**Mixed Strategies**

Further complications can arise as in some cases no single deterministic strategy leads to an optimal result. A well-known result in game theory states that in such cases an optimal mixed strategy must exist. A small change in the lay-out of the last example illustrates this:

♥ K 10 8 7 |

♥ Q 3 2 |

What is the best matchpoint play for this suit? The line of play that maximises the expected number of tricks 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.

Again, this play is *not* optimal in terms of matchpoint objective, as it gets beaten by the following line of play: take a deep finesse by playing to the eight. If the eight loses to the nine, next play the ten and finesse the jack. 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. A similar analysis as in the previous example shows that the line of play that starts with a deep finesse in 31.43% of the cases leads to more tricks than the line of play starting with a finesse. The reverse result holds only in 23.18% of the cases.

Interestingly, the above line of play starting with the deep finesse also fails to optimise the matchpoint objective as it gets beaten by another line of play. In turns out that there are a total of eight lines of play that are non-transitive: the eight lines of play can be thought to be placed on a circle such that each line of play beats its left neighbor. As a result, the optimal approach in the context of the matchpoint objective corresponds to a so-called mixed strategy and is probabilistic in nature: the declarer has to select randomly one of the eight lines of play.

Read more about this topic: Suit Combinations

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