Utility Function u = x/(x + 1)

u(x) = x/(x + 1) is the simplest utility function with diminishing risk aversion limiting to zero. It provides us with a simple model like the model of pendulum in mechanics. One can use it to explain for example how mutually motivated insurance contracts are possible. In the present book the mathematical features of that utility function are investigated. It has been argued that this function is unique and is between Bernoulli’s logarithmic utility and the utility with constant risk aversion. This book is about utility theory which is a part of decision theory. It is addressed for the mathematicians and economists but some ideas concerning physics are suggested as well.