Academic journal article Psychological Test and Assessment Modeling

Mathematical Foundations of and Empirical Investigations into the Dynamic of Top Positions: Stabilization Effect, Reversed Matthew Effect, and Heraclitus Effect

Academic journal article Psychological Test and Assessment Modeling

Mathematical Foundations of and Empirical Investigations into the Dynamic of Top Positions: Stabilization Effect, Reversed Matthew Effect, and Heraclitus Effect

Article excerpt

Abstract

The performance differences between successively ranked individuals tend to increase towards the top. However, the mathematical foundations of this effect are still largely untapped. This article will focus on developing such foundations. It will also be shown that the effect is stable for various natural distributions of eminent achievements. Three new predictions about the dynamics of top positions are formulated and tested with two samples from the world of sports: the best male chess players (individual sport) and male national soccer teams. The stabilization effect describes the phenomenon that the stability of ranks is higher among the top ranks. The reversed Matthew effect asserts that achievement gains among elite players and elite teams are positively correlated with their ranks (i.e. diminishing towards the top). However, in contrast, the Heraclitus effect predicts that the performance gains among the top ranks are nevertheless bigger than what can be mathemat-ically expected from the position in the ranking. All three effects can be empirically corroborated.

Keywords: Top positions, Eminence, Stabilization Effect, Reversed Matthew Effect, Heraclitus Effect

(ProQuest: ... denotes formulae omitted.)

Following Francis Galton (1849), human attributes have long been seen as generally normally distributed according to the Gaussian bell curve, with most people found in the medium range. Leaving this range, the number of people dwindles quickly until it asymptotically tends to zero. Galton defined the concept of giftedness by distinguishing those who are located in the rightmost part of the bell curve, however, he did not specify an exact cut-off point. Terman (1925), who contributed the first large longitudinal study to the research of giftedness, advocated taking the highest 1% according to intelligence.

While the distribution of intelligence displays a high degree of symmetry between very high and very low intelligence, there is a marked asymmetry between very high and medium intelligence that has often given cause to rethink the question of top perfor- mance.

Indeed for a long time in history, it was commonly assumed that a few eminent persons would disproportionately contribute to top achievements while the majority would rarely do so or even not at all.

This notion is often traced back to the Greek philosopher Heraclitus of Ephesus (c. 535 BC - c. 475 BC) who famously said, "One man is worth ten thousand others if he is the best." More recently, Simonton (1984, 1988, 1997, 1999, 2003) established in various studies that the differences in the level of performance between successive ranks increase the closer one gets to the very top. He stressed that top achievements are not at all dis- tributed like a bell curve but that a few excellent achievers contribute a disproportionate- ly large part of all high achievements (Simonton, 2004, 2009a). Systematic evidence for this was first collected by Dennis (1954a, 1954b, 1955). Additionally, there are several obvious examples, even in everyday life. For instance, there is a large number of direc- tors producing good movies, but among the very best of movies there are some directors' names which recur particularly often. Similarly, there is an immense number of authors but some of them appear several times when only considering the greatest novels. Scien- tific confirmation of these everyday observations can be found in the literature (see Hu- ber, 2000; Murray, 2003; Walberg, Strykowski, Rovai, & Hung, 1984).

Following Simonton, we propose that the current state of research on the contribution of eminent persons to top achievements can be summarized in five statements (Simonton, 2000, 2004, 2009a, 2009b):

1. The differences in performance between successive achievers or groups of achievers increase towards the top.

2. Only a few persons are responsible for a large proportion of top achievements.

3. …

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