Does Restricted Choice Work Against Robots?

[Below is an excerpt from a series of Substack articles I'm working on. It wasn't until I started writing this article that the conclusions I draw concerning playing against robots occurred to me.]

In the April 1954 issue of Contract Bridge Journal, Alan Truscott concluded a discussion of a bridge deal with the words "This line of thought, allowing for the defenders having had a choice of plays, crops up in many disguises."

The line of thought Alan was referring to had been introduced to the bridge world three years earlier in an article in Bridge Magazine by the mathematician A. O. L. Atkin. But it did not become widely known until Terrence Reese discussed it in his book The Expert Game, in 1958. There he devoted an entire chapter to the concept and gave it the name The Principle of Restricted Choice.

The applications of Restricted Choice go well beyond the game of bridge. It can provide intuition for understanding a number of apparent probability paradoxes. But its proper application can be tricky at times. In this series of articles, I shall examine some of these "paradoxes" and show how Restricted Choice can shed light on them.

[At this point, I introduce Restricted Choice, using Atkin's original example:

	NORTH ♠ K 10 7 6 2
	SOUTH ♠ A 9 6 3

After explaining enough of the rules of play to allow the non-bridge player to follow the argument, I show how, when East drops the jack under ace and West follows low at trick two, Restricted Choice shows that finessing is better than rising by a factor of 2 to 1.]

There is an issue one might raise with applying Restricted Choice to this problem. Let’s consider what happens on trick two. You lead the three and West plays the eight. If West has both the queen and the eight, he has a choice of cards to play. Thus, when he plays the eight, you might reason that he is only half as likely to have queen-eight as to have just the eight. We already decided that Case QJ is half as likely as it was before play began. Now the same is true of Case J. So it’s even money whether to finesse or rise.

What’s wrong with this argument? It doesn’t hold because of two constraints that were not explicitly stated but that a bridge player would assume: 

Constraint 1: The defenders (East and West) have a goal of preventing declarer from taking five tricks. 

Constraint 2: The defenders assume that declarer cannot see their cards.

When declarer leads toward dummy’s ace-ten and West holds queen-eight, West’s only hope of scoring a trick is to play the eight and hope declarer plays the ace. He has no way to win by playing the queen. Because of Constraint 1, his choice is just as restricted as East’s but for a different reason. So there is no adjustment to the likelihood of Case J.

Constraint 2 is also important. If West thinks declarer can see his cards, he will think it makes no difference which card he plays. So if we remove Constraint 2, West has no reason to prefer one card over the other, and his choice is no longer restricted.

We took these constraints into account when we specified the rules for the defenders’ play in the proposed simulation. It’s worth noting, however, that Constraint 2 does not necessarily hold any more. We now have robots that play bridge. While robots are programmed to try to thwart declarer in his goal (satisfying Constraint 1), they make an assumption humans do not: that declarer can see their cards.

Why? Because programming computers to play bridge is hard, and no one has yet found a suitable algorithm that does not require this assumption. The code for robot play is not open source, so I can’t say for sure that a robot West would randomize his play with queen-eight. But I have seen them rise with the honor in this situation, so I believe they do. If so, then the odds for rising and finessing are approximately the same. 

I say approximately because, as we said earlier, Case QJ is slightly more likely than Case J. The difference didn’t matter when the odds were 2 to 1, but now it does. So rising is actually the percentage play by a small margin.

It's worthwhile emphasizing the reason Restricted Choice applies differently to robots and to humans. Some assume, before hearing my analysis, that I am going to claim they don't randomize properly. As we saw in the Prior Strategy discussion [part of the redacted section above] whether an opponent in fact randomizes his play doesn't matter. What matters is that he might. 

The critical factor in deciding whether to apply Restricted Choice is whether an opponent has a choice of plays or whether his play is restricted. This restriction could be because he has no other card to play, or it could be because his alternative is proscribed by the logic of the situation. The latter consideration is what matters here. Because of a glitch in the way robots "think" about bridge, they have a choice of plays in situations where a human does not. That's where the difference lies.