Expectation-Based Strategy in Gambling
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Maximizing the expected value of a bet by player’s choices and moves before and during the game is the main mathematical criterion of the optimal strategies in gambling.
Although mathematically correct, expectation-based optimal strategies rarely materialize in overall profit in reality, even if the player stick with them and apply them correctly.
Dr Catalin Barboianu, Mathematician and Philosopher of Science PhD
We will see in this article for what games and in what conditions the expectation-based strategy is actually effective and what the alternatives are for other situations.
What is the Expectation-Based Strategy in Gambling?
An optimal strategy in a game is a strategy based on mathematical certainties, which is theoretically better than any other possible strategy with respect to the goals of playing that game (winning it, making profit, or gaining any advantage for a further play).
This “better” should be understood still in mathematical terms, as optimizing is one of the roles of mathematics applied in the real life. Game theory deals with the conception, analysis and evaluation of the optimal strategies.
In gambling, which is governed by randomness, uncertainty, and luck, optimal strategies rely on the basic concepts of probability theory, namely that of probability and expected value.
The mostly used optimal strategy in gambling is the expectation-based strategy, sometimes called ‘advantage play’ in gambling slang.
It is a strategy that follows the principle of maximizing the gain and minimizing the loss in the long run, which actually means to maximize the expected value of every bet, if possible.
This maximization is done by a set of rules that the player has to follow regarding their choices and moves before and during the game.
Expected Value and Its Fluctuation
The expected value of a bet (the expectation) is an abstract notion, defined as:
- EV = probability of winning the bet x net profit if won + probability of losing the bet x loss if lost*
It is abstract first because the notion of probability is abstract. As a consequence, the EV is a kind of average, but this average is not arithmetical, but statistical, more as a limit.
This means there is not a finite number of trials (rounds of playing) that this average is computed on, but an infinite one.
Hence the EV reflects the net profit associated with that bet if it is hypothetically placed repeatedly and in the same conditions over and over again, to infinity.
Casino games and other games of chance are so conceived to have a negative expectation of any of their bets (the opposite in sign of the house edge), for the house to have the guarantee that they always win overall in the long run.
This goal is attained by choosing the payout rates of the bets so that the EV is negative.
However, there are circumstances in some games where the probabilities associated with the bets change from the prior probabilities, depending on the current configuration of the game (the available information).
By choosing the right action (with respect to the outcome to bet on, wagering, and even holding the bets) a player can maximize the EV of their bet and implicitly reduce the house edge.
There are circumstances (rare indeed) when the EV even becomes positive, which have to be exploited if occurred. This is what the expectation-based strategy is about.
Games Allowing Expectation-Based Strategies
Not every game of chance allows optimal strategies, including expectation-based ones, but a very few of them.
It is about those games that have multiple stages, in which the configuration of either the partial outcomes of the game or the wagers change.
As we saw, the general formula of the expected value of a bet aggregates both probabilities and payout rates (because the profit and loss are computed using these rates, applied to the wagers). This is the primary reason why the conditions for a game to allow expectation-based strategies are those I stated before.
Obviously, card games are more entitled to obey such conditions. Among them, blackjack is the most known for its strategic complexity, which includes optimal strategies. A game similar in some respects with blackjack is baccarat, where in some circumstances an expectation-based strategy can be theoretically applied.
Another card game with multiple stages and fluctuant wagering, played against opponents this time, is poker. The game of poker allows optimal strategies, which sometimes are combined with other non-mathematical strategies, falling within the psychology of poker.
How about sports betting? This is a special case, as subjective evaluations of the likelihood of an outcome are involved in strategies not mathematical probabilities. Yet handicapping is a form of chasing positive expectations, hence I will include sports betting in the list and explain this in a future section.
You may be surprised to find that slots, a game of pure luck, offer its own circumstances and moments where an expectation-based strategy can be theoretically applied.
Dr Catalin Barboianu, Mathematician and Philosopher of Science PhD
But is the expectation-based strategy really effective when applied in these games? Does it look as good in reality as it looks on the mathematical paper? We will answer that by seeing see how such strategy is applied and works in every game I mentioned.
Expectation-Based Strategy in Blackjack
In blackjack, the optimal strategy comes in two main forms, known as ‘basic strategy’ and ‘card-counting strategy’. Both strategies are based on maximizing the expected value of each bet by exploiting the information given by the cards exposed or dealt. These strategies are the result of hard mathematical work pursued since the 1960s with the decisive contribution of mathematician Edward O. Thorpe.
The basic strategy indicates the right move for the player (stand, hit, split, or double down) depending on their current total and the dealer’s first card.
The card-counting strategy uses also the information given by the individual cards that were dealt at that table, including for other players, for a more accurate estimation of the probability of receiving certain cards if hitting.
It worth noting that applying such strategies in blackjack assumes high skills, especially for the card-counting strategy.
The question arises as to how effective are such strategies in blackjack, given that expectation-based strategy is supposed to work only in the long and very long run for an optimized EV to get materialized overall. Would a highly skilled player be in profit after, say 100 or 500 rounds?
Why the optimized EV usually gets materialized
History says that when Edward Thorpe first tested his theory in practice (in two US casinos) made a profit of $11,000 in that weekend, with a $10,000 investment.
This would make one incline to a ‘yes’ for our previous question, but we must be circumspect about this, as the world of gambling is one of randomness, where the luck factor is always decisive.
Still, in theory, blackjack house edge can be reduced from a standard 0.5% for the classical game down to 0.1% if played with basic strategy (the lowest house edge among the casino games). At the beginning of the game, the player starts with an apriori probability of 42.43% of winning it, which is a big one.
This probability evolves with every new card dealt and the increases and decreases may be significant in some situations. The possibility of such fluctuations is in fact the element that increases the effectiveness of the expectation-based strategy (especially the card-counting one), because such fluctuations have to be exploited and the game of blackjack offers them.
Moreover, the mathematical constitution of this blackjack strategy by Thorpe also employed the Kelly criterion, which is a strategy of wagering a succession of bets, which optimizes the growth rate of the profit over the optimized play. Hence we have in fact a double optimal strategy applied – one for the play itself and one for the profits.
All these theoretical factors together can explain why the skilled blackjack players are in general successful; however, the answer to our question above can be formulated still in probabilistic terms: It is very likely that an expectation-based strategy will be effective for a skilled blackjack player sticking with the strategy, in a reasonable number of playing rounds. This is also the reason why casinos ban card counters.
Expectation-Based Strategy in Poker
In poker there is a form of expectation-based strategy, which was devised over time in the poker community by poker theorists and skilled players rather than mathematicians. It is about the so called ‘pot odds – card odds’ strategy.
Pot odds at any moment of the game are the ratio between the amount of the current pot, including your stake, and your stake. It reflects the share of the pot that is your contribution as the stake of your bet in that round.
The strategy works this way: If your pot odds are estimated to be lower than the probability of winning (expressed as card odds), you should call the bet. If not, you should fold. Making your call will bring you advantage in the long run (not necessarily in that particular round), while folding will minimize your loss. In statistical terms, the expected value of your bet is positive under that condition.
While most of the successful poker players declared that they use the ‘pot odds – card odds’, an in-depth analysis of the strategy indicates that the eventual actual advantage is more likely to come from exploiting the strength of the hands (in probability terms) and the psychological skills rather than materializing any positive expectation in the long run.
Why the optimized EV is unlikely to get materialized
First, consider the bluff element. Reading a bluff correctly creates for the skilled player an advantage which cannot be quantified mathematically nor assumed as random.
For Texas Hold’em, the current most popular version of poker, a study that analyzed 103 million online games found that 75.7% of the games were decided on the basis of the bids alone, without a showdown. In 49.7% of these showdowns, the hand of a player who had folded would have been the winner.
Second, poker generates far more card combinations as possible outcomes in the game than blackjack does. In Texas Hold’em there are 2,598,960 possible card combinations for the configuration consisting of your own hole cards and the flop cards and 20,358,520 for that consisting of the hole cards and the community board in the turn stage.
With such a huge number of combinations, it is expected that certain configurations will not occur frequently. This combinatorial complexity poses another important issue: it renders probability computations very difficult, many times impossible. And card odds are extremely important, as they change dramatically with very new stage of the game.
In statistical theory, the real gain will approach the mathematical expectation of a bet only if the trials are performed in identical conditions and their outcomes are entirely random.
It is not the case with poker: The bluff alters the random character of the outcomes and the huge number of possible configurations does not ensure a high frequency of identical situations for the strategy to have sufficient continuity with respect to conditions.
In addition, the strategy alone does not reflect the possible changes in the expectation conditional on how the rest of the hand is played. These issues indicate that a more effective optimal strategy for poker should be a probability-based (rather than EV-based) one, evaluating the strength of a hand in terms of card odds.
However, the challenge is – again – the difficulty of computing all the complex probabilities associated with poker and that of memorizing them as well. It is not accidental that constant successful poker players are those mastering the advanced math of poker.
Expectation-Based Strategy in Sports Betting
The case of sports betting is special relative to any discussion about optimal strategies, just because a mathematical probability (and implicitly an expectation) is not definable in this case.
When we talk about the likelihood of an outcome of a sports event, we actually refer to a subjective quantification and not a mathematical measure (like probability is) of the possibility of that outcome.
This is because the multitude of physical factors and conditions determining the outcomes are not quantifiable and the elementary events are not equally possible, like in casino games. Although we use the term ‘implied probability’ in sports betting, this is just another way of expressing the payout odds offered by the bookie and not a probability in the mathematical sense.
In sports betting, analysis of information about an event and choosing those bets where your personal assessment differs from the bookie’s is a form of expectation-based strategy.
Handicapping is in fact a chase for positive expectation, supposed to work not only in the long run, but also for individual bets – as long as your estimated chances for an outcome are higher than its implied probability, you take that bet as having a positive EV.
In this interpretation, and under the reserve relative to the meaning of the mathematical terms, the expectation-based strategy is trivially effective in sports betting, just because this is what the game is about – predicting correctly the outcomes as frequent as possible, for the bets picked especially with this purpose. Moreover, the Kelly criterion – which is another form of expectation-based strategy – can be applied for the bets assumed to have a positive expectation.
In the long run and over all kinds of bets and markets, a bettor’s advantage can also be assessed relative to the bookie’s vigorish rather than probabilities, since the odds offered include that commission (when you sum up together the odds for the outcomes of an event as decimal odds, the result is more than 1, as it would be if those odds were probabilities; that difference reflects the vigorish).
Statistical studies found that a 53-55% win rate over the bets is sufficient for the player to be declared successful, meaning they are in overall profit.
Worthless Expectation-Based Strategy in Baccarat and Slots
In these two games, the application of an expectation-based strategy is possible, but marginal. There are very narrow ranges of circumstances where such strategy is theoretically applicable. An optimized expectation is actually impossible to get materialized in these games.
In baccarat, additional information to use for improving your bet is only available when multiple hands are played at the table before the shoe is shuffled; it is a situation when you can keep the track of the cards dealt before placing your bet.
With every card dealt, the odds of Banker, Player, or a Tie change. Keeping track of the cards and using a counting algorithm, you may find when the odds incline in a side or another.
Such a strategy reduces the house edge, but not that much. For example, the house edge of the bet on the Banker comes to 0.99% compared to the house edge of 1.06% for that bet when not counting cards. But the real downside is the requirement to hold your bets until a favorable situation occurs, to optimize your gain.
This assumes that both the player and the casino are not going to mind if the player will only make a bet once every tens or hundreds of hands. It doesn’t worth the bother, right?
In slots, an expectation-based strategy is only applicable when we chase the jackpot at a must-hit-by progressive jackpot machine, provided that you know the rate of the meter rise of that system and its RTP.
A simple formula provides you with the breakeven point of that system, namely the value of the jackpot meter which is theoretically associated with zero expected value of your bet. The strategy consists of playing the machine only after the meter passes the breakeven point, for keeping a positive expectation.
Why such strategy would be ineffective? First, chasing the jackpot is a risky choice itself for your bankroll. The probability of hitting the progressive jackpot is minute, because it is far lower than the probability of hitting the required base prize (in by-win progressive systems, you have to hit the base prize and your hit to be selected by the triggering algorithm for the progressive jackpot).
That final probability could be in the order of 1 in millions or tens of millions. Besides that, the correct interpretation of the advantage given by the expectation-based strategy is: If you are lucky enough to hit the jackpot in this life (given its probability), when this will happen, it is very likely that your cumulated loss will not exceed the jackpot.
Conclusion
The expectation-based strategies optimize the expected value of your bets, in a mathematical sense. However, in the real life of gambling, an optimized expectation rarely gets materialized, whatever the skills of the players. Abstract notions as probability, expected value, and other statistical averages are defined in mathematics as limits toward infinity.
In general, their application and interpretation in the real life do not match their mathematical definitions and properties; our gambling life is finite, while expected value is an infinite sum.
Dr Catalin Barboianu, Mathematician and Philosopher of Science PhD
Mathematics says that optimal strategies in gambling are better than others and we should trust that, as a theoretical fact. However, the materialization of an optimized expectation is circumstantial.
The games where such materialization is the most likely to happen are blackjack (by a significant house edge reduction) and sports betting (where optimal play is synonym with optimal pick and informed prediction).
In poker, the expectation-based strategy is hardly effective and only if it is backed up with strong math skills. In baccarat and slots, application of such strategy is marginal and ineffective.
For the card games, a strategic alternative to an expectation-based strategy is the probability-based strategy, focusing on exploiting the probabilistic strength of your hands rather than following a long-term discipline of wagering with uncertain results.
This is also applicable to blackjack, for players who don’t have the required skills for a card-counting strategy.
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