The Future is 2D

A friend of mine returned from business school, joking that she learned every problem can be reduced to a 2x2 matrix. It turns out, this is a more common forecasting method than you may imagine, beyond the simple BCG Matrix.

I attended and spoke at Notre Dame’s business foresight conference in 2015. The roster was impressive, and the sheer brainpower present was obvious when a principle from The Institute for the Future led a room full of consultants and strategists through an exercise based on this trusty 2x2. They would plot different variables on the X and Y axis, to describe possible futures. For the sake of example, call X “interest rates” and Y “economic growth.” You can then label the quadrants, plot futures, and broaden your ideas about how the future could look — low rates, low growth; high rates, low growth; etc.

These possible futures are great for considering change, generally. They also leave out a critical component: time. A 2x2 matrix is just a snapshot of some future at some undetermined time. Enter: the cone of plausibility, futurists’ handy device for charting possible futures, over time.

The cone introduces the idea that possible futures broaden as time passes. The further you go into the future, the more facts can change and compound. In a certain sense, this can be thought about as a time series of 2x2 matrixes. Over time [T₀, Tn] the range of possible X values changes—[-X₀, X₀] goes to [-Xn, Xn]. The same happens to the Y axis. The result is a larger 2x2 matrix with a radial boundary of plausible futures.

Now we’ve accounted for time, but the model is still extremely limited. Change is rarely linear or constant, but perhaps second order rate of change is constant, as we see in Moore’s Law’s consistent exponential change. This captures an interesting truth about change itself: change compounds. The possible futures don’t just broaden over time, they do so exponentially.

In this model, the range of possible X values doesn’t just go from [-X₀, X₀] to [-Xn, Xn] over time. Rather, the possible range goes from [-X⁰, X⁰] to [-Xⁿ, Xⁿ]. Only some variables will track the borders of this exponential plausibility “cone,” but the point is that any variable could. What if they don’t?

Many scenarios do not unfold with a normal distribution. This may be well and good for pricing options, but few things take “a random walk.” In many cases, the vector will self-correct, constraining futures. We could call this an “anti-change” future. The more a variable changes, the more pressure there is against further change, approaching a limit.

By the same token, some variables will actively track towards change. Going into an election, we don’t know what the following year will bring, but we often know that candidates are promising change. Peter Thiel argues that any prediction about the future should have two qualities, to have any chance of being accurate: it should be concrete and it should be different from the present. (source: link) We can call this a “pro-change” future. These pro- and anti-change futures may still be simplistic, but they’re better than using a normal distribution!

But wait, it’s even more complex. Variables are rarely independent. To use our original examples of interest rates and economic growth — they’re incredibly intertwined. The cost of capital is inextricably linked to companies’ investments and the return on those investments are key to the Federal Reserve’s monetary decisions.

And what’s with all these smooth curves? Variables don’t change at smooth rates — exponential or otherwise. Change often compounds to create runaway scenarios. It can happen slowly, then all at once. This is the entire premise behind fat tails and black swans. X might not wait until Tn to jump directly to Xⁿ. As the theory goes, a model-breaking event without historical precedence could happen over a very short period of time or even in a moment (e.g., a “flash crash”).

Over just a few dozen months, entire markets can disappear. If our X axis is “price people will pay for CDs” then what exponent should Barnes and Noble have used? It might be fair to say “infinity.” The rise of the internet and the fall of the Soviet Union were shocks to the world, and other black swans happen much quicker — the assassination of Archduke Ferdinand and September 11th are among the single days that shaped history, seemingly without warning. In this sense, the “cone” of plausible futures has no outer bounds.

If tails are in fact nearly infinitely fat, then the real solution might be to avoid rate of change assumptions when outlining the bounds of plausible futures. Instead, perhaps it’s most prudent to accommodate extreme outcomes by designing “win, don’t lose” scenarios that capitalize on convex volatility (we’ll cover these in the next post).

So if the tails of the distribution for tomorrow [T₁] expand in all directions, then the cone’s edges are actually closer to [-∞, ∞]. In this case, the cone collapses, the future is 2D, and you should just use the 2x2 matrix after all…