Small numbers, big bubbles: Gambler's fallacy and hot-hand traders in a heterogeneous-agent market

Financial markets are often imagined as vast, rational engines where prices reflect the true worth of an asset, calculated with cold precision. Yet history is littered with episodes where prices soar far beyond their value, only to crash back down, leaving a trail of confusion and loss. These events, known as bubbles, have occurred for centuries, from the tulip mania of the 17th century to the dot-com boom of the early 2000s. To understand why they happen, economists have long debated whether they are the result of rational calculations or human error. One powerful idea in this debate is that people are not perfectly logical statisticians. Instead, they often suffer from a specific cognitive bias called the "law of small numbers." This is the mistaken belief that a very short sequence of random events must look like the long-term average. For instance, if a coin is flipped and lands on heads three times in a row, a person influenced by this bias might wrongly assume the next flip is almost certainly tails, expecting the short run to immediately correct itself to match the long run.

A new study by Kirill Proskurin explores how this single psychological quirk can explain the entire life cycle of a market bubble, from its sudden birth to its violent end. By building a computer model of a market populated by different types of traders, the researcher demonstrates how a simple misunderstanding of chance can cause prices to spiral out of control. The model does not rely on complex, unexplained rules or the assumption that investors are irrational in a general sense. Instead, it shows that if traders believe price changes are like drawing balls from a small, known bag, their natural reactions to those draws can create massive instability. The study finds that when enough of these traders are present, a single change in the underlying value of an asset can trigger a chain reaction, causing the price to overshoot its true value, crash below it, and then bounce back in a series of smaller waves, all while trading volume spikes in perfect sync with the price errors.

The researcher constructed a virtual market containing four distinct groups of people. First, there are fundamentalists, who know the true underlying value of the asset and try to buy when it is cheap and sell when it is expensive, though their ability to do so is limited by the risk they are willing to take. Second, there are liquidity traders, who buy or sell simply because they need cash, regardless of the asset's value. The other two groups are the behavioral traders, who do not know the true value and instead try to guess the future based on past price movements. These traders all share the belief in the law of small numbers, viewing price changes as if they were drawing balls from an urn. However, they interpret the urn differently. One group, the contrarians, knows the urn started with an equal mix of "up" and "down" balls. When they see a price rise, they believe the urn has run out of "up" balls and that a fall is now inevitable. This is the classic gambler's fallacy, the belief that a streak must reverse. The other group, the momentum traders, does not know the starting mix of the urn. When they see a price rise, they assume the urn is now full of "up" balls and that the rise will continue. This is the hot-hand fallacy, the belief that a streak will keep going.

In this simulated world, the interaction between these groups creates a dynamic that mirrors real-world bubbles. The story begins when the true value of the asset changes. In the model, if the fundamental value jumps up, the price immediately jumps with it, but no trading occurs yet because the behavioral traders have not had time to update their beliefs. In the very next moment, the momentum traders see the price rise and, convinced that the trend will continue, rush to buy. This buying pressure pushes the price even higher, far above its true value. The contrarians, seeing the rise, begin to sell, expecting a reversal, but their influence is often too weak to stop the momentum traders at first. As the price climbs, the gap between the market price and the true value widens, creating a bubble. The model shows that this bubble does not last forever. Eventually, the price stops rising as quickly, and the momentum traders lose their optimism. They begin to sell, and the price falls. Because they are still influenced by the law of small numbers, they overcorrect, pushing the price down below its true value. This creates a crash, followed by a smaller rebound, and then another smaller fall, as the market slowly settles back to equilibrium.

The simulations reveal that the size and duration of these bubbles depend on a few key factors. If the market has more fundamentalists who are willing to trade against the mispricing, the bubbles are smaller and shorter. If the fundamental risk of the asset is higher, or if there are more momentum traders, the bubbles become larger and more volatile. The study also shows that trading volume behaves in a very specific way: it peaks exactly when the price is furthest from its true value. This happens because the disagreement between the groups is greatest at the top of the bubble, with momentum traders buying aggressively while fundamentalists and contrarians sell. This finding aligns with historical data, such as the high trading volumes seen during the peak of the dot-com bubble. Furthermore, the model demonstrates that bubbles can be triggered not just by changes in the asset's value, but also by a sudden influx of cash from liquidity traders. Even without any news about the asset's worth, a large buy order can start the same chain reaction, causing a bubble that looks identical to one caused by fundamental news.

The research provides a clear, mathematical explanation for why markets can be so unstable, grounding the phenomenon in a well-known psychological bias rather than abstract market forces. It shows that the same bias that makes people think a coin is "due" for a tail can also make them think a stock is due for a continued rise, depending on what information they have. The model proves that these behaviors can lead to a unique stable state where the price equals the true value, but that the path to getting there is rarely smooth. Instead, the market tends to oscillate, overshooting and undershooting the target. The study concludes that while these bubbles are temporary deviations, they are a natural consequence of how human minds process short sequences of data. By understanding the specific conditions that make these oscillations grow or shrink, such as the proportion of different trader types, we gain a clearer picture of why markets sometimes seem to lose their minds, and how they eventually find their way back to reality.