and scientists and the absence of such fluctuations in other professions. 1. Introduction ... ple of the first kind: he wrote exactly one comedy every two years in the last ..... The first consequence of our conjecture on the relationship between sat-.
The Role of Extrinsic Motivation in the Dynamics of Creative Professions Sergio Rinaldi and Francesco Amigoni Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy
We analyze in this paper the dynamics of the production rates in creative professions from a purely theoretical point of view. The analysis is an extension of a previous work in which self-esteem was assumed to be the unique source of motivation. By contrast, in this paper we assume that the individual is stimulated by a mix of intrinsic and extrinsic motivation and we show that extrinsic motivation is a stabilizing factor. Our theory explains the uctuating patterns of production frequently observed in artists and scientists and the absence of such uctuations in other professions. Summary.
1. Introduction
This paper deals with the dynamic characteristics of the production rate in creative professions. The study of career trajectories of creative people has been undertaken almost two centuries ago (see Quetelet (1968), originally published in 1835). Up to now, the attempts to model creative production have tackled the evolution of mean productivity along the whole life of an individual, based on data averaged over many people (Lehman (1953), Dennis (1966), Stephan and Levine (1988)). Thus, most ups and downs have been ltered out and the dynamics have been related to the evolution of the productive ability due to learning and aging. In particular, the pattern of mean productivity was modelled by Simonton (1997) through a linear model, giving rise to a double exponential pro le: productivity rst reaches a maximum, and then declines, because of the exhaustion of the initial creative potential. As pointed out in Marchetti (1985), the basic pattern of the current average productivity can also be modelled by means of a logistic equation. In this paper, by contrast, we model the medium term uctuations observed in the career of many artists and scientists. Figure 1 reports, for example, a twenty years segment of the production rate of Mozart and Poincare and suggests the existence of a limit cycle contaminated by random noise. A systematic analysis of the production of numerous artists has pointed out three main patterns: stationary, cyclic, and random. Ibsen is a typical example of the rst kind: he wrote exactly one comedy every two years in the last 20 years of his life. Cyclic behavior (of course, not in a strictly mathematical sense) is quite common. Besides the two examples given in Fig. 1, it can be detected in many other famous artists and writers (e.g. Van Gogh, Dewey, and Lovecraft) but also in a great number of much more normal professions.
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Sergio Rinaldi, Francesco Amigoni
W. A. Mozart
1780
1785
1790
30
25
20
15
10
5 1890
1895
H. Poincaré
1900
1905
1910
Readers in their fties (or more) are invited to spend a few minutes to portrait their production rate in the last twenty years. This can be simply done in the following way: rst partition the set of publications into three categories, say low quality (weight 1), standard quality (weight 3), and high quality (weight 5); then divide the value of each paper by the number of authors and nally sum up the scores, year by year, and plot them. The reader should not be surprised if his graph will point out almost regular ups and downs like those detectable in Fig. 2, which refers to Gustav Feichtinger, to whom this paper is dedicated, and to the rst author. Random patterns, in which no mark of regularity can be (even qualitatively) detected are also possible. Stravinsky and Jung are examples of this last category. 50 45 40
35
30
25
20 15 1775
1984
G. Feichtinger
1989
1994
1999
1979
1984
S. Rinaldi
1989
1994
1999
The above spectrum of production dynamics suggests that the creativity of an individual has basically an endogenous origin and is characterized by a stationary or a cyclic regime, but that random exogenous events can hide
Fig. 2. Normalized production rates of Gustav Feichtinger and Sergio Rinaldi over a period of twenty years (see text for details on the evaluation of the production rate). Both diagrams point out recurrent ups and downs
1979
Fig. 1. The production rates of W. A. Mozart and H. Poincar e during twenty years (data taken from Lampson (1998) and Archives Henri Poincare (1999) )
10 1770
Published works
Production rate
Compositions
Production rate
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3
these regular dynamics. Here we conjecture one potential reason for such an endogenous origin, by extending a previous contribution on the subject (see Rinaldi et al. (2000)). In our opinion the dynamics of creative professions can be captured, at least qualitatively, by the interactions among three state variables. The rst one, called creativity, is a measure of the uency with which new ideas are conceived and, consequently, new results are achieved. The two others are self-esteem and reputation, which are a result of the past achievements. The current trend of a suitable combination of these two variables is assumed to stimulate creativity. This approach is in accordance with modern psychological research on motivation and creativity (Halphin and Halphin (1973), Kramer and Bayern (1984), Wells (1986)) and points out explicitly the dichotomy exisiting between intrinsic and extrinsic motivation (see, for example, Solomon and Corbit (1974), Landy (1978), Guastello (1987)). Of course the relative weights used in the combination of self-esteem and reputation can be rather di erent from case to case. People depending very much upon the opinion of the others (as, for example, managers and politicians) should be expected to be often motivated by reputation, while artists and scientists, or at least some of them, might be more sensitive to self-esteem. The paper is organized as follows. In the next section we present our conjecture in the form of a third-order nonlinear dynamical model. The parameters of the model interpret psychological and behavioral characteristics of the individual and are assumed to be constant in time. This means that our conclusions should not be applied to the initial and nal phases of a long career because adaptation, learning, and performance deterioration due to senescence might be relevant in those periods. Then, in the third section the model is analyzed and shown to have two possible asymptotic regimes, corresponding to stationary and cyclic production patterns. Finally, in the last section the results of the analysis are interpreted by focusing, in particular, on the stabilizing role of extrinsic motivation. 2. The Model
As already announced, our conjecture is now formulated in terms of a thirdorder continuous-time model. The three state variables, namely C = creativity S = self-esteem R = reputation are non-negative: C (t) = 0 represents a state of complete catatonia at time t, while S (t) = R(t) = 0 only if no result has been achieved before time t. The ow of production, indicated by P (t), is instantaneously related to creativity, i.e.
P (t) = f (C (t))
(1)
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Sergio Rinaldi, Francesco Amigoni
where the function f () satis es the following properties (see Fig. 3a)
f (0) = 0
f 0(C ) > 0
f 00(C ) < 0
lim
C !1
f (C ) = Pmax
(2)
Such properties can be formally derived (see Rinaldi et al. (2000)) by subdividing the working time of each individual into time c spent for conceiving new ideas and de ning projects, and time r spent for realizing such projects and by further assuming that c is negatively correlated with creativity. g
f
Mmax
Pmax
0
C
0
. S
(a) (b) The production rate P as a function f () of creativity C (a) and the motivation M as a function g () of the trend of satisfaction _ (b) Fig. 3.
Self-esteem, as well as reputation, are simply exponentially discounted integrals of the past achievements, i.e. the following two linear di erential equations hold for them
S_ (t) = sS (t) + s P (t) (3) R_ (t) = r R(t) + r P (t) (4) where s and r are di erent forgetting coeÆcients. In the following, we will assume that
s > r
(5)
since reputation raises and decays very slowly with respect to self-esteem. However, if the production rate P (t) is constant, self-esteem and reputation coincide, after transient, to production rate. This means that at equilibrium self-esteem and reputation are two unbiased measures of productivity. As far as creativity is concerned, we assume that it is an exponentially discounted integral of motivation M (t), i.e.
C_ (t) = cC (t) + M (t)
(6)
Equation (6) is somehow an abstract de nition of motivation, so that our conjecture is not fully speci ed until we do not specify how M depends upon
Motivation in Creative Professions
5
the state variables. Loosely speaking, we could say that motivation M is related with the degree of satisfaction that an individual feels as a mix of his self-esteem and reputation. In formulas, if we de ne satisfaction as a convex combination of S and R, i.e.
(t) = R(t) + (1 )S (t) (7) we can then specify how M depends on . This is, undoubtedly, the most
delicate point of the model, since many assumptions re ecting di erent opinions are virtually possible. For example, one could imagine that motivation is simply proportional to satisfaction. But one could also assume (and we believe that this is more realistic) that motivation is mainly determined by the trend of satisfaction _ , i.e. M (t) = g(_ (t)) (8)
where the function g () is as shown in Fig. 3b, namely monotone increasing, rst convex and then concave, and saturating to a maximum value Mmax . A consequence of equations (6) and (8) is that when satisfaction rapidly increases, motivation is high and creativity has a sharp increase, while when satisfaction rapidly decreases the individual looses motivation. Since _ = 0 at equilibrium, the motivation in steady state conditions is simply g (0). For a more detailed discussion on the relationship between satisfaction and motivation the reader might refer to Rinaldi et al. (2000), where a speci c functional form of g (_ ) is also derived through a microfounded model. We summarize this section by showing the structure of the model in Fig. 4 and by rewriting the model equations in a more compact form (after substitution of equations (1), (7), and (8) into equations (3), (4), and (6)) S_ = s S + s f (C ) (9)
R_ = r R + r f (C ) C_ = cC + g(R_ + (1 )S_ )
(10) (11)
Model (9-11) with the constraint (5) and with the functions f () and g () as in Fig. 3 is analyzed in the next section. 3. Analysis of the Model
First notice that model (9-11) is a positive dynamical system, since S (0); R(0); C (0) 0 implies S (t); R(t); C (t) 0 for all t > 0. Then, observe that there is only one equilibrium given by
S = R = f (C )
C = g(0) c
and that such equilibrium does not depend upon s , r , and .
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Sergio Rinaldi, Francesco Amigoni
. S(t)
g
trend 0 of satisfaction
M(t) . S
motivation
lc
C(t) creativity
S(t) 1-m d dt
S(t)
f
P(t)
0
C
production rate
ls
self-esteem
+
satis+ faction
m
R(t) reputation
lr
The structure of the model pointing out the relationships among production rate, self-esteem, reputation, satisfaction, motivation, and creativity Fig. 4.
The stability of this equilibrium can be studied through linearization, namely through the analysis of the Jacobian matrix J evaluated at the equilibrium. Such a matrix is given by 2 3 s 0 s f0 5 0 r r f0 J =4 0 0 0 0 s(1 )g r g c + f g (r + (1 )s ) where f0 and g0 are the derivatives of f and g evaluated at C and 0, respectively. The characteristic polynomial of J is a third order polynomial
() = det(I J ) = 3 + 1 2 + 2 + 3 and the three coeÆcients i , i = 1; 2; 3, turn out to be given by 1 = s + r + c s (1 )z r z 2 = s r + r c + c s s r z 3 = s r c where z is the product f0 g0 , i.e. z = f 0(C)g0(0)
(12) (13) (14)
The necessary and suÆcient conditions for the asymptotic stability of the Jacobian matrix are
Motivation in Creative Professions
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1 > 0 1 2 3 > 0 3 > 0
(15) (16) (17)
Condition (17) is always satis ed (see (14)), while conditions (15) and (16) can be easily discussed because 1 and 2 depend linearly upon z (see (12) and (13)). Figure 5a shows, in particular, that the Jacobian matrix is asymptotically stable if and only if
z < z where z is the lowest root of the equation 1 2 3 = 0 z = z*
a1a2 a3 z
(16) (15)
(16)
(18) (19)
z* m=1
1 + lc / ls 1- m
m 1 + lc /ls m = 0 0
lr /ls
1
(a) (b) The ranges of z where conditions (15) and (16) are satis ed. For z < z both conditions are satis ed and the Jacobian matrix is asymptotically stable (a). The function z (; r =s ) for a generic value of and for the two extreme values = 0 and = 1. The function satis es the ve properties (20-24) reported in the text (b) Fig. 5.
Moreover, the well-known Routh criterion applied to third order polynomials says that whenever equation z = z is satis ed, together with inequalities (15) and (17), two roots of the polynomial are purely imaginary and one is real and negative. Thus, when z is slightly smaller than z the Jacobian is asymptotically stable and has one negative eigenvalue and two complex conjugate eigenvalues with negative real part, while for z slightly bigger than z the Jacobian is unstable and has one negative eigenvalue and two complex conjugate eigenvalues with positive real part. In other words, when a parameter is varied in such a way that z becomes bigger than z , the equilibrium of the system (9-11) undergoes a Hopf bifurcation (Strogatz (1994)), i.e. from stable it becomes unstable and surrounded by a stable limit cycle (in order to fully support this statement we should also show that the Hopf bifurcation is supercritical). Thus, in conclusion, under condition (18) the equilibrium is
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Sergio Rinaldi, Francesco Amigoni
asymptotically stable, while under the opposite inequality, i.e. for z > z , the attractor of system (9-11) is a limit cycle. Of course z depends upon all parameters and this dependence could be explicitly speci ed solving equation (19) while taking equations (12-14) into account. Here we limit our discussion to the dependence of z upon and the ratio r =s , which vary in the unitary interval. In particular, we prove in the Appendix that the following ve properties of z (; r =s ) hold (see Fig. 5b)
z(0; rs ) = 1 + sc z(; 1) = 1 + sc dz > 0 d z(1; rs ) = 1 +
c s r s
z(; 0) = 1 1 (1 + sc )
(20) (21) (22) (23) (24)
4. Interpretations of the Results and Conclusions
The formal results obtained in the previous section are now interpreted. The rst consequence of our conjecture on the relationship between satisfaction and motivation is that the production rate of an individual should be either stationary or cyclic. Indeed, the model we have analyzed has a unique global attractor which is either an equilibrium or a limit cycle. This means that our assumptions on the relationships between creativity, production rate, self-esteem, reputation, satisfaction, and motivation have not the power to explain chaotic production rates. Thus, if we believe that our model captures the main mechanisms involved in creative professions, we are forced to conclude that the deviations from pure cycles or equilibria that we observe by looking at the production pro les of many artists and scientists have exogenous sources. This is actually quite plausible since we know that social relationships, health, family life, and many other factors can have a great in uence on motivation, and, hence, on creativity. The condition for stationary production rates, namely z < z (see (18)), and the opposite condition z > z for uctuating production rates can be interpreted in terms of characteristic psychological and behavioral parameters of the individual. For example, recalling that z = f 0 (C )g 0 (0), we immediately recognize that a high sensitivity to variations of satisfaction (i.e., high g 0 (0))
Motivation in Creative Professions
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implies cyclic production patterns. By contrast, people rather insensitive to the variations of satisfaction should be expected to have stationary behaviors. But more can be said by looking at z , which is the other term of the critical inequality z ? z . In fact, low values of z favor cyclic productivity, while high values favor stationary productivity. In this respect, Fig. 5b is rather helpful. It shows that a high value of , i.e. high extrinsic motivation, implies high values of z , and, hence, lower chances to have ups and downs of creativity. This is particularly true, if r is small, i.e. if the environment reacts very slowly to the achievements of the individual. This situation describes quite well the case of people strongly dependent (in the success of their career) upon their reputation and working in slowly reacting environments. Politicians, directors of large public institutions, union leaders are perhaps classical stereotypes of this class. By contrast, low values of , i.e. great emphasis to intrinsic motivation (as opposed to extrinsic motivation), give higher chances to have ups and downs of performance. This should be typically expected in people who are only interested in pursuing their ideas, or in people who have already reached the top of their careers and are, therefore, inclined to look for intrinsic motivations. Poets, painters, and musicians are undoubtedly stereotypes of this class, but university professors can also easily fall in the same class. In conclusion, our analysis has shown that our conjecture can explain typical production patterns of creative professions and support classical stereotypes explained until now by empirical observations. The most interesting property emerging from this study is that extrinsic motivation is a stabilizing factor. References
Archives Henri Poincare (1999): Henri Poincare: List of Published Work. http:// www.univ-nancy2.fr/ACERHP/bibliopk.html#Interrogation. [Accessed July 5, 1999]. Dennis, W. (1966): Creative Productivity Between the Ages of 20 and 80 Years. Journal of Gerontology 21, 1-8. Guastello, S. J. (1987): A Butter y Catastrophe Modeling of Two Opponent Process Models: Drug Addiction and Work Performance. Behavioral Science 29, 258262. Halphin, G., Halphin, G. W. (1973): The E ect of Motivation on Creative Thinking Abilities. Journal of Creative Behavior 7, 51-53. Kramer, D. E., Bayern, C. D. (1984): The E ects of Behavioral Strategies on Creativity Training. Journal of Creative Behavior 18, 23-25. Lampson, L. D. (1998): Kochel's Catalog of Mozart's Works. http://www.classical.net/music/composer/works/mozart/index.html. [Accessed July 5, 1999]. Landy, F. J. (1978): An Opponent Process Theory of Job Satisfaction. Journal of Applied Psychology 63, 533-547. Lehman, H. (1953): Age and Achievement. Princeton University Press, Princeton, NJ.
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Marchetti, C. (1985): Action Curves and Clock-Work Geniuses. Technical Report WP-85-74, International Institute for Applied Systems Analysis, Laxenburg, Austria. Quetelet, L. A. (1968): A Treatise on Man and the Development of his Faculties. Franklin, New York. [Original Work published in 1835]. Rinaldi, S., Cordone, R., Casagrandi, R. (2000): Instabilities in Creative Professions: A Minimal Model. Nonlinear Dynamics, Psychology, and Life Sciences, To appear. Simonton, D. K. (1997): Creative Productivity: A Predictive and Explanatory Model of Career Trajectories and Landmarks. Psychological Review 104, 66-89. Solomon, R. L., Corbit, J. D. (1974): An Opponent Process Theory of Motivation I: Temporal Dynamics of A ect. Psychological Review 81, 119-145. Stephan, P. E., Levine, S. G. (1988): Measures of Scienti c Output and the AgeProductivity Relationship. In Handbook of Quantitative Studies of Science and Technology, Elsevier-North Holland, Amsterdam, 31-80. Strogatz, S. H. (1994): Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistery and Engineering. Addison-Wesley, Reading, Massachusetts. Wells, D. H. (1986): Behavioral Dimensions of Creativity. Journal of Creative Behavior 20, 61-65.
Appendix
Proof of property (20).
z(0; rs ) = 1 + sc
The proof is very easy: put = 0 in (12) and verify that z = 1 + c =s is the lowest root of equation (19). Proof of property (21).
z(; 1) = 1 + c s
Again the proof is just a check: put r = s in (12), (13), and (14) and verify that z = 1 + c =s is the lowest root of equation (19). Proof of property (22).
dz > 0 d
By de nition, equation (19) gives z = z (). Thus, we can write ( But
@ 1 + @ 1 dz ) 2 + 1 ( @ 2 + @ 2 dz ) @ @z d @ @z d
@ 1 = (s r )z > 0 @
(
@ 3 + @ 3 dz ) = 0 @ @z d
(I)
Motivation in Creative Professions
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@ 1 = s(1 ) r < 0 @z @ 2 = 0 @ @ 2 = sr < 0 @z @ 3 = 0 @ @ 3 = 0 @z Thus, taking into account that 1 and 2 are positive for z = z , equation (I) gives dz =d > 0. Proof of property (23).
r z ; ) = 1 + s (1
c s r s
As for properties (20) and (21), the proof is a simple check: put = 1 in (12) and verify that z = 1 + c =r is the lowest root of equation (19). Proof of property (24).
z(; 0) = 1 1 (1 + sc ) Again, put r = 0 in equations (12-14) and verify that z = (1 + )=(1 ) is the lowest root of equation (19).
c s
