Expected Value Part II

In the previous article, the concept of EV was introduced and applied to an actual playing situation. In this article, we will once again look at a couple of starting hands and identify the best strategies available based on EV.

Please note that in the examples below we will assume that the card that you get during the first trick is a jack of a non-trump suit if the deck is not closed right at the start.

Practice Situation #1

Your hand is At-Tt-As-Ts-Ks. The face-up trump is the Qt.

This hand is worth 46 points, so you need an additional 20 points to get to 66 card points. Since you have the highest 2 trump cards and the 3 highest cards of another suit, you can be sure to take all the tricks if you decide to close the deck.

Is it actually a good idea to close the deck right away? As your opponent is going to have less than 20 points 18% of the time, she will score 3 game points 18% of the time. The remaining 82% of the time, you will score 3 game points. Thus, the EV of this strategy is 1.9 game points.

What other options do you have? To my mind, it would not be sensible to play a trump card as you would give up the edge you have as far as trumps are concerned. Consider these numbers: After the first trick, your opponent is 15% to have the two remaining trump cards. If you close the deck no matter what card you get after the first trick, you will not be able to get enough points the whole 85% of the time. In total, your winning percentage will have dropped below 82%.

Another alternative would be playing the ace or ten of spades, but this does not seem reasonable, either: Your opponent will have at least one trump 60% of the time, and since it should be a player's objective to secure as many high cards as possible (as noted before) she would likely decide to trump. Then, the maximum of game points you could score is 2.0 if everything goes your way, which is just 0.1 points above the EV computed for closing the deck right at the start.

That leaves playing the Ks: This is not a bad option: Your opponent cannot have any higher spades and she is unlikely to trump this early in the game. So, you will probably take the trick and have the chance to improve your hand with the next card you will get: A total of 6 of the remaining 14 cards will put you in a highly favorable position: the 2 remaining trumps, the 2 remaining spades, and the 2 remaining aces. After getting one of these cards, closing the deck will lead to 3 game points 98% of the time. Let's assume that if you receive one of those cards which do not exactly improve your hand, you still decide to close the deck. If you get a ten or a king, you will succeed 76% of the time. With a queen, you will get to 66 card points 75% of the time; with a jack, you will still succeed 65% of the time.

In short, you will score 3 game points 84% of the time. The EV has risen to 2.0 game points. Please note that so far we have assumed that your opponent will not under any circumstances try to take the first trick. This simply is an unrealistic assumption. Thus, your actual EV is not going to be higher than the EV of closing the deck at the beginning. The prime reason for this is that this hand cannot get much better than it already is.

This example illustrates an important fact: In Schnapsen, if you have the edge, you must immediately try and capitalize on it. Waiting and hoping for "better" cards will not pay off, especially if there are not many cards that would significantly improve your hand.

So far, we have only looked at what to do when you are to lead. But how should this hand be played if your opponent was to open the game? Obviously, if she announces the trump marriage, you have to take the trick, close the deck, and try to get to 66 card points. If she plays a spade, you take the trick, close the deck, and again try to get to 66. If she plays a high card, you should ask yourself why she would do that. She simply has to have all four remaining cards of high value and a single trump. This situation rarely occurs, and you should continue as in the previous two instances.

The most interesting situation occurs when your opponent plays a low card of a suit that you do not have: Should you trump or should you play the Ks? First of all, note that 3 marriages are still live meaning that if you let your opponent stay in command, she might sooner or later get one of those marriages and get over the 33 card point mark. However, your hand is just too good to just score a single game point with. Furthermore, you can be certain that she does not have two trumps since she could and should have announced the trump marriage. Factoring in that if she has one trump at the beginning of the game, in which case she will improve to two trumps just 8% of the time after the first trick, your overall winning percentage is 80% if you trump and then close the deck. This is equivalent to an EV of 1.8 game points. Compared to playing the Ks, which puts you in a situation in which you might score but a single point, this clearly is the superior strategy.

Practice Situation #2

Your hand is At-As-Ts-Ks-Ah. The face-up trump is the Qt.

This hand is worth 47 points, so you need an additional 19 points to get to 66 card points. The chance that the cards in your opponent's hand are worth less than 19 points is 13%. Also note that you have but a single trump. If your opponent has more trumps than you do, which is the case 27% of the time, there is almost no way for you to score enough card points. If you decide to close the deck, you will fail to win the game 37% of the time (note that you cannot just add the 2 previous percentages up as you would count several hands twice). Still, the EV of this decision is 0.8 game points.

What alternatives do you have? First of all, it should be obvious that playing the At without closing the deck is out of the question as it would be difficult to win the game later on without trump cards. Since your opponent is 77% to have at least one trump, playing the Ah does not seem like a good idea either. A similar argument can be made against playing the As or the Ts, although this would be a slightly better strategy: As your opponent is more likely to have a heart than a spade, you might get the chance to take a trick later on by playing the Ah.

That leaves you with one final option: playing the Ks. Here, the reasoning would be the same as in practice situation #1: Unless your opponent has all 3 remaining trump cards, he is unlikely to trump. Therefore, you get the chance to take a card from the deck that improves your hand. The only question is: What cards do in fact improve your hand? Unfortunately, the only cards you really want are the other trump cards: 3 of 13 cards or 23%. Then, if you decide to close the deck, you will win the game 90% of the time, given that you play the cards in the correct order (if you get the Tt: At-Tt-As-Ts-Ah; if you get the Kt or Qt: At-As-Ts-Ah; your opponent might take a trick earlier, then play would become automatic). If you do not improve, you could still close the deck as long as the new card would allow you to take all the tricks in case your opponent has no more than one trump card. Those cards would be the Js, Qs, Th, Ac, which would be 31% of the remaining cards. Then, you would win the game 64% of the time (the actual percentage is slightly higher as you might get to 66 even though your opponent has more trumps, but 64% seems like a good enough estimate).

46% of the time, you will get a card that does not allow you to close the deck as your chances of winning would be less than 50%. Furthermore, you would likely have to play a card which enables your opponent to take the trick. This would also mean giving up the lead, which becomes an ever more important issue towards the end of a game. As you cannot be sure to have enough trumps, regaining the lead might become tricky. Moreover, at this point there are three live marriages, still.

Factoring in all these facts, it might be possible to slightly increase your EV by playing the Ks first, but as you can see, judging this decision is very complicated. Additionally, your opponent's playing style should be considered when thinking about the possible outcomes. Therefore, in my opinion, it would still be the better decision to close the deck right at the start.

This hand confirms what we saw in practice situation #1: If you have the edge early on, you should try and take advantage of it. This is especially true if you are weak as far as trumps are concerned and the other cards in your hand give you a shot at winning the game.

What if your opponent has the lead and you have the starting hand of practice situation #2? Actually, this hand becomes a lot easier to play: If you can take the first trick by playing the Ah or As, you will do just that. If your opponent plays a low club, you should definitely play the Ks. If you get the chance to take a trick that is worth more than 20 points, go ahead and trump. There really is not a lot you can do in this situation. As the game progresses, however, you should be aware of the possibility of your opponent getting the trump marriage, so you might sooner or later have to spend your trump card on a trick that is worth but few points.