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Probabilistic Models and Profiles Generation
A profile is the set of agents with their preferences over the set of alternatives. What is a probability of a given profile? The two most widely used probabilistic models are Impartial Culture (IC) and Impartial Anonymous Culture (IAC).
Under IC it is assumed that all preferences are equally likely. If there are \(m\) alternatives, \(n\) agents, and each agent has one of \(m!\) preferences, the total number of profiles under IC is equal to \({(m!)}^n\).
Under IAC it is assumed that all preferences are equally likely with respect to anonymity of agents. The total number of profiles under IAC is equal to \(C_{m!+n-1}^n\).
It can be noticed that the number of profiles for both IC and IAC grows exponentially. That is why we do not generate all possible profiles (it will be impossible for large \(n\) and \(m\)). Instead, we generate 1,000,000 random profiles.