Wilderness and the Adjacency Mathematics of Bathroom Tiles

A Fanciful Journey in the Equations of Risk Management

By Morgan Hite

In the ongoing quest of wilderness education to express the conditions under which we labor, one very enterprising fellow at the school where I work has suggested that we might view risk mathematically, and he has proposed the following equation:

(1) risk = likelihood ´ consequences

He is not a mathematician and so of course did not realize the question he has raised by writing this equation down. Naturally, one looks at it and asks, "What are the units?" I shall attempt to explore this question.

Likelihood is a probability of a discrete event occurring within a finite time period. The event is discrete because it can happen or not, such as slipping on a smooth rock, or being hit by a meteorite. There is no middle ground. The time period is finite as a matter of convenience, just as food containers describe the number of calories per "8 oz. serving." If you eat twice as much you consume twice the calories. In our case, if you remain in the risky situation twice as long, you incur twice the likelihood of the event occurring.

Likelihood (L) therefore obeys the following corollary equation:

(2) L = t ´ P

where t is the duration of time and P is the probability of the event in that time.

This model assumes that risk remains constant over some finite period of time. In practice this must be true, but the periods of time are so small (on the order of a few seconds) that overall likelihood (and therefore risk) is best expressed as a sum of discrete likelihoods, Lt.

(3) L = å Lt = å Pt ´ D t

and as the limit of time divisions approaches zero, we are left with

(4) L = _ò Pt dt

Probability is usually expressed as a real number between 0 and 1. A probability 0.74, for example means that in one hundred possible trials the event is expected to happen, on the average, seventy-four times. Likelihood is a sum of products of time and a dimensionless real number, and therefore assumes units of time.

Consequences is a far more difficult quantity for which to determine units. Consequences can be emotional, monetary, and/or practical. If someone slips off a ledge there will be fear, medical bills, and a potential premature end to the expedition. Each of these realms of consequence in turn influences the other, and this suggests three dependent variables linked in a differential equation. How then does one measure consequence?

L.S. Chandarakar of the University of Bangalore has suggested in his Mathematics of the Complex Plane (New Dehli, 1974) that certain quantities with characteristic internal dynamics may be satisfactorily modeled using multi-dimensional variables. When the quantity under study assumes certain finite states, he continues, one may envision these states as polygons, and use an adjacency matrix to express the behavior of the overall variable. He compares this to building a mosaic in bathroom tiles: although each tile (polygon) represents a discrete and internally consistent color (state), the overall pattern may be interpreted independently. If one instead imagines a television screen as a finite matrix of colored dots whose positions are constantly changing, one can begin to imagine how a quantity such as consequences can be modeled using such an adjacency matrix of independent variable states.

The resultant matrix of course is subject to any standard matrix manipulation, and therefore we must ask, in reference to (1), whether Likelihood is a scalar, or is it in fact a vector? One suddenly realizes that we are contemplating that risk itself could be a vector, or even a matrix, and this seems very frightening indeed!

One potential impact of risk being a vector or matrix would be to immediately invalidate all past, present and future discussion of such concepts as "minimizing risk." It would need to be decided what was to be minimized (the determinant? the length of the vector?). One interesting possibility of risk being a vector (R) is the notion that there could be an ideal risk state, R_i, and the minimizable index of real risk (m) defined by the cross-product:

(5) m = R ´ R_i.

Finally, it would be irresponsible to end this paper without mentioning the theory, developed by V.I. Chernikoff of New Zealand Outward Bound Wilderness Centre, that likelihood, consequences, and risk are all quantities of fractal dimension. She makes quite an elegant argument for being able to include them in the roots of a Mandelbrot set:

(6) z = 4i + m

Chernikoff's proof, which is flawed, is based upon the analogy between consequences and cloud boundary layers (both are nebulous). She is quite right, however, in pointing out that consequences do in fact have a behavior characteristic of fractals; that is, the closer they are scrutinized, the same structures, albeit at smaller dimensions, recur. Without an explicit scale present, one is essentially lost in a field of consequences. There is no unambiguous referent to determine whether one is dealing with extremely small scale consequences or large ones. In light of this, the theory that consequence is of fractal dimension certainly cannot be discarded.

Lander, Wyoming, February 1993