Gambler's Fallacy

Research Paper on

Gambler’s Fallacy (29th October, 2009)

Submitted by: Prashant Sahu

Index Gambler’s Fallacy…………………………………………………………………………………3 Explanation of the bias…………………………………………………………….………………3 Studies………………………………………………………..……………………………………4 Explanations for the Bias………………………………………...………………….…………….6 Implications for Investment Decisions……………………………..……………………………..7 Active Investing……………………………………………..……………………………8 Fund Flows……………………………………………………..…………………………9 Equilibrium………………………………………………………………………………..9 References…………………………..……………………………………………………………11

Gambler’s Fallacy 2

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that if deviations from expected behavior are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future. For example, if a fair coin is tossed repeatedly and tails comes up a larger number of times than is expected, a gambler may incorrectly believe that this means that heads is more likely in future tosses. Such an expectation could be mistakenly referred to as being "due". In other words, gambler’s fallacy is the mistaken belief that random sequences should exhibit systematic reversals. An individual who holds this belief and observes a sequence of signals can exaggerate the magnitude of changes in an underlying state but underestimate their duration. The gambler's fallacy implicitly involves an assertion of negative correlation between trials of the random process and therefore involves a denial of the exchangeability of outcomes of the random process.

Explanation of the bias Faced with a streak of events and the necessity to make a choice, people may make one of three possible inductions: (1) that the streak is irrelevant, (2) that the streak will continue, or (3) that the streak will stop. If people generally accepted the first of these inductions, then when faced with a forced choice, they should predict the next event with a probability equal to its base rate (e.g., 50% for a fair coin flip). However, what is often observed is a bias toward one of the other two inductions, even when the events are independent. The induction that the streak should continue is known as the hot hand fallacy. The third induction is often known as the gambler’s fallacy, a tendency to believe that a streak of events is likely to end. Laplace (1814/1951) first wrote about this phenomenon. The Gambler's Fallacy is committed when a person assumes that a departure from what occurs on average or in the long term will be corrected in the short term. The form of the fallacy is as follows: 1. X has happened. 2. X departs from what is expected to occur on average or over the long term. 3. Therefore, X will come to an end soon. There are two common ways this fallacy is committed. In both cases a person is assuming that some result must be "due" simply because what has previously happened departs from what would be expected on average or over the long term. The first involves events whose probabilities of occurring are independent of one another. For example, one toss of a fair (2 3

sides, non-loaded) coin does not affect the next toss of the coin. So, each time the coin is tossed there is (ideally) a 50% chance of it landing heads and a 50% chance of it landing tails. Suppose that a person tosses a coin 6 times and gets a head each time. If he concludes that the next toss will be tails because tails "is due", then he will have committed the Gambler's Fallacy. This is because the results of previous tosses have no bearing on the outcome of the 7th toss. It has a 50% chance of being heads and a 50% chance of being tails, just like any other toss. The second involves cases whose probabilities of occurring are not independent of one another. For example, suppose that a boxer has won 50% of his fights over the past two years. Suppose that after several fights he has won 50% of his matches this year that he has lost his last six fights and he has six left. If a person believed that he would win his next six fights because he has used up his losses and is "due" for a victory, then he would have committed the Gambler's Fallacy. After all, the person would be ignoring the fact that the results of one match can influence the results of the next one. For example, the boxer might have been injured in one match that would lower his chances of winning his last six fights. There are many scenarios where the gambler's fallacy might superficially seem to apply but does not. When the probability of different events is not independent, the probability of future events can change based on the outcome of past events (see statistical permutation). Formally, the system is said to have memory. An example of this is cards drawn without replacement.

Studies Experiments documenting the gambler’s fallacy are mainly of three types: 1) Production tasks, where subjects are asked to produce sequences that look to them like random sequences of coin flips, 2) Recognition tasks, where subjects are asked to identify which sequences look like coin flips, 3) Prediction tasks, where subjects are asked to predict the next outcome in coin-flip sequences. In all types of experiments, the typical subject identifies a switching (i.e., reversal) rate greater than 50% to be indicative of random coin flips. The most carefully reported data comes from the production-task study of Rapoport and Budescu (1997). Table 1 below shows the subjects’ assessed probability that the next flip of a coin will be heads given the last three flips. According to Table 1, the average effect of changing the most recent flip from heads (H) to tails (T) is to raise the probability that the next flip will be H from 40.1% (= (30% +38% +41.2% 4

+51.3%) /4 ) to 59.9%, i.e., an increase of 19.8%. This corresponds well to the general stylized fact in the literature that subjects tend to view randomness in coin-flip sequences as corresponding to a switching rate of 60% rather than 50%. Table 1 also shows that the effect of the gambler’s fallacy is not limited to the most recent flip. For example, the average effect of changing the second most recent flip from H to T is to raise the probability of H from 43.9% to 56.1%, i.e., an increase of 12.2%. The average effect of changing the third most recent flip from H to T is to raise the probability of H from 45.5% to 54.5%, i.e., an increase of 9%.

Evidence that people believe in mean reversion across random devices comes from horse and dog races. Metzger (1994) showed that people bet on the favorite horse significantly less when the favorites have won the previous two races (even though the horses themselves are different animals). Terrell and Farmer (1996) and Terrell (1998) showed that people are less likely to bet on repeat winners by post position: if, e.g., the dog in post-position 3 won a race, the (different) dog in post-position 3 in the next race is significantly under bet. Gold and Hester (2008) found that belief in mean reversion is moderated when moving across random devices. They conduct experiments where subjects are told the recent flips of a coin, and are given a choice with payoffs contingent on the next flip of the same or of a new coin. Subjects’ choices reveal a strong prediction of reversal for the old coin, but a much weaker prediction for the new coin. Explanations for the Bias Estes (1964) suggested that the negative recency observed in these experiments was a habit learned from life that was revealed in the laboratory and, in longer experiments, was gradually 5

extinguished. Yet, if the gambler’s fallacy is a habit learned in everyday life, where might it be learned? Plainly, it is an inappropriate response to situations where there is conditional independence between the successive outcomes of a random process. Nonetheless, outside of gambling casinos and psychology laboratories, there are few—if any—circumstances where one can safely assume conditional independence of a succession of events. Perhaps, then, the gambler’s fallacy reflects adaptation to uncertain situations where negative recency is exhibited. One obvious candidate is where a finite population of outcomes is sampled without replacement. Under these circumstances, expectations with negative recency have some validity because observing a particular outcome lowers the chances of observing that outcome the next time. Accordingly, a number of authors have suggested that the experience of negative recency in life might be responsible for the gambler’s fallacy in experimental tasks where subjects are asked to generate or recognize random sequences (Ayton, Hunt, & Wright, 1989, 1991; Lopes, 1982; Lopes & Oden, 1987; Neuringer, 1989; Triesman & Faulkner, 1990). A rather different account of the gambler’s fallacy was offered by Kahneman and Tversky (1972), who presented a cognitive explanation of the gambler’s fallacy in terms of the operation of the representativeness heuristic. They argued that people expect the essential characteristics of a chance process to be represented not only globally in an entire sequence of random outcomes but also locally in each of its parts. Thus, despite their statistical inevitability, long runs of the same outcome lack local representativeness and are thereby not perceived as representative of the expected output of a random device. Consequently, subjects will expect runs of the same outcome to be less likely than they are. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays close to 0.5 in any short segment more so than would be predicted by chance. The hot-hand fallacy tends to arise in settings where people are uncertain about the mechanism generating the data, and where a belief that an underlying state varies over time is plausible a priori. Such settings are common when human skill is involved. For example, it is plausible–and often true–that the performance of a basketball player can fluctuate systematically over time because of mood, well-being, etc.

Implications for Investment Decisions Gambler's fallacy can have a wide range of implications, in investment decisions being taken in financial markets. Below is the case showing how Gambler’s fallacy can create perceptual difference between individual investors and investment professionals. 6

Panel A is based on a UBS/Gallup survey and represents individual investors’ expectations about returns on the S&P 500 Index for the next year (graphed on the y-axis) against what the S&P 500 return was the previous year (graphed on the x-axis). Panel B is based on Livingston Survey data and represents the same type of relationship but for investment professionals. The regression line for Panel A would be positively sloped, and the regression line for Panel B would be negatively sloped. Graph shows that when forecasting market returns, individual investors are prone to the hot-hand fallacy and that investment professionals are prone to gambler’s fallacy. If one were to graph the relationship between actual year-over year returns to the S&P 500, the slope would be fairly close to zero. There is very little predictive power in last year’s return to the S&P 500, but representativeness leads both investment professionals and individual investors to attach too much predictability to the previous year’s returns.

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For most individual investors, the process generating the market returns is unknown, predisposing them to the hot-hand fallacy. For most investment professionals, experience with stock market history leads them to be familiar with the process generating the market returns, thereby predisposing them to gambler’s fallacy.

Active Investing A prominent puzzle in Finance is why people invest in actively-managed funds in spite of the evidence that these funds under-perform their passively-managed counterparts. This puzzle has been documented by academics and practitioners, and has been the subject of two presidential addresses to the American Finance Association (Gruber 1996, French 2008). Gruber (1996) finds that the average active fund under-performs its passive counterpart by 35-164 basis points (bps, hundredths of a percent) per year. French (2008) estimated that the average investor would save 67 bps per year by switching to a passive fund. Yet, despite this evidence, passive funds represent only a small minority of mutual-fund assets.

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Explanation can be given as an investor prone to the gambler’s fallacy is uncertain about whether expected returns are constant, but is confident that if expected returns do vary, they are serially correlated (ρ > 0). Hence the investor ends up believing in return predictability. The investor would therefore be willing to pay for information on past returns, while such information has no value under rational updating. Turning to the active-fund puzzle, suppose that the investor is unwilling to continuously monitor asset returns, but believes that market experts observe this information. Then, he would be willing to pay for experts’ opinions or for mutual funds operated by the experts. The investor would thus be paying for active funds in a world where returns are independent and identically distributed (i.i.d.) and active funds have no advantage over passive funds. In summary, the gambler’s fallacy can help explain the active-fund puzzle because it can generate an incorrect and confident belief that active managers add value.

Fund Flows Closely related to the active-fund puzzle is a puzzle concerning fund flows. Flows into mutual funds are strongly positively correlated with the funds’ lagged returns (e.g., Chevalier and Ellison 1997, Sirri and Tufano 1998), and yet lagged returns do not appear to be strong predictors of future returns (e.g., Carhart 1997). Ideally, the fund-flow and active-fund puzzles should be addressed together within a unified setting: explaining flows into active funds raises the question why people invest in these funds in the first place. Explanation is that returns are negatively correlated with subsequent flows when flows are measured over a short horizon, and are positively correlated over a long horizon. The negative correlation is inconsistent with the evidence on the fund-flow puzzle. It arises because investors attribute high returns partly to luck, and expecting luck to reverse, they reduce their investment in the fund. Investors also develop a fallacious belief that high returns indicate high managerial ability, and this tends to generate positive performance-flow correlation, consistent with the evidence. But the correlation cannot be positive over all horizons because the belief in ability arises to offset but not to overtake the gambler’s fallacy.

Equilibrium Suppose that investors observe a stock’s i.i.d. normal dividends. Suppose that investors form a sequence of generations, each investing over one period and maximizing exponential utility. Suppose finally that investors are uncertain about whether expected dividends are constant, but are confident that if expected dividends do vary, they are serially correlated (ρ > 0). If investors are rational, they would learn that dividends are i.i.d., and equilibrium returns would also be i.i.d. If instead investors are prone to the gambler’s fallacy, returns would exhibit short-run momentum and long-run reversal. Intuitively, since investors expect a short streak of high 9

dividends to reverse, the stock price under-reacts to the streak. Therefore, a high return is, on average, followed by a high return, implying short-run momentum. Since, instead, investors expect a long streak of high dividends to continue, the stock price over-reacts to the streak. Therefore, a sequence of high returns is, on average, followed by a low return, implying long-run reversal and a value effect. These results are similar to Barberis, Shleifer and Vishny (1998), although the mechanism is different. One another application concerns about trading volume. Suppose that all investors are subject to the gambler’s fallacy, but observe different subsets of the return history. Then, they would form different forecasts for future dividends. If, in addition, prices are not fully revealing (e.g., because of noise trading), then investors would trade because of the different forecasts. No such trading would occur under rational updating, and therefore the gambler’s fallacy would generate excessive trading. In a study conducted by Barberis, Shleifer and Vishny (BSV 1998), investors do not realize that innovations to a company's earnings are i.i.d., but believe them to be drawn either from a regime with excess reversals or from one with excess streaks. If the reversal regime is the more common, the stock price under- reacts to short streaks because investors expect a reversal. The price over-reacts, however, to longer streaks because investors take them as sign of a switch to the streak regime. This can generate short-run momentum and long-run reversals in stock returns, consistent with the empirical evidence (surveyed in BSV).

References The Gambler’s and Hot-Hand Fallacies: Theory and Applications (February 28, 2009) By Matthew Rabin and Dimitri Vayanos http://personal.lse.ac.uk/vayanos/Papers/GHFTA_RESf.pdf 10

The Gambler’s and Hot-Hand Fallacies: Theory and Applications (January 5, 2007) By Matthew Rabin and Dimitri Vayanos http://eprints.lse.ac.uk/24476/1/dp578.pdf

Behavioral Finance: Biases, Mean Variance Returns, and Risk Premiums By Hersh Shefrin and Mario L. Belotti http://www.ifa.com/pdf/BehavioralFinancecp.v24.n2.pdf

Randomness and inductions from streaks: “Gambler’s fallacy” versus “hot hand” By Bruce d. Burns and Bryan Corpus http://www.psychology.siu.edu/bcs/facultypages/young/JDMStuff/Burns.pdf

The hot hand fallacy and the gambler’s fallacy: Two faces of subjective randomness? By Peter Ayton and Ilan Fischer http://www.staff.city.ac.uk/~sj361/p1369.pdf

Gambler's fallacy http://en.wikipedia.org/wiki/Gambler's_fallacy

Gambler’s Fallacy http://ntur.lib.ntu.edu.tw/bitstream/246246/84706/1/27.pdf

Gambler’s Fallacy http://www.nizkor.org/features/fallacies/gamblers-fallacy.html

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