Relay Primer

Despite not being very popular, relay bidding systems are likely the most accurate bidding systems, especially in the game and slam zones. The idea behind them is that only one hand communicates information about itself while the other hand keeps bidding alive by bidding the next step. This process is called the relay, and it stops when the asker knows enough to place the final contract. Even though you may not want to base your entire bidding structure on relays, it is useful to understand some basic ideas, in my opinion.

I will now prove a fundamental result that helps you gauge what is possible within so much bidding space.

How Much Information?

Suppose the responder (the giver of information) has the job of telling asker which one of k distinct types of hand he has. Also suppose that only n steps remain in the bidding ladder. For the purposes of this discussion, we will assume that the final contract lies beyond these n steps and that asker must always bid the next step.

Let f(n) denote the maximum number of types of hand that can be distinguished within n steps. f(1) = 1. This is because responder is forced to bid and there is only one bid he can make, so he cannot distinguish at all. f(2) = 2. This is because the second step must necessarily show one type (because no room is left to resolve any ambiguity). The lower step must also show one type only, because asker can do no better than bidding the second step, leaving no room for further bidding.

If n > 2, the the auctions that ensue can be broken into two mutually exclusive and exhaustive sets. One consists of auctions where responder's first bid is the second step or higher, and the other consists of those auctions where responder's first bid is the first step. The number of elements in the first set is obviously f(n − 1). When responder bids the first step, asker bids the second step to continue the relay, leaving responder n − 2 steps (obviously, the first step doesn't pinpoint one hand type). That means f(n − 2) auctions lie in the second set. Therefore, for n > 2, f(n) = f(n − 1) + f(n − 2).

As you can well see, this sets up a Fibonacci series: 1, 2, 3, 5, 8, 13, 21, …

What Do I Do With This?

Note first of all that the number of items responder can distinguish between is at least the number indicated in the series. It is possible to design a system where asker has the flexibility to ask other questions, by bidding something other than the next step. Therefore, the series only indicates a minimum efficiency, which can always be achieved. Take, for example, the Ogust responses after a Weak Two in spades (2♠). Responder (to the relay) distinguishes between only four hand types using the four steps that are available to him (3♣ through 3♠), which is one lower than the minimum guaranteed to be achievable.

If you're in the habit of tinkering with your bidding system, you undoubtedly come across ask situations. With the clear mathematical foundation that you have here, you can ensure that your ask–answer structure is “tight”. Don't ever say that there's too little information left to convey. There's always information. I remember SP telling me that in the famed Guthi-SP Precision system, a sequence like 1♣-1♦-2♥ initiates an ask. Responder's replies to 2♥, if I remember correctly, consume more than one complete level. The discussion above tells you that whenever you're using up four or more steps, the structure had better be relay.

I will soon post a sample application of this: a relay structure after a Jacoby major-suit raise