Underbidding in preferans: does it deserve a penalty?

HardReading time: ~6 minutes

At many tables, underbidding (perezaklad) is punished almost like breaking the rules: a player declares six while holding a sure seven, eight or even nine tricks. The idea is simple: the defenders fall short of their required tricks and get a penalty to the heap. Let us see whether underbidding actually pays, and whether it makes sense to penalize it at all.

What underbidding is

Underbidding means declaring a lower contract with a hand that is sure to take more: for example, declaring six when you can take seven, eight or even nine tricks. Formally this is allowed: tournament rules do not forbid declaring less than your cards allow. It is a normal move in the trade and may simply come from caution. But underbidding, unlike plain caution, has a different aim: the player lowers the contract not because they doubt their seven, but on purpose, so that the defenders whist and get penalized for falling short of their required tricks.

Tournament rules and underbidding

The official codes of preferans rules have no separate clause on underbidding. Professional tournament rules do not penalize a low contract. A penalty for underbidding, where it exists at all, is an agreement within a particular group, never a rule.

SmartPref conventions have no penalty for underbidding: we follow tournament rules.

The numbers: the real contract against underbidding

Take a deal where player A is sure to take 7, 8 or 9 tricks, and defenders B and C whist. The numbers are worked out for all three conventions, Sochi, Rostov and Leningrad; a separate article covers how they differ. For simplicity, the defenders' entries are shown as a total.

Sochi

Seven, Sochi

Contract	Score sheet	A's result
7	A: 4 to the pool B+C: 4×3=12 whists on A no undertricks, nothing to the heap	pool: 4×10 - (4×10)/3 = 26.67 whists: -12 total: 14.67
6	A: 2 to the pool B+C: 2×3=6 whists on A +2 to the heap of the defender who fell short	pool: 2×10 - (2×10)/3 = 13.33 whists: -6 heap: 10×2/3 = 6.67 total: 14.00

Eight, Sochi

Contract	Score sheet	A's result
8	A: 6 to the pool B+C: 6×2=12 whists on A no undertricks, nothing to the heap	pool: 6×10 - (6×10)/3 = 40.00 whists: -12 total: 28.00
7	A: 4 to the pool B+C: 4×2=8 whists on A no undertricks, nothing to the heap	pool: 4×10 - (4×10)/3 = 26.67 whists: -8 total: 18.67
6	A: 2 to the pool B+C: 2×2=4 whists on A +4 to the heap of the defender who fell short	pool: 2×10 - (2×10)/3 = 13.33 whists: -4 heap: 10×4/3 = 13.33 total: 22.67

Declaring 7 (18.67) is the worst of the three options here, worse even than the full underbid of 6 (22.67). On a seven the defenders take exactly their required tricks, so nothing goes to the heap, yet the whists are still paid at the higher seven rate. Undertricks, and with them the heap, only appear on a contract of 6.

Nine, Sochi

Contract	Score sheet	A's result
9	A: 8 to the pool B+C: 8×1=8 whists on A no undertricks, nothing to the heap	pool: 8×10 - (8×10)/3 = 53.33 whists: -8 total: 45.33
8	A: 6 to the pool B+C: 6×1=6 whists on A no undertricks, nothing to the heap	pool: 6×10 - (6×10)/3 = 40.00 whists: -6 total: 34.00
7	A: 4 to the pool B+C: 4×1=4 whists on A +4 to the heap of the defender who fell short	pool: 4×10 - (4×10)/3 = 26.67 whists: -4 heap: 10×4/3 = 13.33 total: 36.00
6	A: 2 to the pool B+C: 2×1=2 whists on A +6 to the heap of the defender who fell short	pool: 2×10 - (2×10)/3 = 13.33 whists: -2 heap: 10×6/3 = 20.00 total: 31.33

Declaring 7 (36.00) gives A more than declaring 8 (34.00). Right at the line between 7 and 8, the defence's requirement doubles in one jump (from 1 trick to 2), so a contract of 7 adds heap points that a contract of 8 does not, and that heap outweighs the difference in the pool. The real contract of nine (45.33) is still the best choice of all.

Rostov

Seven, Rostov

Contract	Score sheet	A's result
7	A: 4 to the pool B+C: 4×3=12 whists on A no undertricks, nothing to the heap	pool: 4×10 - (4×10)/3 = 26.67 whists: -12 total: 14.67
6	A: 2 to the pool B+C: 2×3=6 whists on A +1 to the heap of the defender who fell short (in Rostov the whist is "half-responsible", so the penalty is halved)	pool: 2×10 - (2×10)/3 = 13.33 whists: -6 heap: 10×1/3 = 3.33 total: 10.67

Eight, Rostov

Contract	Score sheet	A's result
8	A: 6 to the pool B+C: 6×2=12 whists on A no undertricks, nothing to the heap	pool: 6×10 - (6×10)/3 = 40.00 whists: -12 total: 28.00
7	A: 4 to the pool B+C: 4×2=8 whists on A no undertricks, nothing to the heap	pool: 4×10 - (4×10)/3 = 26.67 whists: -8 total: 18.67
6	A: 2 to the pool B+C: 2×2=4 whists on A +2 to the heap of the defender who fell short (half penalty)	pool: 2×10 - (2×10)/3 = 13.33 whists: -4 heap: 10×2/3 = 6.67 total: 16.00

Nine, Rostov

Contract	Score sheet	A's result
9	A: 8 to the pool B+C: 8×1=8 whists on A no undertricks, nothing to the heap	pool: 8×10 - (8×10)/3 = 53.33 whists: -8 total: 45.33
8	A: 6 to the pool B+C: 6×1=6 whists on A no undertricks, nothing to the heap	pool: 6×10 - (6×10)/3 = 40.00 whists: -6 total: 34.00
7	A: 4 to the pool B+C: 4×1=4 whists on A +2 to the heap of the defender who fell short (half penalty)	pool: 4×10 - (4×10)/3 = 26.67 whists: -4 heap: 10×2/3 = 6.67 total: 29.33
6	A: 2 to the pool B+C: 2×1=2 whists on A +3 to the heap of the defender who fell short (half penalty)	pool: 2×10 - (2×10)/3 = 13.33 whists: -2 heap: 10×3/3 = 10.00 total: 21.33

In Rostov the halved heap penalty (the "half-responsible" whist) does not have the same effect as in Sochi: declaring 7 (29.33) is below declaring 8 (34.00), and the result keeps falling steadily, with no jump up.

Leningrad

Seven, Leningrad

Contract	Score sheet	A's result
7	A: 4 to the pool B+C: 8×3=24 whists on A (twice as expensive: double rate) no undertricks, nothing to the heap	pool: 4×20 - (4×20)/3 = 53.33 whists: -24 total: 29.33
6	A: 2 to the pool B+C: 4×3=12 whists on A +1 to the heap of the defender who fell short (half penalty)	pool: 2×20 - (2×20)/3 = 26.67 whists: -12 heap: 10×1/3 = 3.33 total: 18.00

Eight, Leningrad

Contract	Score sheet	A's result
8	A: 6 to the pool B+C: 12×2=24 whists on A no undertricks, nothing to the heap	pool: 6×20 - (6×20)/3 = 80.00 whists: -24 total: 56.00
7	A: 4 to the pool B+C: 8×2=16 whists on A no undertricks, nothing to the heap	pool: 4×20 - (4×20)/3 = 53.33 whists: -16 total: 37.33
6	A: 2 to the pool B+C: 4×2=8 whists on A +2 to the heap of the defender who fell short (half penalty)	pool: 2×20 - (2×20)/3 = 26.67 whists: -8 heap: 10×2/3 = 6.67 total: 25.33

Nine, Leningrad

Contract	Score sheet	A's result
9	A: 8 to the pool B+C: 16×1=16 whists on A no undertricks, nothing to the heap	pool: 8×20 - (8×20)/3 = 106.67 whists: -16 total: 90.67
8	A: 6 to the pool B+C: 12×1=12 whists on A no undertricks, nothing to the heap	pool: 6×20 - (6×20)/3 = 80.00 whists: -12 total: 68.00
7	A: 4 to the pool B+C: 8×1=8 whists on A +2 to the heap of the defender who fell short (half penalty)	pool: 4×20 - (4×20)/3 = 53.33 whists: -8 heap: 10×2/3 = 6.67 total: 52.00
6	A: 2 to the pool B+C: 4×1=4 whists on A +3 to the heap of the defender who fell short (half penalty)	pool: 2×20 - (2×20)/3 = 26.67 whists: -4 heap: 10×3/3 = 10.00 total: 32.67

Across all three conventions, three hand strengths and every intermediate contract between the real one and six (27 combinations), underbidding to 6 never beats the real contract. The gap varies: sometimes, as with a sure seven in Sochi, it is almost zero (14.67 against 14.00, the declarer loses just 0.67 whists), and sometimes, as with a nine in Leningrad, it is huge (90.67 against 32.67). Intermediate contracts (say, seven instead of a real nine) do not always fall in order either: sometimes a particular in-between step turns out worse than both the real contract and the full underbid at once, as the notes above show.

All these numbers assume the defenders whist, since that is what underbidding "for undertricks" counts on. If both defenders simply pass instead, the declarer gets only the value of the game in the pool, and the gap to the real contract grows even wider, because the declarer gets no share of the defenders' heap, which is usually more than the whists written against them.

Sochi uses the "Stalingrad" convention: a contract of six spades obliges the opponents to whist. If the underbid happens to be six spades, the defenders can no longer pass, but even a forced whist does not put underbidding ahead: the difference stays negative.

Our conclusion

The difference in whists between underbidding and the real contract always favours the real contract. So it is better to declare what your hand is worth. Still, in some cases, for example when you do not want the current contract to end the game, underbidding can be a real alternative.

Frequently asked questions

Is underbidding against the rules?

No. No code of preferans rules, the tournament code included, forbids declaring less than you could. A penalty for underbidding, where there is one, is an agreement within a particular group, not a generally accepted rule.

Does SmartPref penalize underbidding?

No, our rules and conventions follow the tournament code.

Should I declare less when I could declare more?

By the numbers, no, although right at the edge the difference almost disappears. On the same deal, with any hand from a seven to a nine, in all three conventions and whatever the defenders do (pass or whist), underbidding brings fewer whists than the real contract. It comes closest to breaking even in Sochi on a sure seven, when a defender whists and falls short: there underbidding loses just 0.67 whists. With stronger hands and in the other conventions the gap grows quickly.

However, if your contract would end the game and you plan to win back your losses, it makes sense to declare less than you need to finish the game.

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