Fighting the Field at Matchpoints

When playing in a competitive game attitude is important. Matchpoint bridge is strangely abstract in that you playing against the players you may never face at the table, so one’s attitude towards the field becomes important. In a recent Regional my partner and I had a healthy lead after the first of 2 sessions in a seniors’ pairs game. The field was not distinguished. The second session began well, and when about a third of the way through my partner scored well in a doubled part score against our main rivals, I felt our position was unassailable. At this point thinking to consolidate our position, I decided to play down the middle and concentrate on reaching the par result. In this I was mostly successful as a later study of the hand sheets confirmed. Yes, there was one hand where I kept the wrong card in the endgame and allowed 3NT to make, but that was only slightly below par as the declarer had given us an opportunity by a misplay. Imagine my surprise when after 11 rounds we found we had fallen behind in the standings. A bad last round where so-so opponents fell into a high scoring minor suit partial (-130) sealed our fate. It was as if we had been riding high, wide, and handsome in a hot air balloon before I turned off the gas burner and set us to drifting gently to the ground, helped along by a nasty downdraft at the end. Our skills had not diminished suddenly, but my attitude had, and this reduced my alertness towards opportunities that presented themselves. Too late I recalled the comment of the late Paul Soloway, ‘I never play for averages.’

It is common for experienced players to advise otherwise. In his book Matchpoints (1982) Kit Woolsey begins by stating, ‘what we are trying to do when we play bridge (or any other game of imperfect knowledge) is to minimize the expected or average cost of being wrong.’ This appears to suggest the need to minimize potential losses by going with the field in close bidding decisions. What many, like me, forget is his following definition, ‘the cost of being wrong is the difference between the result of the action in question and the optimal possible result which would be achieved if the winning action were taken.’ The cost is not calculated against the field action but against the odds present in the lie of the cards. One cannot afford to miss opportunities for a good score.

‘There is no security on this earth, only opportunity’ – General Douglas MacArthur

It is not difficult to treat the problem theoretically where there are just 2 alternative contracts possible. So we might think of the bidding outcomes as a game or a partial, or a slam and a game, or a grand slam and a small slam. Competitive bidding situations commonly involve more than 2 possible contracts, but that can wait for now.

What to Expect in Theory

The number of players doesn’t affect the theoretical results concerning the expected scores which depend solely of the probabilities of 2 events: PB, the probability that a given contract will be bid by the opposition, and PM, the probability that the given contract will make. You will score 1 for every opponent whose score is below yours, score ½ for each opponent who has the same score as yours, and score 0 for each opponent whose score exceeds yours. For ease of explanation we often illustrate the problem in terms of a slam being bid, but the reader may keep in mind the method applies to the other situations mentioned above.

There are 4 possible situations: 1) you bid slam and it makes; 2) you bid slam and it fails; 3) you stay in game and slam makes, and 4) you stay in game and slam fails. The probability of slam making is PM, so the probability of its failing is 1 – PM. The probability of the slam being bid is PB, so the probability that it won’t is 1 – PB. The expected scores under each situation are as follows:

 

Situation 1	PM – ½ PM x PB		Situation 2	½ PB – ½ PB x PM
Situation 3	½ PM – ½ PM x PB		Situation 4	½ + ½ PB – ½ PM – ½ PM x PB

 

We denote the sums, S1, S2, S3, and S4, respectively. The sum S1 + S2 equals the expected score when one chooses to bid the higher level contract; S3 + S4 equals the expected score when one chooses to stay in the lower level contract.

 

S1 + S2	=	PM + ½ PB – PM x PB
S3 + S4	=	½ + ½ PB – PM x PB
So that		S1 + S2 – (S3 + S4)	= PM – ½

 

Clearly, one should choose to bid to the higher level if there is a greater than 50% chance of making it, regardless of what the field is doing. This is a nice result for idealists as one should in theory bid according to an evaluation based on the cards alone. An accurate evaluation requires accurate information, and inevitably there is uncertainty due to the inadequacies of the bidding system. The worse one’s bidding, the more inclined one is to go with the field. It helps in this regard if everyone bids according to the same rules.

The field has an expected score of average (½), so no matter how crazy the crowd, if you follow the crowd, you can expect an average score. That is part of the survival kit of the mediocre player. One might imagine that the probability of bidding a contract should reflect the probability of its making, that is, the field will tend to bid contracts where PM>½ and avoid those where PM<½. The critical decisions will occur where PM is somewhere in the vicinity of ½, that is, where there is maximum uncertainty as to whether the contract is more likely to make than not. However, the field has its preferences and tends overbid games. A 50% chance of making a major suit game requires more than a simple finesse, as one must take into account the possibility of a 4-1 split, an eventuality most players ignore both in the bidding and in the play. On the other hand the field tends to underbid minor suit slams, as most pairs do not have the methods to explore for slam and stop in 4NT. Why, then, are so many good players affected by what the field is bidding?

Baseball and Bridge

Over the long run of a baseball season it is a part of the game that a team experiences ups and downs. So it is at matchpoint bridge: one doesn’t score above average on every hand, and sometimes a disaster occurs at random. The baseball strategy employed for making the playoffs is to beat up on the poor teams while breaking even with the rivals. It is hard to win a matchpoint event if one gets a string of averages against pairs who are handing out tops to others. On the other hand, one should be content to achieving an average against the best pairs, as that means one hasn’t fallen behind them in the race for a top position. To achieve an average one bids as the field bids.

Matchpoint games differ from team games in the same way that the baseball season differs from the playoffs. To get to the playoffs a team needs home run hitters who are considered good if they hit a home run once in 20 at bats. They hit mistakes. They are like the players who take advantage of poor pairs. Once a team gets to the playoffs, the game changes, as a team must face another good team. Now accuracy and consistency are most important, and an ability to bunt may become critical. Often the heroes are steady players who never made the highlights during the season. So it is at bridge. Tactics vary with the players at the table and depends on the quality of the field.

Strategic Bidding and Maximum Uncertainty

At matchpoints sometimes one may wish to minimize the potential loss and sometimes to maximize the potential gain. The 2 strategies can be analyzed mathematically as follows.

The difference (S1 + S2) – (S3 + S4) can be broken down into the following 2 components:

S1 – S4	representing the difference in gains for being in the right contract, and
S2 – S3	representing the difference in losses for being in the wrong contract.

 

One may attempt to maximize the gain for bidding correctly, or attempt to minimize the loss for bidding incorrectly. A very important condition is a probability of ½, which represents a condition of maximum uncertainty as to which contract the field will prefer (PB=½), or which contract will make (PM=½). In such cases, maximizing the gain or minimizing the loss are contrary strategies: as S4 goes up, S3 must go down. If S4 is greater than S1, then S3 must be less than S2. Here are some numerical illustrations.

 

Conditions	I	II	III	IV	V
	PM = ½	PM = 3/8	PM = 3/4	PM = ½	PM = ½
	PB = ½	PB = ½	PB = ½	PB = 1/3	PB = 2/3

 

S1	3/8	0.28	0.56	0.42	0.33
S2	1/8	0.16	0.06	0.08	0.17
S3	1/8	0.09	0.19	0.17	0.08
S4	3/8	0.47	0.19	0.33	0.42

 

Condition I is the condition of total (legitimate) confusion. There is symmetry with regard to bidding slam or staying in game. It matters not one iota, on average, whether one bids on or not. The gains are the same, the losses are the same.

Condition II represents the situation where half the field overbids to a contract with a poor chance of making. The worst possible result is got by not bidding the popular game and it makes (Situation 3). So to minimize the loss one bids the game the field favors even though the chances of making it are poor. To maximize the gain, one sensibly avoids a game that has a poor chance of success.

Condition III represents a situation where half of the field misses a very good slam, possibly stopping in 3NT. To maximize the gain, one should bid the slam, even though one gets the worst score if it happens to fail. To minimize the loss, one avoids the slam, but is likely to score poorly.

Condition IV represents the field’s obvious blind spot, avoiding a so-so minor suit game. The best score is got by bidding and making it, the worst by going down. It doesn’t cost much to stay in a partial (and there may be a bonus when some go down in 3NT).

Condition V is the reverse situation in which the field is eager to bid a 50-50 game. In this case the highest expectation is for bidding against the field, staying in a partial, and making it, while the lowest expectation is for not bidding the popular contract that happens to make.

Minimizing the Effect of Being Wrong

Given a choice of mistakes, under what conditions is it better to bid a contract that fails (Situation 2) than to not bid a contract that succeeds (Situation 3)? The expected advantage for overbidding versus underbidding is ½ (PB – PM). If one is making a mistake, it is better if one has lots of company. The worst outcome possible is got by bidding an unlucky but unpopular slam, as represented by Condition III.

When in doubt one may decide to bid on the basis of what the field may be doing. That will minimize the potential loss, but it may not get one to best contract, as illustrated by Conditions IV and V. Under these conditions the best and the worst scores are got by bidding against the tendency of the field. Under Condition IV, PM + PB < 1, whereas under Condition V, PM + PB >1. In the next blog we shall show this is an important distinction in a team game.

Maximizing the Effect of Being Right

There are those of us who prefer to back their own judgment rather than follow a field that is too often flawed in its approach. The question for us is this: which correct decision is likely produce the highest score on average: bidding a makeable slam (Situation 1) or by staying out of a slam that doesn’t make (Situation 4)? In what way does the decision depend on the probability of slam being bid by the field? The difference in the expected scores for staying safely in game or bidding successfully to slam (S1 – S4) is given by the following expression:

 

Expected Gain	= ½ (3xPM – 1 – PB)
	= PM – ½ when PM equals PB

 

The expected gain for bidding on is a positive quantity when 3xPM is greater than 1 + PB. (Condition IV) This tells us that one may justify bidding a slam that is less than 50% successful if the field tends to avoid it. For example, if 2/3 of the field will avoid a particular slam, one needs only a 45% chance of making it to gain more by bidding slam than by stopping in game. This is swinging bridge, as it risks a large loss albeit in a good cause. On the other hand, if the opponents are more likely than not to bid the higher scoring contract, one need a better than 50% chance of success in order to justify staying low as well. Thus, if 2/3 of the field is expected to bid a slam, one maximizes the expected gain by avoiding the slam with less than 5 chances in 9 of making (PM=56%). Here is an example of this phenomena.

 

Beating the Field

			Torturous	Exchange
♠ KQ85	♠ A93		2 NT	3 ♣
♥ AQ3	♥ KJ5		3 ♠	4 ♠
♦ K543	♦ A107		4 ♥	4 NT (invites)
♣ AK	♣ J432		5 ♦	5 ♥
21 HCP	13 HCP		5 NT	Pass

 

Most responders would not hesitate to bid 6NT immediately on the basis a faith in HCP totals and a near certainty that the field will do the same. Normally with 33 HCP between the 2 hands, 6NT would be a favorite to make, but there are some abnormalities to be noted on a double dummy basis. Most importantly with regard to probabilities, the distribution of HCPs is not consistent with the length of the suits held. One would expect the opener to have better diamonds and worse clubs. The ♣ AK doubleton is a bad feature. The responder’s longest suit is headed by the ♣ J, so there is little prospect for establishing a long trick in the suit. The ♥ J is wasted, the 1 HCP it represents would be much better placed if added to the ♣ J to make it the ♣ Q. In addition the division of sides is an unpromising 7-7-6-6 rather than the more usual 8-7-6-5.

The probability of making slam is less than the probability of bidding it, so the hands represent a situation where one might profit hugely by going against the field. There are 10 tricks off the top, so some luck is needed to move the total up to 12. If spades split 3-3 one can get a functioning squeeze going in the minors when the player with the ♣ Q holds 4 diamonds. Declarer might get lucky if a defender without the ♣ Q discards a diamond from 9862.

A well-informed player has good reasons for downgrading, however, the knowledge needed to make a fine judgment requires an informative sequence of several revealing bids that convey doubt. Given a choice either partner might decide not to bid the slam and the pair will probably score very well indeed by merely taking their 10 tricks off the top. However, most players would not bid in this way – either they consider it dangerous, or they like to take charge, or they are not capable. If the slam just happens to make, a pair that stays out of it will score poorly. Maybe unlucky, but one mustn’t complain.

Bidding Maps

It is helpful to visualize the decision making process by way of a bidding map with PM and PB as the co-ordinates. The simplest version is a flat map that displays the decision to bid on or not without indication of the elevations involved. Here are the flat maps for matchpoint decision as to whether to bid on (Yes) or not (No) or Flip a Coin ( –).

 

Minimize Loss

PM/PB	.45	.50	.56	.60
.45	—	Yes	Yes	Yes
.50	No	No	—	Yes
.56	No	No	—	Yes
.60	No	No	No	—

 

Maximize Gain

PM/PB	.45	.50	.56	.60
.45	No	No	No	No
.50	Yes	—	No	No
.56	Yes	Yes	Yes	Yes
.60	Yes	Yes	Yes	Yes

 

The minimum loss map merely reflects the symmetrical rule, ‘bid on when PB>PM, don’t when PM>PB’ This yields a bad decision when poor games are being bid throughout the room, but many players go with the field in this situation, compounding the error. This is a situation where a brave judgment to pass based on poor trump quality can result in a good score.

The maximize gain map reflects the rule, ‘bid on if 3xPM is greater than 1+PB’. The maximize map yields the better approach as it conforms more closely to the optimal rule of ‘bid on if PM>½’. When PM equals 0.5 (maximum uncertainty as to whether the higher contract makes) and PB is greater than 0.5 (a suspicion most will bid it), one maximizes the gain by stopping short and minimizes the loss by bidding on.

When I hear a player reflect, ‘I thought we might make slam (or game or NT), but there won’t be many in it’, I think, ‘there goes a player so good he can afford to pass up golden opportunities.’ At the end of the game I find he usually scores above average, yes, but he is not near the top. Such a player does better at teams. In the next blog we shall investigate why this is so.

Afterword: Hugh Kelsey’s Advice

In his book Match-Point Bridge (1970) Hugh Kelsey presented the conservative view with regard to bidding close games. He wrote, ‘the game should normally be bid, for most players are healthily aggressive in their bidding habits and a fifty-fifty game will be bid more often than not. If there appears to be any reasonable chance of success you should wish to be in game.’ His advice is equivalent to minimizing the loss when one makes the wrong decision. He notes, ‘The good player does not like to gamble on close bidding decisions…. He therefore chooses to play down the middle on such boards, relying on superior judgement on the competitive hands to pull his score above average.’

This advice seems to me to be poorly argued. There is a limit to the accuracy one can obtain from a bidding sequence, true, but that accuracy may be lessened by interference, so there is more scope for error, not less. We see that all the time. Bidding space is reduced. Superior judgement depends partly on the information made available by unreliable opponents, so one can be mislead, whereas bidding in an uncontested auction depends on the information provided by one’s usually reliable partner who has your best interests at heart.

After the opening lead is made, declarer has a firmer grasp on the probabilities involved. He may feel more in command of the opportunities presented. That is an essential psychological factor. No one argues that declarer should do what the poorer players are doing, finesse at every opportunity, rather one concentrates one’s efforts on besting the average players. One must be prepared to take advantage of a particular lie of the cards that provides an overtrick, even if there is some risk involved due to the fact that most players will not make the same play. For example, a partial elimination and endplay is a common way of achieving this, but one must rely on fairly even split in the side suits, otherwise one may suffer an untimely defensive ruff resulting in a poor score. The difficulty with adopting the same positive attitude during the auction is that the uncertainty when bidding is greater than the uncertainty when one sees 26 cards. Nonetheless there may be more clear mistakes made in play than in bidding.

The argument that one should bid with the field does not even assert that the risk outweighs the gain, for as we have shown, there may be great profit obtained from staying out of popular unmakeable contracts. No, the advice reflects what Pliny the Elder observed 2000 years ago, that the best plan is to rely on the mistakes of others. If one is a good player in a bad field, that cynical approach can be successful, but remember what happened to Rome – it fell to the barbarians in 476 AD.

Afterword 2: Matchpoints as Democracy

Matchpoint scoring is a democratic process. On every hand each pair has a vote on the best contract. Rarely is there complete agreement, but usually a consensus is achieved. The plebiscite may be worded as, ‘does such-and-such a contract have a better than 50% chance of success?’ A player votes ‘yes’ by bidding it. Insufficient punishments are handed those who made a mistake when they joined the majority, whereas the system comes down hard on dissenters who got themselves in trouble even though their motivation was sound and they would be correct more often than not. Their rewards are posthumous. In theory if a self-interested majority votes for a particular contract, it is most likely to be a sound one, but in practice a large number of voters may not have the foggiest idea, so they just goes along with what they think others think. The average player feels safest in the middle. He reacts conventionally the way he has been taught. Often he is left to ask himself, ‘what went wrong?’ That happens a lot, not just in politics.

The majority know that much of what they are being told comes under the euphemistic heading of ‘Wishful Thinking’, but they continue to have faith that vast improvements can accrue from small adjustments here and there. They tend to blame themselves for bad results, not the system, accept punishment for their supposed misdeeds, and feel that next time they’ll get it right. But the next time around another problem that requires fixing crops up unexpectedly somewhere else. They listen to those who propose patches that will make things better. Filling out a convention card is like filling out an income tax form – legal advice is required to take full advantage of the loopholes offered. Most react unfavorably to lying, because they themselves have never been taught how to do it properly. The majority abhors fundamental change and embrace expediency. They sacrifice the future to the past. They keep coming back for more, because it’s a great game.