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.

Basketball Paper Update

Everyone by now knows about this paper I wrote a few months ago: http://arxiv.org/abs/1604.05266

Using data science / machine learning methodologies, it basically showed that the most important factors in characterizing a team’s playoff eligibility are the opponent field goal percentage and the opponent points per game. This seems to suggest that defensive factors as opposed to offensive factors are the most important characteristics shared among NBA playoff teams. It was also shown that championship teams must be able to have very strong defensive characteristics, in particular, strong perimeter defense characteristics in combination with an effective half-court offense that generates high-percentage two-point shots. A key part of this offensive strategy must also be the ability to draw fouls. 

Some people have commented that despite this, teams who frequently attempt three point shots still can be considered to have an efficient offense as doing so leads to better rebounding, floor spacing, and higher percentage shots. We show below that this is not true. Looking at the last 16 years of all NBA teams (using the same data we used in the paper), we performed a correlation analysis of an individual NBA team’s 3-point attempts per game and other relevant variables, and discovered: 


One sees that there is very little correlation between a team’s 3-point attempts per game and 2-point percentage, free throws, free throw attempts, and offensive rebounds. In fact, at best, there is a somewhat “medium” anti-correlation between 3-point attempts per game and a team’s 2-point attempts per game. 

The Mathematics of The Triangle Offense, Continued…

In a previous post, I showed how given random positions of 5 players on the court that they could “fill” the triangle. The main geometric constraint is that 5 players can form 3 triangles on the court, and that due to spacing requirements, these triangles are “optimal” if they are equilateral triangles.

Given that we now know how to fill the triangle, the question that this post tries to address is that how can players actually move within the triangle. The key is symmetry. Players must all move in a way such that the equilateral triangles remain invariant. Equilateral triangles have associated with them the D_{3} dihedral symmetry group. They are therefore invariant with respect to 120 degree rotations, 240 degree rotations, 0 degree rotations, and three reflections.

There are therefore six generators of this group:
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right), \left( \begin{array}{cc} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right),\left( \begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right), \left( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right),\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right),\left( \begin{array}{cc} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \end{array} \right).

In fact, the Cayley graph for this group is as follows:

cayley1

For now, I will discuss how players can move within the action of 120 degree rotations. As in the previous posting, let the (x,y)-coordinates of player i be represented by (x^{i}, y^{i}), where i = 1,2,3,4,5. Then, under a 120 degree rotation, the player’s coordinates get shifted according to:

\boxed{x^{i}_{t+1} = \frac{1}{2} \left(-x^{i}_{t} - \sqrt{3}y^{i}_{t}\right), \quad y^{i}_{t+1} = \frac{1}{2}\left(\sqrt{3}x^{i}_{t} - y^{i}_{t}\right)}

This is a discrete dynamical system. In fact, it can be solved explicitly. Let x^i_{0}, y^{i}_{0} represent the initial coordinates of player i. Then, one solves the above discrete system to obtain:

\boxed{x^i_t =\frac{1}{2} e^{\frac{1}{3} (-2) i \pi t} \left[\left(1+e^{\frac{4 i \pi t}{3}}\right) x^i_0+i \left(-1+e^{\frac{4 i \pi t}{3}}\right) y^i_0\right], \quad y^{i}_{t} =\frac{1}{2} e^{\frac{1}{3} (-2) i \pi t} \left[\left(1+e^{\frac{4 i \pi t}{3}}\right) y^i_0-i \left(-1+e^{\frac{4 i \pi t}{3}}\right) x^i_0\right]}

Now, we can simulate this to see actually how players move within the triangle offense, forming equilateral triangles in every sequence:

20160911_124208

This is running in continuous time, that is, endlessly. In future postings, I will update this to include the other symmetries of the dihedral D_{3} group. However, the challenge is that this symmetry group is non-Abelian, so it will be interesting to implement pairs of consecutive symmetry operations in a simulation that would still result in invariant equilateral triangles.

Hopefully, this post also shows why teams cannot really run “parts” of the triangle, as one player’s movement necessarily effects everyone else’s. This is something that Charley Rosen also mentioned in an article of his own.  

The Possible Initial States of The Universe

Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.

What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter \gamma . In particular, some examples are:

  • \gamma = 0: Vacuum energy
  • \gamma = 4/3: Radiation
  • \gamma = 2: Stiff Fluid

Note: Click the image below to access the simulation!

In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:

  1. dS: de Sitter universe – Inflationary epoch
  2. M: Milne universe
  3. F: spatially flat FLRW universe – our present-day universe
  4. E: Einstein static universe

Note how by changing the value of \gamma , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of \gamma is F a saddle or stable node.

2016 Real-Time 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 cloud-based 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 real-time and applies my PDE-solver / 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 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:

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 year-to-year 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 1982-1988 and 2010-2016 respectively, comparing data from the same length of time. 

For Reagan, we obtain: 


For Obama, 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.