The principle of „restricted choice“ in the card game Bridge made it to Wikipedia. The article is somewhat mysterious, however, and obscures what is really happening. I never infer with Wikipedia or add something to it because I have made very bad experiences doing so, even in my very own professional topics of mathematics. This is especially true if you are not in consent with the previous author.

Let me explain the example for restricted choice in Wikipedia. I assume you are familiar with Bridge. Your suit is divided as

North: A, J, 10, 9 6
South: 8, 7, 5 4

You are missing K, Q, 3, 2.

Assume, you play low from South, West following with the 2, North adds the J, and E winning the trick with the Q, i.e., the trick goes 8, 2, J, Q to East. You get the lead somewhere else and end in South, playing the 7, where West plays the 3. Do you finesse for the Q or not?

If you knew that East always plays the Q from KQ, you are now slightly better by playing the Ace, not finessing. The reason is that the one missing card, the Q, is more likely to be with the player who has one card more. This assumes that you know nothing else about the cards of the opponents. That is rarely the case. If East bid a long suit preemptively, you will rather give West the Q.

But these simple arguments do not take into account that Bridge is a play of strategic planning. East needs to make a decision what card to play in case East is confronted with KQ. And he needs to make that decision before the game even starts. Maybe East is the clever kind that tries to confuse the declarer and selects the K each time. This will mark W with the king if East has the singleton Q and shows it in the first round. Playing the Q from KQ all the time is bad when the singleton K is dealt to East. In fact, any plan favoring one or the other is not a good one. They need to be chosen randomly.

This argument is maybe not clearly expressed in the Wikipedia article. But making a plan fixing the probabilities for each choice ahead of the game is a basic ingredient of mathematical game theory.

What does this mean for the declarer seeing the Q in the first trick? If declarer assumes that East is selecting between K and Q randomly, the finesse in the second trick has a much higher chance than the Ace. The reason is that declarer sees the Q in 1/2 of the cases for KQ only, but sees it always when the Q is singleton. Since both cases happen about equally often, the finesse is about twice as good.

This is a misleading, but correct argument. In fact, there are only two reasonable plans for the declarer:

  • Finesse as often as necessary and it makes sense. This loses two tricks only in 11.6% of all possible deals.
  • Finesse once, and then take the ace, if it makes sense from what you see. This loses in 17.2% of all cases.

Probably, you are like me, and you play in clubs all the time where all players always follow with the lower card from KQ. Then, these thoughts are in vain. But with good opponents, they matter.

Strategically planning happens all the time in Bridge. Selecting a bidding system or a marking are such decisions, and we make them beforehand. Opening with a 5-card major only marks us with 4 or less cards in both majors if we don’t in spite of opening strength, or with less than opening strength if we have 5 cards in one major. It will help the declarer if he records such details in the bidding of his opponents. On the other hand, precise values for each bidding help the partner. It is a strategic decision to judge what is more important. As an example, intervening with a weak suit often only helps the declarer on the other side see the points. So, we don’t.

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