This would be a stable matching (after allowing these changes). Since Nash is guaranteed here, there would be no reason to lie on the original form, unless ther

This would be a stable matching (after allowing these changes). Since Nash is guaranteed here, there would be no reason to lie on the original form, unless there's some weird mixed equilibrium (that's not pure). It'd be interesting to see what conditions we must have on it (if it even exists! I'm not sure to be honest), but the only pure equilibrium is certainly a stable matching.