If you knew that your partner would never bid again, but your opponents were free to keep bidding if they chose, how would you decide what action to take?

I got interested in this question because it was a tractable special case of a more general question about optimal bidding systems., But unlike most other simplified bidding problems, this one actually happens in real life. If you open out of turn when your partner or your left-hand opponent is dealer, your partner "must pass whenever it is his turn to call" (Law 31B) for the rest of the auction. The director will explain to you that your partner is barred, and that you are allowed to make whatever call you wish in an effort to recover and reach your side's best contract -- you are not expected to stick to your normal opening bid. People who open 1NT out of turn, for instance, commonly choose to bid 3NT instead of 1NT, gambling that their silenced partner has the 8 to 10 points required to make game a good prospect.

But most of us only face this decision a few times in our lives. There is no body of literature on what to bid when your partner is barred. We are usually stabbing blindly in the dark when we guess whether our hands are good enough to blast game (or even slam) without any input from our partners.

This is an ideal problem to turn over to a computer: given your hand, it can deal a large sample of possible hands for your partner and opponents, calculate what contracts are makeable double-dummy on each deal, and tell you what action is most likely to be best.

Rather than going through this exercise 635 billion times for every possible hand, I went through it 15,152 times — for every possible combination of hand pattern and point count — and then took a sample of at least 100 random deals where South had the desired shape and strength and the other three hands were completely random.

How badly do you hurt your score by opening out of turn?

Four IMPs.

The fine print: if you optimally select one call with no input from your partner, and your opponents then bid and play double-dummy (always bidding their best contract if they still can, always doubling you if your contract fails, always passing it out if you bid a makeable but inferior contract), then you will lose an average of 4.03 IMPs per board, compared to double-dummy par, if both sides are vulnerable.

Following an optimal strategy, which bids are the most and least likely?

1♠ (8.1%), 1NT (6.5%), 2♥ (4.7%), 1♥ (4.6%), and 2♠ (4.4%). The most common game bid is 4♠ (3.0%), followed by 3NT and 4♥ at 2.7%. The only calls you should never make are 2NT and 5NT.

This should accord with your common sense: all else being equal, spades is the most desirable suit to bid, outranking all the others (and coming ahead of notrump, since notrump contracts suffer expensive sets opposite too-weak partners).

The full list:

Contract	Frequency	Contract	Frequency	Contract	Frequency
Pass	46.7%	3NT	2.7%	6♣	.05%
1♠	8.0%	4♥	2.7%	6♥	.04%
1NT	6.5%	3♣	1.3%	5♥	.03%
2♥	4.7%	3♦	1.2%	5♠	.013%
1♥	4.6%	5♦	.29%	6NT	.003%
2♠	4.4%	5♣	.28%	7♠	.0015%
2♦	3.9%	3♥	.19%	7♥	.0011%
1♣	3.1%	3♠	.13%	7♣	.0007%
1♦	3.1%	4♣	.07%	7♦	.0007%
4♠	3.0%	4♦	.06%	7NT	.0004%
2♣	2.8%	6♦	.05%	4NT	.00008%

OK, what do I bid on which hands?

The following is going to be a bit oversimplified from the full spreadsheet, to make it a bit more manageable.

Balanced Hands: Generally, Pass with 0-12; 1NT with 13-17; 3NT with 18-27; 6NT with 28-30; 7NT with 31+. The exact cutoff between 1NT and 3NT, and between 3NT and 6NT, varies by a point or so according to the exact distribution. In the 13-15 point range, it is better to bid 1 of a major in some cases, especially if you have 4 spades. (Balanced hands with 5 of a major usually bid the major unless strong enough for game; balanced hands with 5 of a minor usually bid notrump.)

Unbalanced hands: Usually bid your longest suit. With two suits of the same length bid the higher-ranking (except that with both minors, there is very little difference between them; bidding the stronger is probably best.) Exactly where the HCP cutoffs are depends both on which suit is your long suit, and on how extreme your distribution in the side suits is.

Compare, for instance: with 5-3-3-2, bid 1♠ with 9-15; 2♠ with 16-18, 3NT (not 4♠) with 19-25; but with 5-4-3-1. bid 1♠ with 8-13, 2♠ with 14-16, 4♠ with 17-24, 6♠ with 25-28, 7♠ with 29-32; 7NT with 33.

As the distribution becomes more extreme, bid game more readily: with 6-1-3-3, pass or bid 1♠ up to 5; bid 2♠ with 6-12; bid 4♠ with 13-23; bid 6♠ with 24-28. With 7-3-2-1, 1♠ on 0-2, 2♠ on 3-8, 3♠ on 9, 4♠ on 10-21, 6♠ on 22-27. With 8-2-2-1, 4♠ from 0-20, 6♠ on 21-26.

The exceptions to bidding your longest suit involve two- and three-suited hands where you bid a shorter major in preference to a longer minor. Typical is 5-1-6-1: pass 0-4, bid 2♦ 5-11 and 6♦ 21-23, but 4♠ with 12-20.

These two- and three-suited hands also involve some of the worst losses: with the huge 5440 and 4441 hands you know you belong in game or slam but you are guessing blindly which game. For instance, the best action with a 0-4-5-4 24-count is to blast 6♦ but at an average cost of 10 IMPs per board.

As a practical matter, you might choose to blast only to game on such a hand, hoping that more than one game makes even if only one slam does, and considering it a "success" to reach a making game when a slam is available, rather than betting everything on picking the right slam. The chosen metric -- average IMPs lost vs. double-dummy par -- makes the gamble-on-a-slam strategy appealing, since playing game when a slam can be made is -13 while failing in slam is only -16. In real life, if the other table is in game, it may be a matter of "+13 or -13" to bid slam, rather than risking 3 to gain 13.

If you want to see what the optimal action on all 15,152 shape/HCP combinations was, based on my sample, this ZIPped comma-delimited text file contains them all.

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