Constrainedness in Stable Matching

In constraint satisfaction problems, constrainedness provides a way to predict the number of solutions: for instances of a same size, the number of constraints is inversely correlated with the number of solutions. However, there is no obvious equivalent metric for stable matching problems. We introduce the contrarian score, a simple metric that is to matching problems what constrainedness is to constraint satisfaction problems. In addition to comparing the contrarian score against other potential tightness metrics, we test it for different instance sizes as well as extremely distinct versions of the stable matching problem. In all cases, we find that the correlation between contrarian score and number of solutions is very strong.