Bob Mackinnon

The A Posteriori Par

Often one reads references to the ‘par result’, which is the optimum result obtainable when one can see all four hands. The bidding is not part of the calculation which depends only on the lie of the cards. There is also a practical par which derives from actual play after the opening lead is made and dummy appears. One might term this the a posteriori or conditional par. The odds have changed from the a priori values and the optimum result may have changed as well. That is what counts and what we must play for.

Imagine you are a pro playing in a PGA event. Your ball is on the edge of the green with the hole located 30’ away on a dangerous downward slope. Lying 2, the task before you is to get the ball in the hole with 2 putts. This is true whether you are playing a par 5 hole or a par 3 hole. On a par 5 hole it would be wrong to think, ‘I mustn’t putt too hard and overshoot the hole’ – that might lead to a cautious 3-putt and a lost opportunity.  On a par 3 hole it would be wrong to think, ‘I need this putt to make up for my error’ – that might lead to ending up on the fringe at the far side. In short, one has to put aside the idea of an a priori optimum result and deal with the real and present situation as best you can.

Everyone realizes they should take advantage of the favorable situations that arise, but not everyone knows that we have to play for the conditional par in what may appear to be an unfavorable circumstance. I know that is true because I am one of those who tend to get disheartened when it appears they may have missed a chance at the theoretical optimum result. My most costly errors come at that time. This is a bad habit, for one’s score depends not on the par result but on what the field can achieve under the limitation of uncertainty. Here is a recent example where my false assumption cost us a tie for first place in the final standings.


Dealer: West
Vul: All













All Pass

It was with great foreboding that I opened one of a minor with 18 HCP. This seldom works for me when I play 2/1 methods, so I began with negative thoughts, always a bad start. The lead was the 5, and when dummy appeared I could see that we had misbid this hand badly. Surely all Wests would raise to 2 over the 1 overcall and East would bid 3NT with a better than sporting chance of making it. So it looked like a bottom score unless 3NT went down on an absolutely filthy break in diamonds.  Continuing to think negatively, I led a second heart to set up the T in dummy, and achieved my objective, +120 on a sequence too painful to recall, must less to admit to. That is what comes from reading too many bridge books where a perfect double dummy result magically appears when one flips the page.

This 120 was a clear bottom. Only one pair reached 3NT, and that declarer made only 9 tricks. If I had kept my cool, I could have made 11 tricks to achieve a score of 10 out of 12. It is safe enough to go to dummy and lead a diamond, ducking on the unlikely event of North playing the K. When North gets in with the K on the second round of diamonds, he has no heart to return. I avoid the spade finesse, and the Q falls in the end giving me 11 tricks in a contract of 1NT. What would be a disaster in theory, in practice would be a good result that was there for the taking on a simple line of play.

When the dummy appeared, I could judge only on the evidence at hand. Even if I had opened a Precision 1, there is no guarantee that we would have reached the theoretical par contract. This combination is better bid using 2/1 methods, given that South will bid as she did, but at other tables South may have preempted 2 making it more difficult to reach 3NT. The point to remember is that seeing the dummy I could judge accurately the a priori par result, but I was unable to judge accurately the conditional par result which depended on the bidding at the other tables.

There is another par contract to be considered at matchpoints, which is the most frequent contract achieved by the field. I call this the mode contract. Arriving at this contract is the objective of many players. They feel they can outplay their opponents so by this means they will gain on a majority of the hands. In a good field the mode contract may yield the a priori par much of the time, but in a poor field with few tables the two may not match that often. Accurate defence against a normal contract may condemn a declarer to a below average score. Against the weaker elements it is better to overbid to a high scoring contract and hope for a poor defence, just as one does in a team game.

Ensuring a Bad Result
At matchpoints, playing safe is like taking out life insurance: someone else ends up the benefits. Most of the time it pays to adopt an aggressive attitude and focus on what can go right. You might say that fortune favors fools. Here is an example where I took out some insurance, and the opponents collected the matchpoints.


















Lead: 3


The AKQ in hearts were a bad placement for half of my 18 HCP, and the 4333 shape further reduced my expectations, so I decided to open 1NT (15 -17 HCP), something an expert would never do. When dummy appeared, I formed a plan, which at our club mostly entails lining up the finesses and arranging to take them in the proper order. I saw that optimistic bidding might have led to 3NT which appeared to have good chances of coming home if the diamonds were favorably located. With a silent curse, I won the Q with my ace and led a diamond towards the dummy hoping for a revealing honor – and I got it, the K! How should I have proceeded from there?

The average player might think to finesse twice in clubs assuming that the percentages favour split honours.  Not me. The appearance of the K marked the LHO with 4=4=1=4 shape or, more likely, 5=3=1=4, so my LHO held 4 clubs and my RHO just 2. It was against the odds she held the Q, but if she did 3NT would make on the club finesse. Pessimistically I thought of what might happen in the likely event I lost a finesse to the Q: a heart would come back, and a defender could hold up on the K so I would be forced into having to play disadvantageously from my hand. 

To avoid an endplay in hearts, I played off the top hearts and exited a spade hoping to force my LHO into breaking the clubs. Making 120 should score well against 3NT going down. The endplay came to pass, but unfortunately the Q was doubleton on my right, so 150 was available if I had gone for it. That would have been an average score as in 3NT some were making and some were going down.

As one cannot beat any pair in 3NT making, one must concentrate at matchpoints on beating those who like oneself are playing in a part score. Although it was against the odds for the Q to be onside, it was even less likely that the defenders would come up with the perfect defence I feared.  That wasn’t Bob Hamman in the red dress and earrings on my left. At matchpoints, if the tricks are there for the taking, you have to take them.  If the opponents defend perfectly, what can you do but tip your cap to them?

Virtue – a Dubious Reward
Last week my humility was given a great boost. The hands did not suit me. In one session we were on opening lead 12 times and scored above average only once, that time being when the opponents were in 2NT making 5, not a top for us as some were in 6NT off 2 aces. One hand stood out as the most interesting I have encountered in some time. Partner opened 1 and the RHO opponent overcalled 1. What is the virtuous bid with the following cards: A94 9632 JT A975?

There are hands which are difficult to get right. One may be virtuous at the beginning but there follow too many chances to go wrong, and one bad decision in a series of choices from either side of the table may lead to an embarrassing result. It may be better to take risks that force the opponents into making the final (bad) decision than it is to bid virtuously then find your defence lets you down. One thereby reduces the problem to a single decision on which success or failure depends.

I was delighted at having been given a chance to test the theory that tells us there are moderate hands where it is best to pass early and await developments. One only gets into trouble, the theory goes, if one pretends the hand can be reconfigured to fit one of the available categories. Partner can balance if distributional, but if he passes with length in hearts you may have virtuously avoided a bad situation – a misfit hand with nowhere to go. So it came to pass, and I was on lead against 1. When defending at the one-level the aim can hardly be to defeat the contract, so it is not the time to pursue an active defence. Both defenders will get a chance, perhaps several, to make the correct switch…. or both players will get a chance to screw it up. Remember, partner may not expect you to have many points. What is your lead?


Dealer: South
Vul: NS


The A would have been a killing lead, but I missed that double dummy choice when I went with the passive J. In the confusion that followed West did not get his spade ruff. Interestingly, West must not cover the J when it is led from dummy. Still, despite very bad defence that allowed 8 tricks, we scored 5 out of 12, above average for us on the day. Most Wests failed to open on their 9-loser 11-count, leaving it to hyperactive NS to bid to a failing contract. At least we beat the sensible ones who stopped in 3 making 140.

At matchpoints it is not right to play for averages, so in retrospect I feel my pass was impractical. This hand was going to be tough for most pairs, so NS could do with a push in the wrong direction. A nebulous negative double might have got a 2 response from partner (it makes) without increasing the danger that NS would find their productive spade fit. It would also have given the defence a better chance to succeed against a higher heart contract. Which raises the question: should West balance with 2 after my pass? The problem there is that my pass with values is a rare occurrence, so West does not relish giving North a second chance. And what of South’s timid pass which allowed no opportunity to improve the contract? So there you have it, contracts at the 1-level are often the most interesting as there are critical choices to be made all ‘round the table.


Jeff LehmanMarch 13th, 2012 at 1:47 pm

Your comment about not declaring in a way that wins only if the opponents conduct a perfect defense was evidenced in a club matchpoint game yesterday.

Bidding unopposed to a normal 3NT contract, declarer saw a lead of the H6 and these cards:


Perhaps fearing a switch to a spade upon losing a club finesse, followed by a spade continuation with split high honors and the SJ after the QT5 holding (losing CQ and at least four spades), declarer proceeded along these lines: won HK, played five rounds of diamonds, two rounds of clubs and two more hearts, ten tricks in all.

But with CQxx in the slot, thirteen tricks were available after the heart lead. +630 scored a 1 out of 7 mps.

Playing all out must be right at matchpoints. Look at all the legs of the parlay to avoid the set: the club finesse must lose, the spade switch must be found, the high spade honors must be split, the SJ must be offside, and declarer’s LHO must own at least four spades!

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AnthonyMarch 21st, 2012 at 11:58 pm

Bob, I just wanted to say that I find your columns excellent and provocative in terms of re-examining my own ideas about how to bid and play. You clearly think quite deeply about the game and I usually learn something from your articles. Please keep it up and know that you have some fans out there!

RobinApril 17th, 2012 at 5:54 pm

Bob, some very interesting stuff here. I sympathize as I am a “thinker” like you but, alas, not an “expert”. I’m thinking about a blog on the implications of getting a damaging lead versus a non-damaging lead. That is perhaps the most important issue determining which “par” you are looking for.

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