The Schnapsen Log
Anything Else? (solution)
Martin Tompa
As always, the first thing to consider, when following to the lead of the last trick before the stock is exhausted, is ducking. If you duck with ♦J, this will be the position, with Katharina on lead:
Katharina: (45 points)
♠ AQ
♥ QJ
♣ K
♦ —You: (15 points)
♠ TJ
♥ AT
♣ —
♦ K
Katharina cannot reach 66 trick points: even if she trumps your ♦K with ♠Q and cashes ♠A, that will only bring her to 65. That means that the outcome will be decided by the last trick, which you will win as long as you trump her ♣K with your ♠T and lead ♦K the first time that you are on lead. Your ♥AT are two sure entries to your hand as long as you don’t lead them yourself. Thus, ducking ♦A guarantees you 1 game point.
Since Katharina had fewer than 33 trick points when she led ♦A, it is worth considering whether you can do better than 1 game point by trumping that lead. Your trick point total after trumping will be 36. The outcome of the deal will then depend on what you draw from the stock.
If you draw ♠A, cashing it and ♥AT will give you plenty of trick points, so you will score 2 game points. At the other extreme, drawing any of ♥QJ or ♣K from the stock means you will be able to win only one or two heart tricks for at most 26 additional trick points, with no chance to win the last trick, because Katharina will have 3 trumps in her hand. Drawing any of these cards will mean a loss of 1 game point for you.
The last possible card you could draw from the stock is ♠Q, resulting in the following position, with you on lead:
Katharina: (32 points)
♠ AJ
♥ QJ
♣ K
♦ —You: (36 points)
♠ Q
♥ AT
♣ —
♦ KJ
If you start with a diamond from this position and continue with your other diamond when you regain the lead, you are guaranteed to take the last trick and score 1 game point.
Putting all this together, the expected number of game points you will gain if you trump Katharina’s ♦A is ⅕(+2) + ⅕(+1) + ⅗(−1) = 0. Thus, you are better off ducking her ♦A and gaining 1 game point.
Are you satisfied with this analysis? Have we missed anything?
The question we haven’t yet asked and answered is whether we can infer anything about the last face-down card in the stock from Katharina’s play. If her play somehow limits the possibilities for that card, or changes the probabilities for the remaining card, then the expected game point analysis we just did isn’t correct.
So consider her play again. She led ♦A from this position:
Unseen cards:
♠ AQ
♥ QJ
♣ K
♦ AYour cards:
♠ T
♥ AT
♣ —
♦ KJTrump: ♠J
Stock: 1 face-down card
Game points: Katharina 4, You 2
Trick points: Katharina 32, You 15
There’s no possible marriage that she failed to declare and no trump exchange she failed to execute, so the previous types of inferences we’ve discussed don’t seem to apply here. Do you see anything else that you can infer from her play?
Is it remotely possible that the skillful Katharina is holding ♠AQ in her hand? Holding those key cards together with ♦A, she would surely close the stock and lead ♠A, expecting to catch your ♠T and easily reaching 66 trick points by cashing ♦A. Even if, from her point of view, ♠T is unluckily in the stock instead of in your hand, cashing ♠AQ and ♦A and ♣K, if she holds that card, is guaranteed to give her enough trick points. (If she doesn’t hold ♣K, then from her point of view you must contribute at least ♦KJ and ♣K on her ♠AQ and ♦A, bringing her trick point total to 67.)
Katharina is such a strong player that it’s impossible she would miss this play while holding ♠AQ. Since she didn’t close the stock, it’s a sure bet that the remaining face-down card is either ♠A or ♠Q. We’ve already seen that, if you trump her ♦A lead, you will gain 2 game points if you draw ♠A from the stock and 1 game point if you draw ♠Q. Therefore, your true expected gain is ½(+2) + ½(+1) = 1.5 game points, not 0 game points as we calculated previously. With this new inference, you are much better off trumping her ♦A than ducking it! (If you happened to be fortunate enough to witness Katharina exchanging ♠J for ♠Q earlier in the deal, then you are guaranteed to draw ♠A from the stock and win 2 game points by trumping her ♦A lead.)
This is by far the subtlest inference we have had to make about the remaining card in the stock. If you made this inference without any help from me, congratulations: with probability 1/2 you have just won 2 game points, enough to win your first game against the champion Katharina!
© 2012 Martin Tompa. All rights reserved.
