Feedback and cycles

Feedback is an important concept in engineering. Negative feedback ensures stability as any deviation from the the desired state is met with a force in the opposite direction of said deviation.  To ensure convergence to an equilibrium, this force must diminish in strength as the deviation shrinks.  On the other hand, positive feedback implies that any deviation is amplified with additional changes in the same direction.  This can cause instability and can lead to chaos. Almost all engineering systems with some dynamical behavior use negative feedback to ensure proper operation. This is very different and almost contradictory to how the terms "negative feedback" and "positive feedback" are used in normal conversation.  In everyday language, "positive feedback" is used in the sense of "constructive feedback", i.e. encouragement whereas "negative feedback" is "destructive feedback".  Another set of terms used in everyday language is "virtuous cycle" and "vicious cycle".  Both are examples of positive feedback (the engineering kind), with the difference between virtuous cycle and vicious cycle being the direction of the deviation.  In a vicious cycle the ever-increasing effect is driven by deviation which is detrimental, whereas in a virtuous cycle the deviation is advantageous.

This is illustrated in the following diagram.  Constructive feedback occupies the top half of the diagram, whereas destructive feedback occupies the bottom half.  A curve from the lower left quadrant to the upper right quadrant can be thought of as a (memoryless) positive feedback algorithm. A curve from the upper left quadrant to the lower right quadrant is a negative feedback algorithm.  A virtuous cycle lives in the upper right quadrant and a vicious cycle lives in the lower left quadrant.

The Monty Hall problem

The Monty Hall problem is one of my favorite problems whose solution is counter-intuitive.  Suppose you are being shown 3 closed boxes by Monty Hall, only one of which contains a prize and the other 2 are empty.  You get to choose a box, and Monty, who knows which box contains the prize, opens an empty box that you did not choose.  He then asks you if you want to change your mind.  Intuitively, one would think that it doesn't make a difference if you switch or not.  But it turns out your odds are doubled if you change your mind.  The reasoning is as follows.  If you have initially picked a box with the prize in it, then not changing your mind will result in you getting the prize.  You have a 1 in 3 chance of initially picking the winning box.  If you have initially picked an empty box, switching will result in you getting the prize, and you have a 2 in 3 chance of initially picking an empty box.  So the odd of winning by switching is double the odd of winning by not switching.