Notrump-raising strategies and hand evaluation
[[ Part 1: introduction, terminology, and evaluation of various
MW-based strategies at total points and MP
]]
Copyright 2006 by Alex Martelli (aleaxit@gmail.com)
This work is licensed under the Creative Commons Attribution 2.5
License (see http://creativecommons.org/licenses/by/2.5/).
The purpose of this article is to initiate a study about how to best
evaluate balanced (BAL) bridge hands (4333 shapes) for the purpose of
bidding the optimal level of Notrump (NT) contract.
Specifically, for any given algorithm for hand evaluation (such as
High-Card Points, HCP: the usual scale, A=4 K=3 Q=2 J=1, commonly
attributed to Milton-Work, and thus also known as MW, or, when
disambiguation is needed, OMW for opener, RMW for responder), a
"responder raising strategy" (RSS) is specified by the RMW range
"low-high" (LH) such that responder will raise a 1NT opening bid to 2NT
with L<=RMW=H points). [[Note the use of an open interval,
to ease the specification of raising strategies that never raise to 2NT,
but rather always either pass or jump to 3NT: such RSS are simply and
naturally represented by a pair (L,H) with L==H]].
Similarly, an "opener acceptance strategy" (OAS) is specified by a
single threshold (T) such that opener will pass a 2NT raise when OMW=T. A "combined strategy" (CS)
is the pair (RSS, OAS) or equivalently the triple (L, H, T). Not all
such triples give distinct results: in particular, when the RSS is an
empty interval (L==H), the choice of T does not influence the result. We
conventionally map all the CS in such equivalence classes to the CS
within the equivalence class with the lowest T, for definiteness.
Similarly, CS where opener always bids 3N on 2N are equivalent to those
in which responder bids 3N instead of 2N with the specified range, and
are similarly mapped to that same CS, not separately considered. So, T
will always be opener's minimum when L==H, and will never be opener's
minimum when L -835840 (-14.1 on 59132 deals)
At V
15-09, bid tr 9 vs 8:
(4+10+101+1305+10397=11817)*(-100) + 28747*(-220) + (15595+2780+192+1=18568)*(+450)
--> +849560 (+14.4 on 59132 deals)
So this is where the -14.1 and +14.4 differences in conditional
expectations come from. In the NV cases, there are 11817 deals in which
not even 2N makes; there, IIO loses 50 points (one undertrick) to IJP.
In 28747 deals, NS make exactly 8 tricks, so IJP's gain for playing 3N
(vs IIO's 2N) is 170 -- 120 for the value of 2N, plus 50 for the
undertrick. In 18568 cases, IIO gains 250 points -- the difference
between the 300-points bonus for a NV game, and the 50-points bonus for
a partscore. The grand total is -835840 over 59312 deals, and division
reveals that the weighted average, i.e., the expectation, is -14.1.
Things are _almost_ identical when V... except for the sizes of the
gains and losses: IIO's losing cases lose 50 points more (because the
undertrick costs 100 instead of 50), but the winning cases gain 200
points more (because the game bonus is 500 instead of 300), which,
overall, is enough to alter the sign of the result.
We don't really need to "play" all the 16*15/2 "matches" at total
points: simply comparing the TP expectations of the various strategies
gives us exactly the results of each "match", in terms of TP won or lost
per hand in average. Here's another table, presenting these results
with higher precision:
TP expectations for all strategies
NV V
HHO 157.83 216.91
HIP 167.15 228.24
HIQ 169.24 227.14
HJP 169.24 226.11
HJQ 164.55 209.20
HKP 160.89 206.74
HKQ 142.49 163.07
IIO 175.73 231.29
IJP 177.82 229.16
IJQ 171.04 213.34
IKP 169.47 209.79
IKQ 148.98 167.21
JJO 165.06 198.75
JKP 156.71 179.37
JKQ 143.00 152.62
KKO 131.32 130.15
We can well expect matchpoint (board-a-match) results to be very
different, since at MP the size of the swing does not matter -- only the
swing's *sign* does. Thus, in particular, there is no reason to
separate the analysis for V and NV, since vulnerability does not affect
the sign of the swing, only its size. On the other hand, only
head-to-head comparisons between strategies matter, so we do need a
full, 16x16 table (of course, we do expect the main diagonal, where a
strategy "vies" against itself, to show perfect equality, while cells of
the table occupying symmetric positions wrt the main diagonal will have
to be exactly complementary). However, to get a general idea of each
CS's general worth, we can show how it fares against the redoubtable
pair of Ron Result and Mary Merchant, who always bid 3N when it's fates
to make and otherwise stop at 1N; it's of course impossible to _beat_
Result-Mechant on any deal, so the performance of each CS can be shown
by the fraction on deals (out of 1000) on which the CS manages to _tie_
the board.
#boards/1000 tied against RM, all strategies
HHO HIP HIQ HJP HJQ HKP HKQ IIO IJP IJQ IKP IKQ JJO JKP JKQ KKO
470 517 540 543 550 524 484 622 647 632 628 566 633 614 567 529
IJP appears best here -- again, IJP is the strategy where responder
passes with 8, raises to 2N with 9 (to 3N with 10), and opener passes 2N
with 15 but goes to 3N over 2N with 16-17. We also see respectable
performances from JJO (responder passes with 8-9, jumps to 3N with 10),
IJQ (as IJP, but opener goes to 3N over 2N only with 17), IKP (responder
never jumps to 3N, but rather passes with 8 and raises to 2N with 9-10;
opener goes to 3N over 2N with 16-17), IIO (responder passes with 8,
jumps to 3N with 9-10), JKP (responder passes with 8-9, raises with 10;
opener goes to 3N over 2N with 16-17). In general, therefore, the
well-performing strategies (for such a "constructive" activity as
bidding game) at MP are more prudent than at IMPs -- again, this is
consonant with traditional bidding theory. However, the "middle of the
road" strategies IJP and IIO, which were optimal at total points,
perform quite fairly at MP, too (particularly IJP).
For head-to-head comparisons of strategies at MP, we will present the
results as fraction of deals (out of 300) in which the first strategy
wins, net of the deals on which it loses (a negative numbers thus shows
that the second strategy wins the overall BM match). [[the number 300 is
chosen somewhat arbitrarily to let the table format readably...!]]
MP matches between all strategies, scores in boards/400
HHO HIP HIQ HJP HJQ HKP HKQ IIO IJP IJQ IKP IKQ JJO JKP JKQ KKO
HHO +0 -43 -62 -65 -81 -62 -60 -60 -82 -80 -79 -59 -65 -62 -44 -23
HIP +43 +0 -18 -21 -38 -18 -17 -41 -63 -61 -60 -40 -46 -43 -25 -4
HIQ +62 +18 +0 -3 -19 +0 +1 -32 -54 -52 -51 -31 -37 -34 -16 +4
HJP +65 +21 +3 +0 -16 +3 +4 -19 -41 -39 -38 -18 -36 -32 -15 +5
HJQ +81 +38 +19 +16 +0 +19 +21 -13 -35 -32 -31 -11 -33 -30 -12 +8
HKP +62 +18 +0 -3 -19 +0 +1 -23 -45 -42 -41 -21 -39 -36 -18 -2
HKQ +60 +17 -1 -4 -21 -1 +0 -34 -56 -54 -53 -32 -54 -51 -33 -18
IIO +60 +41 +32 +19 +13 +23 +34 +0 -21 -19 -18 +1 -4 -1 +16 +37
IJP +82 +63 +54 +41 +35 +45 +56 +21 +0 +2 +3 +23 +5 +8 +26 +47
IJQ +80 +61 +52 +39 +32 +42 +54 +19 -2 +0 +0 +21 +0 +2 +20 +41
IKP +79 +60 +51 +38 +31 +41 +53 +18 -3 +0 +0 +20 +2 +5 +23 +39
IKQ +59 +40 +31 +18 +11 +21 +32 -1 -23 -21 -20 +0 -21 -18 +0 +14
JJO +65 +46 +37 +36 +33 +39 +54 +4 -5 +0 -2 +21 +0 +3 +21 +41
JKP +62 +43 +34 +32 +30 +36 +51 +1 -8 -2 -5 +18 -3 +0 +17 +34
JKQ +44 +25 +16 +15 +12 +18 +33 -16 -26 -20 -23 +0 -21 -17 +0 +15
KKO +23 +4 -4 -5 -8 +2 +18 -37 -47 -41 -39 -14 -41 -34 -15 +0
The salient result here is that IJP "wins" all matches, albeit with very
narrow margins from IJQ, IKP, JJO. IJQ ties with IKP and JJO, and IKP
edges out JJO by a narrow margin. Essentially, this confirms the
set of well-performing strategies, their rough ranking, and the
closeness among the few best ones, that emerged from the one dimensional
table where we compared each strategy against the artificially perfect
scores of Result-Merchant.
In future installments...: comparisons at IMPs (V and NV); other ways to
evaluate high-card strength, and their comparison with MW.