Political Calculus

Disclosure:  This article is primarily mathematical in nature but the very act of discussing politics makes it difficult to fully remove bias.  I feel obligated to disclose that I'm a member of the Green Party.  While I'm neither a Republican or Democrat, I tend to lean to the north-west section of the Nolan chart.  However, I do intend to try my best to make this analysis as neutral as humanly possible.

During my regular social media browsing the other day, I came across two posts of interest.

The first was a statement from the Green Party of Virginia about why they are not endorsing Bernie Sanders ahead of the primary.  While I had expected this to be the case, there was a section of this statement that really caught my attention: "Whether individual Greens choose to vote for Sanders on March 1st is a choice that will depend on their own calculus of what is best for the country" (emphasis mine).

Since one of the co-chairs of the GPVA is a mathematician, I could reasonably assume that the reference to calculus was intended to mean exactly what it says. The problem is that the general population doesn't usually look at elections from this perspective.  People tend to vote based on gut feelings rather than mathematical analysis. For this reason, I disagree with the GPVA's decision. I feel that they have the responsibility to provide party members with information on how to maximize their influence on the election and calculus isn't a strong point for most voters. If the GPVA refuses to take sides in the primary, then I feel obligated to do so in their place.

The second was a data visualization of how various primary candidates would fare against each other in a general election:

With "Super Tuesday" fast approaching, this was exactly the kind of information that I needed!  This effectively provides a payoff matrix for the primary candidates to which I can apply my "political calculus".

Since this is essentially a zero-sum game (a Democratic Win is a Republican Loss and vice versa), I defined my Payoff function using positive pay-off values for Democrats and negative for Republicans:

 Payoff(\begin{bmatrix} Clinton \\ Sanders \\ Trump \\ Cruz \\ Rubio \\ Kasich \\ Carson \end{bmatrix}) =
\begin{bmatrix} Clinton & Sanders \end{bmatrix} \begin{bmatrix} 2.0 & -0.8 & -4.7 & -7.4 & 1.3 \\ 6.0 & 4.7 & 6.0 & 0.5 & 1.3 \end{bmatrix} \begin{bmatrix} Trump \\ Cruz \\ Rubio \\ Kasich \\ Carson \end{bmatrix}

Just by looking at the payoff matrix from Real Clear Politics, and no other information about the election, I would play a minimax strategy.  Applying this approach, I'd come to the following conclusions:

  •  The Democrats should run Sanders.  Even in his worst match-up, versus Kasich, he still wins by +0.5.
  • The Republicans should run Kasich.  Even in his worst match-up, versus Sanders, he only loses by -0.5.  It's a loss, but still a far closer race than with any other candidate.

However, the current polling data suggests a completely different picture.  In the Virginia Primary, FiveThirtyEight is predicting Clinton to win the Democratic primary with a 99% confidence level and Trump to win the Republican Primary with a 77% confidence level. Using these probabilities, the expected general election payoff is +1.36 in favor of the Democrats.

Now, this data has me kind of puzzled from an outsider standpoint.  Neither party, Democrat nor Republican, is playing an optimal strategy.  In fact, they're both leaning towards the candidate with the lowest chance of winning a general election! To make things even more complicated, Virginia's primary system allows me to vote in either party primary! This is great considering that I'm not really a member of either to begin with...

My questions about this are the following:

  • Suppose I want the Democrats to win the general election. Is it in my best interest to vote for Sanders (the optimal candidate according to my minimax strategy) or do I attempt to sabotage the Republican primary by voting for Trump?
  • Suppose I want the Republicans to win the general election. Is it in my best interest to vote for Kasich (the optimal candidate according to my minimax strategy), vote for someone else in the GOP primary more likely to beat Trump, or attempt to sabotage the Democratic primary by voting for Clinton?

To answer these questions, I need to bust out some calculus! I'm going to take the gradient of my payoff function. What this does is calculate how much the overall payoff changes with an infinitely small increase in any candidate's direction. Given my vote versus the rest of the state, "infinitely small" sounds just about right...

 \nabla Payoff( \begin{bmatrix} Clinton \\ Sanders \\ Trump \\ Cruz \\ Rubio \\ Kasich \\ Carson \end{bmatrix} ) =
 \begin{bmatrix} 1.3 Carson-.8 Cruz-7.4 Kasich-4.7 Rubio+2 Trump \\ 1.3 Carson+4.7 Cruz+.5 Kasich+6. Rubio+6 Trump \\ 2 Clinton+6 Sanders \\ -.8 Clinton+4.7 Sanders \\ -4.7 Clinton+6. Sanders \\ -7.4Clinton+.5 Sanders \\ 1.3 Clinton+1.3 Sanders \end{bmatrix}

You'll notice that the value of voting for either Democrat changes with how the Republicans vote and vice versa. I still need to plug in some information about how I expect the rest of the population to vote!

First, I used the probabilities from FiveThirtyEight directly. I figured that this would provide the closest to the true outcome.

 \nabla Payoff( \begin{bmatrix} 0.99 \\ 0.01 \\ 0.77 \\ 0.22 \\ 0.01 \\ 0.00 \\ 0.00 \end{bmatrix} ) = \begin{bmatrix} 1.317 \\ 5.714 \\ 2.04 \\ -0.745 \\ -4.593 \\ -7.321 \\ 1.3 \end{bmatrix}

This supported my initial hypotheses about the optimal candidates. However, I was also concerned that it was overestimating the lead of the front runners. Out of curiousity, I ran the gradient again but using the aggregate polling percentages rather than the probabilities:

 \nabla Payoff( \begin{bmatrix} 0.635 \\ 0.331 \\ 0.396 \\ 0.297 \\ 0.146 \\ 0.075 \\ 0.065 \end{bmatrix} ) = \begin{bmatrix} -0.6023 \\ 4.7699 \\ 3.256 \\ 1.0477 \\ -0.9985 \\ -4.5335 \\ 1.2558 \end{bmatrix}

While there are some differences between these figures and the probability based ones (namely that voting for Clinton helps the Republicans and voting Cruz helps the Democrats...), the overall results seem pretty clear.


From the standpoint of a 3rd party interfering in someone else's primary, here are my conclusions based on the present data:

If you want the Democrats to Win

Vote for Sanders. Voting for Trump is actually the next best option, but the odds are already in his favor. Sanders needs your help more than he does.

If you want the Republicans to Win

Vote for Kasich. The Republicans need to coalesce around a more moderate candidate and fast if they want to win in November. The next best option would be Rubio, but if you split the vote between them then you risk handing the primary to Trump anyway.


Please note that this is a very chaotic system and polling data is notoriously unreliable. New information could swing the results in any number of directions.

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