Math in Elections Part 5 — There are Alternatives!

“There have been 45 presidential elections since 1828. In at least five, the race went to the second-most-popular candidate because of a spoiler. That’s an 11 percent rate of catastrophic failure. Were the plurality vote a car or an airliner, it would be recognized for what it is — a defective consumer product, unsafe at any speed.”

— William Poundstone, Gaming The Votes

There is an argument to be made that Mr. Poundstone is right that the US voting system has failed profoundly at the biggest level a very significant number of times.  You can also make the argument that if you increase your sample size to include elections from all around the world with a similar system to our own, that the system fails with even greater likelihood.  But I won’t dwell on arguing this point.  We have talked about the issues with our primary and general election systems.  Now let’s talk about some alternatives.

In this post I want to talk about some alternative voting systems when there are several influential candidates in the field.  This includes the presidential primaries, PTA elections, governing board elections at your church, and all other emotional affairs where a voter could be drawn to tactical voting if necessary.

— State-Based Alternatives — 

Let’s start with how we elect our president.  First, if we still want to follow the states-focused path of America’s (non-mathematician) founding fathers, then one state-based alternative is a weighted popular vote, where the weight of a vote is determined by which state it comes from.  One way to implement this is to let each state still have its fixed number of contributing electoral votes, but the votes are awarded proportionally to the candidates.  That still allows each American voter to be heard, yet would cap the influence of a state, since each state would have the same number of contributing electoral votes regardless of how many of its voters turned out on election day.  I don’t think this is a great replacement, but I do think it is an improvement.

I will note that two states, the first being The Great State Of Nebraska, and the second being Maine, have a system similar to this.  They award their electoral votes by congressional district, not as a whole state.  This caused Obama to win one of the five electoral votes from the very red state of Nebraska (second most consistently red state in the union over the past half century).  This was the furthest Obama reached into red territory in 2008.  The problem with this process, though, is seen in Nebraska GOP’s response to Obama winning this electoral vote.  They responded by redrawing district lines.  They shifted democratic neighborhoods from the second district, which Obama won, into the more republican first district, which was able to safely absorb these voters without risk of turning blue.

This process of strategically bunching together voters of opposing party identification to yourself in order to contain their influence is called Gerrymandering.  It leads to ridiculously shaped districts like North Carolina’s 12th congressional district, seen below.

State lines, however, will not be changed.  Therefore to avoid these blatantly non-democratic procedures, I believe that it is best to divide up the electoral votes dependent upon state vote tallies, not the tallies from these fluid congressional districts. has a nice feature where you can scroll through time to see how congressional districts have changed.  It’s pretty ridiculous.

A final method that maintains the state-focused viewpoint would be to have a dual system where half the vote was determined by some state-based system (like the electoral college, but ideally a better one), and the other half by a popular vote.  Our Congress is divided up into two houses, one with each representative speaking for roughly the same number or citizens, and the other with a pair of senators representing a single state.  So it would be consistent with the founder’s vision in that regard to have such a balance between the people and the states when electing our president.


Now I want to talk about two voting systems that can be used in any election setting.  They’d work well for the primaries, for instance.  These are Instant Runoff Voting (IRV), and Range Voting (RV).  I prefer the latter to the former.  And so I will save that for my grand finale.  Let’s start with IRV.

— Instant Runoff Voting — 

The idea behind IRV is straightforward.  Each voter gets a list of all the candidates and ranks any of them that (s)he wishes to.  Like so.

We all agree that if a candidate is the top choice for over half of all citizens, then that candidate deserves to be elected.  Therefore, if a candidate is the number 1 choice for over half of the voters, then IRV dictates that that candidate is the winner.  However, if there is no such candidate, then we might agree that there should be some sort of runoff election.  IRV is then designed to immediately simulate a runoff election by people’s other preferences.

After the first round, we create a ranking of all of the candidates, as determined by the number of first place votes they received.  Since we have all this voter data provided to us, let’s talk about what is the best possible way to simulate a runoff election.  It might not make sense to eliminate everyone except the top two (like many runoff elections), since possibly all of the top three got very similar results, and maybe even the third-place candidate is ranked as people’s number 2 choice far more often than the first or second finishing candidates.

So let’s be safe and only eliminate from the race the candidate that finished in last place.  Then, let’s redistribute his/her votes according to the next preference on each ballot that placed him/her in first place.  Then we check again whether we have a candidate who has surpassed 50% of the vote.  If not, we repeat.  The flow chart looks like this:

We continue this process until we have a winner.  The algorithm (with some tiebreak rules) will always give us a single winner.

This system has many advantages.  It’s criteria for declaring a winner is a pretty good one.  And having many runoff elections (but without paying to keep running them, losing voter turnout, etc.), and only eliminating the worst candidate each time, seems like a good idea.  One possible point for discussion is whether having the least number of first place votes does qualify you to be “the worst”.  There could very well be many other more sensible metrics for determining which candidate is the worst representation of society.

There are, however, some unfortunate consequences of IRV.  The way votes are redistributed allows for some pathological behavior.  Here are some examples.

  • It is possible for a candidate to be capable of beating every other candidate in a head-to-head election, but yet fail to win IRV.
  • If you were to divide the electorate into two groups, and run the election on both groups, it is possible for both halves to elect the same candidate but yet the election being run on the entire electorate would select a different candidate.  (All preferential voting systems are susceptible to this pathology.)
  • It is possible that placing a candidate higher on your preference list will actually hurt his/her chances of winning.  One way this can manifest itself is that it can change who the “last place” candidate is, which could change which ballots are redistributed.
  • IRV is susceptible to strategic nomination.  Meaning that placing a candidate in the race that is incapable of winning, can still alter the outcome.

IRV is currently being run in many places.  Many other countries use it as their primary voting system.  But it is also being used in America.  Many cities elect their mayor through IRV.  Oakland’s 2010 mayoral election ended in one semi-pathological instance of IRV (although there are many more examples abroad).  Ms. Jean Quan was elected mayor, but yet was initially far from first place based on the first place voting.  It took 10 runoff rounds to widdle down the field of 11 candidates until finally Ms. Quan obtained 50% of the votes and was crowned the winner.  Ms. Quan had not lead in any round before the last one.  In fact, after round 9 she had earned just 30.94% of the vote, and in the final round just barely crossed the finish line, ending up with 50.96% of the vote.  The table below is a summary of the results, you can see them in better resolution here.

These facts don’t necessarily mean that the system was unfair or that she did not deserve to get elected, but the complexity of the decision process suggests that some very slight, seemingly reasonable and inconsequential adjustments to the voting system or candidate field could have generated a different result.  And therefore this and the pathological examples above suggest that although IRV is much better than our primary system, maybe it is still not best.


And, in my opinion, you’d be right.  I will conclude this post with one more (but almost surely still not the best) voting system.

— Range Voting —

For range voting, each voter is given a table containing each candidate’s name with the numbers 0 through 9 and NO OPINION following each one.  The voter must shade in how much (s)he supports that candidate.  A 0 means that you do not like that candidate at all, a 9 means you really like them.  I will leave it as an exercise to determine what NO OPINION means.  If a candidate is not given a rating, it counts as a NO OPINION. Below is an example of a completed ballot.

So how is a winner decided?  Among all numerical scores that a given candidate received, those scores are averaged.  The candidate with the highest average score is the winner.

Simple enough.  The astute reader may have immediately realized a problem with this system.  What about the random Joe Schmo that no one has ever heard of?    Won’t nearly everyone rate him NO OPINION except himself and his beer-drinking buddies?  Won’t that allow him to get a high average for no good reason?  With the minor candidates getting rated far less often than others, won’t there be an imbalance where some candidate’s friends and family will play a huge influence in helping out their average?  Well, in theory, yes.  But I believe, in practice, no.  If you strongly want your candidate to win (which most people do), there is a good chance you’ll put a 0 down for everyone else, just to make sure.  In practice, I don’t think that this concern would ever come about, especially with any notable races.

So what does this system have going for it?  Well, first of all, it reduces the “spoiler effect”, where similar candidates split their ideological vote, giving the win to a candidate who is not the most popular.  There have been many instances of this in past presidential elections, and certainly many, many more in lower-level American elections and in elections abroad.

Also, observe that in this system, voters do not submit a preference list.  They instead say how much they like or dislike each candidate.  And several candidates can be rated at the same level.  Note that this disqualifies us from applying the Gibbard-Satterthwaite theorem.  That theorem requires preference lists.  Well, fine, but the issue was not whether we could apply the theorem, the problem was the result of the theorem.  So must RV have a dictator, could a candidate be incapable of winning, or is the system susceptible tactical voting?  Nope on all three counts!  The bad things we had said that we wanted to avoid, are indeed avoided!

For these reasons, it is my belief that RV, or some variant of RV (such as dealing with the NO OPINION option in a different way), would make a very good voting system.  If nothing else, it is worth being discussed on the national level to determine if it is indeed as much better of a system as my first impressions think it is.


Well, that’s it for this post.  Turned out to be a longer post than I intended.  So thanks if you stuck with it all the way until here!  Stop by tomorrow to hear me tell you just how insignificant you are!



About Jay Cummings

I am a fifth year mathematics Ph.D. student at UCSD.
This entry was posted in Math in Elections, Math in the "Real World". Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s