Decisions: stochastic or deterministic?

We believe we know the answer to the question of whether decision-making is a stochastic process or a deterministic process. It is clearly not deterministic since we, as human beings, have free choice. It is equally clear that some outcomes are far from surprising; in which case the decisions that led up to those outcomes must have played a decisive role. In physics there are similar situations.

For example if I hope to buy carpet for a room, I first take measurements. In one sense the measurements that I get are random because when I do the measurements multiple times, I usually get multiple answers. In another sense however, we have agreed in our society that the room doesn’t change its dimensions at random and ascribe the multiplicity of measurement answers to measurement error. We take the room dimensions as attributes that exist in the “real world” and our measurement errors as attributes of a flawed process. We make no attempt at making a theory of the flawed process but instead focus on theories that deal with what we perceive to be the real world. We justify our approach in noting that the measurements cluster around some average value with a variance that reflects the accuracy of our measurement tool.

A similar point can be made about decision-making. If we focus on the actual choice made at any one point in time, we may be focusing on a process that has sufficient variability from one person to another that a theory of such processes is not practical. However we may focus instead on the frequency with which a person makes choices from a variety of options. In decision process theory, I argue that these frequencies are part of the physical world with enough regularity that we can profitably construct a theory of their behaviors in time. The real world suggests that such behaviors vary continuously in time as well as along all the dimensions that measure preferences.

Since these assertions are about the world we live in, it should be possible to validate whether decision behaviors from one moment to another occur in a deterministic way. A similar question has been asked about certain biological processes, most notably the behavior of heart beats. Since the biology of the heart is very complex and intimately connected with living organisms as opposed to inert organisms, one might conclude that the heart behavior would be stochastic. This turns out not to be the case and is illustrated by considering “recurrence plots” of the time sequence of heart beats f\left( t \right). One would expect that a contour plot of the heart beats \left| f\left( t \right)-f\left( {{t}'} \right) \right| for different pairs of time would show no structure; one would expect it to look like noise. In fact it shows distinctive structure suggesting that future behaviors of the heart depend on the past behaviors in an organized and continuous fashion.

I propose we look at decision processes in the same way. To illustrate behaviors you might see, I start with a very simple model that displays some of the structures I have seen in numerical evaluation of decision process theory. The simple model emphasizes that the structure I hope to see is not at all esoteric but one that simply has not been noticed much. The important point is that we focus on the behaviors that might follow deterministic rules as opposed to composite behaviors that are mixtures of both deterministic and stochastic effects.

In the simple model I assume two variables, one reflecting time t and the other reflecting some decision preference z. For continuous values of the preference z there is a time sequence, which I take to be given by

g\left( z,t \right)=\cos z\cos t+\sin 2z\sin 3t

There is nothing magic about this choice other than I imagine that in the real world there will typically be multiple frequencies in time and that there will be corresponding variations in preferences. The resultant recurrence plot is provided below as a CDF insert. One can vary the preference and see that the structure will in general depend strongly on the preferences, based on the model.

[WolframCDF source=”http://decisionprocesstheory.com/wp-content/uploads/2012/11/ToyDecisionProcessModel.cdf” width=”805″ height=”723″ altimage=”http://decisionprocesstheory.com/wp-content/uploads/2012/11/ToyDecisionProcessModel.cdf”] CDF Figure: Time sequence recurrence plot for toy model for various preference values starting at  z=0.

I draw attention to two cases. The first is for z=0 in which the single frequency dominates. Seeing periodic behavior in the time behavior of the decision choices would be clear evidence of deterministic behavior. However, it is much simpler than what we would expect from complex structures. The behavior need not be so simple, as can be seen by looking at other values, which display more complex behaviors. To experiment with these behaviors, move the “slider” on the above CDF figure.

This by no means exhausts what we would expect from the real world and by no means exhausts what a realistic theory provides. Decision process theory takes as input time and preference behaviors at some known point. It substantially transforms these inputs into behaviors at all other points of time and preference. Like with physical and geometric theories of this type, the transformations may be essentially linear, in which case the new behaviors look like the old. The transformations may also be non-linear in which case the new behaviors exhibit new phenomena. A simplified process model has been studied extensively by others investigating such new behaviors, as well as looking for complex behaviors, termed creative bios, in physical and biological theories. They have looked for such complex  empirical behaviors in physical and biological processes, such as cosmological behaviors of the early universe and the behaviors of heart beats. Causal behaviors (simple and complex) seem to be more common than commonly believed.