Ethereum has also introduced the idea of programming blockchain operations through a technology called the smart contract. A smart contract is an automated script written in Ethereum's own scripting language, Solidity, that allows an individual to exchange a specified good or service. A popular comparison for smart contracts is the vending machine. If a user of the smart contract gives the vending machine a certain fee, and a product comes out. Accordingly, if a user inputs some cryptocurrency into a smart contract, it executes an exchange of goods or services. As smart contracts are also automated, they erase the need for a middleman. Smart contracts, if programmed properly, can be used for a variety of applications, such as vote automation or tax collection. Building an application on top of a smart contract creates a decentralized application (DApp). A DApp is completely decentralized (no single owner) and automated by its associated smart contract. Currently, there are around 1,539 DApps on the Ethereum Blockchain [10] .
We study the behavioral dynamics of gamblers on a DApp known as Etheroll. Etheroll simulates a virtual dice gambling game where all bets are made in Ether and published on the Ethereum blockchain. Etheroll has an associated smart contract on the Ethereum network which specifies house edges, payouts, and dividends to investors [11] . To begin the dice game, the gambler chooses a number between 2 and 99 (inclusive). The probability that the gambler wins is the number he or she chooses, minus 1, meaning that the gambler can choose between a 1% to 98% chance of winning. The payout (P ) formula, if the house commission per bet is e = 1%, probability of winning is p, and initial wager is W is:
The smart contract then simulates a hundred-sided dice roll. If the result of the dice roll is any number smaller than the number the gambler chose, the gambler wins. After the transaction between the smart contract and the gambler processes, the gambler receives a payout (in Ether) directly to their Ethereum wallet which is inversely proportional to the probability they bet at. Naturally, lower probabilities of winning have higher payouts, and higher probabilities of winning have lower payouts. Regardless of their chosen winning probabilities p, the expected payout of gamblers is negative, E[P ] = −eW , due to the house commission fee charging e percentage for each bet W .
These transactions are publicly available on the Ethereum blockchain. Due to the massive amount of verifying nodes on the Ethereum network, we can be sure about the validity of these transactions. We will explore this data for all four of Etheroll's smart contract updates from April 17th, 2017 to December 12th, 2017. Obtaining real life gambling data, especially data from gambles in casinos is very difficult, if not impossible to obtain. Because of this, mathematical models pertaining to gambling are almost entirely theoretically based. Every bet from Etheroll consists of the bet's Timestamp, Wager, Probability of Winning, UserID, and Profit. With this data, we shall empirically explore gambling behavior and risk attitudes in light of the cumulative prospect theory [12] .
This dataset has many other interesting properties. Having access to timestamps allows us to identify possible changes in strategy influenced by gambling results over time, in their gambling patterns. The fact that gamblers are able to tune their own betting probabilities is also crucial. The ability to tune the effective odds in a wager allows us to evaluate probable risk profiles of certain gamblers. Additionally, we focus on characterizing the entire risk attitudes of the entire gambling ecosystem as a whole. We are also able to evaluate the existence and usage of staking gambling systems (path-dependent strategies). The unique completeness and continuity of this data also allows to us empirically evaluate some famous psychological frameworks, such as the cumulative prospect theory [12] . We also characterize and quantify the effect of a gambler's cumulative "signal", or scaled cumulative profit on their winning probability distributions and betting strategies. This scaling allows us to model the lessened effect of losses and gains over time. 
This population of gamblers on the Ethereum blockchain allows us to empirically observe the tendencies of gamblers in a casino-like environment for the first time. The minimum bet-sizing of 0.1 Ether (4 -53 USD in this dataset) simulates casino-like stakes [13] . The game these gamblers play is simple, parameterized only by the probability of winning they chose and their wager size. We will first characterize the types of gamblers in this online casino through these two variables, the overall distribution of these two variables, and the paired cohort of winning gamblers and losing gamblers. We will also look at how gamblers behave when conditioned on the previous bet. To do this, we will measure the absolute and relative changes in both the probabilities they chose and their wager sizing. We will also measure the cumulative profit of gamblers -to understand how many gamblers actually end up winning anything. Lastly, we will look at gambling strategies and gamblers of interest.
To first characterize this population of gamblers, we visualize the total bet frequency distributions of each gambler. Using histograms, we track each gambler's total gambles per smart contract, and the corresponding frequency of occurrence. In doing so, there is a very pronounced right skew in the distribution of the amount of gambles of each gambler. In fact, in each smart contract iteration, the gamblers who gamble only 1 to 10 times comprise approximately 60-65% of the entire population. This heavy right skew shows that most gamblers in this DApp are mainly recreational gamblers who place anywhere from 1 to 10 bets (See Tab. 1, Fig. 1 ). An interesting qualitative feature of these distributions is that throughout each contract iteration, the relative bet frequencies of these gamblers remained relatively constant. Another interesting feature of the data is the existence of a tail of gamblers who bet at high frequency. The "Whale Bettors", or bettors who bet more than a hundred bets and contribute most of the actual bets on the website comprise only a small fraction of the actual gamblers. Due to the gambler's ruin theory, we see that these whale bettors, who frequently gamble, must be more risk-taking. In contrast, the gamblers who gamble less must be more risk-averse.
We see a very similar right skew in the bet size distributions of Contracts 1, 2, 3 and 4 (Fig. 2) . However, Contract 1 displays a surprising amount of gamblers that are willing to gamble at large bet sizes (Fig. 2a) . Additionally, there are always a few gamblers willing to bet at significant sizings (> 80 ETH, as shown in Figs. 2a-d). Possible reasons for this were probably due to the relatively low price of Ethereum (approximately 1 ETH : 50 USD). Additionally, there were only 90,000 total transactions on the Ethereum network at the time. Many of these gamblers probably did not expect the prices to exponentially rise to 500 USD/ETH.
In observing the overall distribution of the probabilities that the gamblers on Etheroll gamble at, we observe two interesting fixations (Fig. 3) . First, gamblers are extremely drawn to probabilities An interesting qualitative feature of these distributions is that throughout each contract iteration, the relative bet frequencies of these gamblers remained relatively constant. Another interesting feature of the data is the existence of a tail of gamblers who bet at high frequency. The "Whale Bettors", or bettors who bet more than a hundred bets and contribute most of the actual bets on the website comprise only a small fraction of the actual gamblers. Due to gambler's ruin, we see that these whale bettors, who frequently gamble, must be more risk-taking. In contrast, the gamblers who gamble less must be more risk-averse. We see a very similar right skew in the distributions of Contracts 1, 2, 3 and 4. However, Contract 1 displays a surprising amount of gamblers that are willing to gamble at large bet sizes. Additionally, there are always a few gamblers willing to bet at significant sizings (> 80 ETH). Possible reasons for this were probably due to the relatively low price of Ethereum (approximately 1 ETH : 50 USD). Additionally, there were only 90,000 total transactions on the Ethereum network at the time. Many of these gamblers probably did not expect the prices to exponentially rise to 500 USD/ETH.
In observing the overall distribution of the probabilities that the gamblers on Etheroll gamble at, we observe two interesting fixations. First, gamblers are extremely drawn to probabilities within the bound of p = 0.4 0.6. This is slightly different from what median Cumulative Prospect Theory preferences specify, as probabilities around 0.35 0.6 are underweighted, rather than overweighted. Lower probabilities have the opposite pattern. Additionally, these gamblers also An interesting qualitative feature of these distributions is that throughout each contract iteration, the relative bet frequencies of these gamblers remained relatively constant. Another interesting feature of the data is the existence of a tail of gamblers who bet at high frequency. The "Whale Bettors", or bettors who bet more than a hundred bets and contribute most of the actual bets on the website comprise only a small fraction of the actual gamblers. Due to gambler's ruin, we see that these whale bettors, who frequently gamble, must be more risk-taking. In contrast, the gamblers who gamble less must be more risk-averse. We see a very similar right skew in the distributions of Contracts 1, 2, 3 and 4. However, Contract 1 displays a surprising amount of gamblers that are willing to gamble at large bet sizes. Additionally, there are always a few gamblers willing to bet at significant sizings (> 80 ETH). Possible reasons for this were probably due to the relatively low price of Ethereum (approximately 1 ETH : 50 USD). Additionally, there were only 90,000 total transactions on the Ethereum network at the time. Many of these gamblers probably did not expect the prices to exponentially rise to 500 USD/ETH.
In observing the overall distribution of the probabilities that the gamblers on Etheroll gamble at, we observe two interesting fixations. First, gamblers are extremely drawn to probabilities within the bound of p = 0.4 0.6. This is slightly different from what median Cumulative Prospect Theory preferences specify, as probabilities around 0.35 0.6 are underweighted, rather than overweighted. Lower probabilities have the opposite pattern. Additionally, these gamblers also a b c d Figure 2 : Distributions of bet sizes for each of the four contracts, panels (a) -(d), from April 17th, 2017 to December 12th, 2017. There exist "whale bettors" who gamble with considerably large bet sizes.
within the bound of p = 0.4 − 0.6. This is slightly different from what median cumulative prospect theory preferences specify [12] , as probabilities around 0.35 − 0.6 are underweighted, rather than overweighted. Lower probabilities have the opposite pattern. Additionally, these gamblers also have a fixation towards probabilities with very high chances of winning, within p = 0. Additionally, we wish to look at the probability distributions of two possible cohorts of the gambling population: gamblers who win, and gamblers who lose. To do this, we segment our data into gambles of gamblers who lose and those who win. Additionally, we wish to look at the probability distributions of two possible cohorts of the gambling population: gamblers who win, and gamblers who lose. To do this, we segment our data into gambles of gamblers who lose and those who win. This showcases these gamblers are qualitatively more risk averse. This is an odd result. First, we observe that theoretically, gamblers in this casino are more likely to be a self-selecting, risk seeking group. Second, these gamblers must have some interest in Ethereum, and are also forced to bet significant minimum bet sizings (5-53 USD). Additionally, we plot the probability distributions in two separate cohorts of the gambling population conditional on: gamblers who win, and gamblers who lose. To do this, we segment our data into gambles of gamblers who lose and those who win.
In all four contracts, the losing cohort of gamblers have very similar losing distributions (see Fig. 4 ). In general, there is a large central mean at p = 0.5. In Contracts 2 and 3, there is a nearly normal distribution in their probabilities. We observe that in every contract, nearly 25% of bets are losing bets at around p = 0.5. Additionally, many of the extremely risky gamblers who bet at p < 0.5 are represented in this cohort. Naturally, gamblers who bet like this will tend to lose more often. The other tail end of the distribution comprises of the gamblers take p > 0.5 gamblers, tending to be less risky and loss averse. However, as p = 1, they are still bound to lose, and with a maximum probability of 0.99, they will still lose at least 1 out of 100 times on average.
Lastly, we observe that the winning cohort also has relatively similar general distribution throughout the four contracts (Fig. 5 ). This is due to the fact that with a large enough sample, individuality is mostly canceled out. At first glance, it is apparent that there is a distinct left skew in the distribution, with a large amount of bets being distributed at p > 0.5. However, there is still a noticeable fixation by these gamblers to bet at p = 0.5. We also notice that these winners probably tend to be more risk averse, as most of the data is accumulated at p > 0.7. Very few winners occur in the region of p < 0.5, which comprises less than 10% of the data on average per each contract iteration. Lastly, an interesting feature in almost every distribution is an aversion to 0.5 < p < 0.7. This may be due to the shape of the perceived probability weighting function, where probabilities that are higher than p > 0.5 are overweighted. An explanation for why p > 0.7 is so popular comes in the loss-aversion formulation of the value function (see more detailed analysis in our Discussions section below). As gamblers generally wish to avoid losses, they tune their probabilities very high to avoid losses. Additionally, we wish to look at the probability distributions of two possible cohorts of the gambling population: gamblers who win, and gamblers who lose. To do this, we segment our data into gambles of gamblers who lose and those who win.
In all four contracts, the losing cohort of gamblers have very similar losing distributions (see Additionally, we wish to look at the probability distributions of two possible cohorts of the gambling population: gamblers who win, and gamblers who lose. To do this, we segment our data into gambles of gamblers who lose and those who win.
In all four contracts, the losing cohort of gamblers have very similar losing distributions (see Figure 4 .3). In general, there is a large central mean at p = 0.5. In Contracts 2 and 3, there is a a b c d Figure 4 : Distributions of winning probabilities of bets that ended up losing for each of the four contracts, panels (a) -(d), from April 17th, 2017 to December 12th, 2017. Gambling with small winning probabilities imposes great risk of losing, and thus losing bets are skewed towards unfavorable winning probabilities that are less than 50%.
Another way we can observe risk attitudes is in evaluating the relative bet sizing of gamblers versus their tuned probabilities of winning (Fig. 6 ). These plots showcase the loss aversion of most gamblers. When staking very large bets, these gamblers exclusively bet at very high probabilities (Fig. 6) . The largest bets are always bet at extremely high probabilities. With smaller sizings, we see a whole range of probabilities of winning. In general, this is not an extremely surprising finding, as we expect most gamblers to be loss averse.
Another interesting function of this is that these bet frequency and probability distributions mostly shared similar shapes throughout all four contract iterations (Fig. 1, and Fig. 2 ). However, the actual value of these bets greatly varied due to the drastic changes in the market price of Ethereum, the cryptocurrency used in the gambling.
Additionally, it is interesting to see the distribution of the cumulative profits at the end of each gambler's gambling time (see Fig. 7 ).
It appears that Contract 1, Contract 4 have a distinctly normal distribution with µ < 0. However, Contract 3 seems to have a left skewed distribution, and Contract 4 seems to have a right skewed distribution. We also can conclude that most gamblers do not really win anything (matching up with the fact that 60% of gamblers are recreational gamblers). We see that the mathematical edge of the casino (the house commission fee charging e proportion of each bet) has effectively shifted the normal distribution to the left, as expected. This implies that most gamblers are net losers, as expected from an edged game. 
nearly normal distribution in their probabilities. We observe that in every contract, nearly 25% of bets are losing bets at around p = 0.5. Additionally, many of the extremely risky gamblers who bet at p < 0.5 are represented in this cohort. Naturally, gamblers who bet like this will tend to lose more often. The other tail end of the distribution comprises of the gamblers take p > 0.5 gamblers, tending to be less risky and loss averse. However, as p 6 = 1, they are still bound to lose, and with a maximum probability of 0.99, they will still lose at least 1 out of 100 times on average.
Lastly, we observe that the winning cohort also has relatively similar general distribution throughout the four contracts. This is due to the fact that with a large enough sample, individuality is mostly canceled out. At first glance, it is apparent that there is a distinct left skew in the distribution, with a large amount of bets being distributed at p > 0.5. However, there is still a noticeable fixation by these gamblers to bet at p = 0.5. We also notice that these winners probably tend to be more risk averse, as most of the data is accumulated at p > 0.7. Very few winners occur in the region of p < 0.5, which comprises less than 10% of the data on average per each contract iteration. Lastly, an interesting feature in almost every distribution is an aversion to 0.5 < p < 0.7. This may be due to the shape of the probability weighting function, where probabilities that are higher than p > 0.5 are overweighted. An explanation for why p > 0.7 is so popular comes in the loss-aversion formulation of the value function. As gamblers generally wish to avoid losses, they tune their probabilities very high to avoid losses.
Another way we can observe risk attitudes is in evaluating the relative bet sizing of gamblers versus their tuned probabilities of winning.
These plots showcase the loss aversion of most gamblers. When staking very large bets, these gamblers exclusively bet at very high probabilities. The largest bets are always bet at extremely high probabilities. With smaller sizings, we see a whole range of probabilities of winning. In general, this is not an extremely surprising finding, as we expect most gamblers to be loss averse. 
nearly normal distribution in their probabilities. We observe that in every contract, nearly 25% of bets are losing bets at around p = 0.5. Additionally, many of the extremely risky gamblers who bet at p < 0.5 are represented in this cohort. Naturally, gamblers who bet like this will tend to lose more often. The other tail end of the distribution comprises of the gamblers take p > 0.5 gamblers, tending to be less risky and loss averse. However, as p 6 = 1, they are still bound to lose, and with a maximum probability of 0.99, they will still lose at least 1 out of 100 times on average.
Lastly, we observe that the winning cohort also has relatively similar general distribution throughout the four contracts. This is due to the fact that with a large enough sample, individuality is mostly canceled out. At first glance, it is apparent that there is a distinct left skew in the distribution, with a large amount of bets being distributed at p > 0.5. However, there is still a noticeable fixation by these gamblers to bet at p = 0.5. We also notice that these winners probably tend to be more risk averse, as most of the data is accumulated at p > 0.7. Very few winners occur in the region of p < 0.5, which comprises less than 10% of the data on average per each contract iteration. Lastly, an interesting feature in almost every distribution is an aversion to 0.5 < p < 0.7. This may be due to the shape of the probability weighting function, where probabilities that are higher than p > 0.5 are overweighted. An explanation for why p > 0.7 is so popular comes in the loss-aversion formulation of the value function. As gamblers generally wish to avoid losses, they tune their probabilities very high to avoid losses.
Another way we can observe risk attitudes is in evaluating the relative bet sizing of gamblers versus their tuned probabilities of winning.
These plots showcase the loss aversion of most gamblers. When staking very large bets, these gamblers exclusively bet at very high probabilities. The largest bets are always bet at extremely high probabilities. With smaller sizings, we see a whole range of probabilities of winning. In general, this is not an extremely surprising finding, as we expect most gamblers to be loss averse. 
Another important way we can understand gambling behavior is to look at the naturally conditional behavior these gamblers bet with. In this regard, we shall be able to quantitfy the levels of risk aversion and risk seeking behaviors that a gambler takes, given only their previous bet. This analysis only focuses on gamblers who have betted more than once in a given contract. Let us define B = {b 1 , b 2 , ..., bn}, P = {p 1 , p 2 , ..., pn}, R = {1, 0} where B is the set of ordered set of bets the gambler takes, P is the ordered set of their chosen probabilities of winning, and R is the final result, where 0 is a loss and 1 is a win. Let us define this dice game as the mapping of all ordered tuples in B and P to R, or B, P → R. Let us define:
This is the absolute difference in both their bet size or betting probability. We will also look at the relative (percent) difference in the bet size:
We will condition both of these on the result being either a win or a loss. Additionally, these measures obviously do not make sense for any gamblers who only bet a single time, so we will only measure these measurements for gamblers who gambled at least consecutively twice in a contract.
When we look at conditionally chosen probabilities of the gambler, it allows us to understand how they subjectively view a prospect given a previous result. Measuring the absolute difference in probabilities allows us a better view into the population level, subjective perception of the 
Another interesting function of this is that these Bet Frequency and Probability distributions mostly shared similar shapes throughout all four contract iterations. However, the actual value of these bets greatly varied. Additionally, it is interesting to see the distribution of the cumulative profits at the end of each gamblers gambling time (See Figure 4.6 ).
It appears that Contract 1, Contract 4 have a distinctly normal distribution with µ < 0. However, Contract 3 seems to have a left skewed distribution, and Contract 4 seems to have a right skewed distribution. We hypothesize that this distribution is most likely transient, meaning that the skew in Contracts 2 and 4 are determined by variance. We also can conclude that most gamblers do not really win anything (matching up with the fact that 60% of gamblers are recreational gamblers). We see that the mathematical edge of the casino has effectively shifted the normal distribution to the left, as expected. This implies that most gamblers are net losers, as expected from an edged game.
Another important way we can understand gambling behavior is to look at the naturally conditional behavior these gamblers bet with. In this regard, we can begin to understand the levels of risk aversion and risk seeking behaviors that a gambler takes, given only his or her previous bet. This analysis will only look at gamblers who have bet mâȂȃore than once in a contract Let us define B = {b 1 , b 2 , ..., bn}, P = {p 1 , p 2 , ..., pn}, R = {1, 0} where B is the set of ordered set of bets the gambler takes, P is the ordered set of their chosen probabilities, and R is the final result, where 0 is a loss and 1 is a win. Let us define this dice game as the mapping of all ordered tuples in B and P to R, or B, P ! R. Let us define: 
Another interesting function of this is that these Bet Frequency and Probability distributions mostly shared similar shapes throughout all four contract iterations. However, the actual value of these bets greatly varied.
Additionally, it is interesting to see the distribution of the cumulative profits at the end of each gamblers gambling time (See Figure 4.6 ).
It appears that Contract 1, Contract 4 have a distinctly normal distribution with µ < 0. However, Contract 3 seems to have a left skewed distribution, and Contract 4 seems to have a right skewed distribution. We hypothesize that this distribution is most likely transient, meaning that the skew in Contracts 2 and 4 are determined by variance. We also can conclude that most gamblers do not really win anything (matching up with the fact that 60% of gamblers are recreational gamblers). We see that the mathematical edge of the casino has effectively shifted the normal distribution to the left, as expected. This implies that most gamblers are net losers, as expected from an edged game.
Another important way we can understand gambling behavior is to look at the naturally conditional behavior these gamblers bet with. In this regard, we can begin to understand the levels of risk aversion and risk seeking behaviors that a gambler takes, given only his or her previous bet. This analysis will only look at gamblers who have bet mâȂȃore than once in a contract Let us define
where B is the set of ordered set of bets the gambler takes, P is the ordered set of their chosen probabilities, and R is the final result, where 0 is a loss and 1 is a win. Let us define this dice game as the mapping of all ordered tuples in B and P to R, or B, P ! R. Let us define: Figure 6 : Scatter plots of wager sizing versus probability of winning for every single bet for each of the four contracts, panels (a) -(d), from April 17th, 2017 to December 12th, 2017. Gamblers tend to be loss averse by tuning favorable winning probabilities for large bet sizes.
probabilities of their bets. We will refer to the absolute difference between the gambler's chosen consecutive bet probabilities as their probability signal. When looking at the summary statistics (Tab. 2), we see a division (but not extreme division) between the previous bet result being conditioned on a win or loss. We notice that as expected, gamblers who lost tend to bet at slightly larger probabilities where as gamblers that won tend to gamble at slightly smaller probabilities -taking less and more risk, respectively.
When looking at the absolute changes in probabilities chosen by the gambler, conditioned on their previous gamble, we notice an interesting pattern in the distribution (Fig. 10 ). It appears that the distribution has an extremely concentrated mean with fat tails (Laplacian). Calculation of the kurtosis of this distribution (controlled for contracts) shows that it is slightly leptokurtic (for all contracts except the case of contract 4 for losers -which is slightly platykurtic). This implies a fat tailed distribution, which implies that extreme events (deviations in probability choice size between bets) are more likely than in a normal distribution. 
This is the absolute difference in both their bet size or betting probability. We will also look at the relative (percent) difference in the bet size:
• 8i, i < n,
We will condition both of these on the result being either a win or a loss. Additionally, these measures obviously do not make sense for any gamblers who only bet a single time, so we will only measure these measurements for gamblers who gambled at least twice in a contract.
When we look at conditionally chosen probabilities of the gambler, it allows us to understand how they subjectively view a prospect given a previous result. Measuring the absolute difference in probabilities allows us a better view into the population level, subjective perception of the probabilities of their bets. We will refer to the absolute difference between the gambler's chosen consecutive bet probabilities as their probability signal. When looking at the summary statistics, we see a division (but not extreme division) between the previous bet being conditioned on a win or loss (result). We notice that as expected, gamblers who lost tend to bet at slightly larger probabilities where as gamblers that won tend to gamble at slightly smaller probabilities -taking less and more risk respectively.
When looking at the absolute changes in probabilities chosen by the gambler, conditioned on their previous gamble, we notice an interesting pattern in the distribution. It appears that the distribution has an extremely concentrated mean with fat tails (Laplacian). Calculation of the kurtosis of this distribution (controlled for contracts) shows that it is slightly leptokurtic (for all contracts except the case of contract 4 for losers -which is slightly platykurtic). This implies a This is the absolute difference in both their bet size or betting probability. We will also look at the relative (percent) difference in the bet size:
• 8i, i < n,
We will condition both of these on the result being either a win or a loss. Additionally, these measures obviously do not make sense for any gamblers who only bet a single time, so we will only measure these measurements for gamblers who gambled at least twice in a contract.
When we look at conditionally chosen probabilities of the gambler, it allows us to understand how they subjectively view a prospect given a previous result. Measuring the absolute difference in probabilities allows us a better view into the population level, subjective perception of the probabilities of their bets. We will refer to the absolute difference between the gambler's chosen consecutive bet probabilities as their probability signal. When looking at the summary statistics, we see a division (but not extreme division) between the previous bet being conditioned on a win or loss (result). We notice that as expected, gamblers who lost tend to bet at slightly larger probabilities where as gamblers that won tend to gamble at slightly smaller probabilities -taking less and more risk respectively.
When looking at the absolute changes in probabilities chosen by the gambler, conditioned on their previous gamble, we notice an interesting pattern in the distribution. It appears that the distribution has an extremely concentrated mean with fat tails (Laplacian). Calculation of the kurtosis of this distribution (controlled for contracts) shows that it is slightly leptokurtic (for all contracts except the case of contract 4 for losers -which is slightly platykurtic). This implies a 
This is the absolute difference in both their bet size or betting probability. We will also look at the relative (percent) difference in the bet size:
• 8i, i < n,
We will condition both of these on the result being either a win or a loss. Additionally, these measures obviously do not make sense for any gamblers who only bet a single time, so we will only measure these measurements for gamblers who gambled at least twice in a contract.
When we look at conditionally chosen probabilities of the gambler, it allows us to understand how they subjectively view a prospect given a previous result. Measuring the absolute difference in probabilities allows us a better view into the population level, subjective perception of the probabilities of their bets. We will refer to the absolute difference between the gambler's chosen consecutive bet probabilities as their probability signal. When looking at the summary statistics, we see a division (but not extreme division) between the previous bet being conditioned on a win or loss (result). We notice that as expected, gamblers who lost tend to bet at slightly larger probabilities where as gamblers that won tend to gamble at slightly smaller probabilities -taking less and more risk respectively.
When looking at the absolute changes in probabilities chosen by the gambler, conditioned on their previous gamble, we notice an interesting pattern in the distribution. It appears that the distribution has an extremely concentrated mean with fat tails (Laplacian). Calculation of the kurtosis of this distribution (controlled for contracts) shows that it is slightly leptokurtic (for all contracts except the case of contract 4 for losers -which is slightly platykurtic). This implies a This is the absolute difference in both their bet size or betting probability. We will also look at the relative (percent) difference in the bet size:
• 8i, i < n,
We will condition both of these on the result being either a win or a loss. Additionally, these measures obviously do not make sense for any gamblers who only bet a single time, so we will only measure these measurements for gamblers who gambled at least twice in a contract.
When we look at conditionally chosen probabilities of the gambler, it allows us to understand how they subjectively view a prospect given a previous result. Measuring the absolute difference in probabilities allows us a better view into the population level, subjective perception of the probabilities of their bets. We will refer to the absolute difference between the gambler's chosen consecutive bet probabilities as their probability signal. When looking at the summary statistics, we see a division (but not extreme division) between the previous bet being conditioned on a win or loss (result). We notice that as expected, gamblers who lost tend to bet at slightly larger probabilities where as gamblers that won tend to gamble at slightly smaller probabilities -taking less and more risk respectively.
When looking at the absolute changes in probabilities chosen by the gambler, conditioned on their previous gamble, we notice an interesting pattern in the distribution. It appears that the distribution has an extremely concentrated mean with fat tails (Laplacian). Calculation of the kurtosis of this distribution (controlled for contracts) shows that it is slightly leptokurtic (for all contracts except the case of contract 4 for losers -which is slightly platykurtic). This implies a We also provide a normal Q-Q plot (Fig. i) . This Q-Q plot exhibits a very interesting kinked curve -we see that there is clear overweighting in the tails and an interesting S-shaped behavior around the origin. This has the implication that people are more willing to have extreme changes in their chosen probabilities when measured at the bet to bet level compared to a normal distribution. Namely, there is greater tendency in extreme behaviors, such as drastically decreasing probabilities between bets or drastically increasing probabilities between bets. The Sshaped behavior around the origin also shows that gamblers have slight preferences for changing their probabilities very closely around a probability signal of 0, either slightly more or slightly less. Overall, we see that people generally prefer to either make very minor changes to their chosen probabilities or extremely major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to understand how they subjectively view the bet sizing of a prospect given a previous result. Measuring the absolute difference in wager sizes allows us a better view into the user level perception of their own prospect (Tab. 3). We also will isolate the control for each separate contract to further understand if there is significant differences between the gamblers of each contract. We will refer to the absolute difference between wagers as the gambler's bet signal so as to complement the analysis of bet probability signal shown in Figs. 10 and i.
When observing the summary statistics for the bet/wager signal, we notice the very similar behavior for users tending to decrease their bet sizes after a victory, but to increase their bet sizes after a loss (Tab. 3). We also notice that this distribution is heavily skewed and highly leptokurtic, no matter the cut (Fig. 10 ). An interesting difference between the contracts can be found in contract 4, which has far less variance and a generally tighter distribution.
When looking at the absolute changes in wagers chosen by the gambler, conditioned on their previous gamble, we notice an extremely concentrated distribution. Even on the log scale, we can see an extreme concentration of values near 0. It appears that the distribution has a classic, We also provide a Q-Q plot in Fig. 11 . This S-shaped plot shows the classic overdispersed, leptokurtic Q-Q curve, with a large amount of data being dispersed between the left and right tails. Additionally, there is some slight S-shaped behavior near the bet size values around 0, again implying some degree of sensitivity towards values near 0. It can be inferred that gamblers that We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kinked curve -we see that there is clear overweighting in the tails and an interesting S-shape behavior around the origin. This has the implication that people are more willing to have extreme changes in their chosen probabilities when measured at the bet to bet level compared to a normal distribution. That is there is greater tendency in extreme behaviors, such as drastically decreasing probabilities between bets or drastically increasing probabilities between bets. The S shape behavior around the origin also shows that gamblers have slight preferences for changing their probabilities very closely around a probability signal of 0, either slightly more or slightly less. Overall, we see that people generally prefer to make very minor changes to their chosen probabilities or extremely major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to understand how they subjectively view the bet sizing of a prospect given a previous result. Measuring the absolute difference in wager sizes allows us a better view into the user level perception of their own prospect. We also will isolate the control for each separate contract to further understand if there is significant differences between the gamblers of each contract. We will refer to the absolute difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice the very similar behavior for users tending to decrease their bet sizes after a victory, but increase their bet sizes after a loss. We also notice that this distribution is heavily skewed and highly leptokurtic, no matter the cut. An interesting difference between the contracts can be found in contract 4, which has far less variance and a generally tighter distribution. We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kinked curve -we see that there is clear overweighting in the tails and an interesting S-shape behavior around the origin. This has the implication that people are more willing to have extreme changes in their chosen probabilities when measured at the bet to bet level compared to a normal distribution. That is there is greater tendency in extreme behaviors, such as drastically decreasing probabilities between bets or drastically increasing probabilities between bets. The S shape behavior around the origin also shows that gamblers have slight preferences for changing their probabilities very closely around a probability signal of 0, either slightly more or slightly less. Overall, we see that people generally prefer to make very minor changes to their chosen probabilities or extremely major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to understand how they subjectively view the bet sizing of a prospect given a previous result. Measuring the absolute difference in wager sizes allows us a better view into the user level perception of their own prospect. We also will isolate the control for each separate contract to further understand if there is significant differences between the gamblers of each contract. We will refer to the absolute difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice the very similar behavior for users tending to decrease their bet sizes after a victory, but increase their bet sizes after a loss. We also notice that this distribution is heavily skewed and highly leptokurtic, no matter the cut. An interesting difference between the contracts can be found in contract 4, which We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kinked curve -we see that there is clear overweighting in the tails and an interesting S-shape behavior around the origin. This has the implication that people are more willing to have extreme changes in their chosen probabilities when measured at the bet to bet level compared to a normal distribution. That is there is greater tendency in extreme behaviors, such as drastically decreasing probabilities between bets or drastically increasing probabilities between bets. The S shape behavior around the origin also shows that gamblers have slight preferences for changing their probabilities very closely around a probability signal of 0, either slightly more or slightly less. Overall, we see that people generally prefer to make very minor changes to their chosen probabilities or extremely major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to understand how they subjectively view the bet sizing of a prospect given a previous result. Measuring the absolute difference in wager sizes allows us a better view into the user level perception of their own prospect. We also will isolate the control for each separate contract to further understand if there is significant differences between the gamblers of each contract. We will refer to the absolute difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice the very similar behavior for users tending to decrease their bet sizes after a victory, but increase their bet sizes after a loss. We also notice that this distribution is heavily skewed and highly leptokurtic, no matter the cut. An interesting difference between the contracts can be found in contract 4, which We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kink see that there is clear overweighting in the tails and an interesting S-shape behav origin. This has the implication that people are more willing to have extreme ch chosen probabilities when measured at the bet to bet level compared to a norm That is there is greater tendency in extreme behaviors, such as drastically decreasin between bets or drastically increasing probabilities between bets. The S shape be the origin also shows that gamblers have slight preferences for changing their pro closely around a probability signal of 0, either slightly more or slightly less. Over people generally prefer to make very minor changes to their chosen probabilitie major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to unders subjectively view the bet sizing of a prospect given a previous result. Measurin difference in wager sizes allows us a better view into the user level perception prospect. We also will isolate the control for each separate contract to further unde is significant differences between the gamblers of each contract. We will refer t difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice th behavior for users tending to decrease their bet sizes after a victory, but increase after a loss. We also notice that this distribution is heavily skewed and highly matter the cut. An interesting difference between the contracts can be found in co has far less variance and a generally tighter distribution. rmal Q-Q plot. This Q-Q plot has a very interesting kinked curve -we erweighting in the tails and an interesting S-shape behavior around the lication that people are more willing to have extreme changes in their en measured at the bet to bet level compared to a normal distribution. ndency in extreme behaviors, such as drastically decreasing probabilities ally increasing probabilities between bets. The S shape behavior around at gamblers have slight preferences for changing their probabilities very ility signal of 0, either slightly more or slightly less. Overall, we see that to make very minor changes to their chosen probabilities or extremely onally chosen wagers of the gambler, it allows us to understand how they et sizing of a prospect given a previous result. Measuring the absolute es allows us a better view into the user level perception of their own olate the control for each separate contract to further understand if there s between the gamblers of each contract. We will refer to the absolute ers as the gambler's Bet Signal. summary statistics for the wager signal -we notice the very similar ing to decrease their bet sizes after a victory, but increase their bet sizes tice that this distribution is heavily skewed and highly leptokurtic, no esting difference between the contracts can be found in contract 4, which We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kinked curve -we see that there is clear overweighting in the tails and an interesting S-shape behavior around the origin. This has the implication that people are more willing to have extreme changes in their chosen probabilities when measured at the bet to bet level compared to a normal distribution. That is there is greater tendency in extreme behaviors, such as drastically decreasing probabilities between bets or drastically increasing probabilities between bets. The S shape behavior around the origin also shows that gamblers have slight preferences for changing their probabilities very closely around a probability signal of 0, either slightly more or slightly less. Overall, we see that people generally prefer to make very minor changes to their chosen probabilities or extremely major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to understand how they subjectively view the bet sizing of a prospect given a previous result. Measuring the absolute difference in wager sizes allows us a better view into the user level perception of their own prospect. We also will isolate the control for each separate contract to further understand if there is significant differences between the gamblers of each contract. We will refer to the absolute difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice the very similar behavior for users tending to decrease their bet sizes after a victory, but increase their bet sizes after a loss. We also notice that this distribution is heavily skewed and highly leptokurtic, no matter the cut. An interesting difference between the contracts can be found in contract 4, which has far less variance and a generally tighter distribution. We also provide a normal Q-Q plot. This Q-Q plot has a very interesting kin see that there is clear overweighting in the tails and an interesting S-shape behav origin. This has the implication that people are more willing to have extreme c chosen probabilities when measured at the bet to bet level compared to a norm That is there is greater tendency in extreme behaviors, such as drastically decreasin between bets or drastically increasing probabilities between bets. The S shape be the origin also shows that gamblers have slight preferences for changing their pro closely around a probability signal of 0, either slightly more or slightly less. Over people generally prefer to make very minor changes to their chosen probabilitie major ones.
When we look at conditionally chosen wagers of the gambler, it allows us to under subjectively view the bet sizing of a prospect given a previous result. Measurin difference in wager sizes allows us a better view into the user level perceptio prospect. We also will isolate the control for each separate contract to further und is significant differences between the gamblers of each contract. We will refer difference between wagers as the gambler's Bet Signal.
When observing the summary statistics for the wager signal -we notice th behavior for users tending to decrease their bet sizes after a victory, but increase after a loss. We also notice that this distribution is heavily skewed and highly matter the cut. An interesting difference between the contracts can be found in co has far less variance and a generally tighter distribution.
When looking at the absolute changes in wagers chosen by the gambler, condi previous gamble, we notice an extremely concentrated distribution. Even on the lo see an extreme concentration of values near 0. It appears that the distribution has normal distribution. Calculation of the kurtosis of this distribution shows that a b c d Figure 9 : Q-Q plots of bet probability signal per contract. Shown are the Q-Q plots of the absolute changes in probabilities of winning tuned by gamblers conditional on previous bet outcomes (positive signs correspond to a win, and negative signs correspond to a loss) for each of the four contracts, panels (a) -(d), from April 17th, 2017 to December 12th, 2017. Majority of gamblers make minor changes to their chosen probabilities of winning, but it is more likely than the tail of a normal distribution that gamblers also make drastic changes at the extreme values.
are on the tails tend to be far more risk seeking (on the positive tail) or far more risk adverse (on the negative tail), either rapidly increasing their bet sizes or rapidly decreasing their bet sizes (Fig. 11 lepokurtic, implying that the existence of outliers for this distribution is far more than the normal distribution.
We also provide a Q-Q plot. This S-shape plot shows the classic overdispersed, leptokurtic Q-Q curve, with a large amount of data being dispersed between the left and right tails. Additionally, there is some slight S-shaped behavior near the Bet Size values around 0, again implying some degree of sensitivity towards values near 0. It can be inferred that gamblers that are on the tails lepokurtic, implying that the existence of outliers for this distribution is far more than the normal distribution.
We also provide a Q-Q plot. This S-shape plot shows the classic overdispersed, leptokurtic Q-Q curve, with a large amount of data being dispersed between the left and right tails. Additionally, there is some slight S-shaped behavior near the Bet Size values around 0, again implying some degree of sensitivity towards values near 0. It can be inferred that gamblers that are on the tails lepokurtic, implying that the existence of outliers for this distribution is far more than the normal distribution.
We also provide a Q-Q plot. This S-shape plot shows the classic overdispersed, leptokurtic Q-Q curve, with a large amount of data being dispersed between the left and right tails. Additionally, there is some slight S-shaped behavior near the Bet Size values around 0, again implying some degree of sensitivity towards values near 0. It can be inferred that gamblers that are on the tails lepokurtic, implying that the existence of outliers for this distribution is far more than the normal distribution.
We also provide a Q-Q plot. This S-shape plot shows the classic overdispersed, leptokurtic Q-Q curve, with a large amount of data being dispersed between the left and right tails. Additionally, there is some slight S-shaped behavior near the Bet Size values around 0, again implying some degree of sensitivity towards values near 0. It can be inferred that gamblers that are on the tails 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in the Barberis' Casino model [14] . However, we will observe that it is impossible for us to observe sophisticated, committed gamblers from an empirical standpoint, as we do not know what devices used by them.
In our dataset, we are unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse their wallet data exactly at the times they bet at. Because of this, we do not have access to their total wealth information, and thus we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Also of interest is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate one of the most common systems: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason why we choose these systems for our analysis is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies. 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casino model. However, we will observe that it is impossible for us to observe sophisticated, committed gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse wallet data at the times they bet at. Because of this, we do not have access to their total wealth, and we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Another interesting system is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate the most common system: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason we choose these systems is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies. 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casino model. However, we will observe that it is impossible for us to observe sophisticated, committed gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse wallet data at the times they bet at. Because of this, we do not have access to their total wealth, and we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Another interesting system is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate the most common system: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason we choose these systems is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies. 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casino model. However, we will observe that it is impossible for us to observe sophisticated, committed gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse wallet data at the times they bet at. Because of this, we do not have access to their total wealth, and we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Another interesting system is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate the most common system: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason we choose these systems is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies. 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casino model. However, we will observe that it is impossible for us to observe sophisticated, committed gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse wallet data at the times they bet at. Because of this, we do not have access to their total wealth, and we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Another interesting system is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate the most common system: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason we choose these systems is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies. 
We are interested in looking at the existence of gambling strategies because it allows us to validate some of the ideas behind theoretical predictions. We search for strategies that are path independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casino model. However, we will observe that it is impossible for us to observe sophisticated, committed gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betting standards, as it is not possible to parse wallet data at the times they bet at. Because of this, we do not have access to their total wealth, and we cannot search for proportions they bet at. In response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional bets. Another interesting system is the betting system in which a gambler continuously bets at a high probability (analogous to the bet everything system). This is one of the most common systems. Additionally, we will be able to evaluate gamblers who bet with negative progression, staking betting systems. We will evaluate the most common system: the martingale. This is because gamblers following the martingale always return to their original stake, making it easier for us to detect the usage of this system. We also aim to see if gamblers have a mixture of systems, such that they transition strategies over some timescale. The reason we choose these systems is because it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fixed probability/high probability) and path-dependent (mixed) strategies.
We are interested in looking at the existence of gambling strategies because it allows us validate some of the ideas behind theoretical predictions. We search for strategies that are pa independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casi model. However, we will observe that it is impossible for us to observe sophisticated, committ gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betti standards, as it is not possible to parse wallet data at the times they bet at. Because of this, w do not have access to their total wealth, and we cannot search for proportions they bet at. response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional be Another interesting system is the betting system in which a gambler continuously bets at a hi probability (analogous to the bet everything system). This is one of the most common system Additionally, we will be able to evaluate gamblers who bet with negative progression, staki betting systems. We will evaluate the most common system: the martingale. This is becau gamblers following the martingale always return to their original stake, making it easier for us detect the usage of this system. We also aim to see if gamblers have a mixture of systems, su that they transition strategies over some timescale. The reason we choose these systems is becau it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fix probability/high probability) and path-dependent (mixed) strategies.
We are interested in looking at the existence of gambling strategies because it allows us validate some of the ideas behind theoretical predictions. We search for strategies that are pa independent and path dependent, allowing us to verify the types of gamblers in Barberis' Casi model. However, we will observe that it is impossible for us to observe sophisticated, committ gamblers gamblers from an empirical standpoint, as we do not know what devices .
Unfortunately, we will be unable to evaluate gamblers that follow proportional betti standards, as it is not possible to parse wallet data at the times they bet at. Because of this, w do not have access to their total wealth, and we cannot search for proportions they bet at. response to this, we will evaluate gamblers that bet at a fixed wager in place of proportional be Another interesting system is the betting system in which a gambler continuously bets at a hi probability (analogous to the bet everything system). This is one of the most common system Additionally, we will be able to evaluate gamblers who bet with negative progression, staki betting systems. We will evaluate the most common system: the martingale. This is becau gamblers following the martingale always return to their original stake, making it easier for us detect the usage of this system. We also aim to see if gamblers have a mixture of systems, su that they transition strategies over some timescale. The reason we choose these systems is becau it allows us to possibly observe time inconsistencies in both path-independent (fixed wager/fix probability/high probability) and path-dependent (mixed) strategies. a b c d Figure 11 : Q-Q plots of the bet signal per contract. Shown are the Q-Q plots of the absolute changes in bet sizes tuned by gamblers conditional on previous bet outcomes (positive signs correspond to a win, and negative signs correspond to a loss) for each of the four contracts, panels (a) -(d), from April 17th, 2017 to December 12th, 2017. Gamblers on the tails of the distributions are more likely to chase the risk after a win (these on the right tail) and to avoid the loss after a loss (these on the left tail).
To find more simple systems, such as fixed probability, fixed wager or high probability betting systems, we will use Python's Pandas package for data analysis. To classify fixed systems, we convert our data, stored in csv files into Pandas Dataframe, and search for gamblers who have zero variance in their probability choice and wager sets (see Dataset and Methods for details). To classify high probability bettors, we choose any bettor who bets only at tuning the probability of winning p > 0.9.
We also will only apply these methods to gamblers who have sufficient gambling data, e.g.,
whale bettors (bettors with over 100 total gambles). Additionally, we assume that the initial reference frame that these gamblers once they enter the online casino is the beginning of the day their betting session starts at. In this way, we will be able to detect possible time inconsistencies in daily data. With this as our reference point, we found many gamblers that behave similarly to the hypothesized path-independent betting, naive gamblers who are unaware of the time inconsistencies. However, we observe some interesting path dependent betting strategies as well (such as the martingales), or some mixture of betting strategies.
The most common and popular "strategy" found in this population is the fixed probability betting strategy. In this strategy, the gambler chooses some probability, usually p > 0. 5 bets. In evaluating gamblers who bet with fixed probabilities or wagers, we can observe what happens when a gambler has no strategic time inconsistency (irrespective to exit strategies). In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5. An example of this strategy is found in the betting history of the individual label as Professional #5619 in our dataset (see Fig. 12 49. This is an example of a simple, path-independent strategy. No matter what his results are, the gambler sticks to his initial chosen betting strategy. Upon entering the casino, the gambler commits to this strategy, and even upon accumulating losses, this individual continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run the expectation runs negative rapidly. To find more simple systems, such as fixed probability, fixed wager or high probability betting systems, we will use Python's Pandas package for data analysis. To classify fixed systems, we convert our data, stored in csv files into Pandas Dataframe, and search for gamblers who have zero variance in their probability choice and wager sets. To classify high probability bettors, we choose any bettor who bets only at p > 0.9.
We also will only apply these methods to gamblers who have sufficient gambling data, e.g. whale bettors (bettors with over 100 total gambles). Additionally, we assume that the initial reference frame that these gamblers once they enter the online casino is the beginning of the day their betting session starts at. In this way, we will be able to see possible time inconsistencies in daily data. With this as our reference point, we found many gamblers that behave similar to the hypothesized path-independent betting, naive gamblers who are unaware of the time inconsistencies. However, we observe some interesting path dependent betting strategies (such as the martingales), or some mixture of betting strategies.
The most common and popular "strategy" found in this population is the fixed probability betting strategy. In this strategy, the gambler chooses some probability, usually p > 0.5 and continually bets. In evaluating gamblers who bet with fixed probabilities or wagers, we can observe what happens when a gambler has no strategic time inconsistency (irrespective to exit strategies).
An No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a time inconsistency in terms of exit strategies, but this is something we cannot empirically deduce. He or she follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We 49. This is an example of a simple, path-independent strategy. No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a time inconsistency in terms of exit strategies, but this is something we cannot empirically deduce. He or she follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We also observe loss-chasing behavior in the third day. This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate. We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of his or her accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts his or her bet size, eventually exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern. Upon large wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. His bet size starts with an initial staking size w 0 = 0.2 Ether, and follows: 49. This is an example of a simple, path-independent strategy. No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a time inconsistency in terms of exit strategies, but this is something we cannot empirically deduce. He or she follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We 49. This is an example of a simple, path-independent strategy. No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a To find more simple systems, such as fixed probability, fixed wager or high probability betting systems, we will use Python's Pandas package for data analysis. To classify fixed systems, we convert our data, stored in csv files into Pandas Dataframe, and search for gamblers who have zero variance in their probability choice and wager sets. To classify high probability bettors, we choose any bettor who bets only at p > 0.9.
We also will only apply these methods to gamblers who have sufficient gambling data, e.g. whale bettors (bettors with over 100 total gambles). Additionally, we assume that the initial reference frame that these gamblers once they enter the online casino is the beginning of the day their betting session starts at. In this way, we will be able to see possible time inconsistencies in daily data. With this as our reference point, we found many gamblers that behave similar to the hypothesized path-independent betting, naive gamblers who are unaware of the time inconsistencies. However, we observe some interesting path dependent betting strategies (such as the martingales), or some mixture of betting strategies.
The most common and popular "strategy" found in this population is the fixed probability betting strategy. In this strategy, the gambler chooses some probability, usually p > 0.5 and continually bets. In evaluating gamblers who bet with fixed probabilities or wagers, we can observe what happens when a gambler has no strategic time inconsistency (irrespective to exit strategies).
An No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a time inconsistency in terms of exit strategies, but this is something we cannot empirically deduce. He or she follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We To find more simple systems, such as fixed probability, fixed wager or high probability betting systems, we will use Python's Pandas package for data analysis. To classify fixed systems, we convert our data, stored in csv files into Pandas Dataframe, and search for gamblers who have zero variance in their probability choice and wager sets. To classify high probability bettors, we choose any bettor who bets only at p > 0.9.
We also will only apply these methods to gamblers who have sufficient gambling data, e.g. whale bettors (bettors with over 100 total gambles). Additionally, we assume that the initial reference frame that these gamblers once they enter the online casino is the beginning of the day their betting session starts at. In this way, we will be able to see possible time inconsistencies in daily data. With this as our reference point, we found many gamblers that behave similar to the hypothesized path-independent betting, naive gamblers who are unaware of the time inconsistencies. However, we observe some interesting path dependent betting strategies (such as the martingales), or some mixture of betting strategies.
The most common and popular "strategy" found in this population is the fixed probability betting strategy. In this strategy, the gambler chooses some probability, usually p > 0.5 and continually bets. In evaluating gamblers who bet with fixed probabilities or wagers, we can observe what happens when a gambler has no strategic time inconsistency (irrespective to exit strategies).
An No matter what his results are, the gambler sticks to his initial strategy. Upon entering the casino, the gambler commits to a strategy, and even upon accumulating losses, he or she continues to gamble at a suboptimal probability. In gambling at p < 0.5, this gambler is taking significant risk -in the long run his or her expectation runs negative. This gambler may have a time inconsistency in terms of exit strategies, but this is something we cannot empirically deduce. He or she follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We also observe loss-chasing behavior in the third day. This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate. We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of his or her accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts his or her bet size, eventually exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern. Upon large wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. His bet size starts with an initial staking size w 0 = 0.2 Ether, and follows: also observe loss-chasing behavior in the third day. This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate. We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of his or her accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts his or her bet size, eventually exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern. Upon large wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. His bet size starts with an initial staking size w 0 = 0.2 Ether, and follows:
also observe loss-chasing behavior in the third day. This gambler begins by betting a large initial wager size of around 2 Ether (which is around the same size as the ma or she bet at in the past two days), and progressively increases bet sizing as losses a observe a two-peak plateau in the bet sizing of this gambler. The first plateau is p gambler. The gambler won two consecutive large bets (20 ETH), recovering a sign of his or her accumulated losses. Immediately, we observe a drastic decreasing in subsequent loss. The gambler continues to bet at high values, putting themselves m negative. After a string of consecutive losses, the gambler adjusts his or her bet s exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increa only resulted in a larger loss. It seems that this gambler followed a gain exit patte wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk see and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler tak stake W 0 , and continually gambles, tuning his probability through wins and loss strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale sy very quickly. However, certain gamblers still follow this system. As observed follows a martingale strategy. His bet size starts with an initial staking size w 0 = follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. In the wager sizing graphs (F observe a very clear return to the initial staking size after every win. These graphs nearly linear staking gains of the martingale for a gambler from the timespan of 5/ (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from this system. However, we no there is a distinct drop in cumulative profits due the gambler having a stop loss afte lost bets. This showcases the dangerous fast divergence of the martingale system also observe loss-chasing behavior in the third day. This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate. We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of his or her accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts his or her bet size, eventually exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern. Upon large wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. His bet size starts with an initial staking size w 0 = 0.2 Ether, and follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. In the wager sizing graphs (Figure 4 .10), we observe a very clear return to the initial staking size after every win. These graphs showcase the nearly linear staking gains of the martingale for a gambler from the timespan of 5/6/17 5/7/17 (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from this system. However, we notice on 5/6/17 there is a distinct drop in cumulative profits due the gambler having a stop loss after 6 consecutive lost bets. This showcases the dangerous fast divergence of the martingale system. We also see also observe loss-chasing behavior in the third day. Th large initial wager size of around 2 Ether (which is ar or she bet at in the past two days), and progressively i observe a two-peak plateau in the bet sizing of this g gambler. The gambler won two consecutive large bet of his or her accumulated losses. Immediately, we ob subsequent loss. The gambler continues to bet at high negative. After a string of consecutive losses, the gam exiting a a loss of 41.44 Ether. Attempting to chase a fe only resulted in a larger loss. It seems that this gamb wins at the end of his or her first sessions, the gamble In a very heuristic way, we can label these kinds and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extr stake W 0 , and continually gambles, tuning his proba strategy never the only strategy the gambler employe
As the Etheroll minimum bet size is quite large, we very quickly. However, certain gamblers still follow follows a martingale strategy. His bet size starts with follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. observe a very clear return to the initial staking size a nearly linear staking gains of the martingale for a gam (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from there is a distinct drop in cumulative profits due the ga lost bets. This showcases the dangerous fast diverge also observe loss-chasing behavior in the third day. This gambler begins by betting large initial wager size of around 2 Ether (which is around the same size as the m or she bet at in the past two days), and progressively increases bet sizing as losses a observe a two-peak plateau in the bet sizing of this gambler. The first plateau is p gambler. The gambler won two consecutive large bets (20 ETH), recovering a sign of his or her accumulated losses. Immediately, we observe a drastic decreasing in subsequent loss. The gambler continues to bet at high values, putting themselves negative. After a string of consecutive losses, the gambler adjusts his or her bet s exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through incre only resulted in a larger loss. It seems that this gambler followed a gain exit patte wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk se and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler tak stake W 0 , and continually gambles, tuning his probability through wins and loss strategy never the only strategy the gambler employed over the timescale of a day
As the Etheroll minimum bet size is quite large, we observe that a martingale sy very quickly. However, certain gamblers still follow this system. As observed follows a martingale strategy. His bet size starts with an initial staking size w 0 = follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. In the wager sizing graphs (F observe a very clear return to the initial staking size after every win. These graph nearly linear staking gains of the martingale for a gambler from the timespan of 5/ (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from this system. However, we n there is a distinct drop in cumulative profits due the gambler having a stop loss afte lost bets. This showcases the dangerous fast divergence of the martingale syste an interesting time inconsistency from this gambler. Looking at the data, we see exact deviation from this strategy at his or her 69-th bet, where the gambler be Ether at probability 0.5 and loses. If this gambler is following the martingale she should bet 2 4 (0.2) = 3.2 Ether at the same probability (to get a 1 : 1 return) gambler disregards this, and gambles the same gamble of 1.6 Ether, and loses aga a sharp decrease in his or her cumulative gains. Interestingly, the gambler make also observe loss-chasing behavior in the third day. This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate. We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of his or her accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts his or her bet size, eventually exiting a a loss of 41.44 Ether. Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern. Upon large wins at the end of his or her first sessions, the gambler stopped betting. In a very heuristic way, we can label these kinds of gamblers as more risk seeking if p < 0.5 and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. His bet size starts with an initial staking size w 0 = 0.2 Ether, and follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. In the wager sizing graphs (Figure 4 .10), we observe a very clear return to the initial staking size after every win. These graphs showcase the nearly linear staking gains of the martingale for a gambler from the timespan of 5/6/17 5/7/17 (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from this system. However, we notice on 5/6/17 there is a distinct drop in cumulative profits due the gambler having a stop loss after 6 consecutive lost bets. This showcases the dangerous fast divergence of the martingale system. We also see an interesting time inconsistency from this gambler. Looking at the data, we see the gambler's exact deviation from this strategy at his or her 69-th bet, where the gambler bets 2 3 (0.2) = 1.6 Ether at probability 0.5 and loses. If this gambler is following the martingale system, he or she should bet 2 4 (0.2) = 3.2 Ether at the same probability (to get a 1 : 1 return). However, the gambler disregards this, and gambles the same gamble of 1.6 Ether, and loses again. This causes a sharp decrease in his or her cumulative gains. Interestingly, the gambler makes a third bet of also observe loss-chasing behavior in the third day. Th large initial wager size of around 2 Ether (which is a or she bet at in the past two days), and progressively observe a two-peak plateau in the bet sizing of this gambler. The gambler won two consecutive large be of his or her accumulated losses. Immediately, we ob subsequent loss. The gambler continues to bet at high negative. After a string of consecutive losses, the ga exiting a a loss of 41.44 Ether. Attempting to chase a fe only resulted in a larger loss. It seems that this gamb wins at the end of his or her first sessions, the gambl In a very heuristic way, we can label these kinds and less risk seeking if p > 0.5.
Rarer is the fixed wager strategy. This strategy is ext stake W 0 , and continually gambles, tuning his prob strategy never the only strategy the gambler employ
As the Etheroll minimum bet size is quite large, we very quickly. However, certain gamblers still follo follows a martingale strategy. His bet size starts wit follows:
Where w i+1 is the i + 1-th bet conditioned on i losses observe a very clear return to the initial staking size nearly linear staking gains of the martingale for a gam (where he bet a total of 172 times). This is because of the guaranteed return of w 0 from there is a distinct drop in cumulative profits due the g lost bets. This showcases the dangerous fast diverg an interesting time inconsistency from this gambler exact deviation from this strategy at his or her 69-t Ether at probability 0.5 and loses. If this gambler she should bet 2 4 (0.2) = 3.2 Ether at the same prob gambler disregards this, and gambles the same gamb a sharp decrease in his or her cumulative gains. Int Figure 12 : Examples of gamblers using a fixed probability strategy. Shown are (a), (b), (c) the wager sizes, (d), (e), (f) cumulative profits, and (g), (h), (i) scaled cumulative profits over a length of consecutive gamblings, corresponding to three occasions, respectively. This gambler consistently fixed the probability of winning to be p = 0.49, but exhibited inconsistent exit strategies (when to stop gambling during the day).
This gambler may have a time inconsistency in terms of exit strategies (cf. Figs. 12d, 12e, and 12f), but this is something we cannot empirically deduce. This individuals follows the betting strategy, but exits once the losses reach some arbitrary stop-loss. We also observe loss-chasing behavior in the third day (Fig. 12f ). This gambler begins by betting at an extremely large initial wager size of around 2 Ether (which is around the same size as the maximum bet he or she bet at in the past two days), and progressively increases bet sizing as losses accumulate (Fig. 12c) . We observe a two-peak plateau in the bet sizing of this gambler. The first plateau is positive for the gambler. The gambler won two consecutive large bets (20 ETH), recovering a significant portion of their accumulated losses. Immediately, we observe a drastic decreasing in bet size, and a subsequent loss. The gambler continues to bet at high values, putting themselves more and more negative. After a string of consecutive losses, the gambler adjusts their bet size, eventually exiting a a loss of 41.44 Ether (Fig. 12f) . Attempting to chase a few initial losses through increasing bet sizing only resulted in a larger loss. It seems that this gambler followed a gain exit pattern (Fig. 12d) . Upon large wins at the end of their first sessions, the gambler stopped betting (Fig. 12e) .
Rarer is the fixed wager strategy as seen in our dataset. This strategy is extremely simple. A gambler takes some initial stake W 0 , and continually gambles, tuning his probability through wins and losses. Oddly, this strategy is never the only strategy the gambler employed over the timescale of a day.
As the Etheroll minimum bet size is quite large, we observe that a martingale system diverges very quickly. However, certain gamblers still follow this system. As observed, this gambler follows a martingale strategy. Their bet size starts with an initial staking size w 0 = 0.2 Ether, and follows:
Where w i+1 is the i + 1-th bet conditioned on i losses. In Fig. 13 , we observe a very clear return to the initial staking size after every win. These graphs showcase the nearly linear staking gains of the martingale for a gambler from the timespan of 5/6/17 − 5/7/17 (where this individual betted a total of 172 times). This is because of the guaranteed return of w 0 from this system. However, we notice on 5/6/17 there is a distinct drop in cumulative profits due the gambler having a stop loss after 6 consecutive lost bets (Fig. 13c) . This showcases the dangerous fast divergence of the martingale system. We also see an interesting time inconsistency from this gambler. Looking at the data, we see the gambler's exact deviation from this strategy at their 69-th bet, where the gambler bets 2 3 (0.2) = 1.6 Ether at probability 0.5 and loses. If this gambler is following the martingale system, they should bet 2 4 (0.2) = 3.2 Ether at the same probability (to get a 1 : 1 return). However, the gambler disregards this, and gambles the same gamble of 1.6 Ether, and loses again. This causes a sharp decrease in their cumulative gains. Interestingly, the gambler makes a third bet of the same amount, and same probability. Probabilistically, this gambler will win this bet on average, but it comes with significant risk. This showcases the tendency of gamblers to chase losses -similar to the idea of gain-exit strategies. It also matches with the concept of the value function (according to the cumulative prospect theory [12] , see our detailed discussion below). Theoretically, when making decisions under risk, gamblers become risk seeking when faced with losses. This helps to explain the "loss-chasing" phenomenon.
Other possible reasons for this sudden time inconsistency in strategy could involve the gambler's total wealth (wallet size), overall bankroll. From observing their wager sizing, we observe that the gambler never bets past 1.6 Ether. Hypothetically, the next bet in the martingale sequence (3.2) could simply be too much for the gambler to continue, forcing the gambler to deviate from the planned strategy.
The trajectory of the scaled cumulative profit of this gambler shows their valuation ("satisfaction") through the losses and gains, framed near the exit time (Fig. 13d, 13e) . We observe that even though this gambler was a net positive (on 5/6/17), their scaled cumulative profit is very faintly positive, representing the effect of the severe loss. In contrast, we see the effect of ending with a large win on 5/7/17. The gambler's scaled cumulative profit is at a global maximum at their exit time, influencing their exit time. This gambler ended up as a net winner, winning 4.34 Ether.