Learning behaviour of individual investors

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1 Preliminary and Incomplete: Please do not cite or quote Learning behaviour of individual investors Evidence from a financial betting market Tomás Ó Briain Abstract If investors do not learn in a rational Bayesian manner but rather suffer from biases set out in the naïve reinforcement hypothesis, rationality assumptions in individual preference models may not hold. We use a unique longitudinal dataset comprising in excess of 1.5 million fixed-odds financial bets, where bettors perform identical, consecutive decisions which mimic financial choices made in a laboratory, but the use of their own funds departs from the artificiality of an experiment. We test whether agents learn in a way consistent with Bayesian or reinforcement learning in both real and simulated financial markets and disentangle competing learning theories. We also examine whether the disposition to avoid losses is driving behaviour. 1 Motivation Individual preference models assume that agents are rational while empirical research in the area of behavioural finance has suggested otherwise. The possibility of irrational agents in a competitive market is accounted for with the following proposals: (a) irrational agents execute trades randomly and their net effect is negligible, (b) irrespective of trading by irrational agents, a subset of informed arbitrageurs insure that prices are efficient or (c) prices approach equilibrium as agents learn by trading. In effect, if investors are not rational from time to time, they learn in a Bayesian manner to be rational and any market inefficiencies that are caused by such biases are eventually traded out. However, Brav and Heaton (2002) note that the empirical research on convergence to rational expectations equilibrium has demonstrated that this will not just happen, even if agents have the possibility of learning their way to it. Therefore, if investors do not learn in a rational Bayesian fashion and instead suffer from a similar bias to that set out in the naïve reinforcement hypothesis, this assumption may not hold. To that effect, as suggested by Barberis and Thaler Correspondence: Tomás Ó Briain, Room 323, University of Edinburgh Business School, 29 Buccleuch Place, Edinburgh, UK. EH8 9JS. tobriain@gmail.com 1

2 (2003), the continued empirical scrutiny of assumed behaviour is essential to validating the claims of behavioural finance theorists. Empirical testing of behavioural finance theory is complicated by a paucity of transactional panel data. As an empirical analysis of investor behaviour ideally entails a study of panel-data involving the same investors over a period of time, any analysis of investor behavioural in the stock market is only possible where such information is disclosed in a fully transparent manner to the regulator. Although scarce, a number of such datasets are available, a notable one of which is data collated by the Finnish Central Securities Depository which includes the details of shareholdings and financial transactions for all investors in Finland 1. A natural response to this is to turn to a laboratory setting as is common in the experimental economics literature. However, respondents in such experiments may not always be sampled randomly, as many have students as the respondents due to their proximity to the location of the lab on campus. In addition, the costs associated with providing respondents with adequate consideration as to make the contingent claims being traded economically significant are prohibitive. An alternative approach used in previous literature on informed trading has used prices to draw conclusions on the activities of informed traders, however with the dataset used in this paper, their behaviour can be observed directly. By using a longitudinal dataset comprising in excess of 1.5 million individual-level fixed-odds financial bets 2 we have a natural-experimental setting with which to test Bayesian and reinforcement learning theories. The sample includes transactions from more than 30,000 customers from an online bookmaker on major stock indices and also on a simulated market (Virtual Market), a similar dataset to that exploited by academics performing empirical tests of behavioural finance theories with brokerage data. In our setting, bettors are performing identical, consecutive decisions which mimic financial choices made in a laboratory, but the use of their own funds departs from the artificiality of an experiment. Also, in contrast with learning in an IPO setting 3, for example, not only is this a clean experiment (i.e. with no hot and cold IPOs or issue-specific characteristics) but there is also a relatively short time between action and response which should facilitate more expedient learning. Indeed, Brav and Heaton (2002) note that learning in experiments requires immediate outcomes while Russell and Thaler (1985) state that without well-structured feedback, learning may be negligible. We assess how behaviour changes according to different learning outcome paths. It constitutes the first analysis of the financial fixed-odds betting market, and in doing so, sheds light on the activities of relatively recent entrants into the market-making sphere: traditional sports bookmakers. We therefore extend the work of Choi et al. (2009), Pastor and Veronesi (2009), Seru et al. (2010), and Strahilevitz et al. (2011), while also shedding light on a heretofore opaque market 1 See Grinblatt and Keloharju (2000, 2001b,a, 2004), Grinblatt et al. (2008), Kaustia and Knüpfer (2008), Keloharju et al. (2008), Grinblatt and Keloharju (2009), Seru et al. (2010), Shive (2010), Kaustia (2010), Linnainmaa (2010), Grinblatt et al. (2011), Linnainmaa (2011), Keloharju et al. (2012) and Linnainmaa and Saar (2012). 2 The bets have a specific strike, maturity and terminal value, and given their discontinuous payoff functions, are analogous to ultra short-term digital, binary or cash-or-nothing options. This specification makes them trivial to price as (similar to short-term interest rates and fixed-income securites) they expire at a pre-determined price and at a pre-determined time in the future. 3 See Kaustia and Knüpfer (2008). 2

3 setting. 2 Literature Review Do agents learn with sufficient speed to ensure that biases do not affect prices in a systematic fashion? According to Chiang et al. (2011), it is important to examine in what contexts individuals can learn their way out of cognitive biases and in what contexts learning exacerbates bias. Are behavioural biases part and parcel of financial markets? Fixed-odds betting markets offer a quasiexperimental setting in which to perform empirical studies of behavioural financial topics. De Bondt and Thaler (1985) find evidence of systematic price reversals for extreme-return stocks consistent with the overreaction hypothesis, in contrast to the rational Bayesian response to new information. De Bondt and Thaler (1987) re-iterate that the strategy tested in that paper was motivated by the premise the investors are poor Bayesian decision makers. De Bondt and Thaler (1990) further articulate the research of Kahneman and Tversky (1973) which states that people overweight salient information and underweight less salient information when making predictions. They also emphasize that behavioural explanations for anomalous stylised facts observed in financial markets should be taken seriously. Proponents of individual preference models suggest that agents are rational. As stated earlier, irrational, non-bayesian behaviour is accounted for with the following proposals: (a) irrational agents execute trades randomly and their net effect is negligible, (b) prices approach equilibrium as the theory of Efficiently Learning Markets (ELM) (Bossaerts, 1999) suggests that agents learn by trading or (c) irrespective of the trading by irrational agents, a subset of informed arbitrageurs insure that prices are efficient. However, Russell and Thaler (1985) show that the view that the effect of noise traders will be rendered insignificant by the presence of informed traders only holds in a limited number of edge cases. Indeed, there are limits to arbitrage, and suggested by Brav and Heaton (2002), rational arbitrageurs necessitates the presence of rational investors in the activities of arbitrageurs which for various reasons may not be the case. Brav and Heaton (2002) summarise this interplay between agents by stating that an inquiry into financial anomalies is essentially an inquiry into the roles that learning and arbitrage play. In a study of individual investors at a large discount brokerage, Strahilevitz et al. (2011) identify patterns in trading by individuals which are affected by emotions i.e. investors repeating actions which resulted in a profit, while avoiding actions which resulted in a loss. Kaustia and Knüpfer (2008) examine the relationship between returns on previous IPO subscriptions and the likelihood of subsequent participation in IPOs. They conclude that personally experienced returns are an important determinant of future activity and that this is consistent with reinforcement learning theory. Di Guisto et al. (2013) show there is persistence in bettor return and that betting improves with more experiences, however the effect is due to attrition by less skilled bettors. In this case, bettors learn about their ability rather than learn by doing. Nicolosi et al. (2009) examine the trading history of US households by means of a dataset with 3

4 includes an end-of-month portfolio position, trade history file and investor characteristics. They note that as investors gain experience, they adjust their trading strategies accordingly and achieve higher returns as a result. Kaustia and Knüpfer (2008) examine the relationship between returns on previous IPO subscriptions and the likelihood of subsequent participation in further IPOs. They conclude that personally experienced returns are an important determinant of future activity and that this is consistent with reinforcement learning theory. The results of the three tests performed by Kaustia and Knüpfer (2008) indicate that individual investors are affected by personally experienced performance and are more likely to participate, more likely to participate sooner and more likely to participate at a higher intensity if they have experienced positive returns. They state that these results are consistent with reinforcement learning, however they do note a number of alternative reasons for this behaviour, including further unobserved differences between investors, portfolio re-balancing, wealth effects, expectations of preferential treatment by investment banks and the existence of a hot issue market during the sample time period. In outlining their contribution to the literature, they hint at implications for the IPO and asset pricing literature, the role of sentiment in economic decision making and empirical tests of the reinforcement learning hypothesis. Chiang et al. (2011) expand the Kaustia and Knüpfer (2008) study by examining whether investors improve their ability by rational learning or whether their performance deteriorates due to reinforcement learning. They also contend that the Kaustia and Knüpfer (2008) results are also consistent with rational Bayesian learning, as those investors who experience positive returns will tend to participate more often than those who have experienced negative returns. While Kaustia and Knüpfer (2008) analyse whether investors participate move if they have experienced positive returns, Chiang et al. (2011) examine what effect this continued participation actually has on returns. Their dataset includes details on IPO subscriptions in the Taiwanese market and in contrast to that of Kaustia and Knüpfer (2008), includes data on both individual and institutional investors. They also expand on the analysis of Kaustia and Knüpfer (2008) by differentiating in all of their tests between individual and institutional investors. They indicate that individual investors (but not institutional investors) are subject to naïve reinforcement learning, as evidenced by their deteriorating returns as they gain experience and in contrast with the rational Bayesian learning hypothesis. Seru et al. (2010) investigate how individual investors are affected by two distinct types of learning: learning about their own abilities and learning by trading. They conclude that investor performance improves with experience, however they highlight that attrition due to investors learning about their lack of ability may be an important factor. The authors stress that the performance of investors who remain active should improve, not just that the performance of the sample in aggregate improves over time. In approaching the problem, they estimate a simple learning model which looks for evidence of learning in the sample in aggregate, assuming that the attrition from the sample is random and that all investors are homogeneous. The overall result of the Seru et al. (2010) paper is that the correlation of both performance and disposition with investor experience and survival rates suggests that investors learn by trading. The authors state that without having 4

5 controlling for endogenous attrition and individual heterogeneity, however, the literature overestimates the effect of experience on learning. They suggest that the existing literature on learning, stating that authors overestimate how quickly investors become better at trading because they ignore attrition from the sample of investors who have learned about their ability. In fact, learning by trading happens slowly, indicating the possibility of persistent market inefficiencies while investors are in this learning phase. The papers above find evidence of two types of learning: learning about ability and learning by doing. Some of the evidence uncovered could be argued to be consistent with both the rational Bayesian learning hypothesis and naïve reinforcement learning. However useful techniques proposed in the Seru et al. (2010) paper suggest a way to disentangle both types and account for the learning that takes place as lower-ability agents naturally leave the sample 4. The following section introduces some relevant concepts in the learning literature and motivates our two main hypotheses. 3 Hypothesis Developement Reinforcement learning theory, or the law of effect, dictates that agents will repeat behaviour that has been associated with positive feedback and avoid behaviour that has resulted in negative feedback. It dictates that agents should stick to given choices as long as they generate rewards, otherwise they should switch. Rational learning incorporates both private signals and public information, updating beliefs about payoffs accordingly. For example, Bayesian learning refers to weighing both experienced and observed outcomes equally, whereas reinforcement learning over-weighs experienced outcomes. In contrast with a pure stay/switch reinforcement model, Bayesian belief-learners rationally learn from experience. In this section, a distinction is made between two types of learning: bettors initially learning about their own ability and subsequent learning by trading. No differentiation is made between traders who have no ex-post ability and those who are successful during the learning about ability phase. As we have a set of heterogeneous bettors, we proceed by dividing the sample into two distinct subsets: (i) those who show a preference for betting only on Financial Markets and (ii) those who show a preference for betting on the simulated market. In order for our results to have broader implications, we first establish that the first treatment in our setting is indeed synonymous with conventional financial markets. We then differentiate between that setting and the simulated market, establishing boundaries for what type of learning can take place and what type cannot. We motivate our two main hypotheses with prior literature and learning theory, and posit predictions for the behaviour we expect on this basis. 4 A note on the methodological techniques used to mitigate the effects of attrition in the sample is presented in Section 6. 5

6 Treatment A: Financial markets While betting on the financial markets, agents cannot strictly observe each others behaviour, but herding and information cascades are possible as bettors aggregate signals from various sources (including their own private signals). While rational learning can manifest itself in a number of ways in this setting, we test for the existence of learning about ability and learning by doing (Seru et al., 2010). It is a well-documented stylised fact that only a very small fraction of individual traders are successful (Barber and Odean, 2000). Therefore, after a certain number of trades or period of time, traders who have had negative returns will infer that they may be part of the majority, stop trading and leave the sample. While we expect to see losing bettors in this market as a result, those who are rational may infer that they are likely to be unskilled and leave the dataset. If we see changes in stake size in Financial Markets, this could be consistent with either rational or reinforcement learning. Treatment A: Simulated market In the simulated market, however, there is no such ambiguity. Again, as agents cannot observe each others behaviour, there can be no rational belief-learning. There is also full information about foregone payoffs, as bettors know after each losing bet how much they would have won had the counterfactual taken place. As market prices are based on a random-number generator, there can be no private information and outcomes are uncorrelated. As a result, there is no herding or information cascades, no tax-loss selling and no observation or imitation, only private signals and random outcomes. As it is not possible to learn about ability or learn by trading, therefore, if we see changes in stake size in the simulated market, this can be consistent only with reinforcement learning. Changing bet sizes as a result of feedback from the Virtual Market constitutes an emotional response to random rewards. If trading on the Virtual Market is essentially an emotional response to random outcomes, then trading on the financial market types should be motivated by one of two types of learning: learning about ability or learning about trading. Rational learning theory, therefore, predicts changes in bet sizes in financial markets but no path-dependent behaviour in the simulated market. We test for the existence of path-dependent behaviour in both settings with the following two hypotheses: H1a. Bettors on financial markets will exhibit path-dependent behaviour (stake size changes) H1b. Bet size in the simulated market is not path dependent Pastor and Veronesi (2009) state that persistent trading in the face of losses can be consistent with rational learning. In effect, losses are the cost of learning about ability before attrition (on the basis of perceived poor ability) or further learning by trading (in the event of success). Seru et al. (2010) also show that learning about ability is more evident than learning by doing and that investors with poor performance are more likely to cease trading. The rational learning 6

7 theory, therefore, predicts that bettors in the financial market will exhibit behaviour consistent with learning about ability as agents trade in the face of persistent losses in order to update their beliefs about subjective ability. We propose the following hypothesis to test this prediction: H2. Losers persist longer (manifested by lower attrition) in Financial markets than they do in the simulated market as they are learning about ability. Learning theory therefore predicts path-dependent behaviour in the financial markets but reinforcement learning only in the simulated market. It also predicts less attrition in the financial markets than the simulated market. Having motivated our two main hypotheses, we now present the dataset in Section 4, compare behaviour in both settings and present the results in Sections 6 and 7. 4 Data The data for the study comes from a database of transactions provided by an online bookmaker. It contains the details of all bets on placed on the company s financial fixed-odds betting product via the webplatform of the bookmaker, as well as those placed over the phone and via mobile devices over a three-year period. All usernames and customer-specific details are obfuscated, however individual accounts can be identified and tracked throughout the entire period. As there are no customer-specific details provided, no demographics on individual customers are available. Table 1 presents summary statistics on aggregate customer activity. Panel A provides a breakdown of total stake (turnover) according to betting product. The three products with the most betting activity are the FTSE 100 Index, Dow Jones Industrial Index and the Virtual Market, a simulated market where the underlying price is generated by a pseudo-random number generator. The median bet size for bets on the FTSE is e12, with bet sizes ranging from the minimum amount of e0.01 to a maximum of e28,800. Bets are offered for trading in stock indices, commodities, currency pairs and a stock lottery (betting on the last digit of a stock index s settlement price). [Table 1 about here.] Panel B shows the distribution by bet type, with the majority of activity in single 5 bets, while Panel C presents the distribution according to bet price 6. Panel D indicates the breakdown according to source, with the focus of betting on bets transacted via the company s online web interface. While the volume of bets transacted over the phone is clearly much lower, the median bet size is much higher at e48 as opposed to e8.87 for internet bets. To proceed with an exploratory data analysis, a number of histograms is presented in Figure 2. The furthest point from the origin in each case does not show the full extent of the data as the 5 A double is a bet which contains two legs (i.e. one on the outcome of a football match and another on the result of a horse race), with the bet only paying off if both legs are successful. 6 As this is a fixed-odds betting proposition, there are only four discrete bet prices offered for financial bets: 5/6, 1/20, Evens and 5/1. The odds of 8/1 refer to the price of the stock lottery product. 7

8 x-axes have been censored for illustrative purposes, but it is clear that the median amounts in each case are quite small. As such, these are mainly individuals and not institutional or professional bettors. Panel A shows the distribution of the total amount bet and total profit for customers. As expected, total profit is centred just below zero. Panel B provides detail on the total number of bets placed and the number of different financial instruments bet on by each account in the dataset, while Panel C show the distribution of both account tenure in days (in the period covered by the dataset) and trading days. [Table 2 about here.] As we proceed, the key points to take from the summary statistics presented earlier is that the mean and median number of bets across all product types in aggregate is quite small at 4.9 and 1, respectively. As most customers have had a small number of bets, we proceed with a clean, focused, natural experiment, and apply controls on the behaviour of the bettors under analysis. While this will reduce the sample size, it may lead to a more convincing argument of the hypotheses presented earlier. Section 5 outlines the steps taken in proceeding with a univariate analysis while Sections 6 and 7 outline the analysis performed in a multi-variate setting. 5 Methodology In order to proceed with hypothesis testing, we first embark on a univariate analysis. While subtle relationships will be missed in this setting, we add a layer of complexity in the next section in a multivariate setting, controlling for a number of bettor-specific variables. Therefore, as bettors are heterogeneous, we divide the sample into two distinct subsets: (i) those bettors who show a preference for betting only on Financial Markets and (ii) those who show a preference for betting on the simulated market. We first analyse betting in these two groups in aggregate, and then control for risk preferences in a later section by further differentiating according to bet price, making eight groups in total 7. If agents are rational, there should be no path dependent change in stake size for those who bet on the simulated market. Those who bet on Financial markets and are successful during this initial period may infer that they have ability and will continue to trade. If this success persists, we may argue that they have been learning by trading and are applying the appropriate weight to their own signals using Bayesian updating. Those who are unsuccessful and do not exit from the sample (i.e. learning about ability), and those who change their bet sizes based on random rewards may have been subject to naïve reinforcement learning and have learned to fail. To examine changes in bettor behaviour across different groups of bettors, we adopt a similar approach to Strahilevitz et al. (2011), using decision trees to present the evolution of mean stake over the first four bets 8. Figures 3 and 4 present the mean stake changes, levels of attrition and 7 We will summarize the results contained in eight decision trees in total in the next section: one for each group of bettors (Financial and simulated) and for each bet price (5/1, Evens, 5/6, 1/20.). The initial analysis here focuses on bet prices in aggregate and contains only two aggregate groups. 8 The reason we focus on the first four bets is that the mean and median number of bets placed is the sample are 4.9 and 1, respectively 8

9 survival rates for the bettors in each group at each of the first four bets. [Table 3 about here.] [Table 4 about here.] We have restricted membership to each of the two main groups of financial and simulated bettors to bettors who transacted only on a single bet price and only on a single product type in the first four bets. While this has reduced the sample somewhat, it should allow for a more focused test. As can be seen in Tables 1 and 2, the data is highly skewed. We therefore standardized each bettor s initial stake size to one and examine the subsequent changes in stake size at each round of betting, however, we do present both parametric and non-parametric tests in the analysis. The first node in Figures 3 and 4 shows the level of attrition in parentheses (100% initially as all bettors have transacted a first bet), the count of bettors in the group, the standardized mean stake (set to one for the initial bet), and the count of losers and winners at that round. Each subsequent node shows firstly the level of attrition in parentheses, and then the number of bettors playing at that round of betting, the mean stake size and the results of both a one-sample mean test and a one-sample median test (in parentheses), testing for a statistically significant difference in mean and median stake size from one. Asterisks indicates significance at the standard levels. As there is asymmetric attrition from the dataset, we account for this by calculating the mean for survivors only and also an attrition-adjusted mean. The attrition-adjusted mean is calculated on the basis that those who left the sample kept betting with a stake size of zero, being allocated to subsequent nodes according to the probability of winning at each bet 9. Attrition-adjusted mean stake changes and statistics are presented in italics in both decision trees. Expected counts and survival rates in each of the end nodes are calculated on the basis of these probabilities. As regards attrition, and focusing on the end nodes, less financial market bettors drop out of the dataset. In the winning nodes of the fourth bet, 60.8% of bettors that are expected to participate in the next bet are present, whereas the figure for simulated market bettors is only 44.0%. For losing financial bettors, 13.9% of the expected count are present, while the figure for simulated market bettors is only 5.4%. In addition, there is a monotonic increase in attrition with the number of losses, which is expected. These results offer support for our second hypothesis that there is less attrition from the financial markets. This may be due to bettors learning by trading, rationally continuing in the face of losses with a view to either improving or indeed learning about ability and subsequently leaving the dataset at some point. We will return to the distiction between these two types of learning in Section 7. As path dependent behaviour is likely to be most salient for those bettors who have experienced a series of wins or losses, rather than those who have had mixed results, we summarize the results presented in Figures 3 and 4 in Table 5, focusing only on bettors in each of the two most extreme nodes at bets three and four. 9 The probabilities inferred from the sample are 0.11, 0.42, 0.5 and 0.93 for 5/1, Evens, 5/6 and 1/20 bets, respectively 9

10 [Table 5 about here.] Using the methodology outlined above, our null hypotheses suggest that mean standardized stake sizes in the simulated market should be equal to one and path-independent. Mean standardized stake sizes for financial markets are expected to be different from one, indicating either reinforcement learning or rational learning, and path-dependent. As shown in Panel A of Table 5, by the fourth (third) bet, losing financial market bettors have increased their stake sizes by a factor of 2.21 (2.39), however losing simulated market bettors have also increased their stake sizes, albeit to a lesser extent, by a factor of 1.75 (1.40). For bettors in the winning nodes, by the fouth (third) bet, financial market bettors have increased their stake sizes by a factor of 3.37 (2.17), while simulated market bettors have increased their stake sized by an equivalent amount to 3.17 (2.30). These standardized stake changes are statistically different from one (i.e. no change). When attrition is accounted for, the stake sizes for losers on the simulated market collapse to 0.04 (0.06) by the fourth (third) bet, whereas losing financial market bettors reduce their stake sizes to 0.20 (0.38). Winning financial bettors increase their stake sizes to 1.84 (1.26), while winning simulated market bettors increase their stake sizes only to 0.52 (0.47). In effect, winners increase their stake sizes irrespective of market setting. When we account for the amount of attrition at each node by including those that dropped out with a stake size of zero, the stake size changes for financial market bettors are still greater than one, while simulated market winners do not increase their bets by the same magnitude. It seems that surviving financial market winners increase their bet sizes, whereas surviving simulated market winners do not. This offers support to our first set of hypotheses. As regards losers, bettors on the financial markets increase their bet sizes, while simulated market bettors also do so, but not to the same extent. When we account for the amount of attrition from each group, the mean standardized stake changes decrease, and more so for the simulated market. Losing bettors who survive bet more on financial markets than on the simulated market. We will examine whether or not this is due to a disposition to avoid losses in Section 7. We have seen path-dependent behaviour in two groups of bettors, both financial and simulated, in aggregate, and we have restricted membership to each of the two groups to bettors who bet only on a single bet price and only on a single betting product during the first four bets. However, these results may be driven by differences in risk preferences and it may not be appropriate to group customers who bet on the highest-risk propositions (Evens and 5/1) with those who show a preference for the more risk-averse betting propositions (5/6, 1/20). In addition, the stake size changes have, on balance, all been statistically significantly different from one, which may not be the case when we add further controls. To examine how differences in risk preferences affect our results, we further partition each of the two groups by bet price, and present the results in Section 6. 10

11 6 Univariate Results In this section, we test for path-dependent stake size changes and attrition by bet price. Rather than presenting a series of eight decision trees (a decision tree for each of the two groups of bettors for each of the bet prices: 5/1, Evens, 5/6 and 1/20), we present the results for the winning and losing nodes in a table. A summary of the changes in behaviour in the last node of the winning and losing domains of the decision trees is presented in Table 6, cross-tabulated by bet price and bettor group with attrition-adjusted statistics also included as before. We find significant differences in bet stake across bet prices and path dependent changes in behaviour, with significant differences in behaviour in both settings. [Table 6 about here.] Not only are changes in stake size path dependent in both settings, but there is a difference in behaviour between the two groups of bettors. Of the bettors that were unsuccessful, those that bet only on Financial markets, and on the bet prices with the most risk, increase the intensity of their betting almost fourfold, while the estimate for those that bet only the simulated market only suggests that bettors increase their bet size by a factor of However, for the Financial markets one-sample mean tests for the difference between these estimates and 1 (i.e. no change) were not significant at the one percent level (one-sample median tests were), therefore we cannot conclude that there has been a statistically significant change in bet size for financial markets. After adjusting for attrition, losers reduce their bets most on the most risky prospects, but this reduction may be caused mainly by the amount of attrition. Those who survive may not have made peace with their losses in the prospect-theoretical sense (i.e. those who survive bet more, but those who leave the dataset accept their losses and stop betting). Only the results of a one sample mean test are reported in Table 6 whereas the one sample median tests shown in the individual decision trees are untabulated. The median test shows significant results, however the means test for surviving winners are not significantly different from 1 at the one percent level. Nonetheless, the results of the same test on the attrition adjusted values are all significant at the one percent level, indicating that attrition from the sample is changing bet sizes across all bet prices and that this change is significant. Taken together, these results indicate statistically significant stake size changes across almost every bet price for surviving winners in both settings (no difference on Evens bets for winners), and statistically significant bet size changes across some bet prices for surviving losers (no difference on 5/1 bets for winners). As regards bet size changes, these results are consistent with both rational learning in Financial markets and reinforcement learning in the simulated market and address the first of our two hypotheses. 11

12 7 Multivariate Results In this section, we formalize the focused test presented previously in both the decision trees and summary tables. We control for risk preferences by including bet price as an independent variable in a regression model estimation and also include further controls such as the variation in stake prior to the fourth bet. For the simulated market bettors, if the first bet is an unbiased estimator of the fourth bet, a simple regression model with the fourth bet as dependent and the first bet as independent should yield a constant of zero and a slope of one. If this is the case, it indicates no path-dependent behaviour in the simulated market group. We therefore firstly estimate a very simple model with only a single independent variable and thereafter add further controls to examine bettor behaviour. We then perform an F-test of the constant and coefficient of interest. The structure of the test is essentially a test of forecast accuracy in the same vein as a Mincer-Zarnowitz Test (Mincer and Zarnowitz, 1969), and is specified as follows: ln(s(4)) = β 0 + β 1 ln(s(1)) + e i (1) where ln(s(4)) is a bettors fourth stake and ln(s(1)) is a bettors initial stake. Our expectation for the simulated market is β 0 = 0 and β 1 = 1 if S(1)i is an unbiased forecast of S(4). We first perform an initial test on winners and losers, and then introduce our financial market and simulated market groups, controlling for bet price and bet stake range. We do this for both the survivor-only and an attrition-adjusted sample. Rather than incorporate all dropouts from the first bet, assigning bettors to nodes according to their inferred probabilities, we simply look at customers who had at least three bets. Those that continued and transacted a fourth bet are included in the survivor sample. Those that dropped out at this point are included in the sample with a stake size of zero 10. Initially, we will use ln(s(1)) as the main coefficient of interest, while in a later section, we will replace ln(s(1)) with ln(be), the amount a losing bettor would need to wager to avoid realising a loss. The estimation results for losers are presented in Table 7, while the results for winners are presented in Table 8. [Table 7 about here.] Panel A of Table 7 shows the results for losing bettors for survivors only, while Panel B includes the attrition-adjusted sample. In Panel A, the initial model ((1)), shows the estimation for all losing bettors, irrespective of market type. The main models of interest in this table are Models (2) and (5), those for losing financial market bettors and losing simulated market bettors. While the constant is non-trivial to interpret on its own, as it indicates the stake size in the fourth bet had 10 The regression models were initally estimated using this expanded attrition-adjusted sample and the results were broadly similar. The only difference was in the counts of observations in each of the attrition-adjusted model estimations. For example, for all losing bettors, irrespective of market type group, the count is 512. Including bettors from the first bet resulted in an attrition-adjusted N of 2537 as opposed to

13 the first bet size been zero, it is instructive to examine the prediction for the full amount bet in the fourth bet, given the first bet size. We plot these estimation results in Table 9. Of particular note is the fact that we can t reject the hypothesis that the constant of the model for simulated market losers is zero and that the slope is one, as the results of the MZ Test and the t-statistics on the coefficient in Model (5) are insignificant. Therefore, this indicates that the first bet is an unbiased estimator of the fourth bet in the case of the simulated market. In effect, there is path-dependent behaviour in the financial markets, but none in the simulated market. Also, when controls are added, including dummies for bet prices, this relationship stays broadly the same. When stake range is added, however, the effect on the other coefficients is significantly reduced. This is expected. A bettor who has already exhibited variability over the first three bets would be more likely to have a bet stake change in the fourth bet. The results for winners are show in Table 8. [Table 8 about here.] Again, Panel A shows the results for winning bettors for survivors only, while Panel B includes the attrition-adjusted sample. In Panel A, the initial model ((1)), shows the estimation for all losing bettors, irrespective of market type. The main models of interest in this table are Models (2) and (5), those for winning financial market bettors and winning simulated market bettors. We can reject the hypothesis that the constant of the model for simulated market losers is zero and that the slope is one, as the results of the MZ Test and the T-statistics on the coefficient in Model (5) are significant. This indicates that the first bet is not an unbiased estimator of the fourth bet in both the financial markets and the simulated market. In effect, there is path-dependent behaviour in both settings. To summarise the proceeding tables, we plot the regression line of the two main estimation results in the previous two tables, for both winners and losers, in Table 9. [Table 9 about here.] We plot a zero and one for the constant and slope, respectively, for losing simulated market bettors as we cannot reject the MZ Test presented in Table 7 for this group. Thereafter, we plot the results of Models (2) and (5) for the survivor-only and attrition-adjusted samples. It is clear that winners in both setting increase their stake sizes, however in the losing domains, the results are not as clear. Statistically, the losing simulated market bettors have not changed their bet sizes, while financial market bettors, on the other hand, have. Of further note is that in the two winning plots, it seems that bettors increase their stakes up to a point, by that larger stake sizes are not associated with increases in the fourth bet. It may be that the decision of what magnitude to place on the fourth bet is not dictated by the initial bet, but rather by the bettors current profit or loss. Although the concepts of learning about ability and learning by doing may be at play in the winning domain, the disposition to avoid losses may be what s driving the participation by losing bettors in the fourth bet. In effect, 13

14 losing bettors who learn in a rational Bayesian sense should be reducing the weight they apply to their own ability in the financial markets and decreasing their stakes accordingly or dropping out, but seem to increase their stake size regardless. In the following section, we replace the first stake with the breakeven profit/loss amount for each bettor to examine whether the current reference point for each bettor in terms of profit or loss is driving the behaviour in the fourth bet. 7.1 The disposition to avoid losses Kahneman and Tversky (1979) list a series of situations in which observed behaviour is contrary to that predicted by expected utility theory and propose a framworke called Prospect Theory as a model to describe decision making under risk. The most relevant for this analysis are the certainty effect and the reflection effect (choices made by respondents in the negative domain were the opposite of those made in the positive domain). The reflection effect describes a situation in which risk seeking choices in the positive domain are associated with the opposite choices in the negative. Shefrin and Statman (1985) examine agents disposition towards holding on to losing stock positions and realising gains relatively early. Their research question centred around whether investors behaviour was at odds with that proposed by Constantinides (1983) who proposed a prescriptive tax avoidance strategy based on the difference between short-term and long-term tax rates for gains and losses. As regards prospect theory, the disposition effect can be framed by agents who face a paper loss (in the editing phase suggested by Kahneman and Tversky (1979)) as a choice between the certain prospect of realising a loss and the risky prospect of the stock rebounding with a certain probability or continuing to decrease at another. Prospect theory dictates that as the reference point (the current paper loss) is in the negative domain, the risky alternative will dominate. The first example in the paper describes the two alternatives open to an investor who is long a stock that has reduced in price by $10: A. Sell the stock now, thereby realizing what had been a $10 paper loss. B. Hold the stock for one more period, given odds between losing an additional $10 or breaking even. In our setting, and with reference to that proposal by Kahneman and Tversky (1979) and Golec and Tamarkin (1998) that agents may based their behaviour on a full evening of betting, agents with negative returns are faced with the following three propositions: A. Stop playing, accepting a certain loss to date with 100% probability. B. Play again with a 50% probability of winning 83% on a 5/6 bet and a 50% probability of losing 100% of the stake. In effect, the choice is between continuing (survival) and quitting (hazard) and entails a number of distinct choices (using the current state of winnings/losses as a reference point). For those in the 14

15 positive domain: a sure gain (i.e. attrition/quitting while ahead) or a risky gamble (i.e. continue to play). For those in the negative domain: a sure loss (ie. attrition/realizing a paper loss or sunk cost) or a risky gamble (i.e. continue to play). In order to frame the next section in the literature, Table 10 presents our assumptions governing betting on the Financial markets and the simulated market. S(1) and S(4) refer to a bettors initial stake size and fourth stake size, respectively. Again, the reason we focus on the first four bets is that the mean and median number of bets placed is the sample are 4.9 and 1. W W W and LLL refer to a series of winning and losing bets prior to execution of the fourth bet. S breakeven is the equivalent amount an agent would have to wager in the fourth bet to neutralise losses incurred before the fourth bet. [Table 10 about here.] As returns on the Financial markets are not uncorrelated, learning by doing is possible and will willing manifest itself in both continued participation (lower attrition) and overconfidence (larger bet sizes) by bettors. However, repeating actions that have previously been associated with positive outcomes is also consistent with reinforcement learning. These assumptions are presented in the topmost node of Panel A. The disposition to realise gains and avoid loses is treated in the inner nodes of the figure. In the lower panel, as it is not possible to learn about ability or learn by trading, and if there are changes in stake size in the simulated market, this can be consistent only with reinforcement learning. Loss aversion and the disposition to realise gains are again treated in the inner nodes. There is no change in the set up for bettor in either group in the winning domain, therefore, we only present estimation results for surviving losers and an attrition-adjusted sample of losers in Table 11. We will summarize the estimation results by using plots, as before, and will update the estimation results for the losing bettors accordingly. [Table 11 about here.] As in the previous section, to summarise the proceeding tables, we plot the regression line of the two main estimation results in Table 12. Table 12 presents the results of the simple model ln(s(4)) = β 0 +β 1 ln(s(1))+e i, where S(1) and S(4) are agents stakes in the first and fourth bets, respectively. In the two figures in losing domain, S(1) has been replaced with S breakeven, the amount an agent would have to wager to neutralise losses incurred up to the fourth bet and avoid realizing a loss. To account for attrition from the dataset, we present regression estimation results for survivors only and also adding customers who dropped out after the third bet with zero stake sizes. ln(s(1)) has been replaced with ln(1+s(1)) in the attrition adjusted sample. In each figure, the dashed (black) line indicates a 45 line. The solid (red) line indicates the survivor-only sample, while the dotted (blue) line indicates the attritionadjusted sample with S(4) ˆ plotted for a range of values. [Table 12 about here.] 15

16 After the replacement of S(1) with S breakeven, the results for the losing simulated market bettors are broadly the same, however the result of the MZ Test is significant, indicating path-dependent behaviour. A breakeven point of e 15 before the fourth bet is associated with an increase above and beyond the breakeven point in the fourth bet as agents attempt to recoup their losses and avoid the emotions assosicated with regret and disappointment. (The plots of the winning domains are indentical, as the specification of the model presented in Table 10 did not change in the winning domain). However, contrary to expectations, including the breakeven point rather than the first bet has resulted in a futher downward shift for losing Financial market bettors in comparision with the first set of plots. This indicates that a breakeven point of e20 is associated with a decrease in bet size for the fourth bet to slighty above e10. This is counterintuitive and does not sit with the framing presented earlier. It may be that the breakeven point becomes more salient as the number of rounds increase. If the evolution of bet stake changes from the 2nd bet through to the 10th bet were plotted it may show an increase in the effect of the breakeven point until it resulted in a higher subsequent stake. It is possible that the disposition to avoid losses does not appear at as discrete a time period as between the third and fourth bets. It may be that losers on the simulated market are subject to the disposition to avoid losses whereas those who show a preference for betting only on the simulated market do not. 8 Conclusion In financial market transactional data, it is not trivial to disentangle the relative magnitude and effect of rational and reinforcement learning. In a simulated market, however, there is no such confound. Empirical tests of learning theories in a laboratory setting infer how investors learn and behave in financial markets. We show that the effect of reinforcement learning may not be any stronger in financial markets than in a simulated market. This suggests that empirical tests may overstate the true level of biased learning in financial markets with ramifications for the existence, prevalence and magnitude of pricing anomalies. We compare learning in a baseline market where there can be no private information, no herding or information cascades, no tax-loss selling and no observation or imitation, (only private signals and randomness) with a market that has all of the preceding characteristics. This may be because agents are boundedly rational or because they lack the probabilistic information about the structure of payoffs necessary to successfully apply Bayes rule (Wiseman, 2009). We find that agents have the same propensity for reinforcement learning in a simulated market treatment where outcomes are uncorrelated and learning by doing is impossible as they do in a treatment where the literature suggests they should rationally learn from their past experiences. In effect, they place higher relevance on their previous success or losses and disregard public information and other signals in a type of reverse informational cascade. Our results have broader implications for learning theories in financial markets. Bettors on the 16

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