A New Interpretation of Dictatorship with Applications in Social Choice Theory

We say that a social choice function is sensitive to collective change if whenever all inputs change, the outcome necessarily changes. Then, we prove that a social choice function is sensitive to collective change if and only if it is dictatorial. This provides a new interpretation of the essential meaning of dictatorship. We provide an intuitive, geometric proof for a special case of the theorem, which reveals the geometric meaning of dictatorship. In addition, the theorem provides a unified framework to prove other dictatorship results in theoretical social choice; to illustrate this, we show how the Arrow and Gibbard-Satterthwaite theorems can be deduced from this theorem