Voting Paradoxes

By Robert Bernstein   |   March 26, 2020

My immigrant wife Merlie is mystified by our complex voting system. She wonders what these “primaries” are all about. “Why don’t you just vote for your choice among all of the presidential candidates at once and be done with it?”

I am going to break Godwin’s Law and invoke the following example. Suppose we had the following votes for the following people on the ballot:

33 Franklin Roosevelt

33 Wendell Willkie

34 Adolph Hitler

If the top vote getter became President, then Hitler wins. Even though it seems that 66% of the voters would prefer anyone but Hitler. There is clearly a problem with this “plurality” voting system, yet that is exactly what we have in our presidential general election. Consider the 1992 election where Ross Perot got 19% of the vote and Bill Clinton won with just 43% of the vote. We don’t know who Perot voters would have taken as their second choice, but it is quite possible his supporters would have preferred Bush to Clinton.

Which leads to one way to avoid this “voting paradox”: Have a runoff election if no one gets more than 50% plus one vote. Santa Barbara County does this with our County Board of Supervisors election. If no candidate gets a majority of votes in the primary election, then the top two candidates run off in November.

But why waste the time and expense of a runoff? Why not allow voters to rank their choices all at once? In fact, 20 cities and the state of Maine now use what is called Instant Runoff Voting (IRV) or Ranked-Choice Voting. This allows voters to vote for their true first choice without risking electing the person they really detest.

In the 2000 election we have no idea how many people had Ralph Nader or Pat Buchanan as their first choice because many feared “throwing away their vote” and giving the other side’s leading candidate a win. IRV would give a true count.

But careful analysis shows it is possible to “game” IRV by putting your “lesser evil” choice first and your hated most-viable opponent last. It can end up being similar to our current system in practice.

The Marquis de Condorcet discovered “Condorcet’s Paradox” in the 1700s that is similar to an Escher Staircase of voting preferences. Google it to see the details.

But some have argued that the worst paradoxes don’t occur in real life because people have a rational ideological spectrum. For example, one could argue that in 2000 there was a left to right sequence: Nader, Gore, Bush, Buchanan. But this is not necessarily true.

In the 2016 election I met quite a few people who had Bernie Sanders as their first choice during the primary season. But when he lost the primary, those people voted for Trump rather than Hillary Clinton in November. Why? It is hard to get inside their heads, but there were clues in what they said. They were looking for an “outsider” who would “shake things up.” For them, Bernie and Trump had more in common than either had with Hillary.

I am a math person, but I would rather avoid the arcane math of voting paradoxes and focus on this human issue. Most liberals I knew in 2016 thought Hillary was the “safe” choice to beat Trump. But “safe” may be the most risky choice when it comes to actual human nature and our current voting system.

Stanford Professor Kenneth Arrow proved in 1952 that no voting system can avoid counter-intuitive outcomes. Suppose you prefer beer to tea. If Coke becomes available it should not make you prefer tea to beer. But no voting system can avoid such absurdities all of the time.

Two other systems out of an infinity of possible systems that have been used are the Borda system and Approval Voting. Borda lets voters assign points rather than rankings. But it can be gamed in the same way as IRV. Approval Voting lets voters vote equally for all candidates that they approve of. Interestingly, this one is not easy to game with covert strategies.

Apart from voting systems are political systems. Our “winner takes all” system means that 20% of voters might be Greens, yet there are no Greens at all in our Congress. Most Western democracies have “proportional representation”. If even 1% of voters belong to the Panda Party, then the Panda Party gets 1% of the representatives.

And most Western democracies have a Parliamentary system: The party with the most representatives runs the country. Our system is unique in its ability to have total gridlock with a President of the opposite party to Congress. And no other system in the world has our “electoral college” that allows a voter in Wyoming to have over 50 times the voting power of a California voter.

One thing is mathematically certain: Our voting system and political system are provably most subject to paradoxes of any system in use. And we have real problems like the climate crisis that need bold leadership. The people need to demand a better system. But we also need to do our best with our broken system to demand better leaders. And most important: The people must lead in order for the leaders to follow. An election is the start of progress, not the end.


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