From: taddiken@ms.uky.edu (scott taddiken)
Subject: Matchpoint Play: go for the overtrick?
Keywords: matchpoints, play, theory
Summary: A correction of Ruebens' analysis.
Date: Sat, 12 Sep 1992 16:23:54 GMT
Organization: University Of Kentucky, Dept. of Math Sciences

I have my own little contribution to make to the theory of matchpoint
play.  In Ruebens' interesting book "Secrets of Winning Bridge," he
offers an analysis of the following (admittedly contrived) situation:

You have arrived in 6 spades.  It is an excellent contract, but not the
easiest one to bid.  6 is cold, but you can play an alternate line which
could make an overtrick, or could put you down 1.  You don't think anyone
will be in 7, but a fair number will be in 4 or 5.  What is the relationship
between your estimate of the percent of the field in the game and the odds
you should demand in order to go for the overtrick?

Mr. Ruebens' reasoning goes something like this:  Against those who bid only
game you are playing for points.  Against those who bid the slam you are
playing for half points.  Thus, you should demand odds which will equal out
this 2 to 1 ratio, i.e. if 1/3 of the field is in only game, that's a 1/3 to
2/3 ratio, however, since you are playing for full points against the 1/3 in
game, you should be willing to take nothing less than 2/3 to 2/3 odds, that
is to say a 50% chance.

This reasoning, while seductive, is false.  In fact you should demand slightly
higher odds.  Let's say that there are 12 matchpoints at stake.  There will be
4 cases, depending on whether you play safely or go for the overtrick, and
depending on whether the field plays safely or goes for broke.  Suppose the
overtrick has probability P.

Case 1: You and the field play safe.  Whether the overtrick is there or not,
        you make 8 matchpoints (4 from those in game, 4 from your companions
        in slam).

Case 2: You play safe, and the field goes for broke.  P of the time, you make
        4 matchpoints (from those in game, none from those in slam, because
        the overtrick was there).  1-P of the time, you make 12.  The expected
        value is thus 12-8P.

Case 3: You and the field go for it.  When the overtrick is there, you make
        8 matchpoints.  When it fails, you make 4.  The expected value is then
        4P+4.

Case 4: You go for it, and the field plays safe.  P of the time, you make 12.
        1-P of the time, you make 0.  Expected value 12P.

So, when the field is playing safe, the expected values for playing safe or
going for an overtrick are equal when 8=12P, that is, P=66.7%.  Similarly, when
the field is playing aggressively, the values equal when 12-8P=4P+4, that is,
P=66.7%.  When the probability of the overtrick exceeds this value, the odds
favor going for it.

Notice that the probability at the point of ambivalence was the same,
regardless of whether the field was playing aggressively or not.  It turns
out that this is always true, even if some intermediate percentage of the
field is playing for the overtrick.  The rule for calculating the relationship
between the percentage of the field I (for idiots) in game and the percentage
chance P that you demand to play for an overtrick is simple (the algebraic
verification is trivial but tedious):

Add 100 to I and divide by 2 to get P.
Or equivalently: Multiply P by 2 and subtract 100 to get I.

I recommend the second form of the rule.  Let's say that you are considering
taking an 80% chance for an overtrick.  Forget those in such outrageous
contracts that they automatically score 0 or 1 against you, through luck
or stupidity or a combination of these.  You would only take that 80% chance
if you felt that less than 80x2-100=60% of the field were in the inferior
contract.

The message for those of us who play in local duplicate games is that you
should by all means make your contract before taking a dangerous finesse.
-----------------------------------------------------------------------
Scott A. Taddiken, University of Kentucky

If I could only remember to keep count of the trumps...