Extreme Gerrymandering

Question

It is election season in Puzzlevania. There are two candidates: the incumbent and a challenger. Of the $20$ million citizens, only $1$ percent support the current president; the other $99\%$ favoring the challenger.

Here's how the election system in Puzzlevania works: There is a list of numbers, $n_1,\dots,n_L$. The entire population is divided into $n_1$ equally sized groups. Each of these groups is divided into $n_2$ equally sized subgroups, each subgroup is divided into $n_3$ equal sub-subgroups, and so on down the line. On election day, each sub-sub-$\dots$-subgroup elects a representative to vote for it at the next level up. Tie votes count as wins for the challenger.

You can think of this system as a many-layered version of the American electoral college, where all states have the same population.

The current president has the power to choose the numbers $L,n_1,\dots, n_L$, and to choose which voters go in which groups. Can he do this in such a way that he gets elected?

Source: http://www.cs.cmu.edu/puzzle/puzzle20.html

Answer 1

The current president should choose the numbers

9($L$), 32($n_1$), 8($n_2$), 5($n_3$), 5($n_4$), 5($n_5$), 5($n_6$), 5($n_7$), 5($n_8$), 5($n_9$).

In the lowest sub-sub-...-subgroup, the president should place voters in such a way that his supporters outnumber his opposers in the ratio of 3:5, till he runs out of supporters, and he can let the remaining sub-sub-...-subgroups vote for his opponent unanimously. So before the first round of voting, he has $200,000/20,000,000$ (1%) votes. After the first round, it becomes $66,666/4,000,000$ (1.66%). The current president needs to keep following this strategy till his percentage of votes exceeds 50%.

In the subsequent round, the same strategy needs to be followed so he can improve his vote share after each round of voting.

Following this strategy from $n_1$ till $n_9$, gives this vote share:

Supporting vote    Total Vote      Vote Share(%)     Voter ratio per round

200,000            20,000,000      1
                                                     3:5
66,666             4,000,000       1.67              
                                                     3:5
22,222             800,000         2.78              
                                                     3:5         
7,407              160,000         4.63              
                                                     3:5
2,469              32,000          7.72              
                                                     3:5
823                6,400           12.86             
                                                     3:5
274                1280            21.41             
                                                     3:5
91                 256             35.55             
                                                     5:8
18                 32              56.25

After the final group of 32 representatives votes, the president gets re-elected and another period of corruption and malpractice ensues in Puzzlevania.

Answer 2

While CodeNewbie has answered it correctly he can do even better--by making each group 5 he can do it with 20,000 supporters. By itself this conveys no advantage but since he has plenty of extra supporters he doesn't need to settle for 3 of 5 in the final round--he can make this 5 of 5 and win by a unanimous vote! Making each round the same size also makes it look fairer.