The Grump and Widen Show: Weekly Puzzle by Dilip D’Souza

The US election is over (I hear you muttering gratefully under your breath). But while it is still on your mind, let me run a thought experiment by you.

In the Benighted Plates of Amberica (BPA), everyone lives in one of two towns: North Plate (NP) and South Plate (SP). For some reason, all of NP’s voters are thought to belong to the Prepubican Party; all of SP voters are thought to belong to the Lameokrat Party.

BPA conducts peculiar elections, in Stages. Only two candidates can run, one from each party. Authorities call in a batch of voters from NP to vote, note which candidate got the most votes, discard the rest and announce the winner and her winning percentage. Then they call in a batch from SP and do the same. The candidate with the higher percentage is the winner of Stage 1. On they go to Stage 2, operating in the same way, and to more Stages as necessary.

The candidate who wins the most Stages becomes the country’s Crazydent. You can pick holes aplenty in this process, but it’s what BPA’s founders, the venerable Pilgrim Platters, spelled out.

In last week’s election for Crazydent, the two candidates were Dough Grump (Prepubican) and Sonal Widen (Lameokrat). Because of a deadly pandemic, there were many fewer voters than usual — only 500 in each town — and so only two Stages to the election.

In Stage 1, 125 NP voters came in. 116 voted for Grump, or 93%. Then 375 SP voters entered; 326 voted for Widen, or 87%. So Grump won Stage 1.

In Stage 2, 274 of 375 NP voters voted for Grump: 73%. As expected, Widen got most of the Stage 2 SP batch’s votes, but less than 73%. So this was a second Grump Stage win — and Grump readied to crown himself Crazydent.

But Widen yelled: “Hey! Wait a minute! I should be Crazydent! Because if you take vote counts, I won more!”

She was right.

Questions: How is this possible? Can you come up with numbers for how the vote went for the second SG batch? And can you explain what happened here?

Scroll down for the solution

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The Weekly Puzzle Solution

This is an example of what mathematicians know as Simpson’s Paradox. A good example is Hilary Clinton losing the Electoral College vote in the 2016 US Presidential election despite winning the popular vote.

In this case, Grump got 390 of 500 NP votes. Widen got more than 390 of 500 SP votes.

An easy way to think of this is that when the larger batch of 375 from each town voted (SP in Stage 1, NP in Stage 2), Widen’s win (87%) was more comprehensive than Grump’s (73%).

To make that even clearer, imagine the election went like this:

Stage 1: 2 NP voters, 2 for Grump, 100%. 498 SP voters, 490 for Widen, 98%. Grump wins Stage 1.

Stage 2: 498 NP voters, 300 for Grump, 60%. 2 SP voters, 1 for Widen, 50%. Grump wins Stage 2.

But Grump actually has only 302/500 votes. Widen: 491/500.

In this puzzle, Widen could have got anywhere between 65 and 91 votes of that Stage 2 SP batch of 125. Then she would have lost Stage 2, but won the overall count.

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