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Fool On The Hill

Monday, April 26, 1999

FOOL ON THE HILL
An Investment Opinion
by Alex Schay

Probability Theory and the Market

Virtually every day, we make decisions in the face of uncertainty. Considering that we do so on such a necessarily consistent basis, its hard to fathom that our decision-making structures do not adhere to some form of accessible logic. If not a logic of the formalized, probabilistic variety, then perhaps one imbued with a more evolutionary imperative. One that works.

In short, it's hard to completely embrace the notion that we are fundamentally irrational -- as it has been "traditionally" defined. However, when the question of rationality is examined in light of the evidence provided by probability experiments, we have a pretty dismal record. That is, the non-statisticians among us do very poorly when asked to measure the likelihood that an event will happen in the future given certain data.

Many have been embarrassed by the "false positive" puzzle -- in random testing you score positive for a disease that yields false positives 5% of the time in a population where one in every thousand people have the disease. By the way, there is a 2% chance that you have the disease, not a 95% chance -- which is undoubtedly the product of a conspiracy by probability theorists to make people feel like idiots. There's an even better popular example though, which is quite a bit more confounding, alternately called "The Prize" or the "Monty Hall Puzzle."

Given three boxes, you are asked to choose one. In two of the boxes there is nothing, in the third there is a prize. After you have chosen, but before your box is opened, another box is opened to reveal no prize, just air. You are then given the option to change your mind and choose the other unopened box. Does the decision really matter? After all, you've just been shown a box with no prize, and there are only two boxes left. One will have the prize and one won't, so there's a fifty-fifty chance you'll get the prize, right? Sorry, you can actually double your chances of winning the prize by switching to the other box.

I came across this probability brain teaser in its "Monty Hall" incarnation in a recent issue of The Economist (the Feb. 20th - 26th, 1999 issue: "Getting the Goat") where it was noted that violence often ensues after reasonable discussions about the puzzle have been exhausted.

Famed psychologists Daniel Kahneman and Amos Tversky have pursued numerous experiments confirming that we are typically "not rational" when the defining element of reason is action in accordance with the laws of probability. Despite this, they have found that our cognitive models do adhere to a readily definable logic. In one study, Kahneman and Tversky found that a majority of subjects will judge a deadly flood (causing a thousand deaths) triggered by a California earthquake to be more likely than a fatal flood (causing a thousand deaths) occurring somewhere else in North America on its own. Again, probability theory rears its ugly head. The majority of the respondents in the study were wrong, because probability theory asserts that the likelihood of any event (A) is always greater than or equal to any event (A), conjoined with event (B).

Commenting on the Kahneman-Tversky results in their recent book, Philosophy in the Flesh, George Lakoff and Mark Johnson note:

"Presumably, most of their subjects had a mental model in which earthquakes occur with reasonably high probability in California, a model in which earthquakes cause tidal waves, and a model in which sudden tidal waves cause a large number of deaths. When these cognitive models are evoked by the question and linked together, they imply that the probability of a California earthquake triggering the highly fatal flood is higher than other well known scenario for such a highly fatal flood, most of which have many fewer than a thousand fatalities because of excellent warning systems for most large floods."

For most of their book Lakoff and Johnson explore the manner in which we employ our cognitive models to contextualize and make sense of situations and information. We reason and understand using metaphors, frames, and prototypes. And while this may be "contextually inappropriate" at times, it has nevertheless been our most basic and successful tool for survival. While the book really poses a challenge to a Western philosophical tradition that has largely ignored input from cognitive science over the last thirty years, it's also an extremely interesting commentary on the theory of rational action in general, and I would wholeheartedly recommend it to anyone.

Understandably, if we are so ill equipped to reason about basic probability questions, then more profound issues regarding how the stock market works should really be daunting. Perhaps the most pedestrian view is that prices are merely set by the sum total of the actions taken in the market. Our intuition tells us that prices move up and down in a concrete process dictated by money flows, where "ending" prices are a function of the sum of all the inputs. However, as with probability theory our intuition is not a helpful guide when presented with the vast, complex entity called the market. In fact its just downright contextually inappropriate.

In a decentralized market view there are no "leaders" that set prices on the margin, but rather it is through the teeming interaction of countless players that a "meta system" emerges. That is, the market carries properties and characteristics "distinct from the agents that comprise it" (see "The Invisible Lead Steer" at CAP@Colmubia). The aggregate becomes more than the sum of its parts, or put another way, the market is "smarter" than any one of its participants.

Yes, the stock market reflects all available information and incorporates what is believed about the present value of the future, but this is not to say that prices are "correct." Correct is a determination that can only be made in the future. The implication here is that the most fruitful endeavor for the investor is to take a harder look at existing prices to determine what is built into the equity. It is only from this vantage point that disjunctions with possible future results can be assessed.

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