Poker is a very competitive game. Ones longevity in the game depends directly on ones ability to compete with opponents. It has been estimated that about 70% of poker players are losing players in the long run. This high attrition rate is due to two factors, the rake and the large discrepancy in ability between expert and novice players. This paper provides a framework to determine whether a given player is likely to fall into the 95% “novice player” losing category or the 5% “expert player” winning category in limit poker, based on past playing history.
Formal statistical testing can help guide us in determining which category a player might fall into. For simplicity we will assume 2 types of players, expert and novice. However, several estimates and assumptions must first be made before we can estimate what category a player may fall into.
Firstly, we will assume that 60 hands are played per hour of limit poker. Secondly, an estimate of how much an expert player is expected to win per 60 hands is required. This is to be used as the base comparison to judge all players by. It is generally accepted in poker circles that one and a half big bets per 60 hands is a reasonable win rate for an expect player in limit poker. For the purpose of this paper we will assume that this figure is correct. The actual dollar value of a big bet is determined by the game limit. For example in a $5/$10 game a big bet is considered to be $10. Finally, we need to determine the variance or standard deviation per 60 hands that an expert poker player is likely to experience in obtaining their one and a half big bets. Thankfully, this figure too has been estimated by several expert players, including Mason Malmuth, to be around 12.5 big bets per 60 hands. This estimate has been derived by actual playing outcomes, and is indeed inline with my own standard deviation calculations of 12.2 big bets per 60 hands. We will assume 12.5 big bets per hour to be a reasonable figure for 1 standard deviation. It should be noted that for tight games this figure would be substantially lower, perhaps in the region of 5-6 big bets per hour.
To summarise let
expected big bet earnings after 60 hands for an expert player = 1.5
standard deviation after 60 hands for an expert player = 12.5
It will also be useful to obtain an estimate of variance for (n) hands. To do this we can make use of the following formula shown by equation 1.
Where standard deviation after n hands
By incorporating the mean, 95% confidence intervals can easily be obtained using the equation below. It should be noted that 2 Standard deviations represents a 95% confidence interval. Upper and Lower confidence interval equations are shown in equations 2 and 3.
To provide a numerical example consider the following. Is it likely that a player can be an expert player and still be losing 110 big bets after 5000 hands? Working through equations 1 through 3 by substituting in relevant values we derive equations 4, 5 and 6.
= 353.2 (5)
= -103.2 (6)
That is we are 95% confident that after 5000 hands an expert player would expect to be winning between -103.2 and 353.2 big bets. To present this another way, after 5000 hands if a given player is losing more than 103.2 big bets than we can be 95% confident that this player significantly worse than an expert player. We would conclude that losing 110 bets provides significant evidence that the player in question is not an expert player.
We can modify this analysis slightly using the same example to test for a break-even player by substituting zero in for the expected earnings per 60 hands. That above equations reduced to
Upper 95% =114.2*2
Lower 95% = -(114.1)*2
That is we are 95% confident that after 5000 hands a break even player would have expected earnings of between –228 and 228 big bets. We could not conclude that a player losing 110 big bets is significantly worse than a break even player in this case.
This might be a more appropriate thing to test for, since a rational investor should be expected to walk away from poker if they are below a break even player. A gambler may still wish to play poker for entertainment value if they are below this breakeven level, but should expect financial loss in the long run.
The above analysis can be used for any number of hands to create confidence intervals and should be used as a guide to determine your standard of play. Of course in practice one will play different game limits and estimates of the number of big bets one is winning or losing may be difficult. Estimating this number as well as the number of hands played will be required.