Expectations at IMPs and Matchpoints

Most players prefer to play in team matches against their peers, bad players because it’s easier to win, good players because it’s harder. An important difference from the matchpoint game is that the opponents one has to beat are not the diverse crowd situated throughout the room, but are the 2 pairs one’s team is facing directly. You must play well to win against a team of 4 good players. A large field makes for a fair game at matchpoints as the distribution of the scores on any given hand tends to reflect normal conditions. In a small field unusual results have more of an effect and there is more need to swing for top scores against the poorer pairs in order to win.

On many hands at teams the gains for bidding a higher scoring contract and equally balanced by the losses one encounters if the contract fails. On such hands the gain-loss ratio reflects the conditions at matchpoints, the difference being that at teams there is more at stake when bidding games. We shall study this important category in detail, the results being applicable to either matchpoint or team play.

At teams one is facing opponents whose tendencies are well known if not entirely predictable. It makes sense to adapt to some degree to the known quality of the team one is facing. Overall, adopting the tendencies of the opponents is equivalent to playing for a tie. The swing hands are those where there is maximum uncertainty whether or not a contract will make and/or whether or not the opponents will bid it. Traditionally the number of tie boards was considered an indication of a well-played match, but recently one finds players attempting frequently to increase uncertainty, thereby creating swings one way or the other, much to the annoyance of idealists who would prefer matches to won through accurate bidding and double dummy card play.

In our previous blog we considered the effect of the field on matchpoint strategy. Here we pursue the theory for team play where the potential gains and losses are not the same for every hand. There are 4 possible situations: 1) you bid a higher scoring contract and it makes; 2) you bid a higher scoring contract and it fails; 3) you stay in a lower scoring contract and a higher scoring contract makes, and 4) you stay in the lower scoring contract and the higher scoring contract fails. The probability of making the higher score is PM; the probability of the opponent bidding it is PB.

As previously we consider hands with just 2 outcomes. If the opponent is in the same contract, we assume there is a tie, so the score on that board is 0. If the opponent is in the alternate contract, either you gain an amount G or you lose and amount L, where G and L are in general different numbers of IMPs. Under the 4 situations listed above the expected scores, S1 through S4, are as follows:

S1	G x PM x (1 – PB)	S2	-L x (1 – PM) x (1 – PB)
S3	-G x PM x PB	S4	L x PB x (1 – PM)

The advantage to bidding the higher scoring contract that provides gain G is:

S1 + S2 – (S3 + S4) = PM x (G + L) – L

This is the gain factor. The optimum strategy is to bid the higher scoring contract if this quantity is positive and not to bid it if the quantity is negative. Let the potential loss, L, be represented by kG, where k is the ratio of L to G. One bids on if:

PM > k/(k + 1)

Under normal circumstances k lies within the limits 0.5<2. If the gain and the loss are equal, k=1 and the optimal condition becomes PM > ½, as with matchpoint scoring. If the potential loss is twice the potential gain optimally one bids on only if PM> 2/3, as in the case of bidding a grand slam at rubber bridge. At teams, the k associated with a vulnerable game is 0.6, so the game should be bid if PM>3/8.

At IMP scoring the gain factor changes from board to board according to the number of IMPs available for making the correct decision. If one bids the higher contract and the opponents don’t, the expected gain is (1 – PB) times the gain factor. If one bids the lower contract and they bid the higher one, the expected gain is PB times the gain factor. Of course, the gain factor turns into a loss factor if one makes the wrong choice and the opponents the correct one.

Minimizing the Loss, Maximizing the Gain

One may aim to minimize the loss when making the wrong bidding decision on a board, in which case one should avoid bidding the higher scoring contract under the following condition:

| S2 | – | S3 | > 0, such that

(1 – PM) L > PB x [ L – (G – L) x PM ], which can be rewritten as

1 > (PB + PM) – r x PM x PB, where r equals (k-1)/k.

Note the symmetry with regard to PM and PB which act interchangeably. If L equals G, r is zero, in which case one should bid the higher contract if the probability of making it plus the probability of bidding it is greater than 1. This is normal for uncontested auctions.

The gain for bidding the higher scoring contract and making it versus the gain got by bidding the lower scoring contract is given by the following expression:

S1 – S4 = G x PM – L x PB + (L – G) x PM x PB.

To maximize the gain, bid on if PM > k x PB – (k – 1) x PB x PM.

If L equals G, bid on if PM > PB, that is, if the probability of making the higher contract is greater than the probability that the opponents will bid it, even if PM is less than ½, which is contrary to the optimal strategy. To minimize the loss, don’t bid on if the probability of the opponents’ bidding the higher contract is greater than the probability it will fail, that is if PB > 1 – PM. To maximize gain and to minimize loss are not incompatible aims at IMP scoring, and a ‘comfort zone’ achieving both ends is possible.

Comfort Zone On boards where the potential loss and the potential gain are equal, bid the higher scoring if

PM + PB > 1 and PM > PB

The Max-Min Diagrams

In a previous blog we introduced text maps as shown below. ‘Yes’ indicates one should bid higher to achieve the aim, ‘No’, that one should not, and the dashes signify a toss up.

Maximize the Gain

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

Minimizing the Loss

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

The boxes that contain a ‘Yes’ in both diagrams are representative of the comfort zone.

The ‘maybe’ boxes clearly lie along diagonals that separate the Yeses from the Nos.

The conditions for maximizing and minimizing can be represented graphically by lines in a more detailed PM/PB diagram as sketched below. When G equals L the maximize line runs diagonally from the upper left corner to the lower right corner. The shaded area to the left of this line represents conditions in which the gain is maximized by bidding higher. The minimize line is a diagonal from the upper right corner to the lower left, the area to the right representing conditions in which the loss is minimized by bidding the higher contract. The diagonals cross at PM=PB=½, the point of maximum uncertainty. When the gain is not equal to the loss the point of intersection is elsewhere, where there is less uncertainty, as will be discussed in a later blog.

A decision to bid the higher contract can be associated by a point in the diagram that reflects the de-facto probabilities. If the point lies within the comfort zone, one has acted both to maximize the gain and to minimize the loss. If the point lies within the ‘no-no’ zone, one has chosen poorly on both counts. If the point lies in one of the other 2 zones, one has acted either to maximize the gain (on the left) or to minimize the loss (on the right). Another viewpoint is that any mistakes that are made are due to a miscalculation of PM, due to a lack of information on how the cards lie, or a poor prediction of the probable action of the opponents, PB. In a double dummy analysis PM is entirely dependent of the lie of the cards, but in practice the defence may benefit from any information received during the auction.

The Full Picture

Below is shown a map of the decision zones over a range of probabilities when the potential gain G equals the potential loss L, the situation that occurs most frequently with constructive bidding both at matchpoints and IMPs. The numbers given are the expected scores times 1000 for the higher contract on the left and the lower on the right. The aggregate for each contract is given below the line. So we have this pattern displayed for each pair of PM and PB:

S1	S4	(expected gain)
S2	S3	(expected loss)
S1 + S2	S3 + S4	(aggregate)

To obtain the expected scores in matchpoints, divide by 2000, add ½, multiply by the number of opponents playing in the same direction. To obtain the expected IMP scores for a nonvulnerable game, divide by 1000 and multiply by 6; for a nonvulnerable slam, divide by 1000 and multiply by 11.

PM		PB =	2/5	PB =	4/9	PB =	1/2	PB =	5/9	PB =	3/5
	–
2/5		240	240	222	267	200	300	178	333	160	360
		-360	-160	-333	-178	-300	-200	-267	-222	-240	-240
		-120	80	-111	89	-100	100	-89	111	-80	80
	–
4/9		267	222	247	247	222	278	198	309	178	333
		-333	-178	-309	-198	-278	-222	-247	-247	-222	-267
		-67	44	-62	49	-56	56	-49	62	-44	67
	–
1/2		300	200	278	222	250	250	222	278	200	300
		-300	-200	-278	-222	-250	-250	-222	-278	-200	-300
		0	0	0	0	0	0	0	0	0	0
	–
5/9		333	178	309	198	278	222	247	247	222	267
		-267	-222	-247	-247	-222	-278	-198	-309	-178	-333
		67	-44	62	-49	56	-56	49	-62	44	-67
	–
3/5		360	160	333	178	300	200	267	222	240	240
		-240	-240	-222	-267	-200	-300	-178	-333	-160	-360
		120	-80	111	-89	100	-100	89	-111	80	-120

The small area in blue italics is the comfort zone wherein the criteria to maximize the gain and minimize the loss both require that the higher contract be bid. If one is to bid the higher contract outside the area, then one is deciding on the basis of either maximizing or minimizing in isolation. Bidding the higher contract if PM>½ results in a positive aggregate score, sometimes at the expense of a greater potential loss, as in the case of PM=5/9 and PB=2/5, a good contract unlikely to be chosen by the opponents. One gambles a loss of 45 to achieve a gain of 155, a good gamble. This is the situation where a superior system of slam bidding is likely to gain IMPs. Bidding the lower contract at PM=5/9 and PB=3/5, achieves a minimization of the loss by an average amount of 55, but at the average cost of 155, a bad gamble.

The area in the upper right represents a popular contract that is likely to fail. We often see the comment, ‘where there are 8 tricks there will be 9’, so it is very common that one rejects stopping in 2NT and moves on to 3NT even if the contract may prove to be a poor one. In theory one shouldn’t follow the field by bidding a poor contract, but one does so in order to minimize the potential loss. The condition of PM=4/9 is good enough for bidding a vulnerable game, but not a nonvulnerable game. When PB=5/9, PB + PM =1, and the potential losses are balanced between bidding or not bidding 3NT, but there is something to gain by staying in 2NT. This is an opportunity to swing some IMPs. In a recent 7-board Swiss Team match nothing much happened at my table, but I knew we had lost as on 3 of the boards the opponents had scored 120’s, all large losses for our side.

The Maximum Uncertainty Border

Many players live in doubt. The horizontal area representing PM=½ is of particular interest as it represents a border area where everyone is of necessity in doubt. The comfort zone lies below, the no-no zone, above. Along the line the aggregate scores are 0 whether one bids on or not. If one is inclined towards maximizing gain, one does better by staying in the lower contract when the opponents are not. If one is inclined towards minimizing the potential loss, one follows the inclination of the opposition to bid on. The area of maximum uncertainty as to the better decision is centered squarely at PB=½, when it doesn’t matter on average what one decides. A lot of mediocre players live in this neighborhood. They look to general rules to provide guidance in a difficult situation. Of course, on any particular hand decisions do matter, as that determines who come out on top, but on average it’s a toss-up. This is a fine characteristic for a game of chance as the best player does not have an advantage in a situation of maximum uncertainty, so the best player doesn’t always win. Under those circumstances the best policy is to lose gracefully.

In close decisions it is often boils down to a matter of hand evaluation, and there the good players have the advantage. Good players attempt to obtain an accurate estimate of PM and will act accordingly. They look at suit quality and the loser count, whereas mediocre players merely count up their points. Good players recognize texture. Based on the bidding, they anticipate the opening lead, and imagine the play from that point onward. Mediocre players may fear the killing lead and hope to escape it by ‘giving nothing away’ during the bidding, or may avoid the problem by staying low. It becomes a matter of personality. Sometimes, not often, the meek players win.