Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ?

Question

From memories of a quantum mechanics class and Wikipedia:

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. For instance, the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

In practical terms the product of the uncertainty in the position and the uncertainty in the momentum is at least of the order of magnitude of the Planck constant.

A reason why an analogous of this principle could be found in social sciences is that (in my understanding) its origins are purely mathematical. It is a Cauchy-Schwarz-like inequality applied to the scalar product Bra-ket under the dynamic of a particular pde. However, the nanoscopic nature of quantic measurements and the ~10^-34 Planck constant are so small that the uncertainty principle is irrelevant in physics of larger particles, not to mention the other sciences.

A common operation in decision sciences is < belief || outcome > (or < price vector || vector of goods > at a larger scale) and the solution concepts in game theory, for example, allow tremblings as small as possible (epsilon > 0) with fixed outcomes! Therefore, Is there (yet) an equivalent of Heisenberg's uncertainty principle in the game theory, decision theory etc. ?

Answer 1

Let me answer in terms of operator theory. The uncertainty principle can be interpreted as some particular inequality (as you say) such as $$ \frac{\hbar}{2}\Vert{u}\Vert^2\le \Vert{D_xu}\Vert\Vert{xu}\Vert, $$ inequality due to the identity $ 2\Re\langle\hbar\frac{1}{i}\partial_x u, ixu\rangle=\hbar\Vert{u}\Vert^2, $ and in fact to the non-commutation of the unbounded operators $D_x=\frac{\hbar}{i}\partial_x$ with $x$ (multiplication by $x$). Incidentally you can note that the non-commutation of bounded operators $A,B$ (say finite matrices) could never lead to $$ [A,B]=I\tag{NC} $$ as it is the case for $A=\frac{1}{i}\partial_x$, $B=ix$. In fact (NC) is not possible for matrices since the trace of a commutator is 0. This is one reason for which the quantization requires (necessarily infinite dimensional) unbounded operators.

Now this non-commutation property is at the source of quantum mechanics and produces the uncertainty principle inequalities. Each time you will have to deal with a non-commutative algebra of operators, you will have some type of uncertainty principle. In particular if your unknown quantities do not belong to a commutative algebra, you will have an uncertainty principle: the paradigmatic example remains the transfer from classical mechanics where the unknown quantities belong to a commutative algebra (functions on the phase space, such as position $x$ or momentum $\xi$), transfer to quantum mechanics where the unknown quantities are quantization of the previous quantities and are operators, such as $D_x,x$.

The uncertainty principle is simply the signature of non-commutativity and is not limited to quantum mechanics.

Answer 2

I believe there might be something similar to Heisenberg Uncertainty Principle in every decision model, particularly multicriteria decision making models.

If we consider a decision model as a set of criteria that rank a few alternatives differently according to some preference agregation function (that agregates criteria weights, for example additively in a linear model) an analogue to Heisenberg's principle precision idea is the discriminative power of the model.

If you want to refine your appraisal of the alternatives you will consider a few more criteria you had not thought of before so that your model is as detailed and complete as possible, and considers some new possibly forgotten value measures. By doing this you increase the discriminative power on the "Criteria" side, but then you lose it on the "Alternatives" side. From the "Alternatives" side the ideal is to have just one criteria, according to which it is easy to see how they all rank against each other. If you start to have more criteria some alternatives will be best according to some criteria and some will be best according to another, so it is not so easy anymore to rank them. You then have to decide on criteria weights - which criteria are more important, and aggregate. With many criteria ultimately every alternative is best according to some of them and besides the subjectiveness of criteria weights determination that induces misjudgment, in some models these alternatives are considered equally competitive. For example, in DEA (Data Envelopment Analysis) criteria weights make no diference in this last cenario - if one alternative (or decision making unit) is best according to one criteria (input or output) it can’t be ranked below the others.

It seems that by enlarging a decision making model with more criteria you increase discriminative power on one hand but lose it on the other. As if there was a minimum of "Uncertainty" attached to your decision making model, which can't discriminate beyond a certain limit. But this is just an idea. I can't quantify it.