Simulating Pinochle Meld Scoring
What is Pinochle?
Pinochle is a team trick-taking card game that my wife’s extended family plays every time they are together. The unique (“strange” or “nonsensical” could also be argued) rules make the game somewhat difficult to learn and remember. This has led to the game being passed on within families — most new players learn through birth or marriage (I learned via marriage).
Each Pinochle hand starts with each player being dealt twelve cards from a special Pinochle deck. This deck only contains cards 9-Ace with two copies of each card, for a total of 48 cards. Players then enter a round of bidding to set the points needed to win the hand and determine the “trump” suit. This bidding is done without knowledge of your teammate’s cards giving you an incomplete understanding of the situation when you place your bid.
Following the bidding process, the players then lay down their cards to show their “meld” points. There are a number of combinations that can earn you meld points, such as a round of Kings for 8 points or a run from 10-Ace for 15 points. A “Pinochle” is the combination of a Queen of Spades and a Jack of Diamonds, worth 4 points (don’t ask why). Additionally, some meld combinations only come into play or are enhanced if you’re on the team that won the bidding. Each team’s meld points are added up and recorded, and if you won the bidding period then your meld points contribute towards the number of points you need to win that hand. For example, if you won the bidding with a bid of 23 and your team has 9 points in meld you need to take 14 points (out of 25 possible) during play or your team will be “set” and you will lose points.
The importance of meld points towards placing and succeeding to meet your bid cannot be understated. By restricting player knowledge of meld points prior to bidding, players risk making a mistake by over- or under-estimating the meld of their teammate or opponents. A better understanding of the probabilities behind meld scoring would allow a player to make more informed decisions and improve their play.
Pinochle, meet Python
To improve my Pinochle play, I wanted to analyze the distribution of meld to get a better sense for what I can expect of my teammate and opponents. I also wanted to create visualizations that I could easily show to my family so that they could easily learn something new, despite their many years of playing the game (many have 30+ years experience more than me!).
I decided to write a simple Python program to simulate the dealing of cards and scoring of meld, then automate enough trials (1 million to be exact) to provide an opportunity for probability to play out. Then I could use readily available Python tools to analyze and graph my results.
I decided to do this project as quickly as possible, since I’m currently in the middle of spending a week with family. In order to complete it despite all the events and have enough time to show my results and try out some of what I learned, I needed to get this done quickly. I’m very comfortable with Python, didn’t need to install anything, could get running quickly, and knew their were good tools for generating graphs of results automatically.
Following the decision to use Python, I made other technical decisions:
- Configurable. I knew that I would want to extend the project in the future, both to further analyze Pinochle but also to make it adaptable to similar games. Thus, I wanted to make it configurable. All configuration is done in config.py, including meld scoring, number of simulations, graph setup, etc.
- Reusable parts. I wanted to do this quickly, but I also wanted to make my components reusable. There is no reason that a Deck (deck.py) and Score Counter/Grapher (counter.py) could not be completely reusable for other unrelated projects. Similarly, the process of determining meld scores should be so configurable as to have very little specific to Pinochle.
- Thorough. In order to truly learn what to expect while playing, multiple kinds of data is useful. I wanted to analyze individual scoring, team scoring, and hand scoring (all 4 players combined). I tried to streamline processes so that we could run 1 million trials and keep track of all this data without the program taking more than a few seconds to run. I also wanted graphs to be automatically generated, allowing anyone to take my code and run it to verify my results or try their own tweaks.
- Simple. No fancy project organization, no package manager, etc. Get it done quickly using nothing but my Chromebook and the Cloud9 in-browser development environment (which I highly recommend!). The simpler it is the faster I could get it done and the easier it would be for someone else to run with the project.
As anyone who has played a lot of Pinochle would know, the results were heavily weighted towards low values. As fun as it is to get a “meld avalanche” (as my wife calls it) of 15+ points, that really doesn’t happen that often. In fact, you can expect to receive more than 6 points less than 50% of the time. Those 6 points are valuable, but with a minimum bid of 21 you still need to take 15 of 25 points during play in order to meet your bid.
It can feel pretty unfair when you’re playing and your opponents get 15, 20, or even 30 points in meld and you are suddenly in a huge deficit. But the data shows that those events are really uncommon — and unlikely to come your way in the same game. If you like to know the high end of outcomes, even if unlikely, you’ll want to notice that the most a team ever got in meld was 110 points. Luckily, scores that high are beyond rare, since it only takes 100 points to win the game!
As for bidding, it seems like you can’t reasonably expect more than 4 points in meld from your teammate. With a median individual score of 2 and a mean of 3.66, anything beyond 4 is actually pretty lucky. This suggests that bidding, as I’ve usually seen, should be done conservatively. Turns out, people can sometimes have a pretty good sense for probabilities without running 1 million simulations.
I did this quickly, so it doesn’t incorporate everything I’d like to. If I have time, I think these improvements would be great:
- Analyze the affect of my own meld on that of my teammate and opponent. If I have low meld, are they likely to have high meld (and vice versa)?. This conditional probability is what I think a player really needs to know.
- Create further visualizations of the data. This data is so heavily weighted towards the 0–10 range that other visualizations might be useful as well.
- Convert standard deviation to sample standard deviation. I used the population value for the sake of speed, but this really should be changed.
You can check out my work on my GitHub. Let me know what you think or if you have any suggestions for future enhancements!