Malik Magdon-Ismail, Lirong Xia
Direct democracy, where each voter casts one vote, fails when the average voter competence falls below 50%. This happens in noisy settings when voters have limited information. Representative democracy, where voters choose representatives to vote, can be an elixir in both these situations. We introduce a mathematical model for studying representative democracy, in particular understanding the parameters of a representative democracy that gives maximum decision making capability. Our main result states that under general and natural conditions,
for fixed voting cost, the optimal number of representatives is linear;
for polynomial cost, the optimal number of representatives is logarithmic.