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Positive feedback

A positive feedback is a self-supporting or self-accentuating process. Something gets more because it is already plentiful, or it gets less because it is little. The rich get richer, the poor get poorer.

Positive feedback hence enhances the dynamic of a development (in the development‘s own direction). Its general scheme is as follows

A produces more of B, which in turn produces more of A

positive feedback

A simple example for a positive-feedback driven system is the Pólya-urn, named after the Hungarian mathematician György Pólya. Imagine an urn, containing a white and red ball, from which you can blindly draw one of the balls in each round of a game. After each draw the ball you chose is complemented by an other ball of the same color and both balls are placed back into the urn. Before the first draw the probability to select a white ball from the set of two balls is 0.5. If this draw really yields a white ball however, the probability grows to 0.66, since there is now two white and just one red balls in the urn. If the next draw again yields a white ball, the probability to draw white again grows further to 0.75, making the selection of the red ball already pretty unlikely. After nine choices the odds to draw white again are 0.9 to 0.1. Red doesn't have much of a chance anymore to be drawn. The initially small bias towards white has feedback-driven grown into a steady disequilibrium.

Lock-Ins

In 1989 the economist William Brian Arthur investigated the then ongoing competition between two Video-Cassette-Recording technologies for market shares. For analyzing he suggested a model in which consumers are envisaged as having a pre-given (may be irrational) inclination towards one of the technologies. Inclinations are symmetrically distributed. Customers randomly enter the market and decide for one of the two technologies. Each decision however, increases the weight of the chosen technology. Subsequent customers orientate their decisions on this weight. The more customers decide for one of the two technologies the more the corresponding weight increases. If the difference between choices exceeds a certain amount, the network effect of the preferred technology starts to outweigh the inclinations towards the other technology. Customers start to decide against their initial inclination and orientate primarily on what the majority of other customers is doing. This increases the weight even further and generates a lock-in-effect from which no way out seems possible anymore. The loosing technology might even be much better, but the feedback-driven market decided against it.

Arthur, W. Brian (1989): Competing Technologies, Increasing Returns, and Lock-In by Historical Events; in: Economic Journal 99, pp 116-131.

Swarms

Another illustrative example for positive feedback in nature is the emergence of grasshopper swarms. Grass hoppers such as the Anacridium aegyptium interact by way of producing the hormone serotonin as a signal for food availability. Increased tactile stimulation of their hind legs causes an increase in serotonin production, prompting the locust to change color, eat more, and breed more easily. Usually grasshoppers live rather solitary, with solitary and gregarious phases previously been thought to be separate species. However, when serotonin diffusion entices other grasshoppers to assemble and, in response to finding food,to produce serotonin in their own turn, the emerging serotonin concentration can reach levels that transforms the locust into its swarming variety. As a consequence large swarms emerge, covering hundreds of square kilometers and consisting of millions of locust.

causal diagramm

Systems scientifically this is an interesting case of so called downward causation. The increased density of interacting grasshoppers causes the emergence of swarms and this, additionally to being an effect of positive feedback in itself, feeds back on the morphology of the animals, generating further reinforcement in terms of increasing reproduction rates. The micro-level interaction (interaction of individual grasshoppers) causes a macro-level phenomenon (the locust swarm) and this feeds back on the micro-level (the morphology of the individual grasshopper).

Negative feedback

Opposite to positive feedback, which boosts dynamics, negative feedback rather tends to alleviate or mitigate dynamics. It can be seen as a counter-dynamic that opposes, and thus mitigates, the effect of an original dynamic or process, such as the activities of a helmsman (a cybernatist or gouvernere, as derived from ancient Greek) who countersteers the currents that veer his ship off from its original course.

Negative feedback hence dampens the dynamic of a development (contrary to the development‘s own dynamic). Its general scheme is as follows

A produces more of B, which in turn produces less of A

positive feedback

The maybe most quoted example for negative feedback in systems sciences is the air-conditioning system, with a thermostat controlling temperature and actuating either a heater if temperature falls beneath a certain minimum or actuating a cooler when temperature exceeds a maximum. In any case, it prevents temperatures to become extreme.

Another famous example for a negative feedback is the progressive income tax, meant to mitigate the differences in income distributions generated by the self-enforcing (i.e. positively back-feeding) dynamics of economic transactions due to the rich-get-richer phenomenon.

Love affairs

An illustrative example for positive as well as negative feedback and their possible interactions is the story of the love affair of Romeo and Julia as recounted by the mathematician Steven Strogatz (1988).

Strogatz suggests to regard the love of Romeo and Julia as interdependent. Romeo is a fickle lover. The more Juila loves him, the more he begins to dislike her. But when she looses interest, his feelings for her warm up. Julia is different. Her love grows when Romeo loves her, and she turns to dislike him, when he dislikes her.

Systems scientifically this interdependence can be expressed in terms of two interconnected differential equations which express the rate of change in love of Romeo and Julia.

\[\frac{dR}{dt}=-a*J\] \[\frac{dJ}{dt}=b*R\]


\(R(t)=\) Romeo's love for Julia at time \(t\)

\(J(t)=\) Julia's love for Romeo at time \(t\)

Positive values of \(R\) and \(J\) indicate love, negative values indicate dislike. The parameter \(a\) and \(b\) are positive. The following model lets you test this interdependence.


Strogatz, Steven H.: Love Affairs and Differential Equations. Mathematics Magazine. 1988, 61: 35