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The Schnapsen Log

November 4, 2015

Homework on Expected Values (solution)

Martin Tompa

(a) How will the deal play out if you duck this trick? Who will win, and how many game points?

The cheapest and most reasonable discard you can make on the Maestro’s A would be ♣J. This trick will bring the Maestro’s trick point total to 20+11+2 = 33. The Maestro will draw the last face-down card from the stock and you will draw the face-up J, which puts the Maestro on lead from this position:

Maestro: (33 points)
K
KJ
♣ AT

You: (19 points)
AJ
AT
♣ —
Q

From here, whatever the Maestro does, you can take the rest of the tricks. All you have to do is use A to pull the Maestro’s last trump and, after that, all your cards are winners. For example, if he leads ♣T from the diagrammed position, you trump with J, cash A, and then cash your remaining 3 cards in any order. Since you will take the last trick, it doesn’t even matter whether or not you reach 66 trick points. You will win 1 game point, since the Maestro already has 33 trick points himself.

(b) How will the deal play out if you win this trick?

The only way you can win the trick in which the Maestro led A is by trumping with A. This will bring your trick point total to 19+11+11 = 41. Since you won the trick, you will draw the random, face-down card from the stock and the Maestro will draw the face-up J. You could draw any of the five cards you haven’t yet seen: K, K, J, ♣A, or ♣T. We can divide them into 2 cases:

Case 1: If you draw any of those three black cards, then you know the Maestro holds both missing hearts, KJ. In that case, you can run AT, which will bring your trick point total to 41+11+10+4+2 = 68. This is enough, and you will win 2 game points, since the Maestro only has 20 trick points.

Case 2: If you draw either of the unseen hearts, KJ, you will be on lead from this position (which assumes you drew J) or one nearly identical (if you draw K):

Maestro: (20 points)
KJ
K
♣ AT

You: (41 points)

ATJ
♣ J
Q

Either of those small hearts KJ is an unlucky draw for you, and there is nothing you can do to win from this position. The only trick you can possibly win is A, capturing the Maestro’s K, but that only brings your trick point total to 41+11+4 = 56. You also have no chance to win the last trick against someone of the Maestro’s skill, because of his two trumps. For instance, you might try leading Q from the diagrammed position. The Maestro will trump with K and immediately lead his K to force out your A. This will leave you on lead from this position:

Maestro: (27 points)
J

♣ AT

You: (56 points)

TJ
♣ J

All the remaining trick are the Maestro’s, and he will win 1 game point.

Combine these appropriately to determine the expected number of game points that you will win.

What we have seen from the two cases above is that there are 3 cards you can draw from the stock (K, ♣A, ♣T) that will result in a gain of 2 game points for you, and 2 cards you can draw (K, J) that will result in a loss of 1 game point for you. If we let the random variable X be the number of game points you win, its probability mass function is therefore p(+2) = ⅗, p(−1) = ⅖. This means the expected value of X is E[X] = ⅗(+2) + ⅖(−1) = 4/5 = 0.8.

(c) Based on your answers, will you duck the Maestro’s A or trump it? Why?

If you duck, the answer to part (a) says that you are guaranteed to win 1 game point. If you instead trump, the answer to part (b) says that you expect to win 0.8 game points. (You will win 2 game points 3/5 of the time, but you will lose 1 game point 2/5 of the time.) You should therefore duck his A, for the greater expected gain.

“Very nicely played!” the Maestro exclaims when the deal is over and he records 1 game point for you. “Ducking my A was a nice safety play, avoiding drawing one of those low hearts from the stock. I can see that you understand how analyzing the expected game points leads you to the best line of play.”

For another Schnapsen problem with a similar analysis, see the column It All Depends. There are lots of other Schnapsen Log columns posted since May 2012 that use this sort of expected game point analysis.

© 2015 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.

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