Ties III

 

Step To The Rear

Will everyone here kindly step to the rear
And let a winner lead the way
Here's where we separate
The notes from the noise
The men from the boys
The rose from the poison ivy.

But what about Ties!

The outcome of every game is NOT win or lose. It is win, lose, OR TIE. Because humans do not like ties, the rules of most games may employ elaborate methods to break ties: overtime, extra innings, shoot outs, sudden death, winning by an advantage, etc.  To entice players into fair games that could end in a tie even when one player has a clear advantage over the other player, handicaps, ratings, point spreads, etc. are maintained by an impartial overseer. 

All of this is to say that ties are statistically probable and are important. But what if we could not see those ties? What if in reality wins and losses were observable but ties were not observable? What if what is perceived as ties are only the observable portion of a mirror of the original game. But a mirror game does not have to begin at the same point. Its mirror ties only have to be equal to the observable games wins and losses. That then means that the unobservable wins and losses of that mirror game also have to be equal to the ties of the observable game. If wins and losses are equally probable as ties, then this is an inevitable outcome. If wins, loses, and ties are all equally likely then each should occur 33 and 1/3% of the time. That means the unobserved ties should be equal to the total of observable wins and losses. And if the observable wins and losses are equal to the observable but mirror ties, then the mirror game has to be twice the size of the original game. And the dividing line between observable wins and losses and observable but mirror ties should occur at 2/3, 66 2/3 % of the total outcomes. This also means that the intersection point of the original game and its mirror must occur exactly at 50% of all games. If it occurs at a different point, for the observer to perceive observable wins and losses to be equal to the observable mirror ties then it may be necessary for a different surface than the normal Euclidean surface to be used.

There are three surface types as indicated by their curvature: Spherical, Flat (Euclidean); and Hyperbolic. The limit of an outcome on a spherical surface is the same as the outcome on a flat surface if the size of the Radius of the Sphere is large enough. This is because of the identity 1=cos²+sin². However the limit on a hyperbolic surface will NOT be the same as that of a flat surface because of the identity 1=cosh²-sinh². Thus if the starting points of the mirrored game and the original game are not identical, their intersection will not occur at the median of each,  and if the area above the intersection appears smaller, then it is most probably a hyperbolic surface. 

If the surface is flat then the observable perception of a 3-D object on that surface is 50%. And thus 50% is unobservable. You can infer the characteristics of the unboreable portion of the object, but you can NOT observe them. On a hyperbolic surface, the perception would be different. Thus while on a spherical or flat surface the perception might be that the original game is twice the size of the original game. If the intersection is NOT at 50%, the mirror of the original game would be perceived as 5/6 of the original game, (or its additive, and symmetric, inverse 1/6), not 2/3, (or 1/3). The fact that there are observable ties that are a mirror of unobservable wins and losses must lead to the conclusion that the intersection is occurring on a hyperbolic surface and the intersection is perceived to be at 5/6, not 2/3. This is only because it is also being perceived from hyperbolic surface. This has an implication on behavior that will be explored in the next blog post.