ith my last post getting some preliminaries out of the way, let’s broach how physics and math place limits on decisions. Observe that a rational investor has an enormous (albeit perhaps countably finite) list of independent investment and trade strategies available to them at every point in time.
Compounding this situation is the combinatorial explosion of possibilities that exist for expressing every thesis. In the investing world, we can think of simple directional trades, pairs trades, trades involving bets on proxy indicators, leveraged trades, trades involving various simple or exotic derivatives, trades involving various types of collateral for funding, trades involving hard assets, mergers, acquisitions, spin-offs, various complex legal structures such as SPVs, trades combined with use of voice (in the sense of the brilliant Albert O. Hirschman), trades whose thesis is predicated on things that are difficult to quantify (e.g., viewpoints or character traits of key executives/board members), etc. And that extensive list is only scratching the surface of the number of means—and permutations thereof—we could utilize to express a single thesis.
To make matters worse, all of these possibilities must be weighed against opportunity costs outside the market: spending a day with the family, exiting the financial markets entirely in favor of retirement, starting a private firm, etc. The list of strategies available to a "rational" agent at each instant might be enumerable and finite, however it remains an incalculably large buffet of choices.
Moreover, at every instant, we have a metadecision problem: is now the right time to make any decision? If so, how should we prioritize the making of those decisions? How will we know when we have a satisfactory understanding of the various problems, predictions of the future, etc. that are necessary to weigh our various options and decide how to proceed?
As Bertrand Russell once said,
I do not pretend to start with precise questions. I do not think you can start with anything precise. You have to achieve such precision as you can, as you go along.
This observation lies at the heart of the problem. Our fatal conceit is to believe that we have the information and knowledge necessary to render decisions in a "rational" manner. "Rationality" falls from grace the second we contemplate the true grandeur of the decision space we must scour in search of optimality. Rationality may seem acceptable under the sterility of what physicists call the gedankenexperiment (thought experiment). This is well and true, but not representative of real-world conditions.
Real-world decisions are never binary. Rather, they are infinite recurses of decisions, meta-decisions, ad infinitum. In this post, my principal aim is to show that our reliance on heuristics and probabilistic (Bayesian) reasoning to make decisions is not only useful and expedient, but essential due to the intractability of producing exact analysis to support our decisions.
Decision: Algorithmic Perspective
If we examine decisions from the perspective of algorithms, each potential strategy we're considering will involve solving some suite of optimization problems to determine, for example, the optimal quantity to buy or sell; the optimal price to transact at based upon current understanding of market conditions; the optimal way to structure the trade, etc. Likewise, to determine which is the best of myriad strategies at our disposal is an optimization problem of costs versus benefits using what economists call a utility function. The solution to each of these optimization problems is itself not free — for each optimization involves some expense, e.g.:
- bandwidth and energy to collect and transmit the raw information necessary to render the decision
- energy to process the information
- physical space for data transmission, information processing, and information storage
- most importantly, time for the data transmission and processing to occur
- amortized expenses of facilities, human capital, education of human capital, etc.
- amortized R&D costs for developing new analytical techniques, information collection systems, etc.
Increasingly, these costs are a function of IT organizations. Yet, it's important to note they are just as relevant in the mental domain. Before any process can be ingrained into what Kahneman calls System 1, we must first undergo a lengthy period of performing the task—at great expense—in System 2.
That the mere act of being "rational" has an intrinsic cost often seems to be missed. This should be astounding. Latency and cost profoundly complicate what is implied by being a "rational" actor. Not only have we established that every human being is faced with nearly infinite choices at every moment of every day, but that each of those decisions would take a nontrivial amount of effort to reach.
My more astute readers are likely already exclaiming: this is an infinite recurse! To wit, if rational decision-making involves expending various types of costs on solving optimization problems—where the mere act of optimizing is itself not free—, then we likely need to perform meta-decisions and meta-optimizations to determine, respectively, which decisions are worth investing energy into, and to minimize the cost of solving the optimization problem(s) required to reach the decisions and meta-decisions deemed worthy of solution.
Likewise, the set of meta-decisions and meta-optimization problems are expensive, so we likely want to perform meta-meta-decision and meta-meta-optimization, and so it continues ad infinitum. And the above is to say nothing of the fact that any one of these decision and optimization problems could be what computer scientists call NP-hard* (or perhaps even its obscure parent class PH).
*This is probably a good place to stop and discuss what NP-hard means, as few outside of the computer science and quantum computing communities understand its significance. Most of the algorithms that help us in our daily activities are of a class that can be solved quickly. Often, the time to execute is linear, or quadratic, in the amount of data being processed. An algorithm that is linear in execution time can be expressed by an equation of the form:
$$f(n) = an+b$$Where $f(n)$ is the time to execute, $n$ is the size of the data set to be analyzed, $a$ is some constant per-datum computational costs, and $b$ is some constant overhead irrespective of the amount of data to be processed. These terms have close analogues in finance. We could think of $a$ is being the same as the variable costs of production, such as raw materials, energy, or factory labor. Similarly, $b$ is equivalent to fixed costs, such as R&D, G&A expenses, or fixed asset investment.
Most of the algorithms we interact with on a daily basis fall into a few basic complexity categories:
These concepts are so common that computer scientists have adopted a succinct notation for the concept called Big-O notation. This notation dispenses with the details you could express in the above equations by only focusing on the dominant algebraic term as the size of the data set to be evaluated goes towards infinity (i.e., the limit concept from calculus).
If we examine the $i(n)$ equation above, we can observe that its growth as $n$ grows unboundedly large is dominated by the quadratic term. For those of you who are math-inclined, the following derivation illustrates how Big-O simplification works:
$$\mathrm{let}\; i(n) = an^2 + bn + c$$ $$\mathrm{If}\;\, \exists M, M \in \mathbb{R}, M \ne 0\;\,\mathrm{such}\,\mathrm{that}$$ $$\limsup_{n \rightarrow \infty} \left| \frac{i(n)}{M n^2} \right| < \infty$$ $$\mathrm{then}\; i(n) = O(n^2)$$In other words, when the amount of data to be processed is enormous, all but the so-called highest-order term can be ignored. Of course, when dealing with smaller quantities of data, it's important to understand that this simplification may prove quite misleading (e.g., if $bn > an^2$, or $c > an^2$).
NP-hard, on the other hand, is a class of algorithms whose equations don't look like the above. Rather, they have variables in exponents, variables in exponents of exponents, factorial functions, and other functions that explode in magnitude faster than a variable raised to a constant power. For example:
For such algorithms, very small changes in the size of the data set to be analyzed profoundly impact the time needed to compute the solution. The classic example is the traveling salesman problem (TSP). This problem involves a salesperson who has some list of waypoints to stop for meetings. Naturally, the salesperson wants to minimize some performance metric, such as travel time, travel distance, or travel cost.
The best currently known exact solution to TSP falls into the NP-hard category. Its time complexity is given by
$$f(n) = O(n^2 2^n)$$With every additional waypoint added to the visit list, the amount of time needed to optimize the itinerary grows exponentially. When the list is small, adding one additional city to the list can considerably more than double the time needed to find the optimal route. As the list gets large, the marginal increase of each additional waypoint asymptotically approaches two times the cost of one fewer waypoints:
It turns out this algorithm is extremely important to many industries: transportation, logistics, circuit board and microchip design, urban planning, etc. Consequently, a cottage industry has formed to research heuristic solutions (techniques that produce solutions within reasonable percentage bounds of the theoretically optimal solution within reasonable time bounds). Therefore, while in theory this algorithm is extremely difficult to solve, there are practical shortcuts that work reasonably well for moderate numbers of waypoints.
While TSP is special in that there are good heuristics available, the key point for us is that there is a large class of algorithms whose solutions can take longer to compute than our lifetimes. The general intractability of finding exact solutions to problems in a reasonable timeframe significantly complicates decision processes. The best we can do in something less than exponential time is to come up with a heuristic that's known to get us within a certain percentage of optimal. To find the truly optimal solution will generally take too much time and resources to be practical.
As the digression in the grey box above illustrates, there is a large class of problems whose exact solutions we cannot reasonably compute. In fact, deciding to compute the solution to such a problem is itself a consequential decision (due to the high computational costs).
This sheds new light on the earlier discussion of heuristics, and probabilistic decision making. We live in a world where the process of optimizing a delivery driver's route is computationally expensive. Until quantum computers are commercially viable, this significantly curtails our ability to optimize for our utility. We can't predicate our decisions on analysis that would take years—or even longer than our universe has existed—to compute. Hence, we are often forced to treat decisions in much the same manner as we have since the very beginnings of human history—as a set of probabilities to be weighed.
As Bertrand Russell was alluding to in the quote, we are constantly in want of precision. How do we know which decisions to evaluate? How do we know how to optimize the various components of that decision? (Or, equivalently, how do we sift through the myriad similar decisions to find the best of the bunch?) If precision can only be attained through time as we go along and learn more, does the conceit of rationality not lead us down an infinite recurse of inquisition (and likely introspection) before we can in fact render any "rational" decision? Is choosing between myriad similar strategies best performed analytically, or via an evolutionary process?
Innovation & Infinity
The concept of an infinite recurse is at the heart of innovation. As society grows increasingly sophisticated, we make determinations about when, where, how, who, and why to expend more resources to find improved solutions. It is an iterative process of finding incrementally less sub-optimal solutions, as time and resources permit. This is the sine qua non of innovation: at first, we find clumsy solutions to prerequisite problems as a means to realize our goals. Later on—assuming we survive—there will be opportunities to refine our processes. Leadership in innovative endeavors requires impeccable humility. There are no guarantees that finite resources will ever solve a problem!
In the primordial stages of the firm, we accept unwieldy processes because they permit us to live to fight another day—the very heart of all evolutionary processes, including the startup. We accept fragility because optionality is its close companion. While we have no rigorous proof that evolutionary (closed loop) processes are the optimal way to devise competitive solutions to practical problems, the fossil record furnishes strong prima facie evidence. Consequently, those of us who dedicate our lives to forming new companies are wed to organizational and technological agility, tight feedback loops, flat organizational structures, and the primacy of choosing only the best people to work with.
Later, the clumsiness of constituent processes becomes an impediment to scaling an economically important process. This leads to a phalanx of executive decision processes by which we identify, prioritize, delegate, and ultimately tackle bottlenecks. By expending resources, we attempt to streamline processes, and thereby improve margins, throughput, or some other critical performance metric. In the realm of physical processes, this is intuitively obvious.
W. Edwards Deming's work on closed loop process optimization, followed by Six Sigma a couple decades later, are classic examples of human knowledge growing from imprecise trial-and-error approaches pre-Deming to rote, formulaic approaches yielding precise process control post-Six Sigma. If a plant manager had been charged in the 1930s with attaining product variability limits routinely attained today under Six Sigma, their task would have been daunting. The project would have involved tremendous greenfield work and the uncertainty attendant in building new things, whereas today society has paid for the privilege of that knowledge.
While less obvious, the exact same iterative, trial-and-error processes employed when improving output quality on an assembly line, are also employed when searching for a better algorithm to forecast sales. There are deep correlations between the processes employed when inventing intangible property and those used in manufacturing. The process of writing an algorithm is itself a vast recurse of decisions, optimizations, and communications. What is "rationality" if not the application of algorithmic processes on a set of circumstances to effect optimal decisions (and thereby optimize our utility)?
Going back to the Bertrand Russell's quote, the key here is that we are attempting to maximize our utility in a world filled with unknowns. There are myriad precursor decisions and meta-decisions we cannot answer with the desired precision because they would require solving algorithms that aren't tractable in the time available. Ultimately, the decision recurse rests on a multitude of fundamental philosophical and academic questions whose resolution may be beyond the mental capabilities of humans. They may have to await automated proof and consistency checkers that reveal internal inconsistencies for which the human mind lacks the memory and/or mental capacity to ascertain.
Decision Fractals
The intricate details of the nexus of optimal decision making and innovation are necessarily obscure, much akin to endlessly zooming in on a fractal. One cannot factor in an infinite number of precursor decisions, and all of their implications, implications of implications, and so on in another infinite recurse.
But, in some sense, innovation is even worse than endlessly zooming in on a fractal in search of truth. Innovation often comes in fits and spurts. One key finding can radically alter the epistemic landscape—forcing us to reconsider many ideas that depended on older, invalidated understanding. Imagine a fractal with quantum properties. Observing the next deeper zoom level might reveal an inconsistency that causes the entire fractal to change before us. This is a world of complexity orders of magnitude beyond what even Stephen Wolfram considers in his research.
At some level, we all realize that the intellectual foundations on which we predicate our decisions are made of sand, not stone. They're probably largely right, but far from absolute truth. Particularly the surface layers are bound to swept away many times over the course of generations, as we gain insight through hard work. The deeper layers are firmer, closer in consistency to bedrock, yet almost nothing is ever given the designation Absolute Truth. As we slowly pile additional layers of derivative ideas on top of old ones, we find internal inconsistencies, discover new ideas, sweep away bad ideas, and iterate our way towards truth.
This evolutionary process is profoundly expensive, yet highly effective. One of the key questions we will have to answer in this series is whether Bayesian reasoning is the true basis of economic rationality. Given that we cannot be sure that our existing edifice of suppositions will remain unchanged (during the period under which we are making predictions needed to optimize our utility), it would seem that not only conditions of the physical world and economy must be assigned probabilities, but also the cognitive frameworks we utilize to assess value.
However, changes in how we reason are not things we can quantify as risk. Rather, they are a form of uncertainty. We can assign probability to an idea being invalidated, but it would seem absurd to assign probabilities to how we will reason about the world after such an event: the mere act of invalidation tells us nothing about how we will think in the future. That is unknowable.
Preview of Part III
Not even idealized homo economicus can violate the laws of computational complexity, at least if we are to treat it seriously. As I'll cover in Part III, there is a famous problem in computability theory called that Halting Problem, which tells us that it is impossible to predict whether a pairing of algorithm and input will terminate (compute its answer) in finite time.
But even lacking an appreciation of the Halting Problem, it should be intuitively obvious that an infinite recurse of algorithms cannot terminate in finite time*. If we were to base decisions on predictions grounded in first principles, an infinite (or at least absurdly deep, albeit finite) recurse would result. This seems to point in the direction of Bayesian reasoning, although at the peril of potentially misapplying heuristics.
*Lest any computer scientists complain, I need to point out I'm stretching the truth in the name of clarity. There are obscure cases where an infinite recurse of algorithms could be solvable in finite time. For that to occur, we'd have to prove that:
- the infinite set of algorithms are in fact identical to a finite set of algorithms,
- that we can run those finite exemplar algorithms a finite number of times to correctly predict the outputs of the infinite invocations the original problem required, and
- the combination of all the combined algorithms could be computed in finite time.
It's the sort of highly-contrived scenario a mathematician could construct, but which would bear no relevance to reality.
We all intuitively understand that analysis paralysis manifests in all of us. Philosophers have contemplated the dilemma of action versus consideration since time immemorial. My principal aim is, by exploring the problem space from the lens of computer science and computability, to show that "rationality" as it is popularly conceived is not only nonexistent, but impossible. Instead, we must rethink what a proper definition of rationality might be.
There remains a vast expanse yet to cover in this endlessly deep topic. Next, I want to explore the conundrum of how to achieve optimality (i.e., maximize utility, for you economists) in light of this infinite recurse of expensive cognitive and/or computational precursor problems. But, let's close here and leave those details for Part III.
⬅ Part I