The Schnapsen Log

It All Depends (solution)

Martin Tompa

The most natural thing to do in this situation is to declare the marriage and lead ♦Q. The marriage gives you 46 trick points. Your sister cannot afford to duck this trick, because she would have to contribute at least 4 trick points, bringing your total to 53. No matter what you would then draw from the stock, ♥A would give you enough to win 2 game points. She knows she’s better off winning ♦Q in order to cross the 33 point threshold. But whatever she does after that, she cannot prevent you from winning the deal by cashing your two aces. The best she can do before handing over the lead is to trump ♦Q with ♥T and cash ♦A, which will bring her only to 52 points. So declaring the marriage is safe and assures you 1 game point.

Can you do better? Given that Emmi has only 24 trick points and you have all that potential in your hand, it seems there should be some chance of scoring 2 game points. In this position, leading the master trump ♥A prevents her from immediately getting more trick points, brings your own count to at least 41 points (depending on what she discards), and exhausts the stock, which has some of the effect you wanted to achieve by closing the stock.

What happens next depends on that last card remaining in the stock. If it is not a club, then you are guaranteed to be able to cash ♣A and show your marriage, which will be sufficient to win 2 game points. If it is ♣T, though, this won’t work. Emmi will discard ♣K on your ♥A, leaving this position:

Emmi: (24 points)
♠ T
♥ TKJ
♣ —
♦ A

You: (41 points)
♠ —
♥ —
♣ AT
♦ TKQ

You won’t take another trick and Emmi will score 1 game point.

What if the last card in the stock is ♣K? Emmi can still discard ♣T on your ♥A to prevent you from cashing ♣A. But this expensive discard brings your trick points to 47, so all you need to win 2 game points is to declare the marriage.

Remember that ♥T cannot be in the stock, because you know it’s in Emmi’s hand, so there are only 5 possible cards remaining in the stock. To summarize what happens if you were to lead ♥A, 4 of the 5 possible cards remaining in the stock (everything other than ♣T) result in you winning 2 game points, and the 5th possibility (♣T) results in you losing 1 game point. Let’s compute the expected number of game points you will gain if you lead ♥A:

⅘(+2) + ⅕(-1) = 1.4.

On average you stand to gain 1.4 game points by leading ♥A, versus a guaranteed 1 game point by instead declaring the marriage. The ♥A sounds preferable, but its downside is that, with probability 1/5, you will lose 1 game point. This is a common dilemma in the Schnapsen endgame: greater expectation (♥A) versus the safety play (♦Q). It is at this time that you must consider the current game point score and the quality of your opponent.

In today’s problem, you currently have 3 game points remaining to win the game. If Emmi has 3 (or more) game points remaining, then I’d suggest playing ♥A for the greater expected gain. You will end up ahead (Emmi 3, You 1) with probability 4/5, and a little bit behind (Emmi 2, You 3) with probability only 1/5. This seems like a very good gamble.

But if Emmi instead has 1 game point remaining to win the game, then I think I’d go for the safety play. You don’t want to take even the small chance (1/5 probability) of handing her the whole game, since you believe that you are the better player and can hopefully win from the score that would result from the safety play: Emmi 1, You 2.

If Emmi’s current score is in between (Emmi 2, You 3), it’s a close decision. I think I’d still lean toward the safety play: I’d rather play against her from a score 2-2 than give her the small chance of being ahead 1-3.

Against a tough opponent who you think plays much more skillfully than you do, I’d suggest the ♥A lead from any of these game point scores. If your opponent is more skillful than you are, you need some luck to win, so you should be happy to take your chances on collecting more game points when the odds are with you.

There is a lot to consider in these situations that rest on probability. Once you have done the expected game point analysis, you have to consider the current game point score and your assessment of your skill level compared to your opponent’s. It’s one of those judgement calls that make the game really interesting.

© 2012 Martin Tompa. All rights reserved.

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