Toward a least-effort principle for evaluating prices of elements as indicators of sustainability

Power-law distributions of the form f( x ) = a x − k have been observed in several physical, biological, financial and social systems, such as the size distribution in cities. 1 , 2 , 3 , 4 , 5 Zipf’s law for word frequency is a well-studied paradigm, for which Zipf coined the term “principle of least effort”, 6 exhibiting a power law that was later derived by maximizing Shannon’s information entropy 9 under a constraint on the mean-value of the logarithm of a variable. Notably, Zipf's law relates the probability of occurrence of a given word to its rank 7 , 8 rather than its occurrence count. More generally, power laws have been used by Mandelbrot to describe word frequencies (and many other social and physical phenomena 9 ) through the related concept of efficiency in communication. The principle of least effort was first discovered in 1894 by a French philosopher Guillaume Ferrero and he discussed this principle in his article entitled "L'inertie mentale et la loi du moindre effort". 10 In 1949, the principle was proposed by George Kingsley Zipf in his book " Human behavior and the principle of least effort: An introduction to human ecology " . 11 Zipf theorized that the distribution of word use was due to the tendency to communicate efficiently with least effort. Hence, the principle of least effort is also known as Zipf's law. Based on the principle of least effort, it is human nature to want the greatest outcome at the least amount of work. The principle of least effort has also been observed in animal behavior. The animal behavior scientist L. S. Tsai stated that “Among several alternatives of behavior leading to equivalent satisfaction of some potent organic need, the animal, within the limits of its discriminative ability, tends finally to select that which involves the least expenditure of energy”. 12 Zipf’s law has been invoked in mining 13 , 14 , 15 , 16 , 17 and geosciences. 18 , 19 , 20