The Relationship Between The Electoral College and Popular Vote

An interesting machine learning problem: Can one figure out the relationship between the popular vote margin, voter turnout, and the percentage of electoral college votes a candidate wins? Going back to the election of John Quincy Adams, the raw data looks like this:

Electoral College Party Popular vote  Margin (%)

Turnout

Percentage of EC

John Quincy Adams D.-R. -0.1044 0.27 0.3218
Andrew Jackson Dem. 0.1225 0.58 0.68
Andrew Jackson Dem. 0.1781 0.55 0.7657
Martin Van Buren Dem. 0.14 0.58 0.5782
William Henry Harrison Whig 0.0605 0.80 0.7959
James Polk Dem. 0.0145 0.79 0.6182
Zachary Taylor Whig 0.0479 0.73 0.5621
Franklin Pierce Dem. 0.0695 0.70 0.8581
James Buchanan Dem. 0.12 0.79 0.5878
Abraham Lincoln Rep. 0.1013 0.81 0.5941
Abraham Lincoln Rep. 0.1008 0.74 0.9099
Ulysses Grant Rep. 0.0532 0.78 0.7279
Ulysses Grant Rep. 0.12 0.71 0.8195
Rutherford Hayes Rep. -0.03 0.82 0.5014
James Garfield Rep. 0.0009 0.79 0.5799
Grover Cleveland Dem. 0.0057 0.78 0.5461
Benjamin Harrison Rep. -0.0083 0.79 0.58
Grover Cleveland Dem. 0.0301 0.75 0.6239
William McKinley Rep. 0.0431 0.79 0.6063
William McKinley Rep. 0.0612 0.73 0.6532
Theodore Roosevelt Rep. 0.1883 0.65 0.7059
William Taft Rep. 0.0853 0.65 0.6646
Woodrow Wilson Dem. 0.1444 0.59 0.8192
Woodrow Wilson Dem. 0.0312 0.62 0.5217
Warren Harding Rep. 0.2617 0.49 0.7608
Calvin Coolidge Rep. 0.2522 0.49 0.7194
Herbert Hoover Rep. 0.1741 0.57 0.8362
Franklin Roosevelt Dem. 0.1776 0.57 0.8889
Franklin Roosevelt Dem. 0.2426 0.61 0.9849
Franklin Roosevelt Dem. 0.0996 0.63 0.8456
Franklin Roosevelt Dem. 0.08 0.56 0.8136
Harry Truman Dem. 0.0448 0.53 0.5706
Dwight Eisenhower Rep. 0.1085 0.63 0.8324
Dwight Eisenhower Rep. 0.15 0.61 0.8606
John Kennedy Dem. 0.0017 0.6277 0.5642
Lyndon Johnson Dem. 0.2258 0.6192 0.9033
Richard Nixon Rep. 0.01 0.6084 0.5595
Richard Nixon Rep. 0.2315 0.5521 0.9665
Jimmy Carter Dem. 0.0206 0.5355 0.55
Ronald Reagan Rep. 0.0974 0.5256 0.9089
Ronald Reagan Rep. 0.1821 0.5311 0.9758
George H. W. Bush Rep. 0.0772 0.5015 0.7918
Bill Clinton Dem. 0.0556 0.5523 0.6877
Bill Clinton Dem. 0.0851 0.4908 0.7045
George W. Bush Rep. -0.0051 0.51 0.5037
George W. Bush Rep. 0.0246 0.5527 0.5316
Barack Obama Dem. 0.0727 0.5823 0.6784
Barack Obama Dem. 0.0386 0.5487 0.6171

Clearly, the percentage of electoral college votes a candidate depends nonlinearly on the voter turnout percentage and popular vote margin (%) as this non-parametric regression shows:

electoralmap.png

We therefore chose to perform a nonlinear regression using neural networks, for which our structure was:

nnetplot

As is turns out, this simple neural network structure with one hidden layer gave the lowest test error, which was 0.002496419 in this case.

Now, looking at the most recent national polls for the upcoming election, we see that Hillary Clinton has a 6.1% lead in the popular vote. Our neural network model then predicts the following:

Simulation Popular Vote Margin Percentage of Voter Turnout Predicted Percentage of Electoral College Votes (+/- 0.04996417)
1 0.061 0.30 0.6607371
2 0.061 0.35 0.6647464
3 0.061 0.40 0.6687115
4 0.061 0.45 0.6726314
5 0.061 0.50 0.6765048
6 0.061 0.55 0.6803307
7 0.061 0.60 0.6841083
8 0.061 0.65 0.6878366
9 0.061 0.70 0.6915149
10 0.061 0.75 0.6951424

One sees that even for an extremely low voter turnout (30%), at this point Hillary Clinton can expect to win the Electoral College by a margin of 61.078% to 71.07013%, or 328 to 382 electoral college votes. Therefore, what seems like a relatively small lead in the popular vote (6.1%) translates according to this neural network model into a large margin of victory in the electoral college.

One can see that the predicted percentage of electoral college votes really depends on popular vote margin and voter turnout. For example, if we reduce the popular vote margin to 1%, the results are less promising for the leading candidate:

Pop.Vote Margin Voter Turnout % E.C. % Win E.C% Win Best Case E.C.% Win Worst Case
0.01 0.30 0.5182854 0.4675000 0.5690708
0.01 0.35 0.5244157 0.4736303 0.5752011
0.01 0.40 0.5305820 0.4797967 0.5813674
0.01 0.45 0.5367790 0.4859937 0.5875644
0.01 0.50 0.5430013 0.4922160 0.5937867
0.01 0.55 0.5492434 0.4984580 0.6000287
0.01 0.60 0.5554995 0.5047141 0.6062849
0.01 0.65 0.5617642 0.5109788 0.6125496
0.01 0.70 0.5680317 0.5172463 0.6188171
0.01 0.75 0.5742963 0.5235109 0.6250817

One sees that if the popular vote margin is just 1% for the leading candidate, that candidate is not in the clear unless the popular vote exceeds 60%.

 

Optimal Strategies for the Clinton/Trump Debate

Consider modelling the Clinton/Trump debate via a static game in which each candidate can choose between two strategies: \{A,P\}, where A denotes predominantly “attacking” the other candidate, while P denotes predominantly discussing policy positions.

Further, let us consider the mixed strategies \sigma_1 = (p,1-p) for Clinton, and \sigma_2 = (q,1-q) for Trump. That is, Clinton predominantly attacks Trump with probability p, and Trump predominantly attacks Clinton with probability q.

Let us first deal with the general case of arbitrary payoffs, thus, generating the following payoff matrix:

\left( \begin{array}{cc} \{a,b\} & \{c,d\} \\ \{e,f\} & \{g,h\} \\ \end{array} \right)

That is, if Clinton attacks Trump and Trump attacks Clinton, the payoff to Clinton is a, while the payoff to Trump is b. If Clinton attacks Trump, and Trump ignores and discusses policy positions instead, the payoff to Clinton is c, while the payoff to trump is d. If Clinton discusses policy positions while Trump attacks, the payoff to Clinton is e, while the payoff to Trump is f, and if both candidates discuss policy positions instead of attacking each other, the payoff to them both will be g and h respectively.

With this information in hand, we can calculate the payoff to Clinton as:

\pi_c(\sigma_1, \sigma_2) = a p q+c p (1-q)+e (1-p) q+g (1-p) (1-q)

while the payoff to Trump is:

\pi_t(\sigma_1,\sigma_2) = b p q+d p (1-q)+f (1-p) q+h (1-p) (1-q)

With these payoff functions, we can compute each candidate’s best response to the other candidate by solving the following equations:

\hat{\sigma}_1 \in \text{argmax}_{\sigma_1} \pi_1(\sigma_1,\sigma_2)

\hat{\sigma}_{2} \in \text{argmax}_{\sigma_2} \pi_2(\sigma_1,\sigma_2)

where \hat{\sigma}_{1,2} indicates the best response strategy to a fixed strategy for the other player.

Solving these equations, we obtain the following:

If

latex-image-32
then,

Clinton’s best response is to choose p = 1/2.

If

latex-image-33

then,

Clinton’s best response is to choose  p = 1.

Otherwise, her best response is to choose p = 0.

 

While for Trump, the best responses are computed as follows:

If

latex-image-34

Trump’s best response is to choose q = 1/2.

If

latex-image-35

Trump’s best response is to choose q = 1.

Otherwise, Trump’s best response is to choose q = 0.

To demonstrate this, let us work out an example. Assume (for this example) that the payoffs for each candidate are to sway independent voters / voters that have not made up their minds. Further, let us assume that these voters are more interested in policy positions, and will take attacks negatively. Obviously, this is not necessarily true, and we have solved the general case above. We are just using the following payoff matrix for demonstration purposes:

\left( \begin{array}{cc} \{-1,-1\} & \{-1,1\} \\ \{1,-1\} & \{1,1\} \\ \end{array} \right)

 

Using the above equations, we see that if 0 \leq q \leq 1, Clinton’s best response is to choose p=0. While, if 0 \leq p \leq 1, Trump’s best response is to choose q =0. That is, no matter what Trump’s strategy is, it is always Clinton’s best response to discuss policy positions. No matter what Clinton’s strategy is, it is always Trump’s best response to discuss policy positions as well. The two candidates’ payoff functions take the following form:

payofffuncs

What this shows for example is that there is a Nash equilibrium of:

(\sigma_1^{*}, \sigma_{2}^{*}) = (0,0).

The expected payoffs for each candidate are evidently

\pi_c = \pi_t = 1.

Let us work out an another example. This time, assume that if Clinton attacks Trump, she receives a payoff of +1, while if Trump attacks Clinton, he receives a payoff of -1. While, if Clinton discusses policy, while being attacked by Trump, she receives a payoff of +1, while Trump receives a payoff of -1. On the other hand, if Trump discusses policy while being attacked by Clinton, he receives a payoff +1, while Clinton receives a payoff of -1. If Clinton discusses policy, while Trump discusses policy, she receives a payoff of +1, while Trump receives a payoff of -1. The payoff matrix is evidently:

\left( \begin{array}{cc} \{1,-1\} & \{1,-1\} \\ \{1,-1\} & \{1,-1\} \\ \end{array} \right)

In this case, if 0 \leq q \leq 1, then Clinton’s best response is to choose p = 1/2. While, if 0 \leq p \leq 1, then Trump’s best response is to choose q = 1/2. The Nash equilibrium is evidently

(\sigma_1^{*}, \sigma_{2}^{*}) = (1/2,1/2).

The expected payoffs for each candidate are evidently

\pi_c = 1, \pi_t = -1.

In this example,  even though it is the optimal strategy for each candidate to play a mixed strategy of 50% attack, 50% discuss policy, Clinton is expected to benefit, while Trump is expected to lose.

Let us also consider an example of where the audience is biased towards Trump. So, every time Trump attacks Clinton, he gains an additional point. Every time Trump discusses policy, while Clinton does the same he gains an additional point. While, if Clinton attacks while Trump discusses policy positions, she will lose a point, and he gains a point. Such a payoff matrix can be given by:

\left( \begin{array}{cc} \{1,2\} & \{-1,1\} \\ \{0,1\} & \{0,1\} \\ \end{array} \right)

Solving the equations above, we find that if q = 1/2, Clinton’s best response is to choose p =1/2. If 1/2 < q \leq 1, Clinton’s best response is to choose p = 1. Otherwise, her best response is to choose p = 0. On the other hand, if p = 0, Trump’s best response is to choose q = 1/2. While, if 0 < p \leq 1, Trump’s best response is to choose q = 1. Evidently, there is a single Nash equilibrium (as long as 1/2 < p \leq 1):

 (\sigma_1^{*}, \sigma_{2}^{*}) = (1,1).

Therefore, in this situation, it is each candidate’s best strategy to attack one another. It is interesting that even in an audience that is heavily biased towards Trump, Clinton’s best strategy is still to attack 100% of the time.

The interested reader is invited to experiment with different scenarios using the general results derived above.

Will Donald Trump’s Proposed Immigration Policies Curb Terrorism in The US?

In recent days, Donald Trump proposed yet another iteration of his immigration policy which is focused on “Keeping America Safe” as part of his plan to “Make America Great Again!”. In this latest iteration, in addition to suspending visas from countries with terrorist ties, he is also proposing introducing an ideological test for those entering the US. As you can see in the BBC article, he is also fond of holding up bar graphs of showing the number of refugees entering the US over a period of time, and somehow relates that to terrorist activities in the US, or at least, insinuates it.

Let’s look at the facts behind these proposals using the available data from 2005-2014. Specifically, we analyzed:

  1. The number of terrorist incidents per year from 2005-2014 from here (The Global Terrorism Database maintained by The University of Maryland)
  2. The Department of Homeland Security Yearbook of Immigration Statistics, available here . Specifically, we looked at Persons Obtaining Lawful Permanent Resident Status by Region and Country of Birth (2005-2014) and Refugee Arrivals by Region and Country of Nationality (2005-2014).

Given these datasets, we focused on countries/regions labeled as terrorist safe havens and state sponsors of terror based on the criteria outlined here .

We found the following.

First, looking at naturalized citizens, these computations yielded:

Country

Correlations

Percent of Variance Explained 

Afghanistan

0.61169

0.37416

Egypt

0.26597

0.07074

Indonesia

-0.66011

0.43574

Iran

-0.31944

0.10204

Iraq

0.26692

0.07125

Lebanon

-0.35645

0.12706

Libya

0.59748

0.35698

Malaysia

0.39481

0.15587

Mali

0.20195

0.04079

Pakistan

0.00513

0.00003

Phillipines

-0.79093

0.62557

Somalia

-0.40675

0.16544

Syria

0.62556

0.39132

Yemen

-0.11707

0.01371

In graphical form:

The highest correlations are 0.62556 and 0.61669 from Syria and Afghanistan respectively. The highest anti-correlations were from Indonesia and The Phillipines at -0.66011 and -0.79093 respectively. Certainly, none of the correlations exceed 0.65, which indicates that there could be some relationship between the number of naturalized citizens from these particular countries and the number of terrorist incidents, but, it is nowhere near conclusive. Further, looking at Syria, we see that the percentage of variance explained / coefficient of determination is 0.39132, which means that only about 39% of the variation in the number of terrorist incidents can be predicted from the relationship between where a naturalized citizen is born and the number of terrorist incidents in The United States.

Second, looking at refugees, these computations yielded:

Country

Correlations

Percent of Variance Explained

Afghanistan

0.59836

0.35803

Egypt

0.66657

0.44432

Iran

-0.29401

0.08644

Iraq

0.49295

0.24300

Pakistan

0.60343

0.36413

Somalia

0.14914

0.02224

Syria

0.56384

0.31792

Yemen

-0.35438

0.12558

Other

0.54109

0.29278

In graphical form:

We see that the highest correlations are from Egypt (0.6657), Pakistan (0.60343), and Afghanistan (0.59836). This indicates there is some mild correlation between refugees from these countries and the number of terrorist incidents in The United States, but it is nowhere near conclusive. Further, the coefficients of determination from Egypt and Syria are 0.44432 and 0.31792 respectively. This means that in the case of Syrian refugees for example, only 31.792% of the variation in terrorist incidents in the United States can be predicted from the relationship between a refugee’s country of origin and the number of terrorist incidents in The United States.

In conclusion, it is therefore unlikely that Donald Trump’s proposals would do anything to significantly curb the number of terrorist incidents in The United States. Further, repeatedly showing pictures like this:

at his rallies is doing nothing to address the issue at hand and is perhaps only serving as yet another fear tactic as has become all too common in his campaign thus far.

(Thanks to Hargun Singh Kohli, Honours B.A., LL.B. for the initial data mining and processing of the various datasets listed above.)

Note, further to the results of this article, I was recently made aware of this excellent article from The WSJ, which I have summarized below:

The Effect of Individual State Election Results on The National Election

A short post by me today. I wanted to look at the which states are important in winning the national election. Looking at the last 14 presidential elections, I generated the following correlation plot:

  
For those not familiar with how correlation plots work, the number bar on the right-hand-side of the graph indicates the correlation between a state on the left side with a state at the top, with the last row and column respectively indicating the national presidential election winner. Dark blue circles representing a correlation close to 1, indicate a strong relationship between the two variables, while orange-to-red circles representing a correlation close to -1 indicate a strong anti-correlation between the two variables, while almost white circles indicate no correlation between the two variables.

For example, one can see there is a very strong correlation between who wins Nevada and the winner of the national election. Indeed, Nevada has picked the last 13 of 14 U.S. Presidents. Darker blue circles indicate a strong correlation, while lighter orange-red circles indicate a weak correlation. This also shows the correlation between winning states. For example, from the plot above, candidates who win Alabama have a good chance of winning Mississippi or Wyoming, but virtually no chance of winning California.

This could serve as a potential guide in determining which states are extremely important to win during the election season!

 

Canadian Federal Election Predictions for 10/19/2015

Tomorrow is the date of the Canadian Federal Elections. Here are my predictions for the outcome:

canelecpredictfinal

That is, I predict the Liberals will win, with the NDP trailing very far behind either party.