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An alternative voting scheme is proposed to fill the democratic gap between a pre- sident elected democratically via universal suffrage (deterministic outcome, the actual majority decides), and a president elected by one person randomly selected from the population (probabilistic outcome depending on respective supports). Indeed, moving from one voting agent to a group of r randomly selected voting agents reduces the prob- abilistic character of the outcome. Accordingly, building r such groups, each one electing its president (elementary bricks), to constitute a group of the groups with the r local presidents electing a higher-level president, does reduce further the outcome probabilistic aspect. The process is then repeated n times to reach a bottom-up pyramidal structure with n levels, rn−1 elementary bricks at the bottom and a president at the top. Agents at the bottom are randomly selected but higher-level presidents are all designated accord- ing to the respective local majorities within the groups which elect them. At the top of the hierarchy the president is still elected with a probability but the distance from a deterministic outcome reduces quickly with increasing n. At a critical value nc,r the outcome turns deterministic recovering the same result a universal suffrage would yield. This alternative hierarchical scheme introduces several social advantages like the distri- bution of local power to the competing minority, which thus makes the structure more resilient, yet preserving the presidency allocation to the actual majority. It also produces an area around 50% for which the president is elected with an almost-equiprobability slightly biased in favor of the actual majority. However, our results reveal the existence of a severe geometric vulnerability to lobbying. It is shown that a tiny lobbying group is able to kill the democratic balance by seizing the presidency democratically. It is suf- ficient to complete a correlated distribution of a few agents at the hierarchy bottom. Moreover, at the present stage, identifying an actual killing distribution is not feasible, which sheds a disturbing light on the devastating effect geometric lobbying can have on democratic hierarchical institutions

Geometric vulnerability of democratic institutions against lobbying: A sociophysics approach