The Schnapsen Log

Point of Order (solution)

Martin Tompa

What is the first thing to consider when on lead at the last trick before the stock is exhausted? Right, you should first consider closing the stock. Even if you have only 19 points in your tricks and only two or three winners in your hand? Definitely: at least think it through before dismissing it.

Even with the stock closed and matters thus simplified, it seems unclear in what order to do things. That spade suit looks like an opportunity for an elimination play. Should you begin by cashing your ♥A, so that it cannot be used as an entry back into your hand? No, the problem with cashing ♥A is that it is the only thing that stands in the way of Tibor cashing his ♣A. He will discard one of those jacks on your ♥A and you will then likely be stuck in a position something like this:

Tibor: (41 points)
♠ TQ
♥ —
♣ A
♦ A

You: (32 points)
♠ AK
♥ —
♣ —
♦ TQ

You are going to have to give Tibor the lead by playing either spades or diamonds. Either way, he will have 55 trick points and will then cash his ♣A to win the deal.

No, your first inclination that ♥A is a dangerous entry back into your hand is unfounded. To see why, let’s look again at the original position:

Concealed cards:
♠ TQJ
♥ —
♣ A
♦ AJ

Your cards:
♠ AK
♥ A
♣ —
♦ TQ

Trick points: Tibor 41, You 19

The only exit card Tibor could use that corresponds to ♥A as an entry in your hand is his ♣A. That trick would give bring your trick point total to 41, and then your ♦T and ♠A, plus the two discards Tibor would have to contribute, would be enough to get you to at least 66 trick points.

This suggests that the first thing for you to do is to lead ♦Q in order to set up your ♦T as a winner. If Tibor does not hold ♦A, then both of your diamonds and both of your aces are winners, and Tibor will have to contribute a ten or ace to one of these tricks, which will give you plenty of trick points. Assume then that Tibor does win the trick with ♦A, leaving him on lead from this position:

Concealed cards:
♠ TQJ
♥ —
♣ A
♦ J

Your cards:
♠ AK
♥ A
♣ —
♦ T

Trick points: Tibor 55, You 19

If Tibor leads ♣A, we have already seen that your three high cards will give you enough trick points to win. If he leads a spade, you will take all four remaining tricks for at least 72 trick points. Thus, we can assume that Tibor is holding ♦J and exits with that card, putting you on lead from this position:

Concealed cards:
♠ TQJ
♥ —
♣ A
♦ —

Your cards:
♠ AK
♥ A
♣ —
♦ —

Trick points: Tibor 55, You 31

Now it is finally time to cash your ♥A. If Tibor holds ♣A, this trick squeezes him. He cannot discard ♣A, for the same reason that he could not lead it for you to trump, so he must discard a spade. Assuming that he does hold ♣A, his discard leaves him with only one spade remaining in his hand. That means you can cash both of your spades (♠A first, of course) and he will have to discard ♣A on the second anyway, giving you at least 73 trick points.

Can anything go wrong with this plan? Yes. It relies on Tibor holding the ♣A. If that is the last card in the stock, both the elimination and squeeze plays will fail. Here is what the position would be when you close the stock:

Tibor: (41 points)
♠ TQJ
♥ —
♣ —
♦ AJ

You: (19 points)
♠ AK
♥ A
♣ —
♦ TQ

Tibor wins the diamond lead, exits with ♦J, discards ♠J on your ♥A, and can still win a trick with his ♠T. You will win only your three top cards, for a grand total of 58 trick points, and Tibor will score 2 game points.

Since one of the six possible cards face-down in the stock causes your downfall, you have a 1/6 probability of failing. This means that your expected gain is ⅚(+1) + ⅙(−2) = 1/2 game point. It’s not a great outcome, despite your 5/6 probability of winning the deal, because you only stand to gain 1 game point when you succeed, but you stand to lose 2 game points when you fail.

Can you do better than 1/2 game point by leaving the stock open? Let’s see. Leading ♥A is unappealing because it relinquishes trump control to Tibor. Opening up the spade suit is equally unappealing, because you want Tibor to do that.

Probably your best lead with the stock open is ♦Q. Tibor cannot afford to win the trick, because that will leave him open to the same elimination and squeeze combination. Here is the position after he wins the trick with his ♦A:

Tibor: (55 points)
♠ TQJ
♥ —
♣ A
♦ J

You: (19 points)
♠ AK
♥ AJ
♣ —
♦ T

Tibor’s only hope is to exit with ♦J. You will then run your trumps, and on the second trump Tibor is squeezed: he must either discard ♣A or bare his ♠T. Either way, you will accumulate at least 71 trick points and win 1 game point.

Since Tibor is already past the 33-trick-point threshold, he does no worse by ducking your ♦Q lead. The outcome if he ducks will depend on what he discards and what you draw from the stock.

If you draw ♠T, ♠Q, or ♦A from the stock, you will have no trouble getting to 66 trick points by running your winners and/or declaring your marriage. If you draw ♣A from the stock, Tibor will have discarded ♠J on your ♦Q and you will be on lead from this position:

Tibor: (41 points)
♠ TQ
♥ J
♣ —
♦ AJ

You: (24 points)
♠ AK
♥ A
♣ A
♦ T

From here, you cash ♥A and ♣A, the latter squeezing Tibor in a different way: he cannot bare his ♠T, so he must discard his winner ♦J. You can then throw him in with your diamond, and he is then endplayed and must open up the spade suit for you. This is a squeeze without the count, because you squeeze Tibor before relinquishing the lead to him.

The other two cards you could draw from the stock, the two jacks, will both turn out to be losing draws for you. If the last card in the stock is ♠J, Tibor can afford to discard ♣A on your ♦Q and you will be on lead from this position:

Tibor: (41 points)
♠ TQ
♥ J
♣ —
♦ AJ

You: (33 points)
♠ AKJ
♥ A
♣ —
♦ T

You cannot win from this position. If you pull his trump, Tibor can cash both diamonds and reach 66 trick points exactly. If you try to throw him in with a diamond before pulling his trump, he will exit by leading ♦J and you will have to open up the spade suit yourself and yield the winning spade trick to him.

If the last card in the stock is ♦J, Tibor will discard ♠J on your ♦Q and you will be on lead from this position:

Tibor: (41 points)
♠ TQ
♥ J
♣ A
♦ A

You: (24 points)
♠ AK
♥ A
♣ —
♦ TJ

You cannot win from this position. If you pull his trump, Tibor can cash both of his aces and win. Your other alternative is to retain your trump and lead ♦J to set up your ♦T. When Tibor wins this trick, he will force you by leading his ♣A. You will be back on lead from this position:

Tibor: (54 points)
♠ TQ
♥ J
♣ —
♦ —

You: (46 points)
♠ AK
♥ —
♣ —
♦ T

You cannot force him back with your ♦T, because that would give him 66 trick points exactly. All that is left is for you to open up the spade suit yourself, and the spade trick will similarly give him enough to win.

What we have seen, then, is that there are 4 possibilities (♠T, ♠Q, ♣A, ♦A) for the last face-down card in the stock that give you a win and 2 possibilities (♠J, ♦J) that give you a loss. Therefore, your expected gain if you leave the stock open is ⅔(+1) + ⅓(−1) = 1/3 game point. This is very close to your expected gain of 1/2 if you close the stock, but not quite as good. Since Tibor is still 4 game points from winning, the 1/6 probability of losing 2 game points if you close the stock is not terribly worrisome. If he were only 2 game points from winning, I would probably opt for the safer play of leaving the stock open, even though it has a slightly lower expected gain.

This turned out to be a very involved analysis, one that would be difficult to complete accurately at the Schnapsen table.

© 2014 Martin Tompa. All rights reserved.

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