This paper extend the link between stochastic approximation and randomized urn models investigated in Laruelle and Pagès (AAP 2013) for application in clinical trials introduced in Bai and Hu (AAP 2005) or Bai, Hu ans Shen (JMA 2002). The idea is that the drawing rule is no longer uniform among the balls of the urn (which contains d colors), but can be reinforced by a function f modeling in some sense aversion to risk. Firstly, by considering that f is concave or convex and by reformulating the dynamics of the urn composition as a standard stochastic approximation (SA) algorithm (with remainder), we derive the a.s. convergence and the asymptotic normality (Central Limit Theorem, CLT) of the normalized procedure by calling upon the so-called ODE and SDE methods. An in-depth analysis of the case d=2 exhibits two different behaviours: a single equilibrium point when f is concave, and when f is convex, a transition phase from a single to a system with two attracting equilibrium points and a repulsive one. The last setting is solved using results on noiseless traps in order to remove the repulsive point and to deduce the a.s. towards one of the attractive point. Secondly, the special case of a Polya urn (i.e. when the addition rule the I_d matrix) is analyzed, still using result form (SA) about traps. Finally, these results are applied to functions with regular variation and optimal asset allocation in Finance.