# Relay Structure

## General Notes

Relayer (referred to hereafter as "R") will normally make the next highest bid ("Step 1") to ask responder (hereafter referred to as "RR") to continue describing the hand. The exception to this is 3NT which is never a relay. Other non-relay bids by R below game are natural and may be invitational or forcing in context. 2NT is almost never a contract after a second relay.

## Responder's hand valuation after a strong opening.

After an opening Pass, responder normally gives a positive response with a hand with 19+ OP, unless that hand has substantially fewer than the number of controls normally held by a hand of that shape. R will eventually be able to ask for controls with A=2 & K=1 or A=3, K=2 & Q=1, of which there are at most 12 and 24 in a deck of cards with 40HCP. The ratio of AK and AKQ points to HCP is 0.3 and 0.6 respectively. When RR's shape is known, R will know the lengths of RR's two longest suits and can find RR's minimum HCP from the definition of OP. In other auctions RR will have a defined HCP minimum. RR's minimum number of AK or AKQ controls (RR's "base") can be deduced by multiplying the minimum HCP by the appropriate ratio (0.3 or 0.6) and always rounding down to the nearest integer. With a hand on or near the HCP minimum with a large amount of the HCP contributed by queens and jacks, RR may have fewer than the putative base number of controls. In such cases RR may exercise his judgement, however it would be exceptionally rare to describe a hand with a positive response that was two AKQ controls below base.
With a hand with 15-18 OP, or too few controls in a stronger hand, RR describes a semipositive hand. With a weaker hand yet, RR describes a full negative hand. Note that again, a hand that meets the semipositive criteria but has too few controls can be described as a full negative hand. The agreements in auctions after a negative response may be found here.

## Styles of relay structure

There exist at least two conflicting objectives in agreeing a relay method to play: a method should be both easily remembered and effective. Symmetric relays are the tool of choice to satisfy the former objective, however most symmetric relay structures have various technical defects. This is unsurprising - there is considerable difficulty in optimizing a method to suit just one of the objectives.
The most common defect of a symmetric relay structure occurs when RR is constrained to bid an unbid suit when holding that suit. RR must, of course, bid something, but it is usually best for RR's structure to avoid bidding suits they hold when those suits are unbid. On a related theme, it is also unsound for RR to bid notrumps when they are unbid and that is a probable final contract. The simplest remedy is to require RR to respond in a "transfer" scheme. This is a signficant improvement, at a moderate cost in memory effort. However, any transfer mechanism is necessarily tailored to the particular bids being made. Commonly, relay structures move up or down steps according to the nature of the auction, particularly after low-level interference. Thus bids that used to be effective transfers may become natural.
In order to improve efficacy in general, we define two conditions under which the "default" relay structure is modified "on-the-fly". Applying these conditions at the table is demanding; partnerships should feel free to ignore either or both for practical play.

### Conditions for relay structure modifications

• When the default relay structure would
• require RR to show an unshown major suit
• by using a bid in that suit,
• when that suit was previously unbid, then
the roles of the two major suits in the default relay structure are exchanged. Further, when the above conditions apply to a minor suit, the roles of the two minor suits are exchanged.
Note that this condition can apply to both balanced and unbalanced relay structures.
• When resolving the relative lengths of a two-suited hand, one of two cases occurs. In both cases a modification to the relay structure can occur.
1. When the form of the relay structure was  Step Meaning n 5+ in suit X, 4 in suit Y n+1 5+ in suit X, 5+ in suit Y n+2 4 in suit X, 5+ in suit Y, 0-1 in a third suit
and when step n happens to be a bid of suit X when that suit was previously unbid, then the roles of suits X and Y in the above structure are interchanged for resolving the shape of the hand.
2. When the form of the relay structure was  Step Meaning n 4+ in suit X, 5+ in suit Y n+1 5+ in suit X, 4 in suit Y n+2 some hand shape that is not two-suited with X and Y
and when step n happens to be a bid of suit Y when that suit was previously unbid, or when step n+1 happens to be a bid of suit X and that suit was previously unbid, then the roles of suits X and Y in the above structure are interchanged for resolving the shape of the hand.

Note that this condition could potentially be applied to any pair of suits given an appropriately level-adjusted auction.
Note the distinction drawn between when a suit is bid and when a suit is shown.
Where relevant, the first condition is applied before the second condition.

### Constrained relay structures

In auctions where shape constraints exist before relays begin (e.g. after opening bids that deny various suits), or in which shape constraints evolve after inteference, the structures will work in the following ways (impossible features will not be shown):
• When a suit has been shown, the following features are shown in ascending order: balanced (and/or three-suited short in a specific suit), an unshown major suit, any unshown minor suits, then single-suited.
• When a suit has not been shown, the following features are shown in ascending order: balanced, any unshown major suits, then any unshown minor suits.
In both cases, where three-suiters in the underlying structure are not shown "with" balanced hands, they will be shown normally, i.e. by showing both majors and then indicating three-suited status, or by a specific first relay response showing both minors and three-suited status.

## Positive shape-showing relays (default)

Note that this relay structure is not used explicitly anywhere. However it is the "archetype" upon which the other structures are based, and thus has illustrative value. After the strong pass, positive and semipositive shape-showing responses are described here. Relay structures used after limited suit openings are described here and here.

### First response

The responding scheme with an unbalanced positive hand is almost "up-the-line"; RR makes the lowest bid that describes the hand held. This structure (level adjusted) is used after Pass 1D; 1H, and any other situation where no shape constraints exist.

 1H Unbalanced, 4+S 1S Unbalanced, 4+H, 0-3S 1NT Unbalanced, 4+D, 0-3S, 0-3H 2C Balanced, including a four- or five-card major 2D Balanced, 2-3H, 2-3S 2H Three-suited with a major suit shortage 2S 6+C, 0-2D (Low shortage) 2NT 6+C, 0-1H (Middle shortage) 3C 2236 (Even shortage) 3D 1336 (High shortage) 3H 1327 3S 1237 3NT 0337 The only exception to the "up-the-line" rule is the 2H response, which takes priority over the 1H and 1S responses.

### Balanced hands

With a balanced hand, RR selects whichever of 2C or 2D is appropriate.

 2C Balanced, including a four- or five-card major 2D Game-forcing relay 2H Any 4432 with 4S 2S Game-forcing relay 2NT 4432 or 4423 3C Game-forcing relay 3D 4432 3D 4423 (with zoom) 3C 4342 3D 4243 3H 4324 3S 4234 (with zoom) 2S Any 4432 with 4H and not 4S 2NT Game-forcing relay 3C 3442 3D 2443 3H 3424 3S 2434 (with zoom) 2NT Any 5332 with 5S 3C Game-forcing relay 3D 5332 3H 5323 3S 5233 (with zoom) 3C 4333 or 3433 3D Game-forcing relay 3H 4333 3S 3433 (with zoom) 3D 3532 3H 3523 3S 2533 (with zoom) This relay structure avoids RR bidding a four-card major that they hold and are symmetric in the 5-card major holdings. 2D Balanced, 2-3H, 2-3S 2H Game-forcing relay 2S Any 5332 with 5D 2NT Game-forcing relay 3C 3352 3D 3253 3H 2353 (with zoom) 2NT Any 5332 with 5C 3C Game-forcing relay 3D 3325 3H 3235 3S 2335 (with zoom) 3C 3244 3D 2344 3H 3343 3S 3334 (with zoom)

### One-suited hands

With a one-suited hand of at least six cards, RR shows that suit and then rebids at 2S or higher to describe the residual short suits. The mnemonic of bidding as low as possible with more cards in higher-ranking suits is valid, and those with experience of other symmetric relay structures will recognise the low-middle-even-high motif.

 2S 0-2 in the lowest-ranking other suit 2NT Game-forcing relay 3C "Even shortage" 3D Game-forcing relay 3H 6322 3S 6232 3NT 7222 2NT 0-1 in the middle-ranking other suit 3C A 6223 hand with the tripleton in the lowest-ranking suit 3D 6133 with high-ranking shortage The shapes not explicitly shown in the structure can be constructed through "symmetry" - for example, all hands of 6331 pattern include a 3D bid, possibly after 2S or 2NT to specify the suit of the singleton. Other hand patterns are resolved similarly, however the hands with "even shortage" are resolved specially as indicated above.

This structure is more effective than the one given above, because it shows the more frequent shapes at two steps lower net, at the cost of two of the rarest shapes each shown one step higher (or two rarer pairs of steps left unresolved) and loss of "symmetry". Partnerships should evaluate this trade-off and choose a one-suited structure accordingly. In this system, this structure is particularly attractive because of the lower level of many of the relay auctions - full 7321 resolution below 3NT is routinely possible. Where the two 7321 shapes are compressed (see below) after the initial response showing 0-2 cards in the lowest-ranking side suits, they will be shown below the 7222 shape.

 2S 0-2 in the lowest-ranking other suit 2NT Game-forcing relay 3C 6322 3D 6331 3H 7222 3S 7321 (however 7321 shapes may compress, see below, and if so they will precede 7222 in this structure) 3NT 7231 4C 7330 2NT 0-1 in the middle-ranking other suit, or 6232 3C Game-forcing relay 3D 6232 3H 6313 3S 7312 (however 7321 shapes may compress, see below) 3NT 7213 4C 7303 3C 6223 3D 6133 with high-ranking shortage 3H 7132 with high-ranking shortage 3S 7123 with high-ranking shortage 3NT 7033 with high-ranking shortage The "balanced" hands are always shown first. 7222 is arbitrarily included in the low-shortage structure, and a step higher than 6331 because of its lower frequency.

### Two-suited hands

With a two-suited hand, RR shows his first suit up-the-line as listed in the positive responses. RR then shows the second suit and resolves their relative length. A bid of 2D always shows at least four cards in the higher-ranking suit and at least five cards in the lower-ranking suit (always clubs, in fact). However a bid of 2H shows precisely four cards in the lower-ranking suit and at least five in the higher-ranking suit. Hands with two suits of at least five cards always start with two bids at or below 2D, and then a bid of 2S.
With both majors, RR continues with 1NT over the 1S relay to the first response. With a major and diamonds, RR continues with 2C (transferring to diamonds!). With any suit and clubs, RR shows the first suit and then bids according to the above scheme - 2D with at least five clubs, 2H with exactly four clubs.
Thus the structure below applies after showing a second suit with 1NT, 2C, 2D or 2H.

 2D Three-suited with both majors 2H 5+ in the higher-ranking suit, 4 in the lower-ranking suit (a "reverser" bid) 2S 5+ 5+ in the two suits (step in) 2NT 0-1 in the lower-ranking other suit 3C A 5422 pattern 3D 5431 with higher-ranking shortage 3H 6421 with higher-ranking shortage 4C 7411
Again the the structure is "symmetric" - all 6421 shapes will include a 3H bid, after clarifying the relative length of the suits and the shortages with previous actions. The 5-5 substructure is also symmetric internally. Note that the 7411 shape is shown "asymmetrically". The only reasonable alternative is to treat it with 5422 as "even shortage", which is grossly inefficient because it costs a step on the greatly more frequent 5422 shapes.

### Showing 5+ 5+ hands

(step out) Game-forcing relay after showing 5+ 5+ in the two suits

 3C 0-1 in the lower-ranking other suit 3D A (65)11 pattern (even shortage) 3H 5521 with higher-ranking shortage 3S 5530 with higher-ranking shortage 3NT (65)20 with higher-ranking shortage

### Three-suited hands

With a three-suited hand with a minor shortage, RR simply shows both major suits and then uses the "empty" 2D bid to describe his holdings. With a major shortage, RR simply responds 2H. Then after the subsequent relay, RR bids as described.

 Step 1 Low shortage (0-1) Step 2 1444 (high shortage) Step 3 0544 (high shortage) Step 4 0454 (high shortage) Step 5 0445 (high shortage) After showing a low-ranking shortage, RR shows his exact shape by using the step responses as above, beginning at 4441.

After the shape is shown, the next relay asks for controls. R has a choice of three asks: Step 1 asks for AKQ controls, Step 2 asks for AK controls and Step 3 asks for aces (Gerber!) subject to the usual stricture that 3NT and game-level bids above Step 1 are not relay asks. Also, the use of 4D as an end signal supersedes the use of that bid as an AK control ask or ace ask.
AKQ and AK controls are evaluated as simple sums. With AKQ controls, an ace counts three, a king two and a queen one. With AK controls, an ace counts two and a king one. When holding a singleton honour:
• a singleton ace is counted normally in the hand's controls,
• a singleton king counts one AK control for an AK control ask and one AKQ control for an AKQ control ask, and
• a singleton queen or lower is not counted at all.
As described in the section above on hand valuation, both players can deduce RR's minimum HCP holding and thus the control base. The manner in which the number of controls is shown depends on the level of the asking bid. Asking bids of 3H and higher (obviously not 3NT) receive step responses such that Step 1 by RR shows they have a number of controls that is at most equal to the base. Step 2, 3, 4, etc. show one, two, three, etc. extra control(s) over that required for the base. When the asking bid is below 3H then the scheme "inverts" to improve efficiency. Bids of 3NT and 3S show base and base+1 controls respectively. Bids below that show increasing numbers of controls, with bids of 4C and higher showing at least base+4 and more controls. Thus when only 3H is available it shows base+2 or base+3 controls. A further relay then requires 3NT with base+2 or a higher response with base+3 (see below). When both 3H and 3D are available then they show base+2 and base+3 controls respectively. Should 3C also be available, then it shows base+4 controls and in this case, 4C and higher will show base+5 and more controls. Should any lower bids be available, then the scheme expands similarly.
The onus lies on R to ask for controls only when there are no replies that are reasonably likely to be awkward, since their next Step 1 is also a relay. An ace ask always begins at zero.
In one case above, a response is made that shows a specific range of controls. When R bids Step 1, RR zooms to Denial Cue Bidding (see below) when holding the highest number of controls. R should be prepared for this zoom by a maximum RR. This principle holds in other situations where a range of controls is known, including where a maximum can be inferred from a known HCP maximum. Such a maximum is realised when holding the maximum number of aces with the remaining HCP contributed by as many kings as possible, etc.

## Zooming

When RR holds the shape that is the highest call defined in the response structure it is efficient that that response follow on to the number of controls held without requiring R to waste two steps in making a subsequent ask. How a zoom occurs depends whether RR is balanced or unbalanced and limited or unlimited.
• Any unlimited hand must zoom past 3NT if either
• the structure shows shapes past 3NT, or
• they are showing base+4.
• A limited balanced hand may only bid past 3NT if the preceding relay was going to resolve the hand shape into a definite pattern, even if there exists shapes that would normally be shown past 3NT or the hand holds base+4 or more. Rationale: sometimes R needs definite pattern information to choose the correct strain, but will be unwilling to explore if that risks an unsuitable contract higher than 3NT. If shape-showing truncation occurs with a 3NT bid, R may relay with 4C to complete the shape description, with a zoom with base+4 for the highest-ranking shape.
• A limited unbalanced hand never zooms past 3NT unless the structure shows shapes past 3NT.
In all cases where bids at or below 3NT are available to show a range of strengths, the conditions below apply.
• If only 3NT is available, then the strengths are shown with it as a range.
• If only 3NT and 3S are available for showing a range of strengths, then the range is split evenly and contiguously between those two bids, with 3NT showing the upper range (and the smaller range in the case of an uneven split).
• If more than those two bids available, then the strengths are shown "decreasing" from 3NT with the lowest bid forming a range if required.
There is one case in non-level-adjusted relay auctions where shape is shown above 3NT. This occurs when a 7411 pattern is shown with 4C. In this case, bids at or above 4D show the shape that was shown with 3NT (7420) with at least base+4 AKQ controls in the normal manner described above.
In all cases where shapes are shown past 3NT, the bids past the end of the structure show at least base+4 controls stepwise, with the shape that would have been shown with a 3NT bid. Thus the shape shown with 3NT bid, rather than the highest shape, is limited to base+3. By limiting the strength held for a 3NT bid, R will continue to ask past 3NT when that is the last makeable contract less often.

When resolving the shape of a hand known to be balanced, the structure will be similar to that used in non-level-adjusted relays, but with relevant bids swapped so that a major suit is not shown by a bid of that suit. Symmetry will be preserved, however.
When a single-suited relay structure is used up one or more steps, the 7321 and 7231 shapes are compressed into a single response and the ambiguity is not resolved.

## King Parity

After shape has been resolved and an AKQ control number has been given (in response to a direct ask or a zoom), the next relay asks for "king parity". Note that a singleton king is not counted as a king here, consistent with the fact that it counted only one in the AKQ control number.
RR bids Step 1 or higher according to the following scheme:
 King Parity AKQ controls Action Even 2, 3, 6, 9, 10, 12, 13, 16, 18, 19, 20 Bid Step 1 Even 4, 5, 7, 8, 11, 14, 15, 17, 21+ Zoom to Step 2 or higher Odd 2, 3, 6, 9, 10, 12, 13, 16, 18, 19, 20 Zoom to Step 2 or higher Odd 4, 5, 7, 8, 11, 14, 15, 17, 21+ Bid Step 1
As can be seen by inspection, RR stops with odd number of AK controls on certain numbers of AKQ controls and does the opposite when holding an even number of AK controls. The rationales are that
• with 2 AKQ controls RR should bid Step 1 with QQ and zoom with K because that is more effective for the subsequent DCB,
• with most multiples of 3 AKQ controls, frequency analysis shows that around two hands with odd AK controls are held for every hand with even AK controls, and zooming should occur with the more frequent hand types, and
• with other numbers of AKQ controls, there is either no significant difference in the ratio of odd:even AK controls, or a distribution with more hands with even AK controls.
One (approximate) mnemonic is that RR with odd kings bids Step 1 with AKQ controls that are (approximately) "prime" with respect to 3. RR also bids Step 1 with both predicates negated. RR bids Step 2 or higher when only one predicate is negated.
Note that this criterion is somewhat arbitrary. The objective of maximising human slam-bidding efficacy is difficult to codify. Merely ensuring a Fibonacci-like distribution of step frequencies is a fair attempt, but there exist many distributions of hands to steps that satisfy this constraint but which do not permit effective bidding. Moreover, some distribution that could be demonstrated to be the most effective for bidding in the abstract might well be impractical for use by a human at the table. Additional work in this area would be of interest, but of little practical value. Using some king parity style is probably more effective than not using one, and even a simple style of bidding Step 1 arbitrarily with even AK controls is probably acceptable.

## Denial Cue-Bidding

(Also known as "spiral scan".)
After controls (AKQ, AK or A) and possibly king parity have been shown, "denial cue-bidding" commences. In denial cue-bidding the suits are assigned an order of priority - firstly by order of length, but if two or more lengths are equal (or have lengths that are unknown), then the higher-ranking suit is scanned first. On the first ask ("scan"), the RR is asked to look for an ace or king in the suit of highest priority. If neither of these is held then RR bids Step 1 to show this. Otherwise, RR then scans the suit of second priority, also for either the ace or the king, bidding Step 2 to deny a top honour. This process continues through all the suits, and eventually may return to the suit of highest priority (i.e. "spiralling"). When this occurs, RR is expected to look for a second top honour in the suit (if one was previously shown) or the queen (if the ace and king were previously denied). This process continues. In principle, jacks could be investigated, but such auctions are usually too high for such an ask to be useful.
A suit of a given length can only be scanned that given number of times - a singleton may only be scanned once, a doubleton twice, etc. The highest possible relay ask is 6C. If a response is made at 6C or higher, then R must be prepared to place the final contract. No relay response higher than 7C may be given. The onus is clearly on R not to make an ask when there is reasonable chance of unfortunate consequences. It is occasionally necessary to finesse a card during the bidding! (Don't tell your teammates, it spoils your image.)

### Exceptions in Denial Cue-Bidding

• When scanning a suit for the last card held (e.g. the first or second respectively for a singleton and doubleton, etc.), RR stops (i.e. bids the current step) when holding the relevant honour, and spirals if the honour is not held. This is much superior to the normal approach, because it is more probable that the relevant honour is not held. It suffers only a 1 step loss relative to a DCB style that never scans a singleton, losing 2 steps only where "unexpected" information is transferred, whereas scanning a singleton normally loses 2 steps most of the time. A case can be made for not scanning singletons at all when RR has shown AK controls, but that method is not used here.
• When holding the AKQ of a suit, that suit is denied on the first scan. The 3 or 5 control discrepancy that is flagged by this denial is almost always apparent to R and recognisable - and it is assumed that R knows what holding RR has. Experience suggests that by the time this suit is scanned again no ambiguity remains, and both hands will know that AKQ has been shown. Thus on a second scan of this suit when holding at least four cards in the suit, the jack is investigated.
Additionally and analogously, the previous exception applies to a holding of exactly four cards to the AKQ; on the second round RR stops with AKQJ and spirals with AKQx. If a three-card suit contains AKQ then there is only one scan made of this suit.
• When RR is known by both hands to have shown all possible aces, kings and queens, then the DCB skips directly to jacks in the first suit in the scan. This has been known to create amusing encrypted auctions when RR has 6 AKQ controls and a holding of AKQ in a suit is possible and R is missing at least 11 AKQ controls. Now after RR denies the suit in which AKQ might be held, R must ask until RR shows a card that R holds in one "version" of the auction, since the assumption that the control discrepancy will be recognisable is no longer as valid. Potential slams missing 5+ AKQ controls when R is missing at least 11 AKQ controls and has two "empty" suits are too rare for this to be damaging, however.

There exists a nontrivial set of hands where R can know at an early point of a game-forcing relay auction that normal denial cue-bidding will not be effective. Freak R hands with voids, or R hands needing only specific high-card gaps to be filled are most likely to employ these agreements effectively - including that set of hands that the field will bid easily with normal Roman Key Card Blackwood agreements.
Before final shape resolution, where R has only Step 1 defined as a normal relay, as described above, higher steps can be used to continue the relay, but with information transfer to RR that a certain suit will be trumps. The four suits may be set with Step 2-5 in the same kind of priority order as normal denial cue-bidding: RR's longest suits, and then higher-ranking suits first.
After controls are shown, the next ask is for the normal five "key cards", being the four aces and the trump king. RR replies Step 1 with 1 or 4 key cards, Step 2 with 0 or 3 key cards, and zooms with 2 key cards. There follows a structure similar to normal denial cue-bidding, with a priority order headed by the trump queen, then kings of side suits in the normal longest and highest-ranking priority order, then the trump jack, then queens of side suits in the normal longest and highest-ranking priority order, etc. Thus Step 3 shows 2 key cards and denies the trump queen, and Step 4 shows 2 key cards and the trump queen and denies the first king, etc.