برندگان جایزه نوبل ایالات متحده : رشد لجستیک در مقابل لوتکا-ولتررا

The logistic-growth equation is a special case of the Volterra–Lotka equations. The former describes competition only between members of the same species whereas the latter describes competition also with other species. In the study of US Nobel laureates considering laureates per population improves the quality of the logistic fit but the Volterra–Lotka approach suggests that a logistic description would be a good approximation for data per unit of time rather than cumulative data. Fitting logistic S-curves on cumulative data – although proven successful in many business and other applications – constitutes treacherous terrain for inexperienced S-curve enthusiasts. The Volterra–Lotka analysis of Nobel laureates reveals other insights such as that Americans and other nationalities are locked in a win–win struggle with Americans drawing more of a benefit, and also that American Nobel laureates “incubate” new Nobel laureates to a lesser extent than other nationalities.

It has been suggested that the competition for Nobel Prize awards can be described by logistic-growth curves [1]. My first attempt fitting a logistic to the cumulative number of US Nobel laureates in 1988 concluded that the US Nobel niche was already more than half full and implied a diminishing annual number of Nobel Prizes for Americans from then onward [2]. Ten years later I confronted those forecasts with more recent data in my book Predictions – 10 Years Later[3]. The agreement was not very good. The forecasts fell below the actual data and despite the fact that there was agreement within the uncertainties expected for a 90% confidence level the discrepancy did not go unnoticed. A technical note published in this journal in 2004 highlighted the inaccuracy of my forecasts and cast doubt in the use of logistics to forecast US Nobel laureates [4]. On my part, I refit the updated data sample with a new logistic pointing to a higher ceiling and began wondering whether there was evidence here for the known bias of logistics to underestimate the final niche size. The new forecast again indicated an imminent decline in the annual number of American Nobel laureates. Years later while preparing a new edition for my book – Predictions – 20 Years Later – I once again confronted forecasts with data. The situation turned out to be the same as ten years earlier, namely the forecasts again underestimated reality and despite agreement with the result of ten years earlier within the uncertainties expected for a 90% confidence level there was now a clear disagreement between recent actual numbers and the original forecasts of twenty years earlier. The situation was reminiscent of the celebrated Michele-parameter episode in experimental physics where a measurement repeated many times over the period of fifty years kept reporting an ever-increasing value always compatible with the previous measurement but finally ending up in violent disagreement with the very first measurement. So in this paper I want to settle the question of the ever-growing ceiling of the logistic curve fitted to the US Nobel laureates once and for all.

Logistic S-curves are special cases of solutions to the Volterra–Lotka system of equations. The Volterra–Lotka Eq. (3) reduce to the logistic Eq. (2) whenever the coupling constants cij become zero. Whereas logistic growth describes competition only among the members of one species, the Volterra–Lotka system of equations handles competition also with other species. It is advisable to consult the Volterra–Lotka approach – whenever possible – even if one is interested only in logistic growth because it can shed light on how to apply the logistic-growth equation. In the US Nobel-laureates study the Volterra–Lotka solution dictates that a logistic S-curve should be fitted on the annual numbers and not on cumulative numbers. Had we done so we would have obtained an answer very close to the black S-shaped curve of Fig. 3. Deciding whether to fit S-curves on cumulative or on per-unit-of-time data is a crucial first step for all logistic-growth applications and constitutes treacherous terrain for inexperienced S-curve enthusiasts. I myself mastered it only later in my career [11]. The forecasts for American Nobel laureates from the Volterra–Lotka approach are stable around an annual average of 6.1, comparable to the number of Nobel laureates won by all other nationalities together. Moreover the fitted parameters give rise to some interesting insights. The competition between Americans and all others for Nobel Prizes is of the win–win type. Locked in a symbiotic relationship both sides are winning but Americans are profiting more by 50%. At the same time, the ability of Nobel laureates to “multiply”, i.e. the extent to which a Nobel laureate incubates more laureates, is lower for Americans than it is for other nationalities. One may ponder whether the roots of this last observation have something to do with the fact that chauvinistic traits tend to be more endemic in cultures with longer traditions. All conclusions need to be interpreted within the uncertainties involved. From Table 1 and Table 2 we see that the quality of the logistic fits worsens as the time window increases. Normalizing to the population improves the quality of the fits. In Table 3 a confidence level of 72% indicates that there is 7 out of 10 chances that the Volterra–Lotka description is the right way to analyze this competition, not very different from the last fit in Table 2. For the intermediate future – ten to twenty years – the logistic normalized to reasonable population projections would result in forecasts compatible with those of the Volterra–Lotka approach. Still, I would choose Volterra–Lotka because it addresses a more general type of competition. In any case, long-term forecasts cannot be reliable and the whole exercise must be repeated with updated data sets in a couple of decades, by which time it may be appropriate to consider more than just two players.