The theoretical Shapley–Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system

In this article, we evaluate asymptotically the probability $$\phi \left( n\right) $$ ϕ n of an election inversion in a toy symmetric version of the US presidential electoral system. The novelty of this paper, in contrast to all the existing theoretical literature, is to assume that votes are drawn from an IAC (Impartial Anonymous Culture)/Shapley–Shubik probability model. Through the use of numerical methods, it is conjectured, that $$\sqrt{n}$$ n $$ \phi \left( n\right) $$ ϕ n converges to 0.1309 when n (the size of the electorate in one district) tends to infinity. It is also demonstrated that $$ \phi \left( n\right) =o\left( \sqrt{\frac{ln(n)^{3}}{n}}\right) $$ ϕ n = o l n ( n ) 3 n and $$\phi \left( n\right) =\Omega \left( \frac{1}{\sqrt{n}}\right) $$ ϕ n = Ω 1 n .

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