### The balance at cross-IMPs scoring.

Up till now we have assumed that the score calculation is based on match points (MP). But what if we use Cross-IMPs as the scoring method? Will our movement still be in balance? One thing is certain: a "perfect" movement is perfectly balanced for all scoring methods. You meet all other pairs an equal number of times as opponents, and they play an equal number of times in the same and in opposed compass directions. But the compensation we used in setting up the score matrix: playing in the same compass direction is equivalent to 1/h direct encounter is no longer self-evident.
 Paar Contract Res. Score XIMPS NZ OW NZ OW 1 3 NT C +600 -1 2 - 1 - -600 +1 3 3 NT C +600 -1 4 - 3 - -600 +1 5 3 NT C +600 -1 6 - 5 - -600 +1 7 3 NT +1 +630 +4 8 - 7 - -630 -4 9 3 NT C +600 -1 10 - 9 - -600 +1

Let us consider the example we used before. In a contest of 10 pairs the contract is 3NT at all tables. Everyone just makes this contract, except one pair who make an overtrick. At MP's this meant a top (8 MP, and a 0 for the opponents.
At cross-IMPs there are 4 scores to compare with, each of them yielding 1 IMP, for a total score 4 for the pair who managed to get an overtrick, and -4 for their opponents. At the other tables all pairs score +1 or -1 IMP. Pairs who play in the same compass direction as the one strong pair score -1, 1/4 of the score of the pair who had a direct encounter with the strong pair.

You probably have already identified the number 4 (i.e. the number of comparisons on which the cross-IMP score is based) as h, half a top at MP-scoring, or, the number of times - 1 the hand is played. Again we notice that the effect of a direct encounter is h times as large as playing in the same compass direction. This is equally true whether the difference is one trick, as in this example, or whether we have a swing hand where both sides can make game. Also, it does not matter whether or not the Cross-IMP score is calculated as an average, i.e. the XIMPS we used here divided by the number of times a hand is played. In all cases we have:

Also in cross-IMPs playing once against an opponent is equivalent to h times playing in the same compass direction. The techniques we used to obtain an optimal balance are also valid for cross-IMPs.

Conclusion: For cross-IMPs just use the same movements as for MP's.