The “three to one” rule in military operations states that when the attacker has a three to one advantage in effective force over the defender, victory is assured.

This rule of thumb in military operations finds a sympathetic chord in the mathematical operation of the discrete analog of the GB model.

A three to one advantage in opposing military forces is equivalent to a net relative advantage of three minus one equals two, which corresponds to the boundary between decision processes that converge on unity and those that do not.

The three to one rule is thus a local application of a law that is universal in nature, and applies to all decisions, not just those that involve physical violence or military conflict.

The same rule is applied under much different circumstances in the U.S. Constitution, where no amendment is approved unless and until three-quarters (three to one) of the states have duly ratified it.

By comparison, removal of federal officers for high crimes and misdemeanors requires “only” a majority (one to one) of the House of Representatives for impeachment and two-thirds (two to one) of the Senate for conviction.

This may help to explain the incredible staying power of the U.S. Constitution, which has endured for more than two centuries with relatively few changes, in despite of vast geographical expansion, enormous population growth, rapid urbanization, amazing changes in science and technology, multiple industrial revolutions, two world wars, and much, much more.

Eight billion zombies can’t all be wrong. Not at the same time anyway. Time to make a difference. Take a stand. Strike a pose. Capture a moment. Or walk on by.

What does the theory and practice of slavery have to do with Decision Space? Everything. And nothing.

Slavery it seems is built into the very structure of the universe we live in. What a terrible discovery! What can it mean? What does it portend? Is there yet hope? Or is all lost, now and forever more?

As far as Decision Space goes, there is both good news and bad news.

First, the bad news. Slavery is quite common in Decision Space under both Special and General Relativity. Not the single most common type of decision, thankfully. That honor goes to Normal Bass decisions under Special Relativity, and Strange Bass decisions under General Relativity. But common enough for government work.

Slavery in the mental universe of Decision Space should not be confused with slavery in the physical universe.

The specific features of slavery in Decision Space include:

a non-zero starting point

suboptimal performance

excessive use of force

These conditions might be explained in more detail, if you are interested.

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.