The probability that democracy works

We are told that a large number of people opting for a certain choice(ex. a candidate in an election) represents near-certainty that this choice meets necessary and sufficient criteria. No particular person needs to understand how a country works if a lot of people who understand different functions of a country cast a vote on the matter…the aggregate decision represents the closest thing to a complete picture. 

Let’s make the following assumptions:
1. The ideal candidate must satisfy  n equally important criteria and we assume that this candidate is present among the existing candidates. 

2. There are  N voters where  N \gg n and each voter has partial knowledge of the  n necessary criteria. In particular, we assume that each voter is aware of at least one criterion and their knowledge of these criteria is given by a uniform distribution.

From the above assumptions it follows that the probability that the correct candidate is chosen is approximately 1-\sum_{k=1}^{n-1} {n \choose k} (\frac{k}{n})^N .  In theory this is good. However, there are some problems with our theory. 

The first problem is that ‘choice’ of criteria is highly correlated within social groups. Second, the definition of equally-important criteria is problematic. Some criteria like the role of government in tech innovation are more complex than others which means that knowledge of criteria is probably given by a gamma distribution rather than a uniform distribution. Finally, the appropriate candidate might not exist among the set of available candidates.

Now, I think the only way for society to move closer to a system where voting works is to change the current education system. Using uniform grading requirements, students are taught to attain knowledge by consensus which merely encourages groupthink. The necessary alternative is to encourage inquiry-driven learning.

This could come in the form of open-ended competitions, like the Harvard Soft Robotics competition that I’m participating in, or working on projects within Fab Labs/Hacker Spaces. In any case, society will have to shift from skill-based employment to innovation-driven employment due to the growing number of tasks that can be automated. People will be paid for their imagination rather than their time and I believe that in 15 years time sheets will all but disappear. 

If we want a democracy that works and not merely the theatrical nonsense that passes for democracy today, radical changes to the education system will be required at all levels. 

Note 1: The probability calculation can be much more complicated depending upon your assumptions.

Note 2: For the reader that’s interested in my opinion on Hacker Spaces, you may read more here

perturbations that preserve constants of motion

cool_fig_3
simulation of three body problem with random initial conditions

 

Within the context of the gravitational n-body problem I’m interested in perturbations of the initial conditions  \{p_i(0),\dot{p}_i(0)\}_{i=1}^n    which leave all the constants of motion unchanged.

It’s clear to me that the linear momentum(P) is preserved under any transformation of the initial position vectors and energy(H) is preserved under any isometry applied to initial position vectors. However, when I add the constraint of conserving angular momentum(L), the general nature of these transformations which would leave (H,L, P) unchanged is not clear.

Granted, time translations would preserve the constants of motion but in a trivial manner. Now, I’m not sure whether the approach I’ve taken can be considerably improved but within the context of the three body problem in the plane, I transformed this problem into an optimisation problem as follows:

  1. We first calculate the constants of motion which are the linear momentum, energy and the angular momentum prior to perturbing the position of one of the n bodies. Let’s suppose that these constants are given by C_1, C_2, C_3 . 
  2. Assuming our bodies are restricted to move in the plane, there are a total of 4n scalars that determine the initial conditions for this n-body system. It follows that the perturbation of one body(q ) generally requires searching for a 4n-2  dimensional vector x  for positions and velocities. This can be done using Nelder-Mead, a derivative-free method, provided that we define a cost function. 
  3. The functions to be minimised are obtained directly from the linear momentum (P), angular momentum (L) and Hamiltonian (H) equations to obtain a single smooth cost function:\begin{aligned} C(x,q) =(P(x,q)-C_1)^2+(H(x,q)-C_2)^2+(L(x,q)-C_3)^2 \end{aligned}
  4. In order to increase the probability of convergence, the Nelder-Mead algorithm was randomly seeded for each perturbed position q .

I think this method can be easily extended to n-bodies.

Pseudo-anonymous forums

After using the stack-exchange(SE) forums for several years, it’s clear that pseudo-anonymous user profiles are synonymous with lousy user experience. Depending on the stack exchange this manifests itself in different ways but I believe the underlying reasons are the same. 

Consider these two questions:
1. Why don’t hexapods gallop?
2. Rigorous derivation of isoperimetric inequality from ideal gas equation?

In the first case you have a pseudo-anonymous user on the Physics SE with a lot of reputation who tries to reformat the question. He basically says that the question has nothing to do with physics. Eventually, I demonstrate that his claim is baseless and other users on the physics stack exchange support my arguments. A different user with less confidence might have responded differently however.

In the second case, we have a clear question on the Math Overflow that gets a clear answer from an identifiable person. Now, if you check the top users on the MathOverflow you’ll realise that almost every user is identifiable. In fact, among the top 20 users the number of identifiable users among these forums stands at 95% and 75% for the MathOverflow and Physics Stack Exchanges respectively. I believe that the fraction of pseudo-anonymous users is an important indicator of the overall forum experience. 

Pseudo-anonymity is not merely a different type of user setting. It leads to fundamentally different forum interactions for systemic reasons:

1. Users are less inhibited because they won’t be identified. 
2. Users feel less pressure to share good content because any social benefits won’t affect their actual person. 

Due to this reward system, pseudo-anonymous users tend to behave like idiots if they can get away with it and they won’t try as hard to share great content. If you’re Terrence Tao on the MathOverflow the situation is very different. In general, I suspect that the reason for the greater fraction of identifiable MathOverflow users is that they are actually proud of what they share. 

While it’s not clear how the Physics Stack Exchange can incentivise users to use their real names I think they can simply require it. I have no doubt that this would improve the user experience.