Paul Samuelson’s Three Flavors of Randomness - Cantor’s Paradise
Paul Samuelson’s Three Flavors of Randomness Cantor’s Paradise


The flavors of randomness Samuelson describe, of white-, red-, and blue noise, can essentially be summarized as three ways of thinking about what will happen in the future, given what we already know has happened. Taking the efficient-market hypothesis as given, they describe how one might go about thinking about the random behavior of e.g. asset prices in an economic market or base hits in baseball.

White noise / True Randomness

“White noise is truly a random walk. The future is independent of the past. Knowing that the stock rose yesterday has no influence on the probability distribution of what it will do percentage wise between today and tomorrow. That’s white noise. Zero serial correlation coefficient in statistical parlance.“ — Paul Samuelson

Randomness interpreted as white noise considers events as truly random and independent of past events. That is, a white noise interpretation reflects the view that, despite the fact that a stock has gone up every single day in the last three months, tomorrow is a new day and its former price movements provide us with no better approximation about its future behavior than would a coin flip. The white noise interpretation of randomness makes explicit the fundamentally unintuitive notion that despite the very low probability that 76 coin flips in a row will turn up heads (0.0000000000000000000013%), after 75 coin flips, the odds of the 76th coin flip turning up heads is still 50% because the coin has no memory.

Red noise / “The Gambler’s Fallacy”

“Red noise (is what) I called regression towards a mean, where there is a negative serial correlation through time. So that if things (went) up a lot yesterday, today you should bet that they won’t go up as much as they would normally do between today and tomorrow. That is red noise. “ — Paul Samuelson

The “pessimistic” interpretation of random events, red noise, also known as the Gambler’s fallacy, is the mistake notion that if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vica versa).

In literature, descriptions of the fallacy trace as far back a 1796 story entitled A Philosophical Essay on Probabilities by Pierre-Simon Laplace, in which he describes the ways in which men calculated their probabilities of having sons:

"I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls."- Excerpt, "A Philosophical Essay on Probabilities", Laplace (1976)

Blue noise / “The Hot Hand Fallacy”

“Blue noise is the opposite. If things went up yesterday, they will go up, in a probability sense, more often tomorrow. “ — Paul Samuelson

The more “optimistic” interpretation of random events, blue noise, also sometimes known as the Hot hand fallacy, is the phenomenon that a person who experiences a successful outcome has a perceived greater chance of success in future attempts. Purported to have been disproved (as a real-world phenomenon) in a 1985 study by Tversky et al (1985), more recent studies have indicated that future performance might not necessarily be unrelated to a previous “hot streak”.

Regardless, the interpretation of truly random events as indications of a higher probability of certain future events still remains a fallacy.

Samuelson’s work on the efficient market hypothesis (and in turn his descriptions of how humans misperceive randomness) is one of many works in neoclassical economics whose lacking accuracy in their predictions laid the groundwork for what later became the field of behavioral economics. Behavioral economics is the study of the effects of psychological, cognitive, emotional, cultural and social factors on the economic decisions of individuals and institutions and how those decisions vary from those implied in classical theory. For more information, I recommend the book “Thinking, Fast and Slow” by Daniel Kahneman.



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