Cuebids Scoring System
- Expected Value & Stars

Emanuel Unge

2024-09-23

If you prefer to watch a video, here is the article in a video format instead:

The scoring system is something very dear to me as it is one of the main reasons I started Cuebids to begin with. I get a lot of questions about the stars and about EV (Expected Value). It's about time we dive into that with this article.

Before we get into our scoring system, let’s talk about bridge for a second. Bridge is a very complex game, and unfortunately for us, the scoring system is arguably even worse. Just understanding what a given contract will score is difficult enough. Minors give fewer points than majors, which give fewer points than NT (but only in the first trick). We have slam bonuses, game bonuses, undertricks, penalties, doubles, redoubles, vulnerabilities… there is a lot!

So if you’re anything like me, you don’t calculate the score, but instead remember it from previous experience. Worst case, you look at the back of the bid card to see what a given contract is worth.

In the absolute worst case, you might glance at your opponents’ score sheet or shamefully ask your partner. Either way, the system is very complex, and eventually, after some effort, you finally write down your number on your score sheet. But if you’re playing in a Teams game, that is NOT enough.

Now, you need to compare your score to your opponents’. You have to subtract the two scores to get a difference. And then, that is NOT enough again! Now you need to look at a lookup table to understand what your IMP score is.

I think you can now understand why we use stars.

It’s very simple.

It’s a simplified, abstracted system.

But you probably want to know more details, so let’s get into it. The first building block of our scoring system is the Expected Value (EV).

Expected Value

Expected Value is what it sounds like. It is the expected value of a given contract.

Let me give you an example:

You are in 3NT and you have 100% chance of making it. The EV of this contract is:

1*400 (or 100%*400) = 400

But what happens if this game is on a finesse? You now have 50% chance of making it. 50% of 400 is 200. But then you also have a 50% chance of going down one, which would score -50. So the EV of this contract is

0.5*400 + 0.5*-50 = 175

175 is the Expected Value of 3NT on a finesse.

This is pretty useful, and this is something we use in Cuebids.

However, unfortunately, again, it is not that easy.

Usually a contract is not just on a single finesse, but on a multitude of variables at the same time. Perhaps we can't handle a 5-1 split in a suit, there is a finesse somewhere. Perhaps there are some endplay chances. Maybe there is even a percentage we can make a squeeze work.

Not it's very hard to assess the percentage of all these things happening at the same time.

So instead of trying to calculate chance, we simulate it!

We lock North and South's hands (you and your partner). Then we take East and West's hands and we shuffle them 1000 times.

Then we use a Double Dummy Solver to estimate if we are going to make each and every simulation.

And in the end, we have a long list of simulations with different scores.

Now we basically average everything up into one number, for each strain, for each level. (It's not exactly like this, but it is a good simplification). Now we have a number for each contract, which is the Expected Value.

Finding Par

Great, what do we use all of the Expected Values for? We need to get to the next step. We need to find the par score, which is where neither side can improve. Bridge bidding always goes up; you always keep moving up until both sides are satisfied, or at least done with the bidding. You might have seen in some bridge tournaments that the par is written on the scores. It can say par contract 4 doubled. The par is the contract where neither side can make more points by bidding more.

We use this as the golden contract. If you can reach that, you get 3 stars. Sometimes, because the robots are not perfect, you don’t have to reach that and can get 3 stars anyway. The par contract has an EV from the simulations. Once the bidding is finished, you’ve ended in a contract. If it is not the par contract, there is a difference between the EV of the par and the EV of your contract. This difference is what we use for the star calculation.

It is very similar to the IMP calculation, but we add a twist because we like to incorporate matchpoints as well. For example, if it is a part score, then smaller differences become more important than if it’s a slam. For instance, if 4 is the par contract, and you’re in 3: 4 has enough chances to make to yield 255 EV, while 3 always makes, sometimes with an overtrick, to net you 155 EV. The difference is 100 EV.

You lose 100 EV by not bidding 4, so you only get 1 out of 3 stars. Exactly where the cutoffs are depends on many factors, so we won’t dive into that here. But that’s how it works in a nutshell.

Pros and Cons

Simplicity

The first obvious pro is that it is very simple. It is very easy to gauge how well you did with a quick glance at the stars.

Speed

The second pro is that it is fast. We can do this calculation before you even bid the hand. Before you even start the hand, we know what the score of every contract is. That is why, when you play on Cuebids, you instantly see how many stars you get and the expected value.

Opponents' Bidding Doesn't Count?

Now, let's address a common issue with this system: the opponents' bidding is not taken into account when running the simulations. For example, if an opponent overcalls 3, the simulation does not reflect that they have 7 clubs. This means that the expected value might differ from reality in such cases. You might expect your right-hand opponent to have 7 clubs, which could affect your contract. It might mean you're never making 3NT, or always making 6 for instance.

However, in Cuebids, you can unfortunately not use that information. It's not that we don't know how to simulate it, but we can't anticipate what the opponents are going to bid. We don't know if the robots will overcall 3, as it may depend on the robot's version or what you opened. For example, if you opened 1, the robot might not be able to overcall 3 due to the fact that it might be artificial now.

The point is, we cannot anticipate the opponents' bidding. This ties into the pro of speed—simulations are done before you even bid the hand. But this doesn't align with the con of not knowing the opponents' bidding, and that's one of the reasons we avoid trying to incorporate it.

Even if we could account for that instantly, there's still the issue of different players having different information. Perhaps you opened 1 and someone else opened 1NT due to playing different systems or evaluating their hand differently. In that case, one player might see an overcall of 3, while the other does not. Can you truly compare these two situations—deciding whether to bid 3NT or not? I don't think so. It's comparing apples to oranges, which is why we have it set up this way.

In the end, we sacrifice slight accuracy in scoring to get instant feedback and easier comparing.

Double Dummy

How good is double dummy compared to real-life declaring? Obviously, double dummy is pretty damn good. It knows exactly how the cards are distributed and will always make the contract if it is possible. But that’s also true for the defense. They will always make the best defense possible and defeat the contract if it is possible.

It has been shown that double dummy play is very close to real life, but it’s not perfect. There are some deals where it’s more obvious that the double dummy solver is too good. One notable scenario is when a queen is missing in a suit.

If you have

you will always find that queen—regardless of where it is—if you are playing double dummy. In real life, however, that’s almost just a 50/50 shot.

The good news is that we are working on a fix! The beta test went live on the 20th of September 2024, and we are very excited about this. We hope you will be too!

Thank you for reading this article and I hope you learned something!
Please share it with your bridge friends!

Emanuel Unge

Founder of Cuebids