Decision making chaos and determinism

This is an inquiry into decision-making and its connection to uncertainty. It is based on the white paper with the same title. Decision making is one of the most human acts and seems to be the most difficult area to formalize into a theory of behaviors that are causal and deterministic. In fact one might think that the very nature of decision-making is one of chance and uncertainty. One issue we think relevant is the general lack of understanding of causal theories and how they deal with uncertainty. Moreover, in our view, there is insufficient appreciation of the sensitivity of the initial conditions that determine future behaviors. When these issues are taken into account, it becomes easier to see the possibility for causal formal theories of human decision-making.

Consider the sensitivity of future behaviors on initial conditions, which has been extensively studied under the general category of chaos and chaos theory. It has been said in the past that chaos represents for humans the way we perceive the world in its un-ordered state. If we had perfect information, so the argument goes, we would have perfect determinism. Slight disturbances on what we think we know, lead to unknown consequences, even in a theory that is strictly causal and deterministic. So “what is deterministic?” It seems unreasonable to believe that because chaos behavior is possible, we must throw out our causal theories. They work very well and explain a host of data. We believe that a more reasonable approach is that we must be more careful about what we claim to learn from these causal theories.

The theories after all reflect our efforts to identify concepts and attributes that don’t change with time or that change with time in an understandable, causal and continuous way. One thing to explore is how we deal with uncertainty in such mundane activities as measurements, which form the basis of all physical theories.

So for example we understand by length, an attribute that characterizes the height, width or length of something. It is a great accomplishment in understanding to separate this concept from the mechanisms by which we perform the measurement. The mechanisms involve a measuring stick and us as an agent; for those of you measuring a basement, you know that multiple measurements yield multiple answers. Yet we are confident that the basement has well-defined dimensions. How did we come to this conclusion and how did we learn to separate out the uncertainties associated with us as agent and the intermediary of a measurement stick from the invariant concept of length? Today, we all agree that the separation has been done and we are comfortable with the idea of length.

Similarly, we are comfortable with the concept of time, despite our dependence of using clocks to make time measurements. From such simplistic considerations, we have adopted over many centuries, physical theories of the behavior of matter that we depend on. For example, we are comfortable with a host of physics problems that relate distances objects travel with time. We believe we understand how a pendulum works because we can predict the behavior starting from a description in which we describe its restoring force as being the source of the acceleration. The behavior is the set of positions of the pendulum over time. We start the pendulum at rest and “drive” it by a harmonic force. We predict from Newton’s theory where the pendulum will be at any future moment. We compare where the pendulum is by measurements against where it is predicted to be and find agreement to a high degree of accuracy.

This model, because of the non-linear behavior of the force, generates unexpected structure. In engineering and business, there are also distinct ways to gain access to a system’s non-linear characteristics. For the pendulum, one can initiate the behavior by varying the initial conditions. Alternatively, one can “drive” the behavior by applying an external force. For example we might impose an external force characterized by a single amplitude and a single frequency. As we vary the frequency and amplitude we stress the non-linear structures of the problem. For sufficiently large amplitudes we generate chaotic structures: we go from a quasi-periodic structure to one that no longer appears periodic. We create behaviors that appear much more erratic and lack the periodic behaviors seen with smaller driver amplitudes. The idea is that these properties may in fact carry over into the realm of decision-making.

We expect that decision-making has attributes that involve imperfect information as well as perfect information. The challenge is to identify each of these, separating out those attributes that have a predictable behavior from those attributes that are inherently uncertain. We adopt game theory (Von Neumann & Morgenstern, 1944) in which an intrinsic view of decisions is a productive starting point where we separate out the pure strategies as things of permanent interest. A pure strategy is the complete set of moves one would carry out in a decision process taking into account the moves of all of the other players or agents in the process along with any physical or chance effects that might occur. It is a complete accounting of what you would do, a complete plan given every conceivable condition. It is furthermore assumed that you can approximate this complete list with a relatively small list of pure strategies.

Just because there are pure strategies, there is no reason to believe that one of these pure strategies is the right choice to make. If you are in a competitive situation, there may be a downside to your competitor knowing that you will pick one of these strategies. The solution is to “hide” your choice by picking the pure strategies with a specific frequency. The theory determines for you what these frequencies are without informing your opponent which choice you actually will make on any given play.

Thus your decision choice is a specific frequency choice and in that sense represents the measurement of “length” despite the fact that in a real decision process, like a real measurement process, there are lots of uncertainties. You would like to determine the frequency choices the other players make and they want to understand your choices. We emphasize that knowing these frequency choices is not the same thing as knowing what you will actually do on a given play. We take this knowledge in the same way we take the knowledge about the size of our basement. We know how to get a good approximate set of measurements. We know that out basement has a size. For each measurement process we don’t know what size we will actually get.

We extend game theory to decision process theory (Thomas G. H., Geometry, Language and Strategy, 2006) in which the strategy frequencies vary with time. This theory predicts future frequency values based on a given set of initial conditions. The theory is causal in this sense, without actually dictating what a player will actually do at any given moment. The basis of the theory has some similarities to physics and even more underlying similarities to mathematical models of physical processes. Just as in physics, there can be external forces that dictate how the rates of change of frequencies change in time. There will be stationary situations in which these rates of change don’t change, in which the forces generating such changes are zero. We equate that scenario with the whole literature of game theory and its consideration of static games: the frequencies are fixed numbers. Static games provide an important foundation for our approach, though our results diverge once dynamic effects are included.

A second scenario is one in which the fields that generate the forces are static, but the flows, the rates of change of the frequencies, are dynamic. The flows may depend on what other players are doing, and so we can distinguish a special subset of flows that are stationary: at a specific “location”, the flow doesn’t change. However, if you follow the streamline of the flow you will follow a path that changes in time. You might think of a weather pattern that is stationary, in which the wind at any position is constant in speed and direction. If you follow a path along the wind by adding smoke, you will see that the smoke follows a streamline that moves with time. These considerations are really rather similar to the pendulum problem.

Based on these considerations and verifying our ideas from a variety of numerical examples, we argue that behaviors from decision process theory are deterministic, yet represent the uncertainty of choices based on frequency. We can separate out from the decision process the uncertain aspect of the decision, whose future behavior is unknown: we don’t know which pure strategy will be chosen. Thus we identify that aspect of the decision process that deprives us of perfect information. We also identify those aspects of decisions processes that might evolve continuously in time and can be determined in a causal manner. These are the numerical frequencies of choice that form the basis of the choice, but don’t actually determine the specific choice at any given time. Our theory is then about the frequencies and not about the choices.

This is not the end of the story. Just because we have a theory that determines future behavior based on knowledge of past behavior, we are not justified in assuming that the predictions will be insensitive to our starting point. Non-chaotic behavior assumes that the future behavior is not sensitive to small changes in the starting point. This often follows from theories that are linear in nature. Chaotic behavior by contrast expects small behaviors to generate large behavioral differences, even if that behavior ultimately stays bounded. Over time however, we expect to see significant deviations. Some of the non-linear behavior is a consequence that preferences can’t grow without limits. We postulate that concept here, but in fact do see evidence for that behavior in the full theory.

We expect that chaotic behaviors can be generated from within, without recourse to external “drivers’, if there are suitable parameters that can be varied. For the pendulum, the suitable variable would be the initial speed. The initial flows and payoffs are suitable variables for decision processes. The chaotic behavior is a result of the non-linear nature of the forces and can be made visible with a “driver” representing external periodic forces. It is then a matter of whether the amplitudes and frequencies excite the underlying structures. In fact from both, seemingly benign behaviors as a steady state need not indicate the lack of interesting structures. The key is how to excite these structures into existence.

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