Psellos
Contemporary Development With Functional Programming

The Schnapsen Log

April 27, 2012

Expected Game Points (solution)

Martin Tompa

As always, it’s easier to first consider what happens if you duck your opponent’s lead of A. You can safely discard your losing T, bringing Katharina up to 46 trick points. After this you will take all the remaining tricks and gain 1 game point.

Since Katharina currently has only 25 trick points, it would be advantageous to trump A with Q if you could reach 66 points before she reaches 33. There are five possible cards you could draw from the stock, and the number of game points you win will probably depend on which one you draw.

Your definite winners without giving up a trick are ATQ and ♣A, which will bring you to at least 60 trick points, including what your opponent contributes to these tricks. If you draw any of the winners T, A, ♣Q, or K, you will win the deal without giving up another trick, and will gain 2 game points. The worst draw for you is K, which will leave you in this position:

Katharina: (25 points)
T
A
♣ Q
KJ

You: (19 points)

TK
♣ A
AT

In this case you will have to be a little careful not to lose control and lose the deal. The key is to hold tight to ♣A, because that will be the entry back to your hand to win the last trick. There are a few ways to play from this position. The most straightforward is to pull all the trumps and then lead K. Katharina can cash T (on which you discard T, bringing her trick points to 60), but you will win the last trick and 1 game point with ♣A.

Time now to compute your expected number of game points if you trump A. With probability 4/5 you will gain 2 game points, and with probability 1/5 you will gain 1 game point. Thus, your expected number of game points is

⅘(+2) + ⅕(+1) = 9/5 = 1.8.

Compared to your gain of 1 game point if you duck A, an expected gain of 1.8 game points seems much better, so go ahead and trump.

There is one more thing to keep in mind when doing these calculations. If you know certain cards are in your opponent’s hand (because you saw one in a declared marriage or in a trump exchange, or were able to deduce it from the way your opponent played), then those cards cannot be in the stock, and this affects the expected number calculation slightly. In today’s deal, for instance, suppose Katharina had exchanged J for K earlier in the deal. Then you know K cannot be the face-down card in the stock, so there are only 4 possibilities for that card instead of 5. In this case, your expected number of game points is

¾(+2) + ¼(+1) = 7/4 = 1.75.

As you can see, this extra knowledge of one card’s placement doesn’t change the expected number by much at all, just from 1.8 to 1.75.

© 2012 Martin Tompa. All rights reserved.


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About the Author

Martin Tompa

Martin Tompa (tompa@psellos.com)

I am a Professor of Computer Science & Engineering at the University of Washington, where I teach discrete mathematics, probability and statistics, design and analysis of algorithms, and other related courses. I have always loved playing games. Games are great tools for learning to think logically and are a wonderful component of happy family or social life.

Read about Winning Schnapsen, the very first and definitive book on the winning strategy for this fascinating game.

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