Posted by: annemartinfletcher | November 21, 2011

Math Proves Why Congress Can’t Compromise

It is almost a mathematic impossibility for our current voting system to elect moderate candidates. All this time I thought the impasse in Congress was due to flaws in the campaign system. Actually, the campaign experts are applying strong knowledge of how our election system works. In a close election, the candidate who is capable of working with both sides will frequently lose.

Most states elect Congresspersons based on a “plurality-with-elimination” method. The candidate with the majority of votes wins the seat. If there is no majority, then a run-off election is held for the top vote getters. Sound fair? All voting methods have flaws, but plurality-with-elimination has several flaws.

The flaw that is disrupting Congress now, is that the first candidates to be eliminated are usually the moderate candidates. Here is an example: assume that Very Conservative gets 81 votes, Moderate gets 75 votes, Unheard Of gets 10 votes, and Very Liberal gets 80 votes. Further assume that voters favoring Very Conservative or Very Liberal would prefer Moderate over their arch rival. None of the candidates received a majority of the votes. Using the “plurality-with-elimination” method, Moderate (as well as Unheard Of), are eliminated from the election and a run-off election is held between Very Liberal and Very Conservative, who are incapable of acknowledging the validity of the other’s viewpoint.

I can prove mathematically why the candidate who would win in a head-to-head match-up with all other candidates, will lose the election. Well, I can’t prove it, but mathematical economist, Nobel Prize winner Kenneth Arrow proved it back in the fifties. He showed that any method for determining election results will always violate a fairness criteria–specifically the “head-to-head criteria” (winner might be a loser if compared head to head with other candidates), “majority criteria” (method will not always elect the person with the majority of win votes), the “monotonicity criteria” (I’ll ignore this one), and the “irrelevant alternatives criterion” (I’ll ignore this one as well.)

The plurality-with-elimination method can violate the “head-to-head criteria,” as well as all the criterion except “majority.”  A candidate who is everybody’s second choice, even in a multi-candidate election, would lose. Consequently, our voting method almost guarantees that elected officials will be the ones least likely to compromise.

Is there another voting method that could ensure more moderate candidates? Of course there is. You see it on the newscast polls. It is called a “Pairwise Comparison.” In this method, voters rank all the candidates. Then the candidate is compared to each of his or her rivals. The candidate who wins the most pairwise comparisons and is therefore favored above all others by most of the voters, is declared the winner. In fact, the “Pairwise Comparison” method violates fewer fairness criteria than any other voting method I know of.

Good luck, however, getting the politicians who were elected to your state legislature by the plurality-with-elimination method, to change your state’s voting method.

Reference: Robert Blitzer, Thinking Mathematically, Annotated Instructor’s Edition, 5 ed., Prentice Hall (2011)



  1. You are absolutely right, Anne. The Pairwise Comparison is also known as Marquis de Condorcet’s Method. However, that method creates cumbersome ballots (especially if we were to try to pick a Republican Presidential candidate from the current field!).

    My personal favorite (used when I’m in a social group trying to decide which movie to see or which restaurant to patronize) is the Jean-Charles de Borda Count. Every voter ranks every candidate from most favored to most disfavored. If there are five candidates, the favorite of a voter gets four points, the next gets three points, and so on (the least favored gets no points). The candidate with the most points wins.

    The de Borda Count is much simpler, and was identified by Univ. of Ca. mathematician Donald Saari (via chaos theory analysis) as the method producing the fewest paradoxes and the most difficult to manipulate. Also studied were Approval Voting (and Disapproval Voting), Instant Run-Off Voting, and Cumulative Voting.

    I hope that when we do our nation-building efforts (in South Sudan, Afghanistan, etc.) that we encourage them to use better methods than ours! I agree that there is zero chance of changing our current system unfortunately.

  2. Hi Dan!
    I assigned this blog post to my freshman math classes as reading. I hope they read your comment, too!

  3. […] the homogenous vs. the “melting pot,” the change vs. the future-phobic. Our voting apportionment laws do not help this. Forget “nation-building” — if our rhetoric does not calm down, […]

  4. Here are some great videos on the problems with what you call “plurality” (what he calls First Past the Post) voting systems by C.G.P. Grey.

    The first is about FPP voting systems

    The second is about Alternate voting systems

    The third is about Mixed-Member Proportional Representation

    While there are still issues (he addresses them) and that voting would be rather complicated, MMP fixes a lot of the fairness problems with our current voting system.

  5. Hi Sam, these are some great videos! I appreciate you commenting and providing these.

    I espescially like the way Grey describes the fairness criteria that each voting method violates. However, we need to clarify our terminology.

    FPP is straight plurality, not plurality with elimination. The alternative vote or “instant runoff” is closest to plurality with elimination. With just a slight change in how that alternative vote is computed, you could use those ballots to do the Borda method, as explained above by Dan Bloemer, and be even more fair. When Grey alludes to the Cordecet method, he is referring to what I called pairwise comparison. Finally, the MMP is a mixture of an apportionment problem for half the representatives, and plurality for the first half. And they still don’t provide a universally fair system.

    Valuable contribution to the discussion!

  6. This was a very interesting blog entry!

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