12 0.988343         16 0.999024
  ----- -------- -- ----- ---------- -- ----- ---------- -- ----- ----------

  : The probability that a tournament is irreducible.[]{data-label="tab:prob_irred_tournament"}

As $M$ tends to infinity, $P(M)\rightarrow1$ and $P(M)\sim1-\displaystyle{\frac{M}{2^{M-2}}}$ hold.

Definitions and structure of the model {#sect:model}
======================================

#### Social decision rules

Consider a population of $\nu$ [*agents*]{}. Each agent $i$ is characterized by a [*system of transitive preferences*]{} $\succeq_i$ over the set of social outcomes $X$. The set of systems of transitive preferences $\succeq$ is denoted by ${\mathcal{P}}$. A [*social decision rule*]{} ${\mathcal{R}}$ is a function: $$\begin{matrix}
{\mathcal{R}}: & {\mathcal{P}}^{\nu} & \longrightarrow & \overline{{\mathcal{P}}} \\
 & (\succeq_1 ,\dotsc,\succeq_\nu) & \longmapsto & \succeq_{{\mathcal{R}} (\succeq_1
,\dotsc,\succeq_\nu)}
\end{matrix}$$ which determines a [*system of social preferences*]{} or [*social rule*]{} $\succeq_{{\mathcal{R}}
  (\succeq_1 ,\dotsc,\succeq_\nu)}$ from the preferences of $\nu$ individual agents. With $\overline{{\mathcal{P}}}$ we denote the set of systems of (non-necessarily transitive) social preferences; as a matter of fact, we note that the social rule $\succeq_{{\mathcal{R}} (\succeq_1 ,\dotsc,\succeq_\nu)}$ is not, in general, transitive anymore.

If $\Delta$ is the diagonal of the cartesian product $X \times X$, the element $\succeq_{{\mathcal{R}}} \in \overline{{\mathcal{P}}}$ defines a subset $${\mathcal{Y}}_{1,\succeq_{{\mathcal{R}}}}=\{ (x,y)\in X\times X \setminus \Delta \mid 
x \succeq_{{\mathcal{R}}} y \}$$ and the set of [*relevant*]{} social outcomes $${\mathcal{Y}}_{0,\succeq_{{\mathcal{R}}}}=\{x 