The Schnapsen Log

Exit Cards, Stoppers, and Optimal Endplay

Martin Tompa

I am again going to break from our normal format, in order to tell you about a fascinating scholarly article that has a bearing on the Schnapsen endgame. The paper is very mathematical, developing a “calculus” of cards that allows you to look at the hand you were dealt, calculate the maximum number of tricks you can take, and determine a strategy for obtaining that maximum number of tricks. Sounds useful, doesn’t it? And, for the mathematically inclined, this calculus involves infinitesimals, which are “numbers” greater than 0 but less than any positive fraction such as 1/100 or 1/1000000000 that you can come up with. It’s all quite beautiful, in a way that only mathematics can be.

The paper is “Two-Person Symmetric Whist”, by Johan Wästlund, which was published in The Electronic Journal of Combinatorics in 2005. The author is a researcher in the Department of Mathematical Sciences at Chalmers University of Technology in Sweden.

Before we begin, though, a few warnings. Although the two-person card game “symmetric whist” that Wästlund studies has connections to Schnapsen, there are some very fundamental ways in which it differs:

1. The goal of symmetric whist is to take as many tricks as possible. This is a common goal in games such as whist or bridge. Unlike Schnapsen, there is no notion of trick points, nor is there a bonus for winning the last trick.
2. Like most good mathematicians, Wästlund is interested in generalization, so his card deck can consist of any number of suits, with an arbitrary number of cards in each suit. For instance, there is one complicated and interesting example in Section 16 of his paper in which each player’s hand consists of 19 cards distributed among 4 suits, and Wästlund shows how to play it optimally. The mathematician’s compulsion to generalize is a good thing, because it means that his results have a bearing on the 20-card Schnapsen deck, the 24-card Sixty-Six deck, the 52-card Bridge deck, and the 54-card, 5-suit Tarok deck.
3. Wästlund assumes that all the cards in the deck are dealt out to two players. There is no stock from which to draw cards, and each player knows exactly what cards the other is holding. This means that the clearest connection to Schnapsen will be in Schnapsen’s endgame when the stock is exhausted.
4. Wästlund is forced to make a restrictive assumption in order to obtain his results: he assumes that, in each suit individually, each player is dealt the same number of cards. For instance, the two players (named West and East) might be dealt the following hands:

   West:	East:
   ♠ KJ	♠ AQ
   ♥ AJ	♥ KQ
   ♣ A	♣ K

   But they could never be dealt the following hands, because the two players hold an unequal number of spades (and an unequal number of hearts):

   West:	East:
   ♠ AQJ	♠ K
   ♥ −	♥ AK

   This restriction explains why Wästlund calls the game symmetric whist: there is symmetry in the two players’ suit distributions.
5. When following to a trick in symmetric whist, you must follow suit, but you are permitted to duck the trick even if you could have won it. This is a common rule in trick-taking games such as whist and bridge, but doesn’t quite match Schnapsen’s rules either when the stock is open or when it is not. Because each player starts with the same number of cards in each suit, you will always be able to follow suit, and the hands will maintain the symmetric property after each trick is played. This also means that it doesn’t matter whether or not there is a trump suit, because neither player would ever have the opportunity to trump.

And one final warning: Wästlund’s paper is technical and complicated.

If you haven’t given up yet, let me give you the good news now. I will do my best to specialize Wästlund’s paper to Schnapsen-like situations. In particular, we won’t worry about situations where each player’s hand contains more than 5 cards, nor situations where any suit contains more than 4 cards. (Because of the restriction of symmetry, each suit must have an even number of cards.) This means that a lot of what we’ll discuss will be much simpler, specialized versions of Wästlund’s very general results.

But to be sure you aren’t misled, let me repeat that today’s goal of maximizing the number of tricks taken rather than the number of trick points means that Wästlund’s theory may not produce the optimal Schnapsen play. Taking more tricks often means collecting more trick points, but clearly this is not always true. And the restriction to symmetric deals means that there are plenty of Schnapsen (asymmetric) endgames that won’t fall within today’s purview.

For today, then, this is The Symmetric Whist Log rather than The Schnapsen Log. Ready to begin?

© 2012 Martin Tompa. All rights reserved.

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