During a single game, you have to make a number of decisions: What cards do you get rid of? What cards do you lead with? Should you close the deck? Ultimately, these decisions should be based on the potential payoff, i.e. the game points you or your opponent might score. In a nutshell, you should always make the decision that maximizes your payoff. This payoff can be determined using the concept of expected value (EV). The EV tells you what a certain decision is worth in the long run. To illustrate this concept, let's look at an example.
Your starting hand is Qd-Jh-As-Ks-Qs. The face-up trump is the Js, you are to lead. What should you do?
First of all, let's try to classify this starting hand. It contains 3 trump cards including the marriage, which is almost as good as it gets. Then, the hand is worth 23 card points, which is below average, but recall that the trump marriage is worth 40 additional points. As for suits, you dominate the trump suit, but you definitely are on the losing side as far as the other 2 suits are concerned.
Now, what options do you have? What are the corresponding payoffs?
Option 1: Closing the Deck
Right at the start of the game, you could close the deck. Please note that you can be sure to reach as many as 58 card points with the cards in your hand alone: The marriage is worth 40 points, plus you will take three tricks including the As (11 points), Ks (4 points), and Qs (3 points). That leaves you 8 points short of the 66 card point mark. So, the three cards you get from your opponent should be worth at least 8 points. The best possible scenario would be your opponent actually having the Ts, which is the case 36% of the time. After closing the deck, you will lead with the As, so if your opponent has the Ts, he has to play that card as he has to follow suit. These ten additional points already are more than you actually need to win the game.
The only time you will not win the game is when the three lowest cards in your opponent's hand are worth less than 8 points. In other words, he needs one of the following card combinations: 3 jacks; or 2 jacks and a queen. In this particular situation, your opponent cannot have 3 jacks, since you hold one jack yourself and another jack is the face-up trump card, so the only scenario you should worry about is the 2J+Q scenario. Your opponent will have this combination 7% of the time.
In short, of the 64% of the time your opponent does not have the Tt, he is 7% to have 2J+Q.
So 4% of the time your opponent will score 3 points. Thus, the EV of this strategy is 2.7 game points.
Actually, I would consider this a conservative estimate for the following reason: Whenever people have to decide what cards to get rid of they tend to keep the higher cards. In our example, you could decide to lead with either the Jh or Qd after the second trick, knowing that you will definitely take the last trick since there is still one more trump in your hand and your opponent will not have reached 66 points by then. This way you might be able to score some extra points.
However, we should try and keep things simple. The EV as we calculated it is a good estimate.
Option 2: Playing the Jh
An alternative strategy would be not to close the deck and to play the Jh. If your opponent has the remaining trump and figures that he is in trouble, he might take the trick and limit you to 2 points at the most. But once again, let's keep it simple, ignore this scenario, and look at the more probable one: Your opponent has at least one heart 87% of the time. What's worse is the fact that he might already have a marriage or get one after the first trick. If he takes the trick with the Ah, which he has 36% of the time, he might even get to 33 points, in which case you will be able to score a single point only. So let's be optimistic and assume that 13% of the time you will still score 3 points, 83% of the time you will score 2 points, and the remaining 4% of the time you will score one point. The reason why this is an overly optimistic estimate is that we assume that you never lose this game (13% + 83% + 4% = 100%), which is impossible (you will almost always find a way to somehow lose a game). Still, your EV is 2.1 game points only.
Despite the optimistic assumptions, the EV of this strategy is 0.6 game points below the EV of the first strategy. This option definitely is inferior to the first one. Note that you will get a similar result for first playing the Qd instead of the Jh.
Option 3: Announcing the Marriage
A third possible strategy would be to announce the marriage without closing the deck. If your opponent has the Tt, he will definitely take the trick. Once again, he might have a marriage and thus get over the 33 card point mark. So, the probability distribution for this part is the same as it was in option 2: 4% of the time, you will score one game point and 32% of the time, you will score 2 game points.
For the sake of simplicity, let's assume that if you take the first trick, you will then close the deck no matter what card you take from the deck. Please note that if your opponent does not have any jacks in his starting hand, you will definitely get to 66 card points after closing the deck at this point. This will happen 23% of the time.
Even if he does have a jack, there is a fair chance that you will get a card that significantly improves your hand: the trump T, one of the remaining aces, or the Kd (giving you the marriage of diamonds). That is a total of 5 cards of the remaining 14 cards, or 36%. If you do not improve your hand, note that your opponent might get the Tt after the first trick (8%). If that does not happen either, you might not reach 66 card points, but only if your opponent has the remaining J and a Q, which will happen 11% of the time. Therefore, your opponent will score 3 game points 3% of the time.
The EV of this strategy is 2.4 game points.
Option 4: Playing the At
One last possible strategy would be to play the As without closing the deck, then announcing the trump marriage after the first trick. That puts you well above the 33 card point mark, so if you somehow manage to lose this game, your opponent can score a single point only. Please note, however, that your opponent is 42% to have the trump ten after the first trick (if you do not have it by then). The outcome of this game depends largely on the additional cards you will get during the game, but obviously you are in a good position. Your opponent might be more likely to make mistakes in a pressure situation like this. All things considered, I would estimate the EV of this strategy to be below 2.0 (without actually having calculated or simulated this situation).
Summary
In this article, we have looked at how to compute the payoffs of alternative strategies. The strategy with the highest EV is the one yielding the best results over the long run. Therefore, in our example, option 1 was the best strategy with an EV of 2.7 game points. Option 3 led to an EV of 2.4 game points. Some people might argue that this difference of 0.3 game points is not a particularly big difference. Actually, that is true and has to do with the strength of this specific hand. However, you should realize that all players get the same hands over the long run and should therefore try to maximize their payoff in each and every game.
Consider the following statistics: From experience, I expect a player to score one game point in 40% of games, another 40% of the time a player will score 2 game points, and in the remaining 20% of games 3 game points will be scored. That means that a player gets 1.8 game points per game won on average. Consequently, you need to win 4 games on average to win a Bummerl. If both players play equally well, a Bummerl consists of about 7 games. Now let's assume that in every single game you give up 0.3 game points in EV since you play suboptimally. In total, you give up 2.1 game points.
That is a lot, considering that you need to score 7 points to win. To compensate for this difference a huge extra effort is needed. As a player scores 1.8 game points per game won, this basically means that you give up one game without your opponent having to make any kind of effort.
All of these small correct decisions will lead to a large chunk of points in the end and might make the difference between winning and losing.
