Points for Tricks
Part I: Introduction
When I spend time with my family, we spend a significant portion of our time eating, drinking, and playing cards (often simultaneously). Our preferred card game is a variation of whist, the origins (and even the correct name) of which are uncertain and seemingly unknowable. For the purposes of this blog, the game shall be known as B-Whist.
How it Works
The game, in brief, goes like this: four players are dealt a single card. Each player looks at their card and considers whether or not they are likely to win the hand. The highest card wins, but spades trump all else. When each player has made their prediction, the hand is played. If the player predicted his or her result correctly, they win points, and a merry time is had by all.
In the next round, each player is dealt two cards and makes a new prediction; this time on how many hands they will win out of the two for some players this number is arrived at though rigorous application probabilistic reasoning, for others it really is more of a wet finger in the air). The predictions are called bids, and bids are made around the table to the dealer’s left. The two hands of the round are then played. Players must follow suit if they have the cards available, unless they choose to play a trump. When a hand is won, the player collects the all cards played in that hand, places it neatly by their person, and temporarily radiates a smug glow; for they have just won a trick. At the end of the round, if a player’s bid is equal to the number of tricks they have won, they awarded with more points, and the merriment continues (at least in theory). The subsequent rounds follow the same formula, with three cards for each player in the third round, four in the fourth, and so on up to thirteen, and then all the way back down to one again. The dealer (and therefore the first bidder) rotates after each round. At the end of the game, the points are tallied up and the ultimate victor is declared.
There is one further devilish detail to note. The final bidder of the four is not permitted to bid in such a way that the total number of bids made is equal to the number of cards held by each player. Say, for example, that in round nine, Player A confidently bids six, Player B meekly bids zero, and Player C bids two. The total bids made so far sum up to eight, and therefore Player D cannot bid one. Any reader who subsequently plays the game for the first time will quickly come to love and hate this rule.
The Perfect Card Game?
While it is, in my opinion, an excellent game (merriment virtually guaranteed), there are some issues that tend to be brought up every time I play, and these all relate to how the game is scored. How many points are awarded for correct bids? Should points be deducted for incorrect bids? Can’t we just keep it simple and say one trick equals one point? Why do you always have to overanalyse and ruin everything? These are all good questions that deserve answers.
The main problem with B-Whist is that as the rounds get progressively larger, the spread between bids is likely to increase, and ultimately the distance between the players on the scoreboards is bound to follow. Point are generally earned in the high rounds, and losses get more and more difficult to recover from as the rounds get smaller again. As a consequence, players can find themselves in a position where they have no chance of winning relatively early on, which can make the remainder of the game a tedious slog (depending on who you ask).
This is, however, not a problem with the mechanics of the game, but rather with the scoring system. The divergence of results described above are amplified or dampened by different scoring systems, some of which lead to results that do not reflect the players’ efforts. Consider the following example using the most simple scoring system, in which a point is awarded for every trick won. We shall call this system Count Tricks Won (CTW):
In this case, Player 2 wins five times, and Player 1 only wins three times, but Player 2 still wins overall. Now consider another example using the same system:
Here we find that both players win, overbid, and underbid the same number of times, but the final scores are wildly different. Player 1 has won the game because they won the important middle-rounds, and the while Player 2 has performed similarly in all respects, they have failed to capitalise on the available points. In both examples, the players are rewarded differently for their outcomes, and the games are won by focusing on the high-scoring middle rounds. This means that the majority of the game is effectively a waste of time.
Another pitfall of the CWT system is the fact that correctly bidding zero yields no points. The obvious solution, therefore, is to award a bonus point for each win, in addition to the tricks won. Let us call this system (or, as we shall see, this family of systems) CWT+Bonus.
CWT+Bonus presents its own problems, however. First of all, a player can just bid zero and sabotage themselves in every round in order to guarantee a final score of 25 points (a bold strategy, as self-sabotage can be prevented by other players — another wonderful feature of B-Whist, but a possibility nonetheless). Secondly, the bonus point does not do enough to prevent over-reliance on the middle rounds. The bonus increases the proportional value of the early round victories (ie., a winning one trick in round won will yield one point under CWT, but two points under CWT+Bonus — a 100% increase; whereas the corresponding increase between the two systems on a win of twelve tricks in round twelve is only 7.69%), the effects of which can be seen when the above examples are rescored:
As the game summaries above show, when a bonus point is given for each win, the final scores converge, and therefore the different round outcomes become less important. This is even more clear when the bonus is increased to, say, ten points:
Under this system, the scores can be said to more accurately reflect the outcomes of the rounds. In example 1, Player 1 wins five times and now wins the game. In example 2, the number of wins, overbids, and underbids are the same for both players, and Player 1 still wins because of the middle rounds, but the overall outcome of the game is far tighter.
Problem solved? Unfortunately, no. In the above examples, we see that the issue of the rapidly diverging scores is alleviated as the bonus was increased, and as the number of wins becomes more important to the overall score. If the bonus is increased further, the wins will become more important than the number of tricks won, and therefore players will not be rewarded for making high bids. This is just another side of the bid-zero-every-time coin discussed earlier.
What, you may ask, are we meant to do with all of this? How are we meant to come up with a perfect scoring system for B-Whist? That would take time, effort, research, and at the very least a short series of blog posts…
The Perfect Scoring System Doesn’t Ex…
As you may have guessed by now, this post is the first of a series which aims to uncover the unicorn of B-Whist scoring systems. One that makes up for the pitfalls identified in the basic CTW and CTW+Bonus systems described above. Over the course of the next few posts, I hope to offer the following:
- An exploration of the mathematics behind the game and how it informs ideal scoring scenarios;
- A set of criteria for judging different systems based on primary assumptions about what makes B-Whist fun in the first place (hopefully without destroying said fun);
- Simulated games across multiple scoring systems, visualised and analysed in reference to these criteria; and
- A conclusion on the ultimate scoring system — tried and tested through data science!
To conclude this introductory post, I briefly present the base conditions for this project; including a preliminary statement of the starting assumptions, and an explanation of the different types of scoring systems to be investigated.
I find B-Whist to be such an enjoyable game because it combines skilful, thoughtful play with good fortune. You must play the hand you are dealt, but you can be clever about it; coming up with a good bid for your hand, timing the play of your cards, and sabotaging the other players’ strategies. It is also exciting because the scoreboard can be tight and dynamic. The leading player can sometimes find themselves at the bottom of the scoreboard in just a few rounds, and therefore cannot simply coast on early round victories. Ultimately, it is a simple game, which is in large part why it is so charming.
With that in mind, I propose the following starting assumptions on what would make a good scoring system:
- The system maintains a tight and dynamic scoreboard throughout the 25 rounds.
- The system rewards intelligent play (and perhaps risk taking, while also potentially penalising conservative play).
- The system is simple enough to memorise without needing to write down.
These, I submit, are not controversial assumptions. As we shall see in future posts, there are a number of different combinations of point yielding (and subtracting) mechanisms that will all result in widely different scoreboards. The most hopeful of these systems will be modelled on simulated games (the number of which is in the order of 2x10¹⁶⁵!) and, hopefully, a winning candidate will ultimately be uncovered.
Finally my family and I can get back to arguing about something else.