Proportionality in political representation is an essential theme for
representative democracy. In Brazil, this debate appears in the context
of non-proportionality between a federative unit’s population
size and its number of representatives in the Chamber of Deputies.
In other words, the number of deputies in a state is not proportional
to its number of inhabitants, which violates the "one man, one vote"
principle.
Discussions around this disproportionality have motivated scholars
to develop empirical research that aims to identify the causes
and consequences of the phenomenon and to analyze the impact
that the rule introduces in the political process. This article seeks
to contribute to this debate by measuring the effective power of
each Brazilian federation’s entity and proposing alternatives of
distribution for the Brazilian Chamber of Deputies.
To this end, we use a mathematical concept from game theory,
called Power Index, which allows quantifying the existing representational
discrepancies. After evaluating several distributions, we
solved the Inverse Power Index Problem (IPIP) to obtain a distribution
of chairs that reduces such disparities. To solve the IPIP, which
is computationally hard, we use an evolutionary heuristic. As an
objective function to minimize the discrepancy, we use the linear
Shapley rule, in which the power index of each state is proportional
to its population.