Realistically speaking, Pre-New 52 Anti-Monitor would own Darkseid. I can only hope that this will remain the same in this battle, as I always thought of Anti-Monitor to be on par with Galactus and other high-end cosmic entities. Where as Darkseid to be on par with Thanos. Powerful, yes, but against someone like Anti-Monitor or Galactus... they're WAY outclassed. Unless there's a HUGE plot-device in the form of an all-powerful weapon or something.
Indeed. The New 52 Darkseid is way ahead his Pre52 counter part. Tbh i do want Darkseid to win this war... and get the respect he deserves.
Note : The New 52 AM is the COIE AM him self but in standard level "Universal"