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Statistically, What is the Best Opening Lead in a World Pairs Event?
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Data is from the Pairs events in Wroclaw, 2016. WBF data is at http://www.worldbridge.org/repository/tourn/wroclaw.16/Microsite/Results.htm. Data used for this article is at http://www.bridgescoreplus.com/data/wroclaw_2016/index.html. The raw data is at http://www.bridgescoreplus.com/data/wroclaw_2016/all_results.csv. It will be easier to read this article if you have the following page open in a separate window - http://www.bridgescoreplus.com/news/wroclaw_lead_card.html - this has the data in sortable table format but don't look up the answers just yet.

Facts:

1. The one card that produces the best matchpoint score was the least led card by the players at the event.

2. A monkey with a typewriter could make better leads.

3. Best card to lead scores 56%, worse card scores 42%. Players picked the worse card more than the best card.

4. Reading this article could improve your matchpoint scores on opening lead by 14%!

5. Should you lead trumps against a major suit contract? What about leading trumps to a minor suit contract? Yes, there is a difference! Works for one but not the other. Do you know which is which?

So... what is the best single card to lead? Pick one from the 52 available before continuing, see how well you do...

First step is to define "the best lead".

My first definition was to compare how well declarer does against double dummy (DD) based on different leads. For the tournament, the average declarer scores 0.152 tricks more than DD.

There are four cards that cause declarer problems. So much so that declarer will do worse than double dummy. One card causes declarer significantly more problems than the others.

The top four best cards for opening leader are, in reverse order (average DD for declarer in brackets):

8 (-0.02), 4 (-0.07), 7 (-0.08) and the worse card for declarer has a declarer DD of -0.34. Do you know what it is?

Hint: Neither Michael Rosenberg nor Barnet Shenkin would want to play it.

It's the 9 - the Curse of Scotland. Leading this card will cause declarer to give up -0.34 tricks (on average) against DD.

What's the worst opening lead? The 10 - gives declarer 0.43 DD, followed by A (0.36), 6 (0.34), 7 (0.31), 5 (0.30).

So, clearly, the best card to lead is the 9.

But a little thought shows that this may not be the best definition of the best opening lead as it requires declarer to go wrong.

A better definition might be the card that is least likely to give up an overtrick. For each lead, we see if declarer did better than, same as, or worse than DD. 

For this tournament, declarer made more than DD 28.6% of the time, made DD 53.9% and did worse than DD 17.4%. We want the card that minimizes the times that declarer makes more than DD.

There are four cards which are clearly better leads. In reverse order:

9 (21.7%), Q (19.53%), K (18.55%) and clearly the best opening lead with 17.18% is...

8.

So if you want the safest lead, you should lead the 8.

But, there is probably a better definition of the best card to lead.

The best card is the one that will maximize your matchpoint score.

For this tournament, declarer averaged 50.19%, the side on lead averaged 49.81%. Surprised me, not that the two numbers added to 100%, but I expected the matchpoint score to be higher for declarer.

Sorting the cards based on final matchpoint score for the opening leader, we have five clear winners. In reverse order:

Q (53.3%), J (53.6%), 9 (54.7%), 10 (55.5%) and the best card to lead with a matchpoint score of 56.1% is...

... this is the card that you really want to lead if it is in your hand.

What do you think it might be...

It is the 9.

Clearly this is the best card to lead.

What was the most popular card to lead?

I did a study some months ago on the most popular lead from the EBTC 2014 championship. This was a team event, so we may expect the results to be different. See http://www.bridgescoreplus.com/news/lead_card.html for details.

In the EBTC, the most commonly led suits were clubs, spades, hearts, diamonds, in that order.

For Wroclaw, the most commonly led suits were hearts, spades, diamonds, clubs, in that order.

In the EBTC, the most commonly led cards, in order, were K, A, A, A, A, K, 2, K.  

In the EBTC, the least commonly led cards, in order, were 9, 10, 9, 8, 8.

In the EBTC, the K was 6 times more likely to be led than the 9.

What would be expect for Wroclaw - a Pairs event?

Well... the most commonly led cards were, in order: K, A, A, A, A, 6, J, 5, K, K.  We see a pattern. Players like to lead the K.  Not sure why - your average score will be less than 50%.  If they don't have the K, the next most likely lead is an ace.  Of the most common 5 cards led, only two of them generated a matchpoint score of more than 50%.  

The least commonly led cards, were, in order, 9, 9, 8, 7, 8, 8.

Again, we see a pattern in that 9s and 8s are the least likely card to lead.

The ratio of most popular to least popular remained similar.  In Wroclaw, the K was almost 6 times more likely to be led than the least led card, the 9.

What if we compared the frequency of the opening lead with the average matchpoint score? The K was led 1,284 times, but averaged 49.9% for the opening leaders. The 9 was led 221 times, but averaged 56.1% for the defending side.

In statistics, we can do a correlation test.

What are the results? How good do you think bridge players are at leading the best card?

There are two common statistical tests - Pearson, Spearman (Wikipedia them for details). Both provide a number from 1..-1 with 1 showing correlation, 0 showing no correlation, -1 showing a negative correlation.

So... we put the numbers into our favorite stats program.

We are comparing the number of leads of each card against the average matchpoint score for leading that card.

The Pearson result is 0.00065. Statistically, this means that there is (effectively) no correlation between the ability for a player to pick an opening lead and their final matchpoint score.

The Spearman result is -0.012. Spearman is Pearson but with the values ranked before applying the test. Statistically, this means there is a slight negative correlation between a bridge player picking their opening lead and their final matchpoint score.

Based on those two numbers, I make the claim that a monkey with a typewriter can do a better job of picking an opening lead than a bridge player.

Why do all this work?

I want to win a (proper) Gold medal at a World Championship event. Unfortunately, Oprah Rona along with the High Level Players Commission has now promised, just like at a 6 year old soccer competition, that everyone will win a gold medal at the next World Championship. "You get a Gold medal, and you get a Gold medal, and you get Gold medal. Everybody gets a Gold medal." So this is no longer a meaningful challenge for me. But I'd done this work anyway in preparation for Orlando 2018 so may as well share it.

I suspect that, just like annual prize giving for a Kindergarten class, there will be Gold Medals for every possible bridge category in the next World Championship. (I used to run a Cub Scout pack, we had ~120 kids in the pack. Every year, at Pinewood Derby we had a certificate for everyone, "Best Use of Color", "Most Spaceship Looking", "Most like a Block of Wood" so that every Cub when home with something. Creating the lists each year was one of the more requested volunteer activities). The known categories for the next Bridge World Championship already include, "Most Hypocritical Use of Latin in a Press Release", "Most Scoring Errors in an Event", but I digress.

After all this work and study, I'd like to win the Gold medal for "Best Leader".

When you think of "Who is the Best Leader in the World?", what is the first thing that comes to mind?

Trump!

That's right. How can you not think about who is the Best Leader in the World without thinking of Trump.

We already know that the opening leader averages a 49.81% score.

There were 1,762 leads of a trump in Wroclaw. The average score was 50.18%.

Therefore Trump is best!

But before I put my tiny hands on the Best Leader certificate, have we missed something?

There were 19,391 leads of "not trump" against a trump contract. The average score for the defenders was 51.31%.

So... "Not Trump" is better than "Trump".

But, those numbers don't add up. How can the average for the defenders be 49.81%?

There were 8,301 contracts defending "No Trump", the defenders averaged 46.2%.

So if you are defending, "Not Trump" is better than "Trump" which is better than "No Trump".

Back to Trump for a moment.

Of the 1,762 trump leads, 1,276 were against the majors (653 for Spades, 623 for Hearts). Only 486 were against the minors (239 for Diamonds, 247 for Clubs). When leading trumps against a major suit contract, defenders averaged 49.6% (Spades) or 48.65% (Hearts); for minors 52.53% (Diamonds), 53.36% (Clubs). There were more minor suit contracts than major suit contracts. The defenders got it wrong. You do better (on average) if you lead a trump against a minor suit contract than a non-trump. You do better if you lead a non-trump against a major suit contract.

Leading a trump against Spades loses 2.0%, against Hearts 3.5% compared to a non-trump lead. Lead a trump against Diamonds wins 1.7%, against Clubs wins 2.6%.

Conclusion: If the contract is a minor suit, lead a trump; if it is a major suit, don't lead a trump. This advice alone is a difference of 2%! Have you ever seen this in any book on opening leads?

Statistically you do better not leading a trump against a 7 level contract (59.4% v 37.8%).

Necessary caveat:

Point 1: The purpose of the artical was semi-humorous, and to have some fun with statistics. It also helps point out some problems with using statistics in bridge. I've posted similar articles in the past. This is part of the same series. Don't take the article too seriously.

I am well aware of the statistical flaws in this article. There are many. If you are a serious statistician you are welcome to point them out and I'll refer you back to Point 1.

What will be interesting, for the serious statisticians out there, is that this work is predictive. We do not have data from the semi-finals for the championships. This is two days of play. The data is coming (I've been told it may take a couple of weeks). Once the WBF publishes the data, I'll do the same analysis with this new data. If you are a serious statistician, what results in this article do you think will hold true for the new data? What is predictive? That will be an interesting test for any statistician. So feel free to complain about the statistical inadequacy of the work in this article, but if you do so, I ask that you stipulate which criteria will no longer hold true on the new data set.

Will the 9, 10, 9 still be the statistical best lead?

Will you average a better score for a trump lead against a minor suit contract, but a non-trump lead against a major?

Will the declarer advantage remain?

Will the average DD remain around 0.15 tricks?

Should you not lead a trump against a 7 level contract?

Will the red eights continue to average the worst matchpoint score?

Will the black nines continue to average the best matchpoint score?

Will the monkey with a typewriter still do better than the average bridge player?

I'm going to add "Hammond Leads" to my convention card. A "Hammond Lead" is the best statistical lead for the event it is played in. For Wroclaw this would mean, barring any other information, that I would lead the first card I had from the following: 9, 10, 9, J, Q, A, 3, 10, A (remainder of cards are on the web page).

I hope this will improve my percentage score when leading. My last partner is replacing me with a typewriter-owning monkey.

Conclusions:

1. This is Fun with Leads, not a serious statistical polemic.

(But... how much of what we are normally taught about opening leads is true?)

2. If you have it, lead a black nine.

3. Never lead a red eight. Remember: BANNER - Black Always Nine, Never Eight Red. Unless you are playing against me.

4. Black fives should be avoided.

5. If you have to lead an Ace, lead a red one - they do much better than the black ones.

6. The 9 might be a great story card, and will confuse declarer the most, but you still only average 48.6% if you lead it.

7. The 8 may be the safest lead in giving up over-tricks, but you still only score 49%.

8. Seek a monkey with a typewriter for advice on opening leads.

To be continued...(when the semi-final data is published).

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