Continuing with yesterday’s theme of gambling, I would like to present a framework for optimal gambling strategy at a casino.
Some Key Assumptions
First, I assume that all possible gambles you can make at a casino have negative expected value. I am not considering counting cards, controlled shooting, cheating, or games of skill like poker. Second, I assume that the player is risk adverse. I realize that this assumption may not be true in reality since some individual players may be risk seeking when it comes to small sums of money at a casino. Still, I think this assumption is reasonable, especially as the amount a player wagers increases. I also think that adopting a risk adverse mentality will lead to better gambling strategy as I explain below.
Gambling Should Be Viewed As An Investment
The key insight I would like to present is that each gamble at a casino should be viewed as an investment. Obviously, all gambles are terrible investments because every gamble in a casino has a negative expected value (a positive house edge), so the optimal decision is to not play at all. But if you do decide to play, the question you should be asking yourself is, “Which gamble is the least worst?”
The typical advice you will hear is to choose games that have the lowest house edge. This is excellent advice, and this should be the first criteria in selecting your optimal gambling strategy. Gambles with a house edge of around 1.5% or lower include baccarat, blackjack, craps, and video poker.
Some of the more popular games with high house edges include roulette, caribbean stud poker, three card poker, texas hold’em bonus, three card poker, and slot machines. Choosing to play these games is a suboptimal decision because of the high house edge. You can view a more complete list of house edges at the Wizard of Odds.
Volatility Should Be Considered
Modern portfolio theory states that investors must be compensated by taking on additional risk with increased expected return. In other words, there is a positive relationship between risk and return. Finance practitioners usually measure risk as the standard deviation of an investment’s historical returns. Putting aside comments on whether this is appropriate measure of risk for investments, this measure of risk is perfect for our discussion of gambling strategy.
Casino games all have a fixed set of rules which means that the standard deviation of each game can be calculated with certainty. The standard deviation also does not change over time. You should be mindful of the standard deviation of games because your the bankroll you bring to a casino is fixed, and you will typically stop playing if you lose your entire bankroll. High volatility is bad for your bankroll and mental health when playing at a casino.
This means that games with high standard deviation are bad because the risk that you lose your entire bankroll is higher. Put differently, if you are choosing between two games with equal house edge and unequal standard deviation, you should choose the game with the lower standard deviation. This is consistent with modern portfolio theory: an investment with an expected return of 10% and a standard deviation of 1% is better than an investment with an expected return of 10% and a standard deviation of 10%.
Optimal Gambling Strategy
The components of our optimal gambling strategy are coming together. You should choose games with low house edge and low volatility. Most gambles at a casino with the lowest house edges also happen to have the lowest volatility. These gambles are typically bets that pay 1 to 1, like blackjack, baccarat, and the pass line for craps.
There is one more level of optimization: volatility over time. Every casino game moves at different speeds and the number of hands (or rolls) per hour varies along with it. This means that the true volatility that you face varies with the standard deviation of the game and the number of hands per hour. You should choose games that have low volatility over time.
An alternative explanation is that you should choose games that move slowly so you can get more gambling time (and more enjoyment) out of a fixed amount of bankroll.
What is the Best Gamble?
To sum everything up, you should choose gambles with low house edge, low standard deviation, and take a long time to complete. Returning to our list of baccarat, blackjack, craps, and video poker, this means that video poker is a bad choice (high standard deviation since you get paid a lot on rare combinations) and blackjack is a bad choice (low standard deviation but high number of hands per hour).
My personal gamble of choice is the don’t pass line with odds in craps. For readers that are unfamiliar with the rules of craps, I recommend reading the basic rules and strategy at the Wizard of Odds. The don’t pass line with odds has the best combination of the three things we care about: around a 0.5% house edge, very low volatility, and low number of rolls per hour.
One thing I failed to mention is that every person has their own preferences on which game is the most entertaining which should be taken into account. I hope this framework is helpful for novice gamblers in choosing the best gamble for themselves. Advanced gamblers can optimize their gambling strategy by looking beyond the house edge to volatility and the level of volatility over time.