Safety in 1NT Interference

• The most likely contract landed in is made.  For the discussions here, this will be 2 or 2.
• The contract goes down 1.  Technically, this also requires the opponents to have a better part score available than setting you for 100, but after 1NT, the only part score they can make for less than 100 is 1NT making exactly. Further analysis could be done to take this into account, but then we’d also need to consider that if the opps are only making 1NT, there is a decent chance you won’t be doubled, and then your -50 beats the -90.
• The contract goes down 2, and either the opponents have a game they are likely to find, or they are not likely to double. For now, we’ll skew the safety coefficient downward by assuming optimal doubling, and by assuming the opponents do not find games with less than 24 HCP between them.
• You go down 3, and the opponents have a game (this is included if we are assuming favorable vulnerability).

For each hand, either it is safe or not (by the above definition) to make the overcall, and we wish to estimate σ, the probability that the hand is safe to overcall, knowing that the hands form a binomial distribution with parameter σ.  We use the Wilson Score Confidence Interval:

Here, σ-hat is the MLE of the distribution, the percentage of hands for which the above conditions hold during the simulation.  The number n is the total number of hands simulated, the z-values are quantiles of the standard normal distribution, and α is the desired level of significance.  For large n, the center of the distribution is close enough to the MLE, so we worry only about the radius, which is the second term above.  This can be bounded by replacing σ with 1/2.  For α =0.01, we can solve for the radius being at most 0.01, and we obtain about n = 17,000.  Thus, if we sample 17,000 hands, we can be 99% confident that the value we obtain for σ is within 0.01 of the true value.  We will round off our estimates of σ to 3 decimal places, and it should be considered that all given values are within ±0.01 of the true value (with 99% confidence). We also attempt to approximate the average number of IMPs gained (or lost) by interfering, assuming the opps allow you to play. We will assume the following:

• If the contract is made, it depends on whether or not the opps make a part score (or a game): If the opps cannot make a part score, then NV there is a 2 IMP gain and V is a push (110 vs. 100). If the opps make a part score, then at either vulnerability there is a 5 or 6 IMP gain. For the purpose of this estimate, we will assume the opps can make a part score but not a game.

• If the contract is down 1, there is a gain of 0–2 IMPS if the opps make a part score, about 8 IMPS if they make a game (NV) and 11 IMPS if they make a game (V). We will assume a gain of 1 IMP for the part-score difference.

• If the contract is down 2, there is a loss of 4 or 5 IMPS if the opps only make a part score. There is a gain of about 3 or 4 IMPS if the opps make a game (NV) and a gain of 7 or 8 if the opps make a game (V).

• If the contract is down 3, there is a loss of about 9 IMPS if the opps only make a part score. There is a loss of about 3 IMPS if the opps make a game (NV) and a gain of about 3 or 4 IMPS if the opps make a game (V).

• If the contract is down 4, we assume the opps can make a game, in which case we lose about 9 IMPS (NV) and about 5 IMPS (V).

For the following simulation, we tested interference with hands with one or both majors.  The lengths of the 2-suited hands was either 4/4 or 5/4, single suited hands were 5 or 6 cards long, and point ranges were either 6-7 or 8⁺.

Summary of Safety Results (Assuming we are NV)

So, from this information, we see (not surprisingly) that it is always a good idea to interfere with a 6-card suit at at least 6 points.  5-card single suits and 6-7 points are very dangerous when opps are NV, and lose IMPS either way (though this requires optimal doubling).  At matchpoints with Vul opponents, however, it is safe 52.5% of the time.

For 2-suited hands, 5/4 hands are always safe to overcall.  4/4 hands are safe with 8⁺ points and any vulnerability by the opponents.  With only 6-7 points, however, it is only safe (but it is safe) when opps are Vul.

We should note that this is an imperfect measure of safety. For each shape, 1 − σ could represent the “danger” of a bid. This is the percentage of the time that making the overcall will present the opponents with an opportunity to double you for more than they could get on their own. There is no guarantee, however, that the rest of the time they’ll let you play in your “better” spot. As an extreme, if you play against perfect opponents, they will double you only when it is in their best interests to do so, otherwise they will just bid on and find their game. In this case, the overcalls are mostly bad because, except in the case that they can’t make their contract at a higher level than yours, they will just play their contract if it’s better. So the overcall will either make no difference or it will hurt you. Of course, most opponents aren’t perfect, but the effectiveness of the overcall to disrupt the opponents, and even their ability to double should be taken into account. Unfortunately, there is no good measure of this. Also, it should go without saying that determining which of these shapes to overcall is up to partnership agreement, and both partners should be sure to be on the same page.