Our new paper examining the dynamics of an asymmetric HawkDove game was published in The International Journal of Differential Equations:
Category: Politics
The Trump Rally, Really?
Today, The Dow Jones Industrial Average (DJIA) surpassed the 20,000 mark for the first time in history. At the time of the writing of this posting (12:31 PM on January 25), it is actually 20,058.29, so, I am not sure if it will close above 20,000 points, but, nevertheless, a lot of people are crediting this to Trump’s presidency, but I’m not so sure you can do that. First, the point must be made, that it is really the Obama economic policies that set the stage for this. On January 20, 2009, when Obama was sworn in, the Dow closed at 7949.089844 points. On November 8, 2016, when Trump won the election, the Dow closed at 18332.74023. So, during the Obama administration, the Dow increased by approximately 130.63%. I just wanted to make that point.
Now, the question that I wanted to investigate was would the Dow have closed past 20,000 points had Trump not been elected president. That is, assuming that the Obama administration policies and subsequent effects on the Dow were allowed to continue, would the Dow have surpassed 20,000 points.
For this, I looked at the DJIA data from January 20, 2009 (Obama’s first inauguration) to November 08, 2016 (Trump’s election). I specifically calculated the daily returns and discovered that they are approximately normally distributed using a kernel density method:
Importantly, one can calculate that the mean daily returns, , while the volatility in daily returns, . Indeed, the volatility in daily returns for the DJIA was found to be relatively high during this period. Finally, the DJIA closed at 18332.74023 points on election night, November 08, 2016, which was 53 business days ago.
The daily dynamics of the DJIA can be modelled by the following stochastic differential equation:
,
where denotes a Wiener/Brownian motion process. Simulating this on computer, I ran 2,000,000 Monte Carlo simulations to simulate the DJIA closing price 53 business days from November 08, 2016, that is, January 25, 2017. The results of some of these simulations are shown below:
We concluded the following from our simulation. At the end of January 25, 2017, the DJIA was predicted to close at:
That is, the DJIA would be expected to close anywhere between 17398.0923062336 and 20158.94121. This range, albeit wide, is due to the high volatility of the daily returns in the DJIA, but, as you can see, it is perfectly feasible that the DJIA would have surpassed 20,000 points if Trump would not have been elected president.
Further, perhaps what is of more importance is the probability that the DJIA would surpass 20,000 points at any time during this 54day period. We found the following:
One sees that there is an almost 20% (more precisely, 18.53%) probability that the DJIA would close above 20,000 points on January 25, 2017 had Trump not been elected president. Since, by all accounts, the DJIA exceeding 20,000 points is considered to be an extremely rare/historic event, the fact that the probability is found to be almost 20% is actually quite significant, and shows, that it is quite likely that a Trump administration actually has little to do with the DJIA exceeding 20,000 points.
Although, this simulation was just for 53 working days from Nov 08, 2016, one can see that the probability of the DJIA exceeding 20,000 at closing day is monotonically increasing with every passing day. It is therefore quite feasible to conclude that Trump being president actually has little to do with the DJIA exceeding 20,000 points, rather, one can really attribute it to the daytoday volatility of the DJIA!
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.
2016 RealTime Election Predictions
Further to my original post on using physics to predict the outcome of the 2016 US Presidential elections, I have now written a cloudbased app using the powerful Wolfram Cloud to pull the most recent polling data on the web from The HuffPost Pollster, which “tracks thousands of public polls to give you the latest data on elections, political opinions and more”. This app works in realtime and applies my PDEsolver / machine learning based algorithm to predict the probability of a candidate winning a state assuming the election is held tomorrow.
The app can be accessed by clicking the image below: (Note: If you obtain some type of server error, it means Wolfram’s server is busy, a refresh usually works. Also, results are only computed for states for which there exists reliable polling data. )
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:
Some Thoughts on The US GDP
Here are some thoughts on the US GDP based on some data I’ve been looking at recently, mostly motivated by some Donald Trump supporters that have been criticizing President Obama’s record on the GDP and the economy.
First, analyzing the real GDP’s average growth per year, we obtain that (based on a least squares regression analysis)
According to these calculations, President Clinton’s economic policies led to the best average GDP growth rate at $436 Billion / year. President Reagan and President Obama have almost identical average GDP growth rates in the neighbourhood of $320 Billion / year. However, an obvious caveat is that President Obama’s GDP record is still missing two years of data, so I will need to revisit these calculations in two years! Also, it should be noted that, historically, the US GDP has grown at an average of about $184 Billion / year.
The second point I wanted to address is several Trump supporters who keep comparing the average real GDP annual percentage change between President Reagan and President Obama. Although they are citing the averages, they are not mentioning the standard deviations! Computing these we find that:
Looking at these calculations, we find that Presidents Clinton and Obama had the most stable growth in yeartoyear real GDP %. Presidents Bush and Reagan had highly unstable GDP growth, with President Bush’s being far worse than President Reagan’s. Further, Trump supporters and most Republicans seem quick to point out the mean of 3.637% figure associated with President Reagan, but the point is this is +/ 2.55%, which indicates high volatility in the GDP under President Reagan, which has not been the case under President Obama.
Another observation I would like to point out is that very few people have been mentioning the fact that the annual real US GDP % is in fact correlated to that of other countries. Based on data from the World Bank, one can compute the following correlations:
One sees that the correlation between the annual growth % of the US real GDP and Canada is 0.826, while for Estonia and The UK is roughly close to 0.7. Therefore, evidently, any President that claims that his policies will increase the GDP, is not being truthful, since, it is quite likely that these numbers also depend on those for other countries, which, I am not entirely convinced a US President has complete control over!
My final observation is with respect to the quarterly GDP numbers. There are some articles that I have seen in recent days in addition to several television segments in which Trump supporters are continuously citing how better Reagan’s quarterly GDP numbers were compared to Obama’s. We now show that in actuality this is not the case.
The problem is that most of the “analysts” are just looking at the raw data, which on its face value actually doesn’t tell you much, since, as expected, fluctuates. Below, we analyze the quarterly GDP% data during the tenure of both Presidents Reagan and Obama, from 19821988 and 20102016 respectively, comparing data from the same length of time.
For Reagan, we obtain:
The only way to reasonably compare these two data sets is to analyze the rate at which the GDP % has increased in time. Since the data is nonlinear in time, this means we must calculate the derivatives at instants of time / each quarter. We first performed cubic spline interpolation to fit curves to these data sets, which gave extremely good results:
We then numerically computed the derivative of these curves at each quarter and obtained:
The dashed curves in the above plot are plots of the derivatives of each curve at each quarter. In terms of numbers, these were found to be:
Summarizing the table above in graphical format, we obtain:
As can be calculated easily, Obama has higher GDP quarterly growth numbers for 15/26 (57.69%) quarters. Therefore, even looking at the quarterly real GDP numbers, overall, President Obama outperforms President Reagan.
Thanks to Hargun Singh Kohli, B.A. Honours, LL.B. for the data collection and processing part of this analysis.