Our new paper examining the dynamics of an asymmetric HawkDove game was published in The International Journal of Differential Equations:
Tag: Politics
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 (%) 
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 nonparametric regression shows:
We therefore chose to perform a nonlinear regression using neural networks, for which our structure was:
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: , where denotes predominantly “attacking” the other candidate, while denotes predominantly discussing policy positions.
Further, let us consider the mixed strategies for Clinton, and for Trump. That is, Clinton predominantly attacks Trump with probability , and Trump predominantly attacks Clinton with probability .
Let us first deal with the general case of arbitrary payoffs, thus, generating the following payoff matrix:
That is, if Clinton attacks Trump and Trump attacks Clinton, the payoff to Clinton is , while the payoff to Trump is . If Clinton attacks Trump, and Trump ignores and discusses policy positions instead, the payoff to Clinton is , while the payoff to trump is . If Clinton discusses policy positions while Trump attacks, the payoff to Clinton is , while the payoff to Trump is , and if both candidates discuss policy positions instead of attacking each other, the payoff to them both will be and respectively.
With this information in hand, we can calculate the payoff to Clinton as:
while the payoff to Trump is:
With these payoff functions, we can compute each candidate’s best response to the other candidate by solving the following equations:
where indicates the best response strategy to a fixed strategy for the other player.
Solving these equations, we obtain the following:
If
then,
Clinton’s best response is to choose .
If
then,
Clinton’s best response is to choose .
Otherwise, her best response is to choose .
While for Trump, the best responses are computed as follows:
If
Trump’s best response is to choose .
If
Trump’s best response is to choose .
Otherwise, Trump’s best response is to choose .
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:
Using the above equations, we see that if , Clinton’s best response is to choose . While, if , Trump’s best response is to choose . 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:
What this shows for example is that there is a Nash equilibrium of:
.
The expected payoffs for each candidate are evidently
.
Let us work out an another example. This time, assume that if Clinton attacks Trump, she receives a payoff of , while if Trump attacks Clinton, he receives a payoff of . While, if Clinton discusses policy, while being attacked by Trump, she receives a payoff of , while Trump receives a payoff of . On the other hand, if Trump discusses policy while being attacked by Clinton, he receives a payoff , while Clinton receives a payoff of . If Clinton discusses policy, while Trump discusses policy, she receives a payoff of , while Trump receives a payoff of . The payoff matrix is evidently:
In this case, if , then Clinton’s best response is to choose . While, if , then Trump’s best response is to choose . The Nash equilibrium is evidently
.
The expected payoffs for each candidate are evidently
.
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:
Solving the equations above, we find that if , Clinton’s best response is to choose . If , Clinton’s best response is to choose . Otherwise, her best response is to choose . On the other hand, if , Trump’s best response is to choose . While, if , Trump’s best response is to choose . Evidently, there is a single Nash equilibrium (as long as ):
.
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 20052014. Specifically, we analyzed:
 The number of terrorist incidents per year from 20052014 from here (The Global Terrorism Database maintained by The University of Maryland)
 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 (20052014) and Refugee Arrivals by Region and Country of Nationality (20052014).
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 anticorrelations 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:
Protected: Breaking Down The Data of Victims of Police Violence
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 righthandside 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 orangetored circles representing a correlation close to 1 indicate a strong anticorrelation 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 orangered 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!