From: desj@ccr-p.ida.org (David desJardins)
Newsgroups: rec.games.bridge,sci.math.stat
Subject: Re: WC Problem - Full Hand
Date: 7 Sep 92 23:24:02 GMT
Organization: IDA Center for Communications Research, Princeton

I've crossposted this article to sci.math.stat, as I thought some of the
statisticians might want to comment on the basic question of strategy in
long matches with cumulative scoring.  Please follow up to an
appropriate group based on the subject of your posting.

Ben Thompson <bjpt@cs.mu.OZ.AU> writes:
>>> You also have to consider the true value of 10 imps vs 2 imps.  It is
>>> considerably more likely that the winning margin will be 3-10 imps
>>> rather than 1-2 imps.  Given that Line 1 is only some fraction of an imp
>>> better than line 2, it is clear that line 2 is more likely to win the
>>> match for you than line 1.

>The psychological argument is a small point and is completely separate
>from the argument you quoted (and which I have requoted).  The objective
>of bridge is NOT to maximize your expected IMPs.  The objective is to
>win.  Line 2 is quite a bit more likely to change the result of the
>match in your favour than Line 1 (if it is changeable, or needs changing).
>Line 2 maximizes your chance of winning.

Suppose that we neglect completely the psychological effects.  This is
equivalent to treating all of the boards as independent.  It is also
equivalent to treating the particular board in question as the last
board of the match.

So, what does it mean to be the last board of a match?  Ignore ties for
the moment.  Then, precisely as you say, an IMP gain or loss is
significant if and only if it is greater in magnitude that the total
score on the other boards, and opposite in sign.

So, a 2 IMP gain will win the match for you only if you are losing by
one or two IMPs.  A 10 IMP gain will win the match for you if you are
losing by any amount between one and ten IMPs.  All other situations are
immaterial.

So, what is the relative probability that you are losing by a particular
number of IMPs?  Of course it depends on a variety of factors, such as
the strength of your team and of your opponents, and the length of the
match.  However, there are some things that we can say under some
circumstances.

Again assuming independence between scores, a lattice version of the
Central Limit Theorem implies, among other things, that in a
sufficiently long match the distribution of total scores will be
unimodal, with the maximum probability located near the expected score,
and the probability declining on both sides.  (There are a few technical
requirements, which are met in this case.)

The practical effect of this is that, in a long match such as the
Bermuda Bowl, if the teams are equal, then the most likely single result
is that the score is tied.  It is slightly less likely that Team A is
winning by 1 IMP, and approximately the same probability that Team B is
winning by 1 IMP.

Applying this information to our original problem, and assuming that the
two teams are equal, then it is certainly true that it is more likely
that Team A is losing by an amount from one to ten IMPs than that they
are losing by an amount from one to two IMPs.  But, since a score of
(say) -7 IMPs is less likely than a score of (say) -2 IMPs, it is less
than five times as likely:

   Pr(-1) + ... + Pr(-10) < 5 Pr(-1) + 5 Pr(-2) = 5 (Pr(-1) + Pr(-2)).

This is what justifies the expected-IMP analysis.  If the expected IMPs
for Line 1 vs. Line 2 are positive, then that means that Line 1 gains 2
IMPs more than five times as often as Line 2 gains 10 IMPs, and thus
wins the match more often.

The same analysis applies if your team is stronger than the opponents.
In that case, the expected score is some positive number of IMPs for
your side, and the scores decrease from there.  So it is still true that
Pr(-10) < Pr(-1), and the above analysis holds.  However, if your team
is weaker than the opponents, it is probably the case that -10 IMPs is a
more likely score than -1 IMP.  In that case, expected-IMP analysis may
not be valid, making a big gain is worth proportionally more than a
small one.  (And a big gain more valuable than the cost of a big loss.)
This is the basic justification for the well-known practice of
"swinging"---plays which do not maximize your expected score but may
increase your probability of victory.

                                        David desJardins
Newsgroups: rec.games.bridge