A Criticism of Doyle’s survey of time preference: A correction regarding the CRDI and CADI families

Doyle’s (2013) theoretical survey of discount functions criticizes two parametric families abbreviated as CRDI and CADI families. We show that Doyle’s criticisms are based on a mathematical mistake and are incorrect

Beyond chance? The persistence of performance in online poker

A major issue in the widespread controversy about the legality of poker and the appropriate taxation of winnings is whether poker should be considered a game of skill or a game of chance. To inform this debate we present an analysis into the role of skill in the performance of online poker players, using a large database with hundreds of millions of player-hand observations from real money ring games at three different stakes levels. We find that players whose earlier profitability was in the top (bottom) deciles perform better (worse) and are substantially more likely to end up in the top (bottom) performance deciles of the following time period. Regression analyses of performance on historical performance and other skill-related proxies provide further evidence for persistence and predictability. Simulations point out that skill dominates chance when performance is measured over 1,500 or more hands of play

Number preferences in lotteries

We explore people's preferences for numbers in large proprietary data sets from two different lottery games. We find that choice is far from uniform, and exhibits some familiar and some new tendencies and biases. Players favor personally meaningful and situationally available numbers, and are attracted towards numbers in the center of the choice form. Frequent players avoid winning numbers from recent draws, whereas infrequent players chase these. Combinations of numbers are formed with an eye for aesthetics, and players tend to spread their numbers relatively evenly across the possible range

Regression Results for Fixed Numbers of Hands.

<p>The table displays the regression results for subsamples of players who played at least 2<i>n</i> hands during our entire sample period, with <i>n</i> = 1,000, 5,000 or 10,000. The dependent variable is the player’s performance over the second <i>n</i> hands, as measured by either the standard performance measure (Panel A) or the performance robustness measure (Panel B). The explanatory variables are calculated over the first <i>n</i> hands. Other definitions are as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0115479#pone.0115479.t005" target="_blank">Table 5</a>.</p><p>Regression Results for Fixed Numbers of Hands.</p

Regression Results.

<p>The table displays the regression results for our subsample of players who played 1,000 hands or more over the first six months of our sample period (Period 1) and at least 1 hand over the second six months (Period 2). The dependent variable is either the standard performance measure (Panel A) or the performance robustness measure (Panel B). The standard performance measure is defined as the average number of big blinds won per hundred hands after correction for rake (bb/100). The performance robustness measure is the average number of big blinds won after correction for rake divided by its estimated standard error. All explanatory variables are calculated using data from Period 1 only. <i>SPM</i> is the player’s standard performance measure. <i>PRM</i> is her performance robustness measure. <i>Hands (log)</i> is the natural logarithm of the number of hands played. <i>Tightness</i> is one minus the proportion of hands in which the player voluntarily wagered money in the first betting round. <i>Aggressiveness</i> is the number of times the player led the betting as a proportion of the total number of times she voluntarily wagered money. <i>Tournaments</i> is the player’s tournament ability rating from the SharkScope database immediately before the start of Period 2. The results reported in Panel A are weighted least squares regression results with the ratio of a player’s number of hands and her sample variance of the number of big blinds won in Period 2 as weight. Panel B presents ordinary least squares results. The <i>p</i>-values are in parentheses.</p><p>Regression Results.</p

Performance Robustness Measure Deciles.

<p>The table ranks all players who played 1,000 hands or more over the first six months of our sample period into deciles by their performance over these six months. Here, the performance measure that is used to rank players is the performance robustness measure, which is defined as the average number of big blinds won per hand after correction for rake divided by the estimated standard error. The estimated standard error is the sample standard deviation of the rake-corrected winnings per hand divided by the square root of the number of hands. The statistics shown for each resulting decile are defined as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0115479#pone.0115479.t002" target="_blank">Table 2</a>. The various panels and correlation coefficients are also identically defined.</p><p>Performance Robustness Measure Deciles.</p

Standard Performance Measure Deciles.

<p>The table ranks all players who played 1,000 hands or more over the first six months of our sample period (Period 1) into deciles by their performance over these six months, where performance is measured as the average number of big blinds won per hundred hands after correction for rake. For each decile, the first columns show the number of included players (<i>N</i>) and their average performance (bb/100) for this ranking period. The next columns show the number of players from the decile who played at least one hand in the last six months of our sample period (Period 2), as well as their average performance for this measurement period and how they rank on average relative to all other players who played at least one hand (Rank). Average decile performance (bb/100) is expressed both as a straight average (unweighted) and as a weighted average, where the weights are either the square roots of players’ number of hands (weighted by √n) or players’ number of hands (weighted by n). Panel A (B/C) shows the results for observations from the small (medium/high) stakes level separately. Panel D shows the results for all stakes levels combined. For each panel, the table shows the Spearman rank correlation between the two periods for the average performance at the decile level (for each weighting method) and for performance at the player level (in Rank column).</p><p>Standard Performance Measure Deciles.</p

Simulation Results.

<p>The figure displays the proportion of times that a selection of <i>h</i> randomly drawn hand outcomes for players who were among the best performing players in the past do better than a similar-size selection of hand outcomes for players who were among the worst performing players in the past. Hand outcomes are randomly drawn from the subsample of hands from the second six months of our sample period for players who ranked among the best or worst performing percentiles (black lines) and for players who ranked among the best or worst performing deciles (grey lines) over the first six months of our sample period. Players are ranked according to the performance robustness measure (solid lines) or the standard performance measure (dashed lines). The lines are smoothed, with each point representing the moving average proportion across the simulation outcomes available for <i>h</i>-100 up to and including <i>h</i>+100.</p