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Contemporary Development With Functional Programming

The Schnapsen Log

April 27, 2012

Expected Game Points

Martin Tompa

Today I would like to return to the very first deal that we played together. There is something important that we didn’t discuss about that deal. You may recall that you were playing at the Black Eagle club against Katharina, the local champion. Here is the situation again:

Unseen cards:
K
QJ
♣ A
KQ

Your cards:
ATQ

♣ T
T

Trump: J
Stock: 1 face-down card
Game points: Katharina 1, You 1
Trick points: Katharina 30, You 17
On lead: Katharina

Perhaps you remember that Katharina showed the diamond marriage, resulting in a new trick point score of 50, and led Q. This was an example of a safety play: if you discard ♣T on this trick, you are guaranteed to win the deal. You may also remember that, if you win this trick with T instead, 3 of the 4 possible cards you might draw from the stock (K, Q, and ♣A) also resulted in a win for you, and only 1 of the 4 possible cards (J) resulted in a loss for you.

What I am interested in exploring today is your expected number of game points if you choose to win the trick. The phrase expected number is just the mathematician’s way of saying the average value over the 4 possible cards that could still be in the stock. In the current deal, you have a 3/4 probability of winning 1 game point (corresponding to the 3 possibilities in the stock that give you a win) and a 1/4 probability of losing 1 game point (corresponding to the 1 possibility that gives Katharina a win). Therefore, your expected number of game points if you win this trick is simply ¾(+1) + ¼(-1), which is 1/2.

Perhaps you are saying to yourself, “This is meaningless. You can’t win 1/2 game point in Schnapsen.” That’s true. But what it tells you is that you expect to win 1/2 point on average. Imagine that you played this deal 4 times, once for each possible card remaining in the stock, and each time you overtook the Q lead with T. In each of three of those deals you would win 1 game point and in the fourth you would lose 1 game point. Over all 4 deals your net gain would be 2 game points. That comes to an average of 1/2 game point per deal, which is what the expected number tells you.

This is another way of measuring the value of winning this trick versus losing it. If you give up the Q trick, you will gain 1 game point. If you instead win that trick, the expected number of game points you will gain is 1/2. Clearly you prefer 1 point to 1/2 point.

So is everything we’ve discussed today a waste of time? You were already convinced that the safety play was the right move to make without having a value attached to the wrong move. But in many deals, things are not so simple. Imagine that Katharina had only 30 points instead of 50 going into the Q trick, and let’s recalculate the expected number of game points you gain by winning the trick. Now you’ll win 2 game points with probability 3/4 and lose 1 game point with probability 1/4. So your expected number of game points is ¾(+2) + ¼(-1), which is 1.25. Suddenly this is slightly more appealing than winning 1 game point by ducking. At this point, though, you must also take the current score into account. In the situation shown in the diagram above, where you only need 1 more game point to win the game, it does you no good to win 2 game points, so you should prefer the safety play of ducking. But if the score was, say, tied at 4 game points each, then you should prefer to win the trick, and expect on average to gain 1.25 game points.

Now it’s your turn to give it a try. You may experience a sense of déjà vu, because you are once again back at the Black Eagle sitting across the table from Katharina. The only change is that her hat tonight is even more outlandish than last time.

Unseen cards:
AT
AK
♣ Q
K

Your cards:

T
♣ A
ATQ

Trump: J
Stock: 1 face-down card
Game points: Katharina 4, You 5
Trick points: Katharina 25, You 5
On lead: Katharina

Katharina leads A. Plan your play for the remainder of the deal. When you think you have a good plan, you are welcome to read my analysis.

© 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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Homework on Expected Values, Apr 26
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