Theoretical research without projects

Abstract

We propose a funding scheme for theoretical research that does not rely on project proposals, but on recent past scientific productivity. Given a quantitative figure of merit on the latter and the total research budget, we introduce a number of policies to decide the allocation of funds in each grant call. Under some assumptions on scientific productivity, some of such policies are shown to converge, in the limit of many grant calls, to a funding configuration that is close to the maximum total productivity of the whole scientific community. We present numerical simulations showing evidence that these schemes would also perform well in the presence of statistical noise in the scientific productivity and/or its evaluation. Finally, we prove that one of our policies cannot be cheated by individual research units. Our work must be understood as a first step towards a mathematical theory of the research activity.

Fig 1. Expected productivity g_i of a research unit as a function of its budget x.

This picture illustrate the three main assumptions: the function is zero for zero budget, it is non-decreasing, and it is concave.

Fig 2. The productivity of a research unit as a function of the funding x.

As a function of x, the scientific productivity g of a rational player cannot have: (a) decreasing regions; or (b) convex regions. In both cases, using the same budget, the research unit can switch to a more favorable productivity function (dashed line) that is increasing and concave.

Fig 3. Optimal distribution of START funds over the N = 6 researchers.

Fig 4. Productivity as a function of the grant call k, for different scientific policies.

The colors green, yellow, blue and red denote, respectively, the modified gradient scheme, the average rates scheme A, the rule of three and the standard scheme. Starting from a random distribution of funds, we study the performance of different policies under increasing amounts of statistical noise. For both the modified gradient scheme and the average rates scheme A, we chose $ϵ=\frac{0.25*X}{{\text{max}}_i(g_i^0/x_i^0)}$. If each term lasts s = 4 years, then, in all cases, the rule of three would require 12 years to steer the community to a configuration where the average scientific production is greater than 90% of the maximal value.

Fig 5. Cosmetic surgery.

We modify g_i(x) + δ from $\widehat{x}$ to 0 such that the new function ${\tilde{g}}_i\left(x\right)$ has an infinite slope at x = 0. Similarly, we modify g_i(x) + δ from $\check{x}$ onwards so that the slope of ${\tilde{g}}_i\left(x\right)$ is zero from $X_i^{+}$ onwards.