Psellos
Contemporary Development With Functional Programming

The Schnapsen Log

August 19, 2013

Revised: August 26, 2013

Expected Game Points and Role Reversal

Martin Tompa

There is a type of analysis that I have been avoiding until now. I must admit that I have been avoiding it partly out of mental laziness and partly out of fear of my own confusion. But it’s been nagging at me to confront it, particularly since the appearance of a recent column entitled “The Last Place You Look”. Today we’ll bite the bullet and confront this analysis together.

Here was the position analyzed in that column:

Unseen cards:
AQ
J
♣ Q
AK

Your cards:
K
K
♣ A
TQ

Trump: Q
Stock: 1 face-down card
Trick points: Emmi 20, You 31
On lead: You

(In that column, you had only 1 game point remaining to win the game, but for today I will assume that neither player is that close to winning the game.)

Uncle Hans advocated leading T and demonstrated the following conclusions:

  1. If Emmi wins the trick, you are guaranteed to gain 1 game point.
  2. If Emmi ducks T, three draws from the stock (Q, K, J) give you 2 game points, two draws (A, A) give you 1 game point, and one draw (♣Q) costs you 1 game point. The expected number of game points you gain, if Emmi ducks T, is 7/6.

Hans summarized with the following words: “… you expect to gain about 1 game point whether she ducks or not.” The implication was that your expected gain lies somewhere between 1 and 7/6. Would it be cowardly of me to lay the blame for this falsehood on Hans, saying that I myself knew better all along? Probably so, and I hesitate to add cowardice to my already admitted character flaws of laziness, fear, and confusion. But the fact is that it is incorrect to infer that your expected gain is between 1 and 7/6, and that’s what today’s column is about.

What is wrong, then, with this inference? The problem is that it rests on the assumption that Emmi will decide either to duck or to win the trick without looking at her cards. But, in fact, it’s very likely that her action on your T lead will depend on the exact 5 cards she is holding. In particular, with some of those holdings that led to you gaining 2 game points, Emmi is likely to win the trick rather than ducking, thus denying you the possibility of that second game point. Because of this, as we will see, your actual expected gain is strictly less than 1 game point, not between 1 and 7/6.

Let’s take as a sample the case where the last face-down card in the stock is Q. We know that, if Emmi were to duck your T lead, you would win 2 game points. But would Emmi duck in this situation? To answer that, we need to stand up from your seat, walk around the table, and take Emmi’s seat. Here is what the position looks like from her side of the table when you lead T:

Unseen cards:
KQ
K
♣ A
TQ

Emmi’s cards:
A
J
♣ Q
AK

Trump: Q
Stock: 1 face-down card
Trick points: Emmi 20, You 31
Lead: T

Notice that the position from Emmi’s point of view is nearly identical to the original position from your point of view, except that the two sets of cards are swapped and Q has been moved from Emmi’s cards to the unseen cards, because we are considering the case when Q is still in the stock. I call this type of analysis role reversal, because you must assume the role of your opponent.

Would Emmi duck your T lead in this position? That is an easy question to answer, now that we’ve laid out the cards, because Emmi knows exactly how the two hands would look one trick later. Her best discard is ♣Q, and this would be the position after that trick:

You: (44 points)
KQ
K
♣ A
Q

Emmi: (20 points)
A
QJ
♣ —
AK

Emmi can easily see that you will cash K, declare the marriage, and win 2 game points. She would do better to win T with A and cross the 33 point threshold. So no, she will not duck if Q is in the stock.

By a similar role reversal analysis, for each of Q, K, J, A, and A as the last face-down card in the stock, Emmi can see that she is certain to lose 1 or 2 game points by ducking (by the exact same lines of play Uncle Hans used to show that you would gain 1 or 2 game points if you drew one of these cards), whereas she will lose at most 1 by winning the T trick, so she may as well win the trick. She will not duck any of these leads.

The only remaining face-down card to consider is ♣Q, which is the one that caused you to lose 1 game point if Emmi ducked. Here is what the position looks like for this case, from Emmi’s point of view:

Unseen cards:
K
K
♣ AQ
TQ

Emmi’s cards:
AQ
J
♣ —
AK

Trump: Q
Stock: 1 face-down card
Trick points: Emmi 20, You 31
Lead: T

Emmi can see that, if she discards Q on your T, the only other trick she wll lose is to your K (likely trumping her K), and she will win 1 game point. The situation in this case is a little more complicated, though, because your trick point total is still below 33. Therefore, Emmi will consider whether she can do better by winning your T lead. (From your point of view, we hope she does decide to win the trick because we know, as the cards lie, that she will lose 1 game point if she does.) Indeed, if Emmi overtakes your T with her A and were to draw any of K, K, ♣A, or Q from the stock, she would win 2 game points! If she were to draw ♣Q (which we know to be the case but Emmi doesn’t), she can see that she would lose 1 game point. Thus, it might seem to a naive Emmi that her expected gain if she wins the trick is ⅘(+2) + ⅕(−1) = 7/5 game points, which looks better than her gain of 1 game point if she ducks.

But there is an additional piece of information Emmi has at her disposal that might rule out some of these optimistic draws from the stock. Namely, would you have led T if any of K, K, ♣A, or Q had been in the stock? To answer this, we have to do a double role reversal! We must have Emmi look at the position from your point of view with one of these cards in the stock. (Perhaps at this point you will appreciate why I kept avoiding this type of analysis out of fear of confusion.) As an example, let’s have Emmi ask whether you would have led T if K were in the stock. Here is what the position would look like from your point of view:

Unseen cards:
AKQ
J
♣ —
AK

Your cards:

K
♣ AQ
TQ

Trump: Q
Stock: 1 face-down card
Trick points: Emmi 20, You 31
Lead: T

Notice that this double role reversal differs from the original position at the beginning of this column just by exchanging K and ♣Q in the two sets of cards.

Would you have led T from this position? The answer is no. It would be easy for you to see that Emmi could gain 2 game points by winning this trick with A, cashing K, and declaring the spade marriage. Even if you thought A might be in the stock, she would still win 2 game points by trumping T, cashing A, and declaring the marriage. You could instead avoid the loss of 2 game points by cashing K or leading ♣Q.

In a similar way, based on your lead of T, Emmi can eliminate any of K, K, ♣A, or Q as the card remaining face-down in the stock, and infer correctly that the remaining face-down card is ♣Q. Based on this inference, she will duck and you will lose 1 game point.

We are finally ready to put all this together and do the expected game point analysis correctly. When you lead T at trick 5, with probability 5/6 (any of AQ, J, and AK remaining face-down in the stock), Emmi will win trick 5 with A and you will gain 1 game point. With probability 1/6 (♣Q remaining in the stock), Emmi will duck and you will lose 1 game point. Therefore, the expected number of game points you will gain is ⅚(+1) + ⅙(−1) = 2/3. Notice that this is less than the gain of either 1 or 7/6 game points that Uncle Hans discussed in the earlier column. The reason is that Emmi uses the information of her own cards and your T lead in order to choose the play that minimizes your expected gain.

© 2013 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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