and the evolution of cooperation. Gottman et al. (2002a, 2002b) used

modeling in both formulating the theory of marital interaction as well as in designing

counseling interventions for unhappy couples. Losada’s model (1999; Losada and

Heaphy 2004) focuses on team interaction. What is common to all of these studies is that

they all use modeling as a tool to understand phenomena related to dynamic interaction.

This paper focuses on Losada’s research on team interaction and in particular, on the

characteristics of his dynamical model.

Meta Learning and Systems Intelligence

Losada (1999) defines Meta Learning (ML) “as the ability of a team to dissolve attractors

that close possibilities and evolve attractors that open possibilities for effective action”.

Attractors of the first type are something that “trap individuals and organizations into

rigid patterns of thinking that inevitably lead to limiting behavior”. The “attractor” that

Losada refers to as closing possibilities is used as a metaphor for behavioral patterns that

“teams get stuck” with.

1 http://www.systemsintelligence.hut.fi

2

The concept of ML is presented in connection with a dynamical model of team

interaction. The development of the model is said to reflect observations of team

interaction and team performance. The measurements were coded using bipolar scales

which were positivity-negativity, inquiry-advocacy and other-self. The coding was based

on the observed verbal communication of teams. The interaction patterns were studied by

analyzing the time series of the data. For this, a measurement called the degree of

connectivity was used which was measured by “the number of cross-correlations

[between the participants’ time series] significant at the .001 level or better” (Losada

1999). It is said that this measure is indicative of “a process of mutual influence” (Losada

1999; Losada and Heaphy 2004). Dutton and Heaphy (2003) interpret the degree of

connectivity as a “measure of a relationship’s generativity and openness to new ideas and

influences, and its ability to deflect behaviors that will shut down generative processes”.

Losada (1999) connects the degree of connectivity with the concept of connectivity of the

elements of a boolean network (see e.g. Kauffman 1993). These interpretations of the

measure still leave the exact definition of the degree of connectivity somewhat open.

The suggested model (Losada 1999) is used to predict the types of dynamics that are

theoretically possible for a team. “These dynamics are of three types: point attractor, limit

cycle, and complexor…” (Losada and Heaphy 2004) The authors argue that point

attractor dynamics correspond to low performance, limit cycle dynamics to medium

performance, and complexor dynamics to high performance of a team. In terms of the

ML ability, the model presents one way of describing some behavioral patterns a team

might get stuck with.

Losada (1999) also reports that his research has had practical implications for his

consulting work. His strategy for organizational interventions is based on identifying

which attractors trap teams into “low performance patterns” and designing interventions

that aim at dissolving these attractors and evolving new ones that open possibilities. SI

research too is interested in identifying such interventions. If there is a valid model of

team interaction, it could serve as a valuable tool for simulating and analyzing the

efficiency of interventions. An important SI research question is: How can an individual

influence the ML ability of a team, i.e. how does a team learn to meta learn?

SI research is interested in the type of micro-behavior in team interaction that Losada

observed. The behavior of the model that is reported in the original papers seems to

reflect the observations. This allows us to derive a working hypothesis that the model

configuration for different teams somehow reflects their ML ability. This paper attempts

to contribute to the deeper understanding of the model behavior and its accordance with

Losada’s observations.

Background of the Model

Losada (1999) collected the data by observing sixty strategic business unit (SBU)

management teams of a large information processing corporation in sessions where they

were developing their annual strategic plans. These sessions were held in a lab designed

for team research. The verbal communication of the teams was coded using the above

mentioned three bipolar scales as follows.

A speech act was coded as inquiry if it involved a question aimed at exploring and

examining a position, and as advocacy if it involved arguing in favor of the

3

speaker’s viewpoint. A speech act was coded as self if it referred to the person

speaking or to the group present at the lab or to the company the person speaking

belonged, and it was coded as other if the reference was to a person or group

outside the lab and not part of the company to which the person speaking

belonged. A speech act was coded as positive if the person speaking showed

support, encouragement, or appreciation, and it was coded as negative if the

person speaking showed disapproval, sarcasm, or cynicism. (Losada 1999)

The coded speech acts “were aggregated in one-minute intervals” to generate the time

series of observations. The teams’ time series were analyzed and the connectivity

measure was determined for each team. (Losada 1999; Losada and Heaphy 2004)

Losada found that the average ratios of the three bipolar scales and the connectivity

correlate with the performance level (high, medium or low) of a team. On the average,

high performing teams had high positivity/negativity ratios as well as inquiry/advocacy

and other/self ratios near one. Low performing teams had low positivity/negativity ratios

as well as low inquiry/advocacy and other/self ratios, i.e. there is more advocacy than

inquiry and more self-orientation than other-orientation. High performance teams had a

high level of connectivity whereas low performance teams had a much lower level of

connectivity. Medium performance teams were found to be somewhere in the middle.

The amplitudes of the time series of the observations were high and nondecreasing for the

high performance teams, whereas the low performance teams’ time series “showed a

dramatic decrease in amplitude…about the first fourth of the meeting and stayed

locked…” (Losada 1999). Again medium performance teams were found to be

somewhere in the middle. (Losada 1999; Losada and Heaphy 2004)

The modeling process is described as a search for a set of nonlinear differential equations

that would produce time series that would match the general characteristics of the actual

time series. Below is a related quote from Losada (1999, p. 182)

Thinking about the model that would generate time series that would match the

general characteristics of the actual time series … it was clear that it had to

include nonlinear terms … One such interaction is that between inquiry-advocacy

and other-self … represented by the product XY. I also knew from my observations

at the lab, that this interaction should be a factor in the rate of change driving

emotional space. …

… I also knew that connectivity … should interact with X and the product of this

interaction should be a part of the rate of change of Y, according to the time series

observed.

In the description of the modeling process and the model itself the meaning and

interpretation of the model parameters are not fully explained, and the reasons for

choosing the model equations remain open.

The Model

Losada’s model of team interaction has three variables, inquiry-advocacy (denoted by

X ), other-self (denoted by Y ) and emotional space (denoted by Z ). Our interpretation

4

of the variables, which is based on Losada’s speech act coding method (Losada 1999, p.

181; Losada and Heaphy 2004, p. 745), is as follows.

At any time t , X and Y are computed by subtracting the amount of advocacy speech

acts from inquiry speech acts at that time and other-referring speech acts from self-

referring speech acts, respectively. Thus, a positive )(tX means that there is more

advocacy than inquiry at time t , whereas a negative )(tX means that there is more

inquiry. Similarly, a positive )(tY means that there are more self-referring than other-

referring speech acts at time t , whereas a negative )(tY means that there are more other-

referring speech acts. The model does not include a variable which would indicate the

difference of the number of positive and negative speech acts. Losada and Heaphy (2004,

p. 757) assume, that the ratio of positive and negative speech acts can be computed from

this variable by using the equation

b

ZtZNP

)0()(/

−= (1)

where )(tZ is the level of emotional space at time t , )0(Z is the level emotional space

at time 0=t and b is a dissipation coefficient of the emotional space variable, see

equation (2) below (Losada and Heaphy 2004, p. 757). Thus, high values of emotional

space correspond to high positivity/negativity ratios, whereas low values of emotional

space correspond to low positivity/negativity ratios. (Losada 1999)

The rate of change of emotional space is assumed to depend on the interaction between

inquiry-advocacy and other-self and on the level of emotional space itself, so that:

bZXYdt

dZ−= , (2)

where b is the proportionality coefficient determining the dissipation rate of emotional

space. In the model b is fixed to 8/3. 2

The growth rate is determined by the product of

inquiry-advocacy and other-self variables. The dissipation term bZ− is directly

proportional to the level of emotional space itself.

Emotional space increases when inquiry-advocacy and other-self are both either negative

or positive and the product of these variables is greater than the absolute value of the

dissipation term. Thus, whether emotional space either decreases or increases, depends on

the quadrant, in the ( X ,Y ) plane, in which the team operates (see Figure 1). The

validity of this assumption is, however, unclear. There can be problematic situations, for

2 One should note that there is a typographical error in Losada (1999): parameters b in equation (2)

and a in equation (4) have been interchanged in the paper. In both papers that present the model

equations, Losada (1999) and Fredrickson and Losada (2005), a = 10 and b = 8/3, but in Losada

(1999), a is in equation (2) and b in equation (4). The equations are correct in Fredrickson and

Losada (2005).

5

example, when speech acts refer to a person or a group in the lab or within the company.

The model predicts that in this case emotional space, and thus positivity, increases only if

speech acts involve arguing in favor of the speaker’s viewpoint rather than exploring and

examining another team member’s position.

Figure 1. The sign of the term XY in each quadrant of the ( X ,Y ) plane.

The rate of change of other-self is assumed to depend on all the three variables and in

addition on the connectivity of the team, so that:

YXZcXdt

dY−−= , (3a)

where c is the connectivity parameter. The first term, cX represents the interaction of

the connectivity constant and the inquiry-advocacy variable. The second term XZ−

represents interaction between the inquiry-advocacy and emotional space variables. The

sign of the term the term depends solely on the sign of X since 0≥Z (for all 0tt > , for

some 0t ) “for all interesting values of the parameters” ( a , b and c ) (Sparrow 1983, p.

535). Other-self dissipates at a rate directly proportional to the level of other-self itself.

The proportionality coefficient of the dissipation rate of other-self is one. Rewriting this

equation gives

YZcXdt

dY−−= )( . (3b)

Note that the term )( ZcX − acts as a positive feedback mechanism, i.e. it increases the

rate of change of other-self as long as )( Zc − remains positive. This rate diminishes as

Z exceeds the level of connectivity ( cZ > ). Thus, the connectivity parameter

introduces a positive and a negative saturation level for the other-self variable. This holds

provided that the inquiry-other variable is bounded.

6

Finally, the rate of change of inquiry-advocacy is assumed to be directly proportional to

the difference between the other-self variable and the inquiry-advocacy variable, so that:

)( XYadt

dX−= , (4)

The related assumption made here, is that inquiry-advocacy follows other-self. If other-

self is smaller or greater than inquiry-advocacy, inquiry-advocacy decreases or increases,

respectively. This assumption seems to reflect the observation that generally self

orientation preceded advocacy and other orientation preceded inquiry in all performance

categories. Parameter a is a proportionality coefficient determining the rate at which

inquiry-advocacy follows other-self. In the model a is fixed to 10.

Since c introduces two saturation levels for other-self, it also introduces two saturation

levels for inquiry-advocacy, due to the fact that inquiry-other follows other-self. As

mentioned above, emotional space increases, when inquiry-advocacy and other-self are

both either positive or negative. The growth rate is proportional to the product of these

variables. Thus, connectivity introduces an upper limit, which is proportional to c , for

the emotional space variable. Following the terminology of population models, one can

also interpret c as the system’s (team’s) carrying capacity of positivity.

The equations describing the dynamics of inquiry-advocacy, other-self and emotional

space are identical to those that Lorenz (1963) presented in his seminal paper entitled

Deterministic Nonperiodic Flow. Lorenz used these equations to approximatively

describe the dynamics of heat convection in a fluid. This is also mentioned in Losada

(1999), Losada and Heaphy (2004) and Fredrickson and Losada (2005). However, it

remains unexplained why the authors ended up with exactly the same equations. The

value 8/3 for the parameter b is said to be used by scholars in many disciplines who use

Lorenz attractors (Losada and Heaphy 2004). Besides this note, the authors do not

explain why the chosen parameter values are the same ones as Lorenz used.

Analysis of the Model

Model Validation

The model was validated by comparing the model behavior with coded observations. The

match between theoretical and empirical data sets was “indicated by the cross-correlation

function at p < .01” (Fredrickson and Losada 2005). This implies that the model can

produce data similar to the original data. When running the model, Losada (Losada 1999,

pp. 183-188; Losada and Heaphy 2004, p. 754-755) used one set of initial values, i.e. (X0,

Y0, Z0) = (1, 1, 16) and three values of c , i.e. 18, 22 and 32. According to Losada and

Heaphy (2004), the initial values “eliminate transient, which represents features of the

model that are neither essential nor lasting”. It is not, however, explained why all teams

should start with more advocacy than inquiry and more self-orientation than other-

orientation. The validation by simulation which was presented in the original papers does

not, in general, guarantee that the model could be used to predict behavior in different

environmental conditions.

7

Analysis of the Model Behavior

Next, the model behavior is analyzed in more detail. Since the model equations are

identical to the ones in the Lorenz model of fluid convection, related research (see e.g.

Yorke and Yorke 1979; Sparrow 1983; Lorenz 1963, 1993; Csernák and Stépán 2000)

can be referred to. The simulations presented here were run with MATLAB3. Following

Losada, a fourth-order Runge-Kutta algorithm with a time step of 0.02 was used for the

numerical integration.

Qualitative Behavior of the Model

With a specific set of parameter values, the behavior of the model depends solely on the

initial values. Losada considers parameters a and b to be constant and the connectivity

parameter, c , to be an adjustable parameter.

The lowest value Losada used for the adjustable parameter, c , was 18 and the highest

value was 32 (Losada 1999). The model displays limit cycle dynamics for large values of

c , see e.g. Fredrickson and Losada (2005) and Sparrow (1983) for details. Here,

parameter values are allowed to range from 0 to 35, as the largest connectivity measured

by Losada was less than 35 (see Figure 4).

The Lorenz system has one to three steady states which are obtained when the time

derivatives are set to zero. This gives the following three steady state solutions (Lorenz

1963).

0=== ZYX , (5)

−=

−±==

1

)1(

cZ

cbYX, (6)

of which the latter two steady states only exist when c > 1.

With the chosen range of parameter values, the system exhibits three types of dynamics:

convergence towards a stable steady state, metastable chaotic behavior and chaotic

behavior. Using the definition of Academic Press Dictionary of Science and Technology

(Morris 1992), chaos refers to dynamic behavior that is deterministic, nonperiodic and

sensitively dependent on initial conditions. Sensitivity to initial conditions means that an

arbitrarily small change in the initial conditions may lead to significantly different future

behavior. One should also require that the trajectories are bounded, i.e. they do not go to

infinity. Otherwise, for example, a system

=

=

22

11

xx

xx

&

&, (7)

3 http://www.mathworks.com

8

could be characterized as being chaotic. For this system, the origin (0, 0) is an unstable

steady state. For any other initial states, the trajectories diverge. Metastable chaos refers

to dynamic behavior that is similar to chaotic behavior, with the exception that the

chaotic behavior is transient, i.e. after some time the system no longer behaves

chaotically and thereafter, for example, converges towards a stable steady state.

Accordingly, this initial phase of chaotic behavior is referred to as a chaotic transient.

When 10 << c , only the origin (0, 0, 0) is stable. This means that the system converges

to the origin from all initial states. 1=c is a bifurcation point which means that when c

passes this value, the qualitative behavior of the model suddenly changes. As a result of

this bifurcation, the origin becomes unstable and two stable steady states – denoted by P1

and P2 – emerge. When 01 cc << , the system converges to one of the steady states, i.e.

to P1 or P2. Both of these steady states have a basin of attraction from which the system

converges towards the steady state. A basin of attraction refers to a set of initial states in

the phase space from which the system evolves to a particular attractor. Another

bifurcation point is at 0cc = . Now, there is a region of metastable chaos. Depending on

the initial state the system either converges to one of the steady states or exhibits a

chaotic transient before eventually ending up in one of the steady states. Yet a new basin

of attraction emerges at another bifurcation point, 1cc = . The two steady states (P1 and

P2) are still stable, but a chaotic attractor now emerges. Depending on the initial state, the

system will converge to one of the two stable steady states or it will behave chaotically

indefinitely. When c passes the last critical value (relevant to this study), crc , the two

steady states lose their stability and the basin of attraction for the chaotic attractor is the

entire ( X ,Y , Z ) space. (Dykstra et al. 1997)

It is to be noted that since parameters a and b are not adjustable, the model assumes

that the dissipation rate of emotional space and the rate at which inquiry-advocacy

follows other-self are identical for all teams. For example, since b determines the

dissipation rate of emotional space, it influences the upper limit and the average of the

emotional space variable, equation (1). Moreover, parameters a , b and c together

determine whether the system behaves chaotically or not (Lorenz 1963) which, in turn,

determines whether the system eventually ends up in P1 or P2 or behaves chaotically

indefinitely. In P1 advocacy and self dominate, in P2 inquiry and other dominate, whereas

when the system behaves chaotically neither inquiry and other nor advocacy and self

dominate.

For parameter values 10=a and 3/8=b the critical values for the connectivity

parameter are 93.130 ≈c , 06.241 ≈c and 74.24≈crc (Sparrow 1983). In Losada’s

model of team interaction, low and medium performance teams’ value of the connectivity

parameter belong to the metastable chaotic region. High performance teams’ value of the

connectivity parameter belongs to the chaotic region.

With randomly chosen initial values, the average duration of the transient chaos depends

upon the initial values and on the difference between 1c and the chosen value of the

adjustable parameter. When the adjustable parameter, c , is close to 1c , the average

9

duration of the chaotic transient is long, and when the adjustable parameter is close to 0c ,

the average duration tends to zero. (Dykstra et al. 1997)

Different types of dynamics with different values of c are presented in Figure 2.

Figure 2. Behavioral regions of the Lorenz model. For low, medium and high

performance teams, the values of c (connectivity) are 18, 22 and 32, respectively.

93.130 ≈c , 06.241 ≈c and 74.24≈crc .

Qualitative Behavior of the Model: three performance categories

The time series of observations of the low performance teams showed a dramatic

“decrease in amplitude for all three dimensions about the first fourth of the meeting and

stayed locked […] for the rest of the meeting“ (Losada 1999, p. 182). When connectivity

is set to 18, the system converges quickly to either P1 or P2, from most initial states. This

type of behavior of the model is in agreement with what was reported of the time series

of observations of low performance teams. Losada and Heaphy (2004, p. 752) refer to

this type of dynamics as the “point attractor dynamics”.

The time series of observations of the medium performance teams in all three dimensions

“tended to have patterns of decreasing amplitude.” (Losada 1999, p. 182) When

connectivity is set to 22, typically a chaotic transient is seen (Dykstra et al. 1997) since

the set of initial states which cause the system to exhibit a chaotic transient is large. After

the chaotic transient, the system converges to one of the stable steady states. This is in

relatively good qualitative agreement what was reported of the time series of

observations. Losada and Heaphy (2004) refer to the dynamics of medium performance

teams as “limit cycle dynamics”. However, as shown in Figure 3 below, the systems does

not end up in a periodic orbit but converges towards either P1 or P2, potentially exhibiting

a chaotic transient before that.

c=1 c0 c=18 c=22 c1 ccr c=32

10

(a)

(b)

Figure 3. The development of emotional space over time for two different initial

conditions and c = 22, (a) (X0, Y0, Z0) = (7.5, 7.5, 21.0), (b) (X0, Y0, Z0) = (1, 1, 16).

Finally, “High performance teams had time series that showed high amplitudes over the

whole duration of the meeting in all three dimensions” (Losada 1999, p. 182). When the

11

connectivity parameter is set to 32, the basin of attraction for the chaotic attractor is the

entire ( X ,Y , Z ) space causing the system to produce time series ( )(tX , )(tY and

)(tZ ) that do not dampen over time. This is in accordance with the observations.

In terms of amplitudes, the model behavior is in accordance with what was observed of

the time series of observations. However, neither these amplitudes nor the chaotic

behavior of the model implies that the real world system of team interaction has the

potential to produce chaotic behavior. The chaos should be detected from the time series

of observations (Kodba et al. 2005).

Analysis of the Model Predictions

The average positivity/negativity ratio is obtained from equation (1), by substituting the

mean of the emotional space time series generated by the model into )(tZ , and letting

16)0( =Z and 3/8=b . Actually, Losada and Heaphy (2004) have computed the

positivity/negativity ratio from equation (1) by letting 1)( −= ctZ . However, to compare

the model behavior and observations, the mean value of the emotional space variable

should be used. This will cause a small deviation from the positivity/negativity ratio

computed the way Losada and Heaphy suggest.

When the model is run with different values of the connectivity parameter, a linear

dependence between connectivity and the positivity/negativity ratio is seen (Figure 4a).

This is in good agreement with Losada’s observations (Figure 4b). However, because of

the few data points presented in the original papers (low, medium and high performance

teams), the question of linear dependence is left open. It is to be noted that, if equation (1)

is used to compute the positivity/negativity ratio when the model is run with a different

)0(Z , the model predicts a positivity/negativity ratio which is substantially different

from the observed ratios. If )0(Z in equation (1) is set to 16, no matter what the initial

values are, the positivity/negativity ratios predicted by the model are close to the

observed ratios.

(a)

12

(b)

Figure 4. Positivity/negativity ratios for different values of the connectivity parameter.

The model predicts a linear dependence between the connectivity parameter and the

positivity/negativity ratio (a). In figure (b) averages of the observed positivity/negativity

ratios are plotted against observed values of the connectivity measure (Losada 1999).

The ranges of variation are presented with error bars.

The initial values of the inquiry-advocacy and other-self variables affect the behavior of

the model. Since for sufficiently small c , the two steady states outside the origin are

stable (Lorenz 1963) teams with relatively low connectivity can end up in one of the

steady states.

When using the initial values presented in the original papers, and when connectivity is

set to 18, the behavior of the model is in accordance with the observation, that low

performance teams have very low inquiry/advocacy and other/self ratios. By changing the

initial values, the model shows qualitatively the same dynamics, i.e. damping oscillation

around a steady state converging towards the steady state. Now the system operates

entirely in the quadrant of the ( X ,Y ) plane in which both X and Y are negative, i.e.

inquiry and other are dominant. This seems to be in disagreement with the observed

findings, as illustrated in Figure 5.

13

(a)

(b)

Figure 5. Phase space trajectory projected onto (a) (X,Z) and (b) (Y,Z) plane when c =

18. Trajectories shown with light gray lines refer to initial values (X0,Y0,Z0) = (-1, -1, 16),

and trajectories shown with black lines refer to initial values (X0,Y0,Z0) = (1, 1, 16).

The range of variation for connectivity in medium performance teams is 22 ± 6.3 per

cent, i.e. 23.386] [20.614,∈c (Losada and Heaphy 2004). The ranges of variation of

inquiry/advocacy and other/self ratios are not given in the original papers but, generally,

there is more advocacy than inquiry and more self-orientation than other-orientation.

When connectivity is set to 20.62 ( 23.386] [20.614,∈ ) and the model is run with the

initial condition given in the original papers, the system ends up in the quadrant of the

( X ,Y ) plane in which both X and Y are negative, i.e. there is more inquiry than

advocacy and more other-orientation than self-orientation. This behavior seems to be in

disagreement with the reported observations (Figure 5).

14

(a)

(b)

Figure 6. The development of inquiry-advocacy (a) and other-self (b) over time, when c

= 20.62 and (X0,Y0,Z0) = (1, 1, 16).

One might consider that connectivity should be an integer since it is a measure of the

“number of cross-correlations” at some of significance. However, similar behavior, i.e.

behavior that seems to be in contradiction with the reported observations, can be seen

when connectivity is set to 22 and the model is run with a different numerical integration

algorithm. While the system exhibits a chaotic transient it is sensitively dependent on the

15

initial conditions. The behavior is similar to that of “full” chaos. Consequently, the stable

steady state to which the system eventually converges, can change due to a small shift in

the initial state or the value of the connectivity parameter, or due to a change in the

numerical integration algorithm. This is why the model displays the behavior described

above and illustrated in Figure 6. However, no reference to this type of behavior is given

in the original papers.

Discussion

Losada’s observations imply that the rigidity of behavioral patterns correlates with excess

negativity, advocacy and self-orientation. High performance teams are high in inquiry but

also high in advocacy. The same applies to positivity and negativity, and other and self. It

would appear that negativity, advocacy and self all have a role in a time, or at least, they

are something inherent in human interaction. For example, advocacy can be seen as an

important element of dialogue (Senge et al. 2004).

What seems to be differentiating high performance teams from low performance teams is

that high performance teams do not get “locked” with negativity, advocacy or self modes,

They are able to dissolve these “attractors”, i.e. they are able to meta learn. This also

implies that the average positivity/negativity, inquiry/advocacy and other/self ratios do

not tell the whole story of team interaction and its relatedness to team performance. SI

research is interested in such capabilities of human agents where one avoids myopic

behavioral schemas which might result in undesired lock-ins, for example, getting locked

with temporary negativity or advocacy.

Saarinen and Hämäläinen (2004) describe another behavioral pattern resulting in lock-in.

The mechanism is called the system of “holding back in return” which refers to myopic

reactive behavior commonly observed in everyday life. Human agents reacting with

reciprocal behavior of “holding back” produces a lock-in system. This can create a

perception that the lock-in cannot be dissolved. Furthermore, this sustains the behavioral

pattern which is beneficial to no one. We see that this behavioral pattern is similar to the

undesired lock-ins in the Losada setting. Avoiding this locking by meta learning is an

expression of SI in teams. Both SI and our interpretation of the ML ability refer to the

abilities of a human agent or a team to dissolve structures of holding back.

SI research strives to search and develop models of social interaction which help reveal

the hidden potential that is inherent in most human systems. There are many social

interaction models, such as the ones introduced earlier. What kinds of modeling tools for

SI are most appropriate in, for example, the analysis of the “attractors” that Losada

observed remains an open question for future research.

16

References

Axelrod R. and Cohen M. D. 1999. Harnessing Complexity: Organizational Implications

of a Scientific Frontier. The Free Press.

Blume L. 1993. The Statistical Mechanisms of Strategic Interaction, Games and

Economic Behavior, Vol. 4, pp. 387-424.

Collins R. 1981. On the Microfoundations of Macrosociology, The American Journal of

Sociology, Vol. 86, No. 5, pp. 984-1014.

Collins R. 2004. Interaction Ritual Chains. Princeton University Press.

Collins R. and Hanneman R. 1998, Modelling the Interaction Ritual Theory of Solidarity

in Doreian P. and Fararo T. (Eds.). 1998. The Problem of Solidarity: Theories and

Models. Gordon Breach, Amsterdam.

Dutton J. and Heaphy E. 2003. The Power of High Quality Connections in Cameron Kim,

Dutton Jane, and Quinn Robert (Eds.). 2003. Positive Organizational Scholarship:

Foundations of a New Discipline. Berrett-Koehler Publishers.

Csernák G., Stépán G. 2000. Life Expectancy Calculations of Transient Chaotic Behavior

in the Lorenz Model, Periodica Polytechnica- Mechanical Engineering, Vol. 44, No. 1,

pp. 9-22.

Dykstra R., Malos J. T., Heckenberg N. R. 1997. Metastable Chaos in the Ammonia Ring

Laser, Physical Review A, Vol. 56, No. 4, pp. 3180-3186.

Fredrickson B. and Losada M. 2005. Positive Affect and the Complex Dynamics of

Human Flourishing, American Psychologist, Vol. 60, No. 5, pp. 678-686.

Gintis H., Bowles S., Boyd R. T., Fehr E. 2005. Moral Sentiments and Material Interests:

The Foundations of Cooperation and Economic Life. MIT Press.

Gottman J., Murray J., Swanson C., Tyson R., Swanson K. 2002a. The Mathematics of

Marriage – Dynamic Nonlinear Models. MIT Press.

Gottman J., Swanson C., Swanson K. 2002b. A General Systems Theory of Marriage:

Nonlinear Difference Equation Modeling of Marital Interaction, Personality and Social

Psychology Review, Vol. 6, No. 4, 326-340.

Kauffman S. A. 1993. The Origins of Order: Self-organization and Selection in

Evolution, Oxford University Press.

Keskinen A., Aaltonen M. and Milton-Kelly E. 2003. Organizational Complexity,

Finland Future Research Centre Publications, 6/2003.

Kobda S., Perc M. and Marhl M. 2005. Detecting Chaos From a Time Series, European

Journal of Physics, Vol. 26, pp. 205-215.

17

Lorenz E. N. 1963. Deterministic Nonperiodic Flow, Journal of the Athmospheric

Sciences, Vol. 20, pp. 130-141.

Lorenz E. N. 1993. The Essence of Chaos. University of Washington Press.

Losada M. 1999. The Complex Dynamics of High Performance Teams, Mathematical and

Computer Modeling, Vol. 30, pp.179-192.

Losada M. and Heaphy E. 2004. The Role of Positivity and Negativity in the Performance

of Business Teams, The American Behavioral Scientist, Vol. 47, No. 6, pp. 740-765.

Morel B. and Ramanujam R. 1999. Through Looking Glass of Complexity: The Dynamics

of Organizations as Adaptive and Evolving Systems, Organization Science, Vol. 10, No.

3, pp. 278-293.

Morris C. (Ed.). 1992. Academic Press Dictionary of Science and Technology, Academic

Press.

Saarinen E. and Hämäläinen R. P. 2004. Systems Intelligence: Connceting Engineering

Thinking with Human Sensitivity in Hämäläinen R. P. and Saarinen E. (Eds.). 2004.

Systems Intelligence – Discovering a Hidden Competence in Human Action and

Organizational Life, Helsinki University of Technology, Systems Analysis Laboratory

Research Reports, A88, October 2004, pp. 9-37.

Senge P. 1990. The Fifth Discipline: The Art and Practice of Learning Organizations,

New York. Doubleday Currency.

Senge P., Ross R., Smith B., Roberts C. and Kleiner A. The Fifth Discipline Fieldbook:

Strategies and Tools for Building a Learning Organization. Doubleday Currency.

Simon H. A. 1952. A Formal Theory of Interaction in Social Groups, American

Sociological Review, Vol. 17, No. 2, pp. 202-211.

Sparrow C.1983. An Introduction to the Lorenz Equations, IEEE Transactions on Circuits

and Systems, Vol. 30, No. 8, pp. 533-543.

Stacey R. 1995. The Science of Complexity: An Alternative Perspective for Strategic

Change Management, Strategic Management Journal, Vol. 16, No. 6, pp. 477-495.

Stacey R. 2001. Complex Responsive Processes in Organizations: Learning and

Knowledge Creation. Routledge.

Yorke J. A. and Yorke E. D. 1979. Metastable Chaos: The Transition to Sustained

Chaotic Behavior in The Lorenz Model, Journal of Statistical Physics, Vol. 21, No. 3, pp.

263-277.

Young P. H. 2001. The Dynamics of Conformity in Durlauf S. and Peyton P. H. (Eds.).

2001. Social Dynamics. MIT Press.