Article from

Actuary of the Future

November 2015 Issue 38 er cards coming out equates ing the deck composition af- An Uncharacteristic to more high cards left to be ter each card that comes out. played in the deck, while a This program, paired with the Application of Actuarial higher count indicates to the model described above, pro- players that the deck might be vides a complete picture of Science—Card Counting in their favor. real-life blackjack games, which I use to model the relative effec- in Blackjack THE MODEL tiveness of various card-count- ing strategies as well as other By Michael Adams Pulling from some of the tools I learned on our preliminary metrics of the game. exams and my extensive work ollege graduates com- brings us to the casino game experience in Microsoft Excel, DATA AND RESULTS ing into the actuarial called blackjack. I used Bayesian and condition- Similar to an actuarial model profession usually have C al probability to develop a dy- used in practice, the user can one goal in mind: to pass their CARD COUNTING IN namic chart of optimal player input various game parameters, exams and gain employment as BLACKJACK moves (hit, stay, double, split, customize the card-counting an actuary. At this level in their Blackjack has been long-studied surrender) based on the player’s strategies, and run the model to career, it’s expected. As with by statisticians due to a unique hand and the card that the deal- produce a rich dataset of game- any professional, actuarial stu- characteristic that the game er is showing. The chart con- play data that can be mined for dents’ career goals and aspira- possesses: being purely chalked tains expected values for each informative game metrics and tions will shift as they grow as move and is based on a set of up to chance, which sometimes strategy performance. Using professionals and are exposed game variants. varies into the player’s favor this output, players can tweak to different types of work. A depending on past cards dealt. student of actuarial science is their strategies and see the Players who can identify inter- With this model and some as- quantitative impact that these very well-suited for these shift- vals of the game during which sumptions, we can perform cal- ing career objectives due to the changes make on their perfor- they have the edge over the culations to obtain a composite mance. fact that preparation for the dealer can increase their bets expected value for the game field involves mastery of many to take advantage of their edge. and a chart of optimal moves The richness of the data and transferrable core skills. The common term for the pro- for a player—information that customizability of the model al- cess players use to obtain infor- is readily available with an In- low the users to answer almost mation about the current state ternet search. However, I em- On the preliminary exams, an of the deck is “card counting.” phasize that this model is dy- any question they have about actuary will learn about proba- namic because it depends on the game. One useful way to bility, statistics, financial math- visually represent the results The process of card count- the user-inputted composition ematics, modeling mortality of the trial is to plot the user’s ing involves players keeping a of the remaining deck, which and other uncertain events. On deck edge against the running running “count” in their head brings me to the Monte Carlo the job, skills such as data anal- count using a particular count- by summing pre-determined simulations. ysis, model development and ing strategy, determined by the values associated with each programming become more user. For example, here are two card that has come up on the SIMULATIONS prominent. A professional with counting strategies: the strat- deck. Generally, lower cards A big part of many actuarial a strong command of these egy most commonly used by have higher associated val- students’ training and on-the- skills is well-positioned for a counters called Hi-Lo (1) and a ues (+1, +2, etc.) and higher job work is programming. A successful career in almost any less popular strategy called Re- cards have lower associated useful programming language analytical field. vere Adv. Plus-Minus (2). While values. There are numerous that complements Excel work counting strategies with dif- is Visual Basic for Applications both use pre-determined val- The purpose of this article is ferent values associated with (VBA). Using my work experi- ues assigned to each card, the to demonstrate how this array the cards. But universally, the ence in various programming values differ between the two of skills can be used to analyze nature of the game is such that languages and in VBA direct- strategies. a complex system of uncertain having more high cards left in ly, I developed a program that events with results applicable in the deck is beneficial to the performs millions of black- In comparing these two strat- the real world—to some. This player. Therefore, more low- jack game simulations, updat- egies on the same simulation

18 | NOVEMBER 2015 ACTUARY OF THE FUTURE (1)

(2)

run (with about half a million Additionally, you can see from If you have any questions, com- game simulations), you can these graphs that even with ments or suggestions, please Michael Adams feel free to reach out to me is an actuarial see that they behave different- playing 500,000+ games, there analyst in San ly, and informed counters can is still a great deal of volatil- at michael.adams452@gmail. Diego, Calif. He adjust their betting strategy ity. This arises partially from com. n graduated from the University of accordingly. A big part of card the “all-or-nothing” nature of California, Santa counting is doing so in a dis- blackjack game outcomes, and Barbara actuarial program (class creet manner. Card counting partially from the fact that card of 2012) and serves as a council member of the Actuary of the is not illegal, but many casinos counting at its best is still only Future Section. that suspect you of counting a weak indicator of player edge. will ask you to leave the casino. With this in mind, one could My intent with this article is not argue that the second strategy to argue that an actuary would in our example is marginally be wise to career-change into better than the first. You can counting cards at casinos. Rath- see that whereas the player’s er, it is to demonstrate that the edge in (1) increases to above array of skills we develop in our 0 percent sharply at count 10, studies and on the job enables us the player’s edge using (2) grad- to conquer a very broad range ually increases above 0 percent of analytical pursuits, not limit- starting at count 7, and stays ed to traditional actuarial work there for a wider count interval. in insurance or consulting. We This enables players using (2) can apply our skills to perform to gradually increase their bets complex analyses that wouldn’t to take advantage of their edge, be possible without our thor- while avoiding the casino’s ough understanding of statistics, arousal of suspicion. modeling and complex systems.

NOVEMBER 2015 ACTUARY OF THE FUTURE | 19