A recent study published in the Journal of Consumer Research  offers evidence for the unsurprising suggestion that the gambler's fallacy doesn't apply only to gamblers: some shoppers are apparently prone to it too, although in their specific case the term “loyalty fallacy” might be a better fit.

What is the gambler's fallacy? It's also called the “Monte Carlo fallacy” or the “fallacy of the maturity of chances,” and it's the fallacy (more specifically, the misunderstanding of probability math) which makes gamblers think they're “on a roll” or “due for a payout” or something similar.

Here's the simplest example: suppose you flip a coin, and bet on whether it will come up heads or tails. Assuming it's a fair and honest coin (as opposed to some trick or weighted coin), any individual flip has a 50%, or 1 in 2, chance of coming up either heads or tails. And if you flip the coin 100 times in a row, recording heads or tails each time, you'll most likely end up with approximately 50 showings of each.

### Unpredictable series

That said: it's almost certain those hundred coin flips won't result in a predictable, repeated heads/tails/heads/tails pattern. Instead, you'd see periods where the coin came up heads several times in a row, interspersed with times where tails repeated itself, and periods where heads and tails clearly appeared purely at random.

So if you're flipping this coin, or betting on its outcome, it might be tempting to think “This coin has come up heads the last five times — that means it'll probably come up heads next time, too! It's on a streak!” Or you might think the exact opposite: “It's come up heads five times in a row — that means it's due to show tails on the next flip.”

But both arguments are incorrect. Fact is (again, assuming an honest coin flipped by an honest person), that coin still has exactly a 50% chance each of coming up heads or tails, regardless of which side appeared in the last five flips.

With that in mind, consider the December 2014 edition of the Journal of Consumer Research, which published a study called “Lucky Loyalty: The Effect of Consumer Effort on Predictions of Randomly Determined Marketing Outcomes.”

According to the abstract,

This research explores how loyal customers, those who have invested relatively high amounts of effort with a firm in the form of past purchases, respond to randomly determined marketing outcomes (e.g., winning a prize in a random drawing). Across five studies, participants exhibit a “lucky loyalty” effect, in which they believe that greater effort (e.g., dollars spent at a retailer or number of nights stayed at a hotel) results in greater likelihood of obtaining randomly determined promotional outcomes. Loyal customers report these higher subjective likelihoods for randomly determined outcomes because they feel they deserve special treatment from the firm. Theoretically, this work demonstrates that individuals appear to believe that they can earn “unearnable” outcomes through effort, even when the effort and outcome are unrelated.

### Loyalty rewarded?

In other words: when a store, hotel chain or any other business holds a random contest promotion – such as “Fill out this card for a random drawing, with a winner to be picked later,” customers who routinely patronize that business think they have better-than-random odds of winning, because they deserve it as a customer-loyalty reward.

Such contests are not to be confused with actual loyalty-reward programs – if, for example, your supermarket offers you a money-off coupon for every \$100 you spend at that supermarket, you do indeed “deserve” such a coupon for every \$100 you spend. The study does make mention of these loyalty programs, in the context of explaining how customers who participate in such programs, where rewards are indeed passed out according to “loyalty” (as measured by how much you spend, or other factors), often become more likely to think they'll win other, random, promotions where loyalty or customer history does not actually play a role:

A variety of companies, from hotels and airlines to different types of retailers (e.g., Macy’s, CVS, Kroger), have loyalty programs that offer customers discounts and other rewards in exchange for repeat business In this research, we conceptualize loyalty as effort in the form of a customer’s past purchases with a firm, and we explore how customers who have invested more effort with a given firm (e.g., spent more money on a particular retail outlet’s credit card, stayed more nights at a particular hotel chain’s properties, etc.) respond differently to that firm’s promotions compared to consumers who have invested little or no effort with the firm. Specifically, we address differences with respect to promotions involving a random element …. customers who have been more loyal to a firm believe they are more likely to receive randomly-determined promotional outcomes from the firm (i.e., that they are “luckier”) compared to customers who have put in little to no effort with the firm in the form of past purchases ….

This attitude is not identical to the gamblers' fallacy belief that they're “on a streak” or “due” for a certain outcome, but it definitely has something in common with it: the erroneous belief that past activity has any bearing on future outcomes.

The study's authors were Rebecca Walker Reczek and Christopher A. Summers from Ohio State University, and Kelly L. Haws from Vanderbilt University. (Despite the December label on the issue, the actual study was published online on Aug. 22. It's available as a .pdf file here).