Nash Equilibriums III

 

All about the Bass

Because you know I'm all about that bass
'Bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass, hey

I beg to differ. It’s all about the Math!

I know that Math is Hard, but bear with me. Math may be the explanation, and the solution, to our current problems. This might sound strange but what mathematicians would call infinity, a scientist might call an absolute, an ethicist might call Truth, and an Evangelical would call God, But all can be analyzed/approached the same way.

Let’s suppose the existence of infinity/absolute/Truth/God. The absence of these things would be death, flat-lined. A flat-line has no Amplitude, no variance. The presence of these things must therefore have an Amplitude according to the formula ½A²=σ², where A is the amplitude and σ² is the variance. Infinity/absolute/truth/God as a wave, the opposite of a straight line, must be ...doh...infinite. A wave is an infinitely repeating function of π, which is merely saying that while a wave is infinite, there might be no difference between its behavior at π, 2π, 3π, …nπ, …∞π. If there is an absolute that repeats as a function of π, it must have a repeating median/mean, μ, of π /2.

Individuals can act like a wave. (Don’t believe me! Have you never seen a crowd at a stadium doing the wave!). So how do you get many individuals to perform like a wave. A normal (The mathematical name for it. Not a moral judgement!) distribution of individuals is the logistics, hyperbolic secant squared,  distribution, 1/4s*sech²((x-μ)/2s). S is a function of the variance, according to the formula σ²=s²π²/3. That function is also known as the Probability Distribution Function, PDF.

While ordinary wave functions repeat with a period related to π,  hyperbolic wave functions, such as sech, repeat with a period related to πi, where i is the imaginary number, i=√-1. ( I know math is hard, but the kids in Algebra know this!). Also for functions of x, f(x), there is a derivative of that function, f’(x), which is the slope of that function, and an integral of that function F(x)=∫f(x)δx, which is the area under that function. The derivative of the logistics distribution is
-1/8s²*sech²((x-μ)/(2s))*tanh((x-μ)/(2s)). The integral of the logistics function, which also goes by the name Cumulative Distribution Function, CDF, is 1/2*tanh((x-μ)/(2s))+1/2. ( again Math is Hard. Feel free to check with the kids taking AP Math!)

All of this to say that a wave function, its derivative, and its integral can be defined for any value of x given only two parameters: the median, μ,  which is akin to the phase of the wave, and the range, s, which defines the amplitude of the wave. There are three behaviors that are of interest. The first is User Optimal, UO,  defining only the median, as μ=0 and allowing any value for s. This can be defined as “Only I matter,” “Second place is first loser”, “Winning is the only thing”, “All’s fair” , etc. The second behavior is System Optimal, SO where you accept infinity/absolute/Truth/God/π and thus the median must be π/2, but you say that  you always are 100% certain. The problem is that since you are only an individual, according to the logistics distribution you must also be 40% certain at either 1/3π or 2/3π, and 6% certain at either 1/6π or 5/6π, etc. This means that you then have to be much more than 100% certain if you add all of these together. While being more than 100% certain sounds great, it is mathematically impossible. In fact the integral, CDF, of the logistic function when μ=π/2 and s=.25, which is consistent with being 100% certain at μ, starts at 50% certain at x=0, while the certainty should not be 50% until  x=π/2.

This is like the scene at the door to the back room in Casablanca. UO behavior is asking ”Do you know who I am?”  SO behavior is throwing all UOs out of the Club. The third behavior, the Nash Equilibrium (for reasons that would make your head hurt) is saying that μ=π/2 and s= 0.51451 and this is like saying “I know who you are. You’re lucky your cash is good at the bar.”     https://www.youtube.com/watch?v=aALbiGJpw7c.  An ideal single wave would be s=.55139 which is consistent with a value of s=.5 (twice the SO value) on a hyperbolic non-Euclidean surface because of the use of the hyperbolic secant, from a perfect multiple wave. The derivative of this function and its integral under all of these behaviors are shown below. 

Figure 1 Derivative, dPDF, of logistic function

Figure 2 Certainty, PDF, of logistic function 

Figure 3 Integral, CDF, of logistics function

Those following a User Optimal, UO,  strategy will  try to get those following a System Optimal,  SO, strategy to join them by saying that "I will act like you are correct ( the quiet part NOT out loud being especially since as a UO, I can accept any s) if you act like my median of μ=0 is correct". Instead if the SO wants to be like perfection, then it should adopt a Nash Equilibrium, NE, strategy. The best strategy is not to win, UO, and not the common good, SO, but to quote mathematician John Nash from Ron Howard’s Oscar winning A  Beautiful Mind, is  “to win for the common good,” a Nash Equilibrium. If you are a SO plan to join with NEs, NOT with UOs, if you want to be perfect.

Rebound

 

Red Rubber Ball

And I think it's gonna be all right
Yeah, the worst is over now
The mornin' sun is shinin' like a red rubber ball

But how does that red rubber ball bounce?

A bounce, rebound, occurs when an object, such as a particle, encounters a discontinuity. That discontinuity can be a physical surface, or it can be merely observational, that is the ability to observe, and measure, may be the actual reason that there appears to be a  discontinuity..

If a particle is moving, and is not acted upon by a force, that particle moves in a straight line. That is Newton’s Law of Inertia. However this is only true if space is flat. It is more proper to say that a particle moves along the geodesic in its space. If the space is flat, then the geodesic is a straight line. But if that space is not flat, for instance is spherical or hyperbolic, then it is non-Euclidean, and only flat space is Euclidean.

We say that the Earth is a sphere, and we live on the Earth’s spherical surface. That is why the shortest distance between two points on earth is more properly a Great Circle Distance. While this is true, it might be only of interest to airplane pilots and others who measure vast distances. When the distances involved are far less than the radius of the Earth, then the solution for the hypotenuse of a triangle on that spherical surface is cos(c/R)=cos(a/R)*cos(b/R), where R is the radius of the Earth/spherical surface, and it is virtually identical to the solution on a classical flat Euclidean surface, cos(c)=cos(a)*cos(b), as can be verified by using the series for the trigonometric functions. Both of these are equal to Pythagoras’ Theorem, c=√(a²+b²). It is therefore customary to say that the distances on Earth  are spherical globally but are flat locally. Might this also be true for space?

The solution for the hypotenuse of a triangle on a hyperbolic surface uses hyperbolic trigonometric functions, cosh(c)=cosh(a)*cosh(b). This has a different solution than the classical solution. The classical solution relies on the circular identity, cos²+sin²=1.  In hyperbolic space the identity cosh²-sinh²=1 applies. Additionally space may not be merely what can be observed, it might be that which can not be observed, i. In this case reality having a coefficient of zero for that which can not be observed can be expressed as a complex number which is reality plus zero imagination, r+0*i. If reality is the solution of a triangle r²=(a²+b²)+0^2*i, then its solution in hyperbolic space is
ln(cosh(√(a²+b²)) ± sinh(√(a²+b²))) because cosh(0²) is 1,  where the ± indicates that there are two solutions. Because cosh is symmetrical while sin is symmetrical, and for small values of a²+b² compared to the size of the universe, sinh(√(a²+b²)), the uncertainty, is also small. This can be also expressed as a single solution, ln(cosh(√(a²+b²))-sinh(√(a²+b²))), if reality is one solution. This is no different than electrical engineering where some solutions have real and imaginary components, and the imaginary component is ignored. The single solution merely says that the real solution has the opposite sign of the imaginary solution, and that the imaginary solution is being ignored.

A rebound in flat space is symmetrical because a straight line is symmetrical about that discontinuity. Hyperbolic motions are NOT symmetrical as real numbers. They are almost linear on one side of the discontinuity and almost parabolic on the other side of the discontinuity. If there is a linear motion on one side of a discontinuity and that discontinuity is NOT a surface and the motion is parabolic on the other side of the discontinuity, this probably is an indication that the motion is hyperbolic, NOT a highly skewed parabola. A parabola requires an imaginary solution if that motion passes through, is rotated by 180º or π. It is suggested that is more reasonable to assume that this discontinuity is because the observable behavior continues as unobservable behavior than it is to assume the behavior has become imaginary.

 

 

Engineers III

 

This is Me

When the sharpest words wanna cut me down
I'm gonna send a flood, gonna drown 'em out
I am brave, I am bruised
I am who I'm meant to be, this is me
Look out 'cause here I come
And I'm marching on to the beat I drum
I'm not scared to be seen
I make no apologies, this is me

And me (sic) IS an engineer!

An engineer has been defined as  someone who is  good at math and socially awkward.  I admit to being socially awkward and I am a Professional Engineer.  As to the good at math, here goes my feeble attempt.

The logistic distribution, also known as the hyperbolic secant squared distribution, is a normal distribution. Its Probability Density Functions, PDF, is f(x)=

1/(4s)*sech²((x-μ)/(2s))

and its Cumulative Distribution Function, CDF, which is the integral of f(x),  ∫f(x), is

½ tanh((x-μ)/(2s)) +½.

The derivative, f’(x), of the PDF is

-1/(8*s²)*sech²((x-μ)/(2s))*tanh((x-μ)/(2s)) = (-1/s)*PDF*(CDF-½).

Each of these are wave functions in hyperbolic space. They each have the same period of πi.  The hyperbolic tangent, tanh, also has a period of πi.  Each wave function has the same phase, μ/2s. For the Amplitude of each of these three waves to be the same, s must be equal to ½, in which case the phase for each wave would be just μ.

The PDF can be considered to be equivalent to momentum in classical Newtonian systems, a spring in a mechanical systems, a capacitor in electrical systems, etc. The derivative of the PDF can be considered to be distance in Newtonian and mechanical systems,  a resistor in electrical systems, etc. The CDF can be considered to energy in Newtonian systems, a dashpot in mechanical systems,  an inductor in electrical systems, etc. Since energy and mass are convertible according to Einstein’s Equation, E=mc², this also has implications for mass via relativity.

If s=½ is taken to be one volume divided into two sheets, then it could be on a two-sheeted hyperboloid. If space is then hyperbolic, not flat, then two Minkowski light cones intersecting  at an origin, could instead be considered not to be light travelling on a flat Euclidean surface where the geodesic is a straight line, but light traveling on a hyperbolic surface, where the geodesic is hyperbolic and therefore non-Euclidean. If, as suggested by Mabkhout , the universe is hyperbolic, it may be just one (observable) sheet of that hyperboloid. For a function to span both the observable and unobservable sheets there must be a transition/discontinuity between the two sheets.

Euler’s Formula is  e^ix=cos(x)+sin(x)*i. This can be viewed as a special case of a transformation of a complex number from cylindrical polar coordinates to Cartesian coordinates, r*e^ix=r*cos(x)+r*sin(x)*i, where there are three dimensions, the  dimension of space and dimension of time, reality r, where r²=(r*cos(x))²+(r*sin(x))²,  and an imaginary dimension, i , when r=1, and x is the angle of rotation of the imaginary axis. If reality has a coefficient of 0 for the imaginary dimension/axis, then both sin(0)=0 and sin(π)=0, but cos(0)=1 while cos(π)=-1. This means that if reality has a coefficient of the imaginary axis of zero, then there are two sheets forming that surface/plane; one sheet which has the opposite sign of the other.

A transition/discontinuity is observed in many applications. At a discontinuity, a particle can rebound from that discontinuity and still remain in the same space/sheet. However if a particle passes through that discontinuity, then it must be transformed, and unobservable from the original space/sheet. It is suggested that for many applications, such as fluid in a channel or pipe, or traffic on a road, a transition occurs at a discontinuity from laminar, uncongested to turbulent, congested conditions. This is most probably the consequence of remaining in the same space and infers the existence of an unobservable sheet to which a transition will occur.

If the discontinuity is physical, then the path after the discontinuity is a rotation by π/2, 90º.  This means that that a path passing through a discontinuity should then be two rotations by π/2, in other words,  a rotation by π or 180º. If a path appears to behave like it is encountering a discontinuity in the absence of a physical discontinuity, it is proposed that this is an observational discontinuity. What is not being observed could in fact pass through the observational discontinuity, as opposed to a physical discontinuity which will prevent passage.

 

 

Leaders II

 

Following the Leader

Following the leader, the leader, the leader
We're following the leader
Wherever he may go

What does it take to be a leader?

A leader is NOT someone who only tells his followers what they want to hear. “I have nothing to offer you but blood, toil, sweat and tears” and  “The only thing we have to fear is fear itself” may not sound like sound like inspiring speeches to all of their followers, but who can doubt that Winston Churchill and FDR were great leaders. Leaders tell the real truth, even if that truth is hard.

Leaders may be great orators, but they say it with few inspiring words rather than lengthy speeches. Abraham Lincoln’s most famous speech, the Gettysburg Address, was less than 3 minutes long.

A great leader is not the most powerful person. He does not have to be a great and powerful wizard, only a good man. https://www.youtube.com/watch?v=-RQxD4Ff7dY.

Choose wisely when you choose a leader. Remember “All that glitters is not gold.” Choose the steak, not the sizzle.

Election 2024 II

 

Here Comes the Judge

Yeah, life! You son-of-a-gun you
Come November, election time
You vote your way, I'll vote mine
'Cause there's a tie, and the money gets spent

So how are you voting this November?

An election is about policy AND trust. A voter may disagree on policy, but pick the candidate who is more trustworthy. This is because while the election is between two political parties. and we are in a two party system because of Duverger’s Law, a contest, such as an election, should have three not just two outcomes. Those outcomes are win, loss and tie, not just win and loss. But if we only have only two candidates, how can we make choices and predictions? The answer is in Games Theory. The outcomes are actually:

1)     Policy 1/Trust;

2)     Policy 1/Don’t Trust;

3)     Policy 2/Trust; AND

4)     Policy 2/Don’t Trust.

Then there are four choices. A tie can then be replaced by 2) Policy 1/Don’t Trust AND 4) Policy 2/Don’t Trust. Then a voter can choose among three outcomes. Let’s say that Policy 1 is Democratic and Policy 2 is Republican. Let’s also say that Democratic policies are favored by 45% and Republican policies are favored by 55%. If only policies are considered in elections then the outcome is clear and the Republican candidate would win.. But let’s also say that the Democratic candidate is trusted by 60% and the Republican candidate is trusted by 40%. On this basis the Democratic candidate would win. But Democratic voters  will probably vote for the Democratic candidate regardless of trust, and Republican voters will probably vote for the Republican candidate regardless of trust. The election will be decided by the swing, independent, unaffiliated voters that should be 1/3 of the electorate. They will pick and choose on policy AND trust. A scientist might say that a win is a true positive, a loss is a true negative and a tie is either a false positive or a false negative.

Let’s say that the Democratic candidate who is trusted is Kamala Harris. Let’s say that a Democratic candidate who is not trusted is Krysten Sinema. Let’s say that a Republican candidate who is trusted is Mike Pence (I am tempted to say Adam Kinzinger, but he endorsed Harris or Nikki Haley, but she endorsed Trump). Let’s the Republican candidate who is Not Trusted is Donald Trump. But only Harris and Trump are on the ballot.

Independents should equally weight trust AND policy. The cross product of trust and policy for Kamala Harris is (45% * 60%= 27%). The cross product for Donald Trump is (55% * 40%=22%). Assuming that Republican voters are 1/3 of the electorate and Democratic Voters are 1/3 of the electorate, the win among the unaffiliated voters will make Harris the preferred candidate. The closest analog in my lifetime is the election of LBJ vs Goldwater. Goldwater did not lose based on swing voters  favoring LBJ’s policies, but because those voters did not trust Goldwater. History may not repeat itself but it sure does rhyme.

Which is why Harris should NOT campaign on policy. As she is doing, she should ignore policy and campaign on trust.  History for $100? Who will win this election? IMHO, Harris if she continues to campaign on trust.

 

Laffer Curve

                                                       

Everybody’s Got A Laughing Place

Everybody's got a laughin' place,
A laughin' place, to go ho-ho!
Take a frown, turn it upside-down
And you'll find yours I know ho-ho!

But nobody needs a Laffer Curve!

The Laffer Curve attempts to explain the revenue from an income tax with the rate of that income tax. It does so by assuming that this curve can be explained by a parabola. IMHO that is why there is a problem. It assumes that the sweet spot, maximum, is where the first derivative of the parabola is zero. IMHO, the curve should NOT be a single parabola but instead should be two intersecting hyperbolas. That is why there is a problem. A parabola requires that for incomes below a zero tax rate, the tax revenue be negative, and the income be imaginary, and at tax rates more than 100% the revenue be negative, and the income be imaginary.  A hyperbola approaches zero revenue, but never becomes zero or negative, and thus the  income never becomes imaginary.

A single parabola can look like two intersecting hyperbolas, but these looks can be deceiving. This behavior is better explained as the lower portion of two intersecting hyperbolas. These hyperbolas are in fact inverses of each other and the point of intersection is also defined. The tax rate should be 16.7% (1/6) of total income. This is the median tax rate, NOT the highest tax rate. If the median is 16.7%, and the lowest effective tax rate is 0%, then the highest tax rate in a normal distribution should be 33.4%. This is the effective rate, not the marginal rate. The effective tax rate is like speed; the marginal tax rate is like acceleration. They are NOT the same and should not be confused with each other. The marginal rate in tax tables depends on the number of equal tax brackets. The marginal tax rate is the percentage of the maximum tax rate which applies.  The effective tax rate is the marginal rate multiplied by that maximum rate. The tax rates should change from the lowest effective rate of 0% to the highest effective tax rate of 33.4% and the lowest income in the highest bracket should be close to the normal  maximum income. If the median household income is $74,580, in 2022 US dollars as reported by the US Census, then the normal maximum income should be twice that, $149,160 in 2023 USD income. According to the U.S. Census Bureau, the mean household income in the United States in 2022 was $105,555. If income was normally distributed, and the median is 50% then the maximum should be twice the median in a normal distribution . To still be normal if skewed, the highest income should not exceed $233,063, 2/ln(2) or ~π Times the Median Income.. The mean income should ideally be half of the highest income. The highest income tax bracket for married filing jointly in the 2023‑24 IRS tax code begins at  $693,750 and there are 7 brackets. The tax brackets using this, as well as the other highest incomes, and the 7 tax brackets are shown in the table below.

The tax brackets are only a linear approximation of what is a non-linear function.  The Census mean income occurs in the 3rd of the 7 brackets in the current IRS tables, but these tables have unequal income brackets. If the income brackets are equalized, but the highest Tax Code income is retained, then the Census mean income occurs in the 2^nd of the 7 brackets . If the highest income is π times the Census median income, then the Census mean income occurs as it should in the 4^th, middle, of the 7 income brackets.

This sounds counter-intuitive, but it is mathematically consistent. Most of the tax revenue comes from the lowest tax brackets. This is because while the rate per taxpayer is low, there are many more taxpayers in these brackets, so that the total revenue from these brackets is very high. The fixation on the rates paid by the few taxpayers in the highest brackets instead of the revenue from the majority of taxpayers in the lowest brackets has been distorting public policy. By confusing marginal (second derivative) and effective (first derivative) tax rates, by having unequal tax brackets, and by using the Laffer Curve, this is inconsistent with other observations, for example observations of traffic in Florida. Those observations of traffic also suggest that the majority of observations occur before the highest volume. Those observations do NOT support a regular ( such as the one proposed by Greenshields) or irregular (highly sewed such as the one proposed by Van Aerde) parabola, but instead intersecting hyperbolas where the intersection  occurs at 1/6^th of the variance, range of congestion ( similar to tax rates in the Laffer Curve.)

Those who don’t learn from history are doomed to repeat it. Those who don’t know math are doomed to be conned.

References

Greenshields, B. (1935). A study of traffic capacity. Highway Research Record, (pp. 448-477). Washington, DC.

Van Aerde, M. (1995). A single regime speed-flow-density relationship for freeways and arterials. Washington D. C.,: Presented at the 74th TRB Annual Meeting,.

 

 

Kamala Harris II

 

Joshua fit the Battle of Jericho.

Know you've heard about Joshua
He was the son of Nun
He never stopped his work until
Until the work was done

God knows that
Joshua fit the battle of Jericho
Jericho, Jericho
Joshua fit the battle of Jericho
And the walls come tumbling down

Joshua meet Kamala.

Kamala Harris, the modern Joshua to Joe Biden’s Moses, tonight accepts the torch as the Presidential nominee of the Democratic Party.  This November, in addition to a glass ceiling shattering, expect the walls to come tumbling down. MAGA better get the name right.  When she fights, she wins!  Not going back!