Probability, Information, and Bidding

In a previous blog we gave a mathematical treatment, complete with numerical charts, of the contrasting bidding strategies of maximizing the gain when one guesses right and minimizing the loss when one guesses wrong in terms of 2 parameters, the probability of making a higher scoring contract (PM) and the probability that the field will bid that contract (PB). It is now time to flesh out the skeleton with some examples that support our assertion that generally the best strategy is to attempt to maximize one’s potential gains. That strategy entails bidding to higher scoring contracts that have a high probability of making and a low probability of being bid by the field. Most contracts of this type will involve distributional hands that are not efficiently handled by those players who depend greatly on HCP evaluations.

The circumstances of a deal that fit this description and prompted criticism from my partner were described in the previous blog. Here are the hands again.

 

			Bob	Pard
♠ QJ96532	♠ A1084		1 ♠	2 ♠
♥ A1072	♥ 95		3 ♥ *	4 ♠
♦ A9	♦ J6		6 ♠	Pass
♣ —	♣ K7652		*HSGT
5 losers	8 losers

 

At the time my jump to 6♠ was a quick, straightforward decision based largely on the losing trick count given that partner had limited his hand while promising support in the required areas. On the club lead I played low from dummy and the RHO nearsightedly put up the ♣ A which provided for a discard of my losing diamond. 12 tricks were made.

Accuracy is most easily achieved if one player is able to describe his holding within a limited context. This allows his partner to make an evaluation of their combined assets, hence a calculated guess of their overall potential. Thus, after I opened 1♠ , responder’s most descriptive bid is 3♠ when holding: ♠ AT84 ♥ 95 ♦ J6 ♣ K7652, showing 4 spade, 3+ controls, and 8 losers. The losing trick count is an important part of the definition with particular relevance to slam bidding in spades. The total HCPs is not. With this information my hand can be reassessed as a 4-loser hand, and a slam try can be made.

A later examination of the scores revealed that making just 650 would be worth only 33% of the matchpoints, that is, 4 out of 12. Making 680 was worth 8 out of 12. Obviously 4♠ was a contract that was hard to defend with double dummy accuracy. A conservative argument might run as follows: ‘I go with the field and stay in game. If I make 12 tricks I score 8 MP. If I bid slam I can score 12 MP, a gain of 4 MP, but if it goes down, I score a zero, so I am gambling a gain of 4 MP against a loss of 8 MP. Therefore, I need 2:1 favorable odds to bid the slam.’ This incomplete argument describes a strategy of minimizing the loss. Taking a wider view, one may calculate the expected scores for bidding a slam and for staying in game when the chance of taking 12 tricks is less than 2 out of 3, let’s say 5/8, which allows for some diminution from the evidence of the deal.

 

Average Game Score		4 x 3/8	+ 8 x 5/8	= 6.5
Average		0 x 1/3	+ 12 x 5/8	= 7.5

 

On the evidence of the actual results produced, it is clearly better on average to bid this slam in the environment in which it was played. Because the opposition faced was less likely than most to find the correct defence and hold declarer to 11 tricks, the argument for bidding slam is strengthened. Of course, many players would be afraid to bid the slam against strong defenders, thus failing to take advantage of a good opportunity.

Now we wish to study further the reasons behind my partner’s discontent. In the following deal the main feature is a long suit that will provide tricks opposite a limited, balanced hand, so the defence needs to be quick if slam is to be defeated. We have all felt the pressure to find the killing lead, so why not apply that pressure on an opponent?

 

				Bold		Brash
♠ AKJ	♠ 32		2 NT	4♣ *		2NT	6♣
♥ AQ43	♥ K5		4♠	5♣ *		Pass
♦ QJ9	♦ 32		5♠	6NT
♣ KJ6	♣ AQ109743		Pass
6 losers	6 losers		* Gerber
20 HCP	9 HCP

 

A HCP evaluation indicates that a slam is not likely as the total of HCP will not reach the requirement of 33 HCP. So the true believer in HCP evaluation will bid 3NT although 2 suits are poorly held and leave it at that. A losing trick count evaluation indicates otherwise. Normally a 2NT opener will deliver 6 cover cards, and a 7-loser hand opposite will deliver enough winners to produce 12 tricks. The question to be answered is whether or not sufficient controls are present. An exchange of information might settle the matter, so that 6NT is bid when all is well, and not when the situation is unfavorable.

The losing trick total from the hands taken separately indicates that 12 tricks should normally be available. Taken together one sees that the defenders can cash the ♦ AK off the top, but they may not do so, so the probability of making a slam is not zero. If responder determines that slam is a fair contract unlikely to be bid by the field, he may reasonably decide to attempt to maximize his gains in the face of uncertainty. It is to his benefit to get to slam in an uninformative manner that will maximize his chances of gathering 12 tricks.

The brash approach of 2NT – 6♣ is not well conceived. Yes, in the face of failure it may not go down as much as 6NT, but that is not the proper way to maximize the score. The opening leader may decide to lead his ♦ A immediately and thereafter set 6♣ whereas against 6NT he would be reluctant to put down the ♦ A possibly to give away the contract.

The better approach, given responder is determined to bid slam, is to bid to 6NT under a seemingly normal circumstances. The chance of the ♦ A and ♦ K being in different hands is roughly 50%, and if the contract is 6NT one is unlikely to receive a lead away from an honor if the bidding indicates general strength. In fact, declarer might receive a passive club lead, ‘giving nothing away’. So, the calmer approach that goes through the motions of ace-asking always with the intention of bidding 6NT has the better chance of success.

To restate the obvious, the probability of making depends on more than where the cards lie, it depends also on the information made available to the defenders. Note that the above 2NT hand is not the best possible distribution of 20 HCPs, because of an overly qualified club suit. On the following more probable hand, 13 tricks might be available.

 

			Bold			Scientific
♠ AK8	♠ 32		2NT	4♣ *		2NT	3♠ *
♥ AQ43	♥ K5		4NT	5♣ *		3NT	4♣
♦ AQ95	♦ K5		5♥	6NT		4♦	4♥
♣ J6	♣ AQ109743		Pass			5♠	6♣
5 losers	6 losers					6NT	Pass
							*forces 3NT

 

Responder begins with an estimate of normal expectation from a limited hand opposite and must decide how much information is enough information. Information comes at a cost, so responder must weigh a possible improvement in accuracy against the cost of perhaps steering the defence along the right path. The point is this: it is a matter of choice how much information responder decides to gather before making a final decision.

Let’s take an everyday situation where most responders are happy to jump to a normal conclusion without looking for possible flaws.

 

			Normal			Informative
♠ Q6	♠ J7		1NT	3NT		1NT	2♣
♥ AK3	♥ QJ7		Pass			2♦	3♣
♦ AJ42	♦ KQ105					3♦	3♥
♣ J954	♣ A1073					3♠	4♣
15 HCP	13 HCP					Pass

 

Despite a combined 28 HCP a 3NT contract is threatened on a black card lead. Responder may discover the spade flaw by initiating an informative auction as shown on the right. Opener’s 3♠ bid expresses doubt with regard to spades. Consequently responder bids the safer 4♣ . However, one would not expect applause for taking that approach, because almost every pair in the field will be taking the uninformative approach, in effect trading a slight risk against the potential cost of giving defenders information. Indeed, 3NT might make off the top if the opening leader underleads from ♠ AKxxx.

As the process here is the same as the process of bidding an optimistic slam, one must ask the question, why was my partner unduly upset by success? In bidding 1NT – 3NT one similarly may be bidding blindly to a helpless contract, but one is assured of company. That means one will not a score a zero. If one is wrong, the loss is minimized. If one is right in theory to bid 4♣ , some days one scores a top, some days one loses to everyone who makes 3NT on a normal but helpful lead. On average it will pay to bid 4♣ , but with regard to variability, there is less volatility in going with the field. Thus I conclude that my partner is greatly adverse to high volatility. He is content to go with the field on most hands, and await gifts on subsequent boards. ‘Count your HCPs and try to match the actions of the field,’ is appropriate advice for a beginner, but if one maintains this approach over the years, one’s bidding doesn’t progress much beyond the novice stage.

One’s primary aim should be to bid higher scoring contracts that make most of the time. If one bids 3NT without further ado, it is because this contract most likely will make, so there is little need to seek or give away further information. Because one is dealing with 2 balanced hands, the HCP evaluation is expected to produce an accurate prediction. However, when an unbalanced hand sits opposite a balanced hand, a HCP evaluation is an inaccurate indicator, so it is a mistake to persist in its use just because most will do so.

If both hands are unlimited, the captaincy is up for grabs. Here is an auction I encountered recently when facing a pair who were infrequent partners yet well experienced in use of the methods of a 2/1 system.

 

1♥	1♠
3NT	4♣		is 4♣ Gerber?
4♥	4♠		looking for more
Pass			missing a Grand Slam in 3 strains

 

A start of 1♥ – 1♠ has proved notoriously troublesome, especially when a jump shift is wasted on showing weakness. Many pairs reached slam, but no one could find a way to reach the optimum 7♣ . Opener demonstrated a fear of being left to play in a part score by jumping to 3NT. If one is making a limited, descriptive bid, it helps immensely if partner can decode it accurately. Responder took 3NT to be ‘gambling’ in the modern sense with long hearts, and a scattering of minor suit honors, shortage in spades. Actually it was intended to show a hand with a strong preference for 3NT with strength in the minors and a weak heart suit, five to the ace, and shortage in spades, here the singleton ♠ K.

Responder was short in hearts, but felt he should make an encouraging move by introducing his second suit, clubs. Dangerously, both players had made bids that were subject to misinterpretation as 4♣ was taken to be RKC Gerber. 4♥ showed 3 key cards, but was taken as a sign-off. This horrible mix-up (worth 2 MPs, nonetheless) might have been avoided in several ways, but the fundamental difficulty was that the pair had quickly reached the lofty heights of 3NT without a sufficient exchange of information.

To overcome a need for openers to jump to show a strong hand, modern bidders allow themselves to beat around the bush in the early going, adopting the third- and fourth-suit forcing route: 1♥ – 1♠ ; 2♣ – 2♦ . Neither partner has yet provided a good description apart from not choosing a limited, descriptive bid on the second round. The aim is not to inform partner but to get through the round safely. Someone will have to come clean sooner or later…maybe. One can see this is a very different situation from the cases discussed above where one player limited his hand early, so that his partner could make a unilateral decision based on probability within the context of his partial knowledge.

My Candidate for the Best Bid Hand

When prizes are given for the best bid hands, the nod typically goes to a pair who have exchanged information through a long series of bids that ends in a cast iron slam. As Linda Lee asked in an earlier blog, why don’t we give prizes for a short auction? I know some of my best calls have been passes in the balancing seat. A quick decision, like 1NT – 7NT, is based on probability rather than actuality. If 13 tricks are taken on an exotic squeeze, one doesn’t expect to be praised for the bidding, only for the play. The best played hand of the year may derive from consideration of the probabilities of the distribution of the opponents’ cards where certainty is not guaranteed. The attitude is that declarer has made the best of the bad situation when placed in a doubtful contract that makes only on a particular lie of the cards. In order to qualify for the bidding prize the defenders must be seen to be helpless. This is not the usual situation in practice.

In reality the play of the cards is linked to the bidding that precedes it. Defenders and declarers operate in a realm of uncertainty where probabilities play a major role. So the best bid hand might be one in which the opening lead has been made more difficult by uninformative bidding, and a defender later makes a fatal discard as he is unaware that his apparently worthless jack-to-five is the key to the defence (as happened recently in the WBF Open Teams). So, the worse the bidding, the worse may be the defence.

 

♠ 52	♠ AK4		1♣	1♦
♥ 43	♥ A7		3♦	3♥	(asks minor controls)
♦ AKQ7	♦ J1098		4♣ (6)	4♥	(minor queens?)
♣ AKQJ7	♣ 10986		5♣ (2)	5♥	(minor jacks?)
19 HCP	12 HCP		5NT (1)	7♣
			7♦

 

Playing Perfecto Club, a 4-card majors system, responder bid 1♦ with the thought that on a heart lead 3NT might be better played from the other side. Opener’s jump was constructive, promising 4 diamonds and at least 4 controls. The following heart bids by responder were artificial asking bids that revealed opener held 6 controls in the minors, 2 queens and the ♣ J. Opener opted to play in the 4-4 diamond fit as his club suit would provide a discard for a potential losing trick in the majors. Beautiful, as 7♣ doesn’t make whereas 7♦ is cold even though diamonds split 4-1 and spades 6-2.

The best bid was not 7♦ , but what followed it – a double asking for a club lead. Fearing that a club ruff was imminent, responder corrected hopefully to 7NT, but down 1. Was the doubler void in clubs? No, they split 2-2. He deduced from the opponents’ skilful bidding that 7♦ was likely to make, so his side was doomed to a very poor score, because no other pair in the field could match its flawless perfection. The only hope was that 13 tricks would not be available in 7NT. It was a gamble that would cost little or nothing if it were wrong, but it was probably right for the opponents would have bid 7NT freely if they were confident of 13 tricks. Their long sequence was just too convincing. Perhaps perfection has its flaws and a little uncertainty is a good thing. So there you have it: a single bid wins the prize.