Tag: games

The House Always Wins: Simulating 5,000,000 Games of Baccarat a.k.a. Punto Banco

The associated GitHub repository with R code.

Past weekend, I visited the casino with some friends. Of all games, we enjoy North-American-style Baccarat the most. This type of Baccarat is often called Punto Banco. In short, Punto Banco is a card game in which two hands compete: the “player” and the “banker“. During each coup (a round of play), both hands get dealt either 2 or 3 cards, depending on a complex drawing schema, and all cards have a certain value. Put simply, the hand with the highest total value of cards wins the coup, after which a new one starts. Before each coup, gamblers may bet which of the hands will win. Neither hand is in any way associated with the actual house or player/gambler, so bets may be placed on both. All in all, three different bets can be placed in a game of Punto Banco:

The player hand has the highest total value, in which case the player wins (Punto);
The banker hand has the highest total value, in which case the banker wins (Banco);
The player and banker hands have equal total value, in which case there is a tie (Egalité).

If a gambler correctly bets either Punto or Banco, their bets get a 100% payoff. However, a house tax will often be applied to Banco wins. For instance, Banco wins may only pay off 95% or specific Banco wins (e.g., total card value of 5) may pay off less (e.g., 50%). Depending on house rules, a correct bet on a tie (Egalité) will pay off either 800% or 900%. A wrong bet on Punto or Banco stands in case Egalité is dealt. In all other cases of wrong bets, the house takes the money.

My friends and I like Punto Banco because it is completely random but seems “gameable”. Punto Banco is played with six or eight decks so there is no way to know which cards will be next. Moreover, the card-drawing rules are quite complex, so you never really know what’s going to happen. Sometimes both Punto and Banco get only two cards, at other times, the hand you bet on will get its third card, which might just turn things around. Punto Banco’s perceived gameability comes through our human fallacies to see patterns in randomness. Often, casino’s will place a monitor with the last fifty-so results (see below) to tempt gamblers to (erroneously) spot and bet on patterns. Alternatively, you might think it’s smart to bet against the table (play Punto when everybody else goes for Banco) or play on whatever bet won last hand. As the hands are dealt quite quickly in succession, and the minimal bet is often 10+ euro/dollar, Punto Banco is a quick way to find out how lucky you are.

Image result for punto banco scoreboard — Examples of Baccarat monitors, often placed next to a table.

So back to last weekend’s trip to the casino. Unfortunately, my friends and I lost quite some money at the Punto Banco table. We know the house has an edge (though smaller than in other games) but normally we are quite lucky. We often discuss what would be good strategies to minimize this houses’ edge. Obviously, you want to play as few games as possible, but that’s as far as we got in terms of strategy. Normally, we just test our luck and randomly bet Punto or Banco, and sparsely on Egalité.

As a statistical programmer, I thought it might be interesting to simulate the game and its odds from the bottom up. On the one hand, I wanted to get a sense of how favorable the odds are to the house. On the other hand, I was curious as to what extent strategies may be more or less successful in retaining at least some of your hard-earned cash.

In my simulations, I follow the Holland Casino Punto Banco rules, meaning a six-deck shoe and a Banco win with 5 pays out 50%. I did adopt the more lenient 9-1 payoff for Egalité though. Several hours of programming and some million simulated Baccarat hands later, here are the results:

Do not play Baccarat / Punto Banco if you do not want to lose your money. Obviously, it’s best to not set foot in the casino if you can’t afford to lose some money. However, I eagerly pay for the entertainment value I get from it.
You lose least if you stick to Banco. Despite having only a 50% payoff when Banco wins with 5, the odds are best for Banco due to the drawing rules. Indeed, according to the Wizard of Odds, the house edge for Banco (1.06%) is slightly lower than that of Punto (1.24%).
Whatever you do, do not bet on Egalité. Because most casino’s pay out 8 to 1 in case of a correctly predicted tie, betting on one seems about the worst gambling strategy out there. With a house edge of over 14%, you are better off playing most other games (Wizard of Odds). Although casino’s paying out a tie 9 to 1 decrease the house edge to just below 5%, this is still way worse than playing either Punto or Banco.

The figure below shows the results of the five strategies I tested using 50,000 simulations of 100 consecutive hands. Based on the results, I was reluctant to develop and test other strategies as results look quite straightforward: play Banco. Additionally, Wikipedia cites Thorp (1984, original reference unknown) who suggested that there are no strategies that will really result in any significant player advantage, except maybe for the endgame of a deck, which presumably requires a lot of card counting. If you nevertheless want to test other strategies, please be my guest, here are my five:

Punto: Always bet on Punto.
Banco: Always bet on Banco.
Egalité: Always bet on Egalité.
LastHand: Bet on the outcome of the last hand/coup.
LastHand_PB: Bet on the outcome of the last hand/coup, only if this was Punto or Banco.

The above figure depicts the expected value of each strategy over a series of consecutive hands played. Clearly, the payoff is quite linear, independent of your strategy. The more hands you play, the more you lose. However, also clear is that some strategies outperform others. After 100 hands of Baccarat, playing only Banco will on average result in a total loss below the amount you wager. For example, if you bet 10 euro every hand, you will have a loss of about 9 euro’s after 100 rounds, on average. This is in line with the ~1% house edge reported by the Wizard of Odds. Similarly, betting only Punto will result in a loss of about 130% of the bet amount, which is also conform the ~1.4% house edge reported by the Wizard of Odds. Betting on Punto or Banco based on whichever won last (LastHand_PB) performs somewhere in between these two strategies, losing just over 100% of the bet amount in 100 hands. Your expected losses increase when you just bet on whichever outcome came last, including Egalité, resulting in around ~-150% after 100 hands. This is mainly because betting on Egalité, which seems about the worst strategy ever, will result in a remarkable 493.9% loss after 100 hands.

Apart from these average or expected values, I was also interested in the spread of outcomes of our thousands of simulations. Particularly because gamblers on a lucky streak may win much more when betting on Egalité, as the payoff is larger (8-1 or 9-1). The figure below shows that any strategy including Egalité will indeed result in a wider spread of outcomes. Betting on Egalité may thus be a good strategy if you are by some miracle divinely lucky, have information on which cards are coming next, or have an agreement with the dealer (disclaimer: this is a joke, please do not ever bet on Egalité with the intention of making money or try to cheat at the casino).

If you want to know how I programmed these simulations, please visit the associated github repository or reach out. I intend on simulating the payoff for various other casino games in the near future (first up: BlackJack), so if you are interested keep an eye on my website or twitter.

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So You Think You Can (A/B) Test?

Decision making under uncertainty is complicated. These days, many business rely on real-life experiments, or A/B tests, to reduce that uncertainty and improve their decision-making. For instance, here’s a presentation on how A/B testing helps improve business outcomes at Etsy.

Lukas Vermeer built So You Think You Can Test, an online simulation game in which you are the decision-maker in a company. You control the backlog and running of experiments and each day you have to decide which tasks to prioritize (or deleted entirely). Your decisions affect the sales of the company, so be wise and use the experimental information to your advantage.

You can play the game here.

Libratus: A Texas Hold-Em Poker AI

Four of the best professional poker players in the world – Dong Kim, Jason Les, Jimmy Chou, and Daniel McAulay – recently got beat by Libratus, a poker-playing AI developed at the Pittsburgh Supercomputing Center. During a period of 20 days of continuous play (10h/day), each of these four professionals lost to Libratus heads-up in a whopping total of 120.000 hands of No Limit Texas Hold-em Poker.

A player may face 10 to the power of 160 different situations in Texas Hold-em Poker: more than the number of atoms in the universe. It took extensive machine learning to compute and prioritize the computation of the most rewarding actions in these situations. Libratus works by running extensive simulations, taking into account the way the professionals play, and figuring out the best counter strategy. Although it is not without flaws, any “holes” the players found in Libratus’ strategy could not be exploited for long, as the algorithm would quickly learn and adapt to prevent further exploitation. The experience was completely different from playing a human player, the professionals argue, as Libratus would make both tiny and huge bets and would continuously change its strategy and plays.

The video below provides more detailed information and also shows the million-dollar margin by which Libratus won at the end of the twenty day poker (training) marathon:

Neural Networks play Super Mario Bros & Mario Kart

Seth Bling calls himself a video game designer, a hacker and an engineer. You might know him from MarI/O: his neural network that got extremely good to at playing Super Mario Bros. The video below shows the genetic approach Seth used to train this neural network. Seth randomly generated a starting population of neural networks where the inputs – the current frame in the Mario video game – were randomly connected to the outputs – the eight buttons to press (jump, duck, up, down, right, left, etc). By giving the neural nets that made it furthest into the game a larger chance to pass on their genes (their input-output relations) to the next generation with slight mutations, Seth automatically generated neural networks that were more and more proficient in completing the game. In short, by evolution, Seth’s neural network learned the most effective response to the changing video game environment.

After MarI/O, Seth this week posted his newest creation: MariFlow. Here, Seth trained a neural network on 15 hours of training data, consisting of Seth himself playing Super Mario Kart. The neural network thus learned what buttons (output) Seth would most likely push when he encountered a certain Mario Kart parcours piece (input). However, due to random chance, the neural net would often get itself stuck in situations that Seth had not encountered in his training sessions (e.g., reversed, against a wall). The neural net would fail miserably in such situations because it had not learned how to behave. Accordingly, Seth had to generate new training data for these situations and he did so using Human-Computer Interactions in Machine Learning: Seth and the neural net would play alternatively for a while, thus generating training data for situations that Seth would not have encountered on its own. After the neural net was trained with these additional data, it became quite proficient in playing Mario Kart (like Seth) often even winning matches! If you want to know more, you can read the manual here or watch Seth’s video below. If you want to replicate or just play with the data, Seth made everything available here.

Seth has active YouTube, Twitch and Twitter channels and I recommend you check them out!

Regular Expression Crosswords

A regular expression (regex or regexp for short) is a special text string for describing a search pattern. You can think of regular expressions as wildcards on steroids. You are probably familiar with wildcard notations such as *.txt to find all text files in a file manager. The regex equivalent is .*\.txt$.

Last week I posted a first tutorial on Regular Expressions in R and I am working its sequels. You may find additional resources on Regular Expressions in the learning overviews (R, Python, Data Science).

Today I came across this website of Regular Expression Crosswords, which proves a great resource to playfully master regular expression. All puzzles are validated live using the JavaScript regex engine. The figure below explains how it works

Via the links below you can jump puzzles that matches your expertise level:

Video: Human-Computer Interactions in Reinforcement Learning

Reinforcement learning is an area of machine learning inspired by behavioral psychology, concerned with how software agents ought to take actions in an environment so as to maximize some notion of cumulative reward (Wikipedia, 2017). Normally, reinforcement learning occurs autonomously. Here, algorithms will seek to minimize/maximize a score that is estimated via predefined constraints. As such, algorithms can thus learn to perform the most effective actions (those that minimize/maximize the score) by repeatedly experimenting and assessing strategies.

The approach in the video below is radically different. Instead of a pre-defined scoring, human-computer interaction is used to assign each action sequence (each iteration/experiment) a score. This approach is particularly useful for complex behaviors, such as a back-flip, for which it is hard to pre-define the constraints and actions that lead to the “most effective” back-flip. However, for us humans, it is relatively easy to recognize a good back-flip when we see one. The video below shows how the researchers therefore integrated a human-computer interaction in their reinforcement learning algorithm. After observing the algorithm perform a sequence of actions, a human actor indicates to what extent the goal (i.e., a backflip) is achieved or not. This human assessment thus functions as the score which the algorithm will try to minimize/maximize.

This approach can be really valuable for organizations seeking to improve their machine learning application. The paper on the principle (Deep Reinforcement Learning from Human Preferences) can be found here. The scholars conclude that this supervised approach based on human preferences has very good training results whereas the cost are similar the simple bulldozer approach of training a neural net from scratch using GPU servers.