Watching with Woolsey

Watching the Resinger Cup final on BBO was a real treat as it featured one of my favorite commentors, Kit Woolsey, who guided viewers through the action. We seldom encounter BAM scoring any more, which is a pity for every trick on every board counts. The IMPs game pales by comparison as an exciting and instructive contest of skill and a demonstration of card play, as there is none of this “next’ being called out after the opening lead, or yawns being expressed in print when the bidding stops at 1NT, a contract that should be one of the most exciting in bridge. My version of The Official Encyclopedia of Bridge states that BAM went out of favor largely because it is the most difficult contest for good players to overcome better players. That tells me that, contrary to some opinions, the luck factor is reduced to a minimum.

A British BBO commentator felt it was easier to sacrifice at BAM than at IMPs as the cost is limited, not like at IMPs where a large penalty risks a large carry-over. If the opponents can make a vulnerable game, a penalty of 800 translates to a loss of 5 IMPs, whereas a penalty of 500 represents a gain of 3 IMPs. Although at BAM scoring the gain and loss are equal, as Woolsey pointed out one shouldn’t be inclined to sacrifice at BAM if one can pass and tie the board without risk as a score of ½ is added to one’s running total. Sometimes sacrificing is 2:1 against the odds – if you are wrong and their contract does not make, you score a zero, whereas letting them play in their doubtful game or slam can have a positive outcome even if it is not theoretically optimal. In a previous blog we applied some mathematics to bidding decisions of this sort, so for clarification I decided to apply the same methodology to the situation to which Kit was referring.

The probability of making slam (or game) we denote as PM and the probability that the opponents at the other table will take the sacrifice is denoted as PB. If we do what the opponents do, the board is tied, the resultant score is ½ (on a scale of 1, ½, 0). There are 8 possibilities to be considered when we assume the sacrifice will cost less than the slam.

 

Condition	Action	Result	Expected Score
Slam makes	we sac, they sac	½	½ PM x PB
(PM)	we sac, they don’t	1	PM x (1-PB)
	we don’t, they do	0	0
	we don’t, they don’t	½	½ PM x (1 – PB)

 

Slam doesn’t make	we sac, they sac	½	½ (1- PM) x PB
(1 – PM)	we sac, they don’t	0	0
	we don’t, they do	1	(1 – PM) x PB
	we don’t, they don’t	½	½ (1 – PM) x (1 – PB)

 

Expected score for sacrificing	½ PB + PM – PM x PB
Expected score for not	½ PB + ½ – PM x PB

The advantage to not sacrificing is the difference: ½ – PM.

Clearly, if the slam has a better than 50% chance of making, on average it pays to sacrifice, regardless of what the opponents are doing. This implies that the decision rests solely with one’s expectations as to the lie of the cards. However, Woolsey points out that one may be making a poor bet to sacrifice when the opposition is not likely to do so. Let’s look at the conditions when we are certain they won’t sacrifice (PB=0).

Expected score for sacrificing = PM

Expected score for not sacrificing = ½

Thus, you are guaranteed an average score by not sacrificing. In order to make an even bet, where risk equals gain, slam must be a certainty. That was Woolsey’s point.

What’s Wrong with this Picture?

Here is a little story. Fred has just inherited $10,000, so he goes to his accountant to ask advice as to how to invest it.

‘Fred, this is your lucky day,’ the accountant says, ‘just this morning I read in the newspaper about this new sport franchise, Mexialleys, whose stock is predicted to rise 20% in the next year. It’s a terrific investment just waiting for someone like you.’

‘Tom,’ Fred says, ‘you mean I should put in $10,000 in order to gain $2000? Seems to me I would need 5:1 odds in order to justify such a gamble. Thanks, but I think I’ll just stuff the cash in my mattress and take out some extra fire insurance.’

Playing bridge is not akin to betting on a horse or putting all your money in one stock, it is like a mutual fund with many variable components, the sum of which serves on average to reduce the overall variability. Every hand is an investment for which there is a cost, a probable gain, and a probable loss. One will inevitably incur losses on some boards, the hope being that the cumulative gains will outweigh the losses by a sufficiently large margin so as to meet the aims set at the beginning. One’s strategy should depends on what wishes to achieve: scoring above average, or competing for a top finish.

Back to the Reisinger

In the qualifying stages there is evidence that indicates one should cautiously accept an average score. In the last 2 sessions the Jacobs team qualified for the final with scores of 52% and 53%. Non-qualifiers included Mahaffey at 35% and 65%, and Hampton at 42% and 61%. I am sure players on those teams wished they had scored a modest 45% on their first rounds, but it is difficult to control violent swings. Compensating for lows doesn’t mean you’ll get to keep the highs, as there is a different strategy involved. As 50% is not qualifying score one cannot be content with an average on every board. One should approach every hand with cautious optimism, not reckless abandon.

With regard to sacrificing against a slam, the even bet limit depends on the probability that the opponents will sacrifice. PB may not be independent of PM, as the more likely the slam is to make, the more likely the opponents will find the sacrifice. Nonetheless, the relationship between the 2 depends on the bidding. If one side opens 2NT at one table, say, it may not become obvious that a sacrifice is a good risk, whereas if the opening bid were a big club, a preemptive overcall could pave the way in that direction. In many cases it will be difficult to guess the auction at the other table. The situation in which one is maximally uncertain as to whether the opponents will sacrifice or not (PB = ½) constitutes a reasonable approach in unexceptional circumstances. Under this assumption, the expected scores are as follows:

Expected score for sacrificing = ½ PM + ¼ . Expected score for not = ¾ – ½ PM.

For PM=5/6 one has an even bet, the potential gain of 1/6 above average equals the potential loss below, but the expected score for not sacrificing is a lowly 1/3. One scores 2/3 on average by sacrificing, so it seems obvious to do so. There is something very wrong with the argument that one should act only when the gain is greater than the risk as only in 1 case out of 6 will it be wrong to sacrifice, and if you don’t sacrifice you are willing to accept a frequent poor score. So Fred may have doubts about Mexialleys, but it would not be reasonable to expect the shares to become worthless overnight. If he trusts the available information, it makes sense to invest for probable gains rather than take losses time after time, but the degree of risk depends on what Fred is trying to achieve.

Just reaching the Reisinger final may be good enough for some, but if one wishes to be near the top one needs to step it up. Here are some 2010 results w/o the carry-overs.

 

Team	Session 1	Session 2	Session 3
Cayne	61%	52%	1
Smirnov	63%	50%	2
Rosenthal	50%	57%	3
Gordon	44%	61%	4
Jacobus	56%	52%	6

 

The top 2 teams were over 60% in first session and held on with slightly above average scores in the second. Rosenthal improved significantly by coming back with a 57% in the second session. Gordon’s recovery was more spectacular but their earlier session was too damaging. The Jacobus results are interesting; they scored well in the finals, but were handicapped by being 2.5 carry-over points behind the leaders after playing against the full field. So, scores in the low 50’s in the qualifying rounds will prove a handicap, however, just getting to the finals is worth 60 or more platinum masterpoints. For most players, being happy with average boards early in the going makes good sense.

If a top team had been capable of 2 final sessions of 57%, they would have won, but, of course, it doesn’t work that way – there will inevitably be random variability. If one’s target is 57% per session, one has developed the winning approach. Put yourself in the West seat in the 1st final session playing a board on which the 2 top finishers went face-to-face. Would you, as the front-runner, sacrifice in a nonvulnerable 5♦ or pass?

 

Dealer: West Vul: NS	North ♠ 8 6 4 ♥ A 7 5 2 ♦ A 9 6 ♣ A 10 7
West ♠ J 9 5 3 ♥ J ♦ 10 7 5 4 ♣ 9 8 4 3		East ♠ A 2 ♥ K 4 ♦ K Q J 8 3 2 ♣K Q J
	South ♠ K Q 10 7 ♥ Q 10 9 8 6 3 ♦ — ♣6 5 2

 

West	North	East	South
Pass	1♣	Dbl	1♥
Pass	2♥	3♦	4♥
???

Would you as West pass or bid 5♦ ?

West for the eventual winning team, Michael Seamon, bid 5♦ on the hope that his partner, James Cayne, could hold the losses to 4 tricks, which he did. This was in keeping with the mathematical analysis given above which indicates that, regardless of the action at the other table, one should be inclined to sacrifice if the probability of their making is close to 1. Seamon thought so, undoubtedly influenced by the presence of the singleton ♥ J and the absence of anything else of much value. This action created a pickup against the second place team whose West took the conservative view of passing, scoring a zero for doing so when 4♥ by NS made easily. This was a swing of 1 full board and put Cayne in the lead. They barely held on in the second session, so this earlier board proved critical to the end result. The result indicates that one should be prepared to take one’s chances as they arise rather than hope to get them near the end after the situation has become desperate and one cannot be as selective in the choice of boards on which to swing.