## Saturday, July 26, 2014

### Bridge bidding arithmetic

[I don't suppose any non-Bridge players will read this. But just in case I'll give some info in square brackets at times. In Bridge players partner in pairs who have to cooperate in bidding. The (pseudo-)auction involves bids by the four players (partners opposite) in order 1c,1d,1h,1s,1nt,2c,...7nt. One way to cooperate is a relay where one player always makes the cheapest bid and the other describes their hand. The hand has 13 cards. We say a hand is 3541 if it has 3 spades, 5 hearts, 4 diamonds and 1 club. "Game" is 4h or 4s or 5c or 5d (or 3nt) and scores a lot extra. Slam (12 or 13 tricks out of 13) scores even more.]

After a relay the asker needs to set the suit below game [to allow slam investigation]. We need to do this below 4h, so it can be done starting at 3s, for example: 4d sets spades; 4c sets hearts; 3s sets a minor and responder always bids 3nt after which 4c sets clubs and 4d sets diamonds.

So lets assume for the moment that we are only showing distributions. How many different distributions can each bid now show:

3h: 1
3d: 1
3c: 1
2nt: 2 (asker then bids 3c and responder can bid 3d or 3h with the 2 distributions)
2s: 3 (asker then bids 2nt and responder can bid 3c or 3d or 3h with the 3 distributions)
2h: 5 (asker bids 2s, responder bids 2nt with 2, 3c/3d/3h with others)
2d: 8 (...)
2c: 13 (...)

Which continues to be the fibonacci numbers (each number the sum of the previous 2): 21,34,55,...

So how many distributions are there:
 dist probability variants cumulative prob/variant 4-3-3-3 0.1054 4 4 0.02635 4-4-3-2 0.2155 12 16 0.01795833333 5-3-3-2 0.1552 12 28 0.01293333333 5-4-2-2 0.1058 12 40 0.008816666667 4-4-4-1 0.0299 4 44 0.007475 5-4-3-1 0.1293 24 68 0.0053875 6-3-2-2 0.0564 12 80 0.0047 6-3-3-1 0.0345 12 92 0.002875 5-5-2-1 0.0317 12 104 0.002641666667 6-4-2-1 0.047 24 128 0.001958333333 7-2-2-2 0.0051 4 132 0.001275 5-4-4-0 0.0124 12 144 0.001033333333 7-3-2-1 0.0188 24 168 0.0007833333333 5-5-3-0 0.009 12 180 0.00075 6-5-1-1 0.0071 12 192 0.0005916666667 6-4-3-0 0.0133 24 216 0.0005541666667 7-4-1-1 0.0039 12 228 0.000325 6-5-2-0 0.0065 24 252 0.0002708333333 7-3-3-0 0.0027 12 264 0.000225 8-2-2-1 0.0019 12 276 0.0001583333333 7-4-2-0 0.0036 24 300 0.00015 8-3-1-1 0.0012 12 312 0.0001 6-6-1-0 0.00072 12 324 0.00006 7-5-1-0 0.0011 24 348 0.00004583333333

Though the cumulative total goes up linearly and fibonacci goes up exponentially, still we don't have a lot of room to cover low probability distributions. And certainly we can't afford to waste much space.

One can consider what order to show distributions in. A simple scheme is: (a) show more balanced hands first (lower number of distribution points [doubleton=1, singleton=2, void=3]); (b) within that constraint show hands with more major cards (i.e. in decreasing number of hearts+spades); (c) within that constraint show in decreasing heart length.

The reason for showing more balanced hands first is that this makes slam less likely, so asker can often quickly bid game after a lower bid, without relaying it out and telling the defenders more than necessary.

One of the objectives of bidding is to find useful trump fits. [It is desirable to have 8 trumps. Hands without an 8 card major (heart/spade) fit are usually best in 3nt if possible.] Relaying wastes space compared to having both partners contributing. The question of how to best use both partners is a difficult problem. A simple approximation/guess is to use binary search. Here's how that works:

Taking a suit at a time in some algorithmically defined order the bidder cuts the length of that suit in half and bids one step with the higher (or lower) range and otherwise goes up a level and repeats.

Suppose you are playing "Romex" and a 1NT opening is 20+ points [ace=4,k=3,q=2,j=1]. One way to play the responses follows. Note that 1 step is towards the middle, bypass shows more extreme.

At any time by either player: 3s/4c/4d set the suit, as described above.
2c: 0-4 points. Higher bids forcing to game.
2d: 0-3H
2h: 4-7H 0-3S -- beyond this we further breakdown the majors
2s: 4-5H 4-7S
2nt: 6-7H 4-5S
3c: 6H 6-7S
3d: 7H 6S 0C 0D made it with a bid to spare

That worked well because narrowing the range in the majors also narrowed the range in the minors. But in other auctions we can find all 8 card major fits but the minors don' get narrowed down (as much). For example here are continuations after 1n-2s:
4c/4d: setting major with 4, otherwise deny 4
2n: 2-3S (i.e. matching the top half of responders 4-7)
3c: 3H 0-1S (ie hitting top of openers 4-5H range)
3d: 0-2H 1S (hearts eliminated)
3h: 0-2H 0S 4-5C 4-7D -- With 11-13 minor cards can be 47 56 65 or 74.
and a few more.

Given the fibonacci connection it might be better to split unevenly with approximately 38% in 1 step and 62% beyond (1:1.62 = 0.62:1 is the golden ration and consecutive fibonacci numbers tend towards that ratio). But this is not nearly enough and low probability distributions have to be lumped together to make it work when starting at 2d. Realistically one also needs to start lower or start with a restriction on the range of distributions. And indeed that is commonly the case for auctions starting lower than 1nt.