The Matter of Frequency

In a previous blog we considered the matter of frequency when choosing a partner to play in a matchpoint game that one wants particularly to win. We decided that in a single session event it is better to choose a partner who avoids tops and bottoms whereas in a longer event it is better to choose a partner whose tops exceed his bottoms. The reasoning behind that rested on the observation that with few opportunities for an appropriate swinging action, a bottom may be unrecoverable, whereas with many boards to come there are sufficient opportunities to a bottom (or two) near the beginning to be overcome. Over many boards the governing factor is not the number of bottoms expected but the probability of the difference between tops and bottoms. Mathematically, over a large number of boards the governing probability distribution function tends to be bell-shaped, symmetric about the median difference, whereas for a few number of boards the probability distribution function is skewed towards the number of bottoms.

What about scoring at Swiss Teams on a victory point scale? There are two differences to consider: the gains to be had from bidding vulnerable games, and the length of the matches. A few months ago I played in a Saturday night Swiss consisting of 4 7-board matches. In the second match before comparing scores with our teammates I knew our team had lost by a considerable margin because my score card showed three boards of -120, 2 of them on boards on which I expected our normally aggressive partners to be in vulnerable games going down. Those adverse swings pretty well ruled out our team’s chances of winning the event, but we finished respectably by winning the next two matches by a large margin. Our opponents’ only win was against us. My experience tells me that in order to win these events one has to bid close vulnerable games in an attempt to maximize one’s gain, even when the odds are against making it, but suppose one is playing for the event with just 7 boards left to play. Should we rein it in, try to avoid minus scores, and bid only those games that have a better than 50% chance of success? Jeff Rubens thinks so.

In the December issue of The Bridge World Editor Rubens has introduced the concept that one’s strategy should depend on the length of the match being played. He argues that with only a few boards remaining to be played, the best strategy is to win more boards than one loses, that is to say, the frequency of success outweighs the potential gain, just as it does throughout a Board-A-Match contest. He maintains that the long run odds in favor of bidding a vulnerable game are an exaggeration in a short match, where it is more important to be right rather than ‘have a good bet for an average result.’ Presumably he would aim to bid only to vulnerable games with at least a 50% chance of success. The idea is the same as for playing for tops and bottoms in a matchpoint game, namely, the odds are against recovering a loss, even when one gains 10 IMPs if a game is successful and loses 6 IMPs if it is not. The context of Rubens’ example is the final 7-board round of National Swiss scored on Victory Points where a 20 IMP victory will see his team place 3rd, and a 5 IMP victory will see them in 12th place.

We assume that a partnership decides before a match which strategy they will pursue. That may depend on what they assume the opponents will do in the same circumstances. Or not.: Rubens is silent on that aspect. Let’s suppose that the partners decide to play according to the long term odds and that they will bid all vulnerable games with at least a 37.5% chance of success, 3 chances out of 8. That means they are prepared to fail on 5 chances out of 8, because the opponents, following Rubens’ advice, have decided not to bid such games. If the game succeeds they gain 10 IMPs (G = 10), and it fails, they lose 6 IMPs (L = – 6 IMPs). Thus succeeding on just 3 deals balances the losses on 5 deals, and if the odds of making game are better than 37.5% a gain is expected in the long run. However, in a short match one will not encounter 8 such deals. Let’s look at the situation with only a few deals provide the opportunity to bid an inferior game with a 3 out of 8 chance of success, the lower limit usually recommended for bidding vulnerable games.

 

One Decision

G frequency 3/8,

or L frequency = 5/8

 

So one will lose IMPs 5 times for every 3 times one gains.

 

Two Decisions	Results	IMPs	Frequency (x64)
	G-G	20	9
	G-L	4	30
	L-L	-12	25

 

Here one gains IMPs 61% of the time and loses 39% of the time, which is highly encouraging for bidders. In terms of the frequency of gaining the advantage, there is a large difference between making just one decision and having the opportunity to decide twice under similar circumstances. This is not the case when G equals L, as in Board-A-Match, where the aggressive bidders will suffer 25 loses for every 9 victories.

 

Three Decisions	Results	IMPs	Frequency (x512)
	G-G-G	30	27
	G-G-L	14	135
	G-L-L	-2	225
	L-L-L	-18	125

 

The most frequent result is a loss of 2 IMPs. This may not matter much in the final victory point tabulation. The significant swings are more frequently to the plus side, 162 versus 125, 56%, for an average gain of 1.5 IMPs on those selected boards. Between gaining 14 IMPs (52% of the time) and losing 18 IMPs, there is an expected net loss of 1.4 IMPs over the long run, but overall there is a 5% chance of scoring 30 IMPs. Thus, with 3 opportunities the bidders will occasionally gain a very large margin of victory at a relatively low cost, as well as gaining significantly more times than losing significantly.

 

Four Decisions	Results	IMPs	Frequency (x4096)
	G-G-G-G	40	81
	G-G-G-L	24	540
	G-G-L-L	8	1350
	G-L-L-L	-8	1500
	L-L-L-L	-24	625

 

We consider 8 IMPs to be a significant amount, so one loses on 52% of the significant deals when 4 decisions are made. In that sense, one approaches a 50-50 split on a frequency basis between gain boards and loss boards. The conclusion is that with close decisions one should bid games, as frequency of failure will not be a major concern.

Next we consider bidding games on hands where there is a 50% chance of success. First assume we face ultra conservative opponents who will avoid such games.

 

One Decision	Results	IMPs	Frequency (x2)
	G	10	1
	L	-6	1
			Net Gain = 2 IMPs

 

Two Decisions	Results	IMPs	Frequency (x4)
	G-G	20	1
	G-L	4	2
	L-L	-12	1
			Net Gain = 4 IMPs

 

Three Decisions	Results	IMPs	Frequency (x8)
	G-G-G	30	1
	G-G-L	14	3
	G-L-L	-2	3
	L-L-L	-18	1
			Net Gain = 6 IMPs

 

Four Decisions	Results	IMPs	Frequency (x16)
	G-G-G-G	40	1
	G-G-G-L	24	4
	G-G-L-L	8	6
	G-L-L-L	-8	4
	L-L-L-L	-24	1
			Net Gain = 8 IMPs

 

Clearly it does not pay to be ultra conservative. With multiple decisions there are more plus boards than minus boards got by bidding.

Let’s assume that the opponents will bid 50% games 50% of the time. Whether the game makes or not is maximally uncertain, as is whether they will bid it or not. Maximum uncertainty implies that success or failure depends on the how the cards were dealt, an unpredictably random process, not on the accuracy of the bidding or on the skill of the declarer. Half the time when, they bid game, they will tie those who always bid game. Otherwise, they lose 10 IMPs a quarter of the time and gain 6 IMPs a quarter of the time, for the net loss of 1 IMP. Does it make sense to play to lose 1 IMP rather than to tie? No.

Let’s suppose both teams are neutrally selective, bidding half of their 50% games. Assume the choices are random and independent and the results are random. Half the time both teams make the same decision, so there is no resultant advantage. On one-quarter of the deals, one team gains 4 IMPs and on another quarter, the other team gains 4 IMPs. Perfect balance has been achieved. The greater the likelihood that one team will bid its 50% games, the greater its advantage over a neutrally selective team. Finally, if ideally one always bid games that have a probability of success greater than 50%, under those conditions one can only match those simple souls who blast away and bid some bad (37.5%) games along with all the good (50%) games. Note that bidding systems are based on probable outcomes and seldom provide enough information for a player to estimate the probability of success with great accuracy. So bidding a 50% game or not is more a matter of inclination rather than science which gets you to the point of decision but doesn’t tell you with certainty which way to go on any particular hand. That is the beauty of bridge as a game. We’ll discuss bidding methods further at the end of this article.

The Expected Number of Decisions

We consider the example of 8-board matches within which one’s side is vulnerable on 4 boards. On half of those boards our side holds the advantage. We assume (without detailed numerical evidence) that the chance of having to make a close decision as to whether to bid game is 1 in 4, an exciting proposition. How many such decisions are expected over 1 match, 2 matches, or 3 matches? That can be calculated using the binominal probability distribution function. Let P(n) represent the probability of n boards requiring a close decision. P(0) indicates the probability of not having to decide.

 

Probability	Over 8 Boards	Over 16 Boards	Over 24 Boards
P (0)	0.32	0.10	0.03
P (1)	0.42	0.27	0.13
P (2)	0.21	0.31	0.23
P (3)	0.05	0.21	0.26
P (4)	0.00	0.09	0.19
P (5)	—	0.02	0.10

 

Over an 8-board match the greatest expectation is for one decision (42%). The probability having to make 2 decisions is half the maximum (21%). The probability of no decisions is 32%, resulting in a skew towards the lower number, characteristic of a few numbers of possibilities. Over 2 matches the greatest expectation is for 2 decisions, and over 3 matches, for 3 decisions. Over 24 boards the probability of having to make 2, 3, or 4 decisions is 78%. At the beginning of the session it makes sense to assume an attitude of bidding all close vulnerable games. The same applies over 16 boards.

The situation similar to that considered by Rubens is that where 8 boards remain to be played, and your team is in contention for an honorable placing, but out of contention for the top spot. The chances are 2:1 that your partnership will have to make one close decision rather than two. If early in the last match you bid game and lose 6 IMPs, that loss may drop your team from 16th place to oblivion as there is little hope of a second chance. If you bid the game and gain 10 IMPs, you reach the dizzying heights of 5th place, and if you get another chance, you may actually gain another 10 IMPs and attain 3rd place, although the odds are well against being so lucky. So what kind of player are you – the one who strives for excellence, or the one who is afraid of dropping out of the money?

Like the ill-fated Icarus of Greek mythology my preference is to rise as high as possible without fear of precipitously losing elevation. So I adopt the strategy of bidding close games, regardless of the point in the match at which they occur. With victory point scoring each board regardless of the order in which it is played contributes to the end result. Many games are pitifully lost when near the end the leading team grows cautious and attempts to sit on their lead. It is a common occurrence in competitive sports that the eventual winners come from behind at the end by vigorously striving to maximize their gains rather than passively waiting the leader to make mistakes. (This weekend the golfer Graeme McDowell demonstrated how to overcome errors and come from behind.)

Consider a 4-match victory point game at our local club. The winning score is usually slightly greater than 60 out of 80. One may attain 60 by the route 15-15-15-15, but that would be unusual as the last match will be against a team that has been playing with luck on their side. If both teams play with caution one’s scores may turn out to be 15-15-15-10, which gives a respectable total, but not one short of a winning total. So to win the event one must try to win the last match by at least 10 IMPs, attaining a sequence of 15-15-15-15. A vulnerable game in the last match is a god-given opportunity to pick up the margin of victory on one board.

Two swings of 6 IMPs each got by not bidding close vulnerable games will also achieve a victory point margin of 10, but the odds against 2 such swings are low. First, one may not get two chances, and second, the chances of gaining 6 IMPs on both is low, being the product of probabilities. Consider the case of a lower limit chance of success.

 

Chance of a 10 IMP swing on 1 board	0.42 x 0.375 = 0.16 ;
Chance of a 12 IMP swing over 2 boards	0.21 x 0.625 x 0.625 = 0.08

 

So it is reasonable when going into a final match to attempt to swing for 10 IMPs if the opportunity arises, rather than hope for 2 chances to gain 6 IMPs by keeping out of close vulnerable games aggressive opposition may bid. One hopes for at least one chance to bid a close vulnerable game and get it right. The first wish has a probability of 68%; the second wish is in the lap of the gods. I look at it this way: if there are no close decisions encountered in a match, my chances of winning the event are reduced.

Judgement and Bidding Systems

Here is the problem Rubens presents: holding ♠ J6432 ♥ AQ85 ♦ 2 ♣ Q96, do you raise to 3♠ after the sequence 1♦ – 1♠ ; 2♠ – ? He judges one should not and gives these hands to illustrate his point.

 

♠ AK5	♠ J6432	1 ♦	1 ♠
♥ K104	♥ AQ85	2 ♠	???
♦ KJ843	♦ 2
♣ 73	♣ Q96
14 HCP	9 HCP
7 losers	8 losers

 

Less than 30% chance of no trump loser on a bad day, only 8 tricks

Whether by HCP evaluation or by losing trick count, the evaluation is that 9 tricks are available, but not 10. So normally one would not bid game. However, we know these are merely approximations, and there must be some probability associated with the claims of the total tricks available. Bidding systems are based on such criteria, so we can expect that these indicators represent a 50% chance of making the specified number of tricks. How can we then arrive at the 37.5% limit for 10 tricks?

Exact statistics would help, but lacking these we have to make a guess. With the losing trick count, we might estimate a deficit of a quarter of a loser, represented by the absence of a well-placed jack. With HCPs we would guess about 24 HCP would represent the normal requirement without shortages. Trump quality is the key. Judgment plays a part as well as the information revealed by the bidding process.

Generally when one raises from 2♠ to 3♠ , the proposition put to partner is, ‘if you have something extra for your bid, please bid game.’ Because no specific information is provided, this calls for a blind evaluation with particular attention paid to the trump situation. On point count opener has shown something between 12 and 14 HCP, so in the example shown above, opener may judge he has the extra required, and bid game. This is what Rubens fears, so he passes. It is bad bidding practice when one cannot invite safely. I would not raise to 4♠ on just 3 trumps without the possibility of trumping effectively in the short-trump hand. I avoid punishing an enterprising partner who is looking at 5 trumps in an 8-loser hand. Both shouldn’t stretch because of the vulnerability alone.

The losing trick count evaluation works better as a blind indicator. Opener is expected to have 7 losers. He has 7 losers, so he does not raise. Yes, the ♦ J may appear to be the little bit extra that is needed, but opener does not hold 4 trumps, as expected in a good system, such as Precision. So even under Precision methods opener should not raise to game even with a near maximum 14 HCP.

Normally with 4 trumps and 6 losers opener will give a jump raise to 3♠ , so responder knows immediately to raise to game with an 8-loser hand. So on a raise from 2♠ to 3♠ one’s best hope is to find opener with a good 7-loser hand, as shown below.

 

♠ AQ75	♠ J6432	1 ♦	1 ♠
♥ K104	♥ AQ85	2 ♠	3 ♠
♦ K874	♦ 2	4 ♠	Pass
♣ 73	♣ Q96
12 HCP	9 HCP	50% chance of	no trump loser
7 losers	8 losers

 

In this case opener has fewer HCPs than in the previous example, and the ♠ K has been demoted to the ♠ Q, yet the chances of making game are much greater due to the fact that a 9-card trump fit has been uncovered. A vulnerable game can be bid with justification. Rubens chooses to play ‘safely’ in 2♠ , and that decision might win 6 IMPs. Non-experts who count points and fear an aggressive partner might do the same.