If you use your own Geisha they will always win but they always die doing it. Remember a Geisha never fails. So... if one wins so does the other one. So if you have to use one of your Geisha make sure you use a low honour Geisha against their high honour Geisha. Of course if they are close to your Heir or Daimyo - hit them with everything.
If you have to use Ninja - use a lot and use the previous advice of having high honour ninja and try to get the Geisha early before they pick up honour.
The new expansion has made it a lot easier to kill Geisha. In the original version killing a Geisha was a 'bitch' if you will excuse the expression.
If you are in trouble make sure you move your Daimyo and Heirs a lot. The one thing about Geisha is that they are visible unlike Ninja.
Sorry I rambled on a bit.