Ponder This Challenge - December 2020 - Maximal losing vote

Let's consider a simplified version of the USA elector-based election system. In our system, the states are represented by a vector pop = [p1, p2, ..., pn] of odd numbers, given the number of eligible voters in any state.

Each state is allocated a given number of electors in the following manner: There are 1,001 electors in total, and they are assigned to states in proportion to their population. If p = p1+p2+...+pn, then state i receives (pi/p)*1001 electors, rounded down. The remaining electors are allocated one at a time for the states, ordered by the size of the remaining fractional elector (e.g., a state that got 3.5 electors will get another one before a state with 10.25 electors).

For example, if pop = [51, 61, 77, 89, 99] then the electors are [135, 162, 205, 236, 263].

There are two candidates in the election. If a candidate receives more than half the votes in a state, she gets all the electors for that state. The candidate with the most electors wins the election.

A vote is a vector [v1, v2,..,vn] that represents the number of votes a candidate received in all the states. The size of a vote is v1+v2+...+vn - the total number of votes for that candidate. A losing vote is a vote that results in losing the election (i.e., not getting enough electors). A losing vote is maximal with respect to a given population vector pop if any vote of a larger size is not a losing vote.

Your goal: Find a population vector p=[p1, p2, p3, p4, p5] on five states with 101<=pi<= 149 and a maximal losing vote v=[v1, v2, v3, v4, v5] such that the size of v is 71.781305% the size of p up to a millionth of a percent.

Present your answer in the following format:
[p1, p2, p3, p4, p5]
[v1, v2, v3, v4, v5]

A bonus '*' for computing the maximal losing vote for the 2020 presidental election (where the populations are the eligble voters for each state, but the electors are computed by the non-simplified, real-world formula).

Solution

• Click here to view the solution

  One possible solution is
  P = [101, 101, 115, 125, 125]
  V = [50, 50, 57, 125 125]

  A solution to the bonus question requires some source of USA voter data; there isn't one canonical source, so multiple answers were accepted. One answer by Florian Fischer used http://www.electproject.org/2020g and gave the result 77.92%, which is typical; the answers mostly range between 77-79%.

Solvers

• Lorenz Reichel (30/11/2020 4:42 PM IDT)
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