The addition of one more dimension of time opens up the possibilities of Decision Space considerably. Fully rational decisions influenced by internal forces representing the goals and aspirations of decision makers become possible for the first time in this dimension.

Dimension 2 produces a wide range of decision classifications under Special Relativity, unlike Dimension 0, which is restricted to Luce decisions, and Dimension 1, which is restricted to Normal Bass decisions.

Dimension 2 cannot produce valid Strange Bass decisions under Special Relativity. This is reserved for General Relativity.

The migration patterns of General Relativity open up yet more of Decision Space. The two most common types of decisions in Decision Space under General Relativity are Strange Bass and Normal Bass decisions, followed by imaginary decisions and a host of other more specialized forms.

Once time is unrestricted in terms of discrete measurement, the practical side of Decision Space begins to emerge into a more recognizable shape. The GB model is at its best in determining the shape of decision curves.

We have seen that when you remove time from the GB equation, free will (goal oriented behavior that makes any kind of difference) completely disappears, leaving nothing but the universe and its impersonal physical laws of nature behind. Welcome to the proper time-oriented definition of predestination.

What happens when we reintroduce time into the equation? The short answer is that free will begins to matter just as soon as we “excite” decision making to its next highest level or dimensional state. Whoa, Nelly! Is that a relief or what? The first dimension of Decision Space is represented by a line, but not a straight line. Shades of slightly non-Euclidean spatial architectures!

Interestingly, all decisions in the First Dimension under Special Relativity occupy a line which is exactly identical to the Outer Limit previously observed in the original DB experiment! How is that for a pure coincidence of a most kindly predisposition?

The characteristics of the simplest decisions in the universe (or at least, in Decision Space) are equally fascinating to observe. When General Relativity is added to the one dimensional equation, we observe that initial starting conditions influence the outcome systematically as a function of the internal influence only. This effect diminishes as we approach the origin of Decision Space, where the universe continues to reign supreme.

When we fully randomize initial starting conditions, we see that Decision Space expands to fill the void, but only up to a certain point. No decision in the First Dimension can ever be perfectly rational, since perfect rationality occurs only at the origin in this dimension.

The most rational decisions in the First Dimension are all dominated by the external influence exerted by the universe itself. All of the decisions in the First Dimension that are dominated by the internal influence representing the goals and aspirations of the decision maker are completely irrational by contrast. This means that the First Dimension represents the door that opens up the possibility of rational decision making in this universe, without offering any possibility for rational decision making by decision makers as a thing in and of itself.

Higher dimensions are required for rationality to appear in conjunction with decision making, goal-oriented behavior, and possibly even life itself.

Unlocking the secrets of the General Theory of Relativity in Decision Space is a bit like solving a certain kind of puzzle. Perhaps it is best to begin at the beginning. The beginning of Decision Space is not what you might expect it to be. It is not located at the mathematical origin, the traditional place for new beginnings expressed in the form of Cartesian coordinate systems.

The mathematical origin of Decision Space represents what I call Stationary decisions, in which there are no net external or internal forces acting on a decision process at all, yet in which the final equilibrium value predicted by the Bass model is either undefined or infinite in scope, neither of which is particularly useful in terms of decision analysis.

The mathematical origin is not the true origin of Decision Space. When decisions are reduced to their simplest form of mathematical expression, which is a dichotomous discrete choice dummy variable which generically represents any measurable difference between two states of being somewhat arbitrarily designated as before and after, the true origin of Decision Space is thus revealed to the world in all of its magnificent if somewhat hazy glory.

The true origin of Decision Space occurs when the external influence exactly equals 1. The internal influence oddly enough can take any arbitrary value between negative infinity and positive infinity when the external influence is exactly equal to 1, because the external influence completely dominates all decisions at this particular point in Decision Space. Weird, huh? Freaky, no less.

It appears that the physical universe can exist quite comfortably without any decision makers at all, and proceed on its merry way in strict conformity with the physical laws that govern the physical universe, with no help or hindrance from Decision Space whatsoever, but only as long as Decision Space remains wrapped up at the single point that defines Dimension 0, and which occurs at {1,0}.

This is the origin of life, or at the very least the origin of Decision Space, where all decision makers and any type of goal-oriented behavior in this particular universe must begin their journey into the light of day as either relevant and meaningful to them or anyone else.

This then is Dimension 0. This then is the origin of decision making in the context of Decision Space. This then is the simplest form any decision can take under the General Theory of Relativity in Decision Space. So simple, the very existence of decision makers is no longer required in order for the physical universe to operate universally in conformity with physical laws such as gravity, strong and weak nuclear forces, and electromagnetism, for example.

But that is only what happens at Dimension 0. What happens when Decision Space extends to higher dimensions? Let us proceed, one dimension at a time, to see how Decision Space unfolds to reveal the operation of decision making in Decision Space at higher levels of dimensional complexity.

Dimension 0 is eerily and aptly described by the words of the immortal William Shakespeare in Hamlet, when the Player King rather provocatively states that:

“Our thoughts are ours, their ends none of our own”

Fortunately, this rather pessimistic conclusion is not always true. So perhaps we should begin to peel that baby back a layer or two?

One of the first things I noticed when I conducted my Deterministic Bass experiment using three time periods of data was a boundary condition:

This condition becomes more obvious when we scan out:

And even more obvious when we scan further out:

But what does it mean? Where does it come from? Is it a theoretical or a practical limitation? What mathematical form might it take, and why?

I soon determined the mathematical form:

The “theory” is in fact a mathematical formula with no theoretical justification other than it seemed to fit the data. But the mathematical fit was less than perfect, and the theoretical justification remained nonexistent, or emergent, to give it a finer sounding moniker.

For the last decade, this situation has remained basically unchanged. There is an observed limit, with nothing else known about it.

We talked about free will mostly in our last post. That’s because sensible people prefer to exercise free will rather than allow themselves to be exposed to the terrible indignity of predestination. Sadly, predestination does occur in the lives of everyone at some point in time, specifically to include the end of existence as we know it in this world, otherwise known as death itself. Predestination mathematically takes the form of a dichotomous discrete choice variable suspended in time between before and after.

As the above graph illustrates, when a step function is separated in time from one time period to the next, there is one step up in time period 1, and no change thereafter. This type of decision is the product of an overwhelming external influence that always explains 100% of the variation in the dependent variable, leaving nothing more than 0% for the internal influence to explain.

This graph and the relationship it explains is unaffected by the internal influence, which can take any value from negative infinity to positive infinity and still fail to explain even the smallest portion of the variance. A rather odd situation, you must agree. This dichotomous discrete choice time variable occupies a single point in Decision Space {1,0}, and thus has a rather limited dimensionality of 0 dimensions.

Who among us is capable of exercising free will in decision making based on their very own needs, wants, desires, and goal orientations? Conversely, which of us may be subject to predestination, and are therefore doomed by fate even before they are born, no matter what they may choose to think, say, or do? The correct answer is both, of course.

Free will and predestination are both built directly into the structure of the universe we live in. Free will and predestination are both explicit functions of time, and nothing else. Free will occurs when you have all of the time in the world. Predestination occurs when you have no time left at all. With rare exceptions.

The moment of death, which takes less than an instant, is the only time in any decision maker’s life when it can truly be said that predestination is fully in control, for time itself has run out for that particular decision maker. No decision maker is ever capable of exercising free will in its purest form unless they literally have all of the time in the world, all of the time in the universe, an infinity of time in other words. For the rest of us mere mortals, decision making is by its very definition a mixed bag, in which free will and predestination both appear as internal and external influences on the necessarily discrete process of decision making.

The above graph is a little rough, and might be improved, but clearly demonstrates the effect of time on free will. The more time one has, the greater the range of possibilities in terms of goal-oriented behavior, otherwise known as free will.

How simple life can be, when you really think about it.

The Butterfly Effect is real. Not just in the physical universe we so much love and respect, but in the mental universe of Decision Space as well. Let’s see if we can figure out how this is possible just for fun.

I would say that the Butterfly Effect in the physical universe is a function of uncertainty in the first instance. Uncertainty emerges as a fundamental property of the fabric of which the physical universe is composed from little things like chaos theory and complex systems analysis. In a sense, the Butterfly Effect is the form that uncertainty takes in relation to our inability to sense the presence of the physical universe or to obtain measurable data regarding the physical universe in any or all of its manifold forms. The modeling implications are rather trivial by comparison, since all you need to model the universe is a perfect copy of the universe itself.

The Butterfly Effect in the mental universe of Decision Space operates on somewhat different terms. Unlike the physical universe, where the Butterfly Effect emerges from chaos as a fundamental limitation on knowing or describing or predicting future changes in the state of the physical universe as a system, in the mental universe of Decision Space the Butterfly Effect represents a wildly different kind of limitation or uncertainty. The Butterfly Effect in Decision Space has more of a dreamlike quality to it, a goal that can never be achieved, but which like all the colors of the rainbow appears to be quite near, and can be captured if one only takes the one more step necessary for its realization in both form and substance. A chimera which is both the only real goal in life, and one that can never be obtained. A paradox, a contradiction in terms, and the fundamental meaning of life itself. No one ever said the universe had to be fair.

The Butterfly Effect in Decision Space is intimately tied to McFadden decisions. McFadden decisions provide the most basic and fundamental goal of civilization in mathematical form. McFadden decisions mark the boundary between two distinct dimensional domains, those occupied by Normal Bass decisions and Strange Bass decisions. There are an infinite number of permutations that can produce McFadden decisions in theory, yet examples of pure McFadden decisions are very difficult to find in decision making practice.

One simple example of a practical McFadden decision is this:

Practical McFadden decisions are interesting in and of themselves, but in general have have little or nothing to do with the Butterfly Effect. Another paradox of traffic flow, ye gods!