allow us to prove results in order to compute the universal basin of attraction of a given social outcome (which is a studied and open problem in economics).

In Section \[sec:probability\] we will start studying the problem of probability on tournaments with the extra module structure. We will compute the maximum number of local optima that a given social rule can have and the probability to have a given number of local optima in a two dimensional social rule. This probability is related to the phenomenon (very important and studied in economics) of the trade-off between [*decidability*]{} (i.e. the possibility of reaching some social optimum in a feasible time) and [*non manipulability*]{} (i.e. the convergence of the social decision process to a unique global outcome that does not depend upon initial condition and agenda). Then it would be very interesting to generalize this result to an $n$-dimensional space of features.

We will also define a function to measure the gain in using Marengo and the second author’s model instead of the classical one by means of the probabilities that a social outcome is an optimum in the two models.

In Sections \[sec:algorithm\] and \[sec:numerical\] we will approach the far more 