The Schnapsen Log

Odds and Expectation (solution)

Martin Tompa

As we’ve often discussed, when you are on lead at the last trick before the stock is exhausted, the first thing you should consider is closing the stock. Let’s do that.

Closing the stock looks very appealing in this position. You can probably just cash ♥T and ♦A, which will provide enough trick points for you to win the deal and 1 game point. If you’re like me, you will think to yourself, “What’s the probability that ♥J just happens to be that last mystery card in the stock? It’s only 1/6.” And you’ll close the stock and lead ♥T without further thought. Only when things go wrong do you give it further thought, tinged with regret. In today’s deal, this will be the position when things go wrong and ♥J is in the stock:

Katharina: (38 points)
♠ —
♥ —
♣ ATK
♦ TQ

You: (41 points)
♠ A
♥ T
♣ Q
♦ AK

Trying to cash ♥T in this situation leads to immediate disaster: Katharina will trump with ♦T and cash ♣A for 72 trick points. You can delay the disaster by instead leading ♣Q. Katharina will win with ♣A and return ♣K for you to trump with ♦K, leaving you on lead in this position:

Katharina: (52 points)
♠ —
♥ —
♣ T
♦ TQ

You: (49 points)
♠ A
♥ T
♣ —
♦ A

You can delay for one more trick by cashing ♦A, but after that you must lose the deal to Katharina’s ♦T. Therefore, in this one case of ♥J in the stock, you are going to lose 2 game points.

Still, that only happens with probability 1/6, so it’s good to close the stock, right? Let’s compute your expected number of game points to answer this question: you expect to gain ⅚(+1) + ⅙(−2) = 1/2 game point by closing the stock. This expected gain is positive, but it’s not impressive. The problem is that you stand to gain 1 game point when you win, but stand to lose 2 game points when you are unlucky. Whenever both players have passed the 33 trick point threshold, as in today’s deal, you should think carefully about whether it’s worthwhile to close the stock. This situation is like a bookie giving you bad odds on a gamble, and it’s our calculation of expected gain that reveals the situation.

Let’s go back and see what happens if you don’t close the stock. If Katharina already holds ♦T, leading any card other than ♣Q is likely to hand her 1 game point. Let’s try leaving the stock open and leading ♣Q. If Katharina wins the trick with ♣A, she will be on lead in this position:

Katharina: (52 points)
♠ —
♥ J
♣ TK
♦ TQ

You: (41 points)
♠ A
♥ T
♣ —
♦ AKJ

There is nothing she can do to prevent you from winning ♥T and ♦A, which will bring your trick point total to 67.

This means that Katharina will duck your ♣Q lead rather than winning it. The only card she can use to duck is ♥J, which is what she wants to use anyway to prevent you from cashing ♥T. When you win this trick and your trick point total is 46, the only suit you can draw from the stock that might not give you an immediate win is a club. The worst club you can draw is ♣T, leaving you on lead from this position:

Katharina: (38 points)
♠ —
♥ —
♣ AK
♦ TQJ

You: (46 points)
♠ A
♥ T
♣ T
♦ AK

Your only immediate winner is ♦A, which isn’t enough. You can’t lead ♠A or ♥T, as trumping that and cashing ♣A would give Katharina enough trick points. But leading ♣T does the trick of throwing Katharina in. Whatever she returns, you will win both of your trumps ♦AK and reach 66 trick points.

Although closing the stock will gain you 1 game point 5 out of 6 times, what we have learned is that not closing the stock and leading ♣Q will gain you 1 game point however the cards lie. This is therefore a safety play, an old familiar story to those of you who have been faithful readers from the beginning.

© 2012 Martin Tompa. All rights reserved.

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