Suit Combinations - Goal Setting

Goal Setting

Although there can be little debate on what is the game-theoretically optimum play of a suit given the suit lay-out and the objective function to be maximised, the choice of what constitutes the right objective function for a given practical situation can be subject of debate. Generally, the specification of the objective function depends on the type of scoring. In team matches with IMP scoring, the objective of maximising the imp score usually corresponds to the goal of maximising the likelihood of obtaining a specified number of tricks from the suit under consideration (see above examples). In matchpoint scoring, one usually assumes that the objective of maximising your matchpoint score corresponds to the goal of maximising the expected number of tricks from the suit under consideration. This assumption is not always correct. The goal for declarer in matchpoint scoring rather is to ensure that his line of play beats alternative approaches in term of scoring more tricks on as many lay-outs as possible. When applying this 'matchpoint objective' to the line of play for a single suit, optimum lines of play originate that may differ from the non-exploitative line of play that optimises the expected number of tricks from the suit. An example illustrates the point:

What is the best matchpoint play? The line of play that maximises the expected number of tricks from this suit is to finesse by playing to the ten. If the ten loses to the jack, you next play towards the king. If the ten loses to the ace, you next play the queen. This approach results in three tricks in 28.7% of the cases, two tricks in 54.4% of the cases, and one trick in 16.9% of the cases. The expectation value for the number of tricks is therefore 2.12 tricks.

However, this play is not optimal in the sense of optimising the above described matchpoint objective. Consider the line of play that starts by taking a deep finesse by playing to the eight. If the eight loses to the nine, next play to the king. If the eight loses to the jack, next let the ten run. If the eight loses to the ace, let the queen run and then finesse over the jack. This play results in 2.09 expected tricks, a results slightly less than the above 2.12 tricks obtained by playing to the ten. Yet, the play that leads to 2.09 tricks on average, beats the play leading to an average of 2.12 tricks in terms of matchpoint objective.

This can be seen by considering the lay-outs on which the line of play that starts with a deep finesse takes more tricks than the line of play starting with a finesse and vice-versa: it follows that the deep finesse beats the finesse in 22.95% of the cases, while the finesse beats the deep finesse only in 18.33% of the cases. In the remainder of the cases (58.72%) both lines of play lead to the same number of tricks.