You can find the Muse of Hope analysis here!
I am not sure if you mean this as a session analysis or an analysis of the general dynamic between the two, but I’ll go with the latter.
Now, the Lord of Doom is defined as one who has complete control over calamity, death, fate, and structure. That on its own lets you know the level of power that we’re dealing with here. The Lord of Doom has complete and utter control over death and destiny, two very, very important themes in SBURB.
The Lord of Doom can change who dies and who doesn’t, possibly bringing the ability to bring people back to life or give others the ability to straight-up ignore fatal blows! They could also have the power to doom others to a terrible fate, prevent calamity from ensuing, or CAUSE calamity instead!
But most importantly, a fully-realized Lord of Doom has the power to completely alter the way that the game is played, taking a welding torch to the game’s mechanics, and has the potential to work the game in their favor. Compared to the Prince of Doom, who zaps out rules he doesn’t like, the Witch of Doom, who bends the rules, the Bard of Doom, who makes the rules fall apart, and the Maid of Doom, who ignores the rules and makes their own…
The Lord of Doom at the height of their power has the ability to throw out the already existing rulebook and write their own. This makes for an extremely dangerous class who can completely change how everything works in the blink of an eye. You might not even be playing the same game by the end of a session with a Lord of Doom!
Now, bearing in mind that a Muse of Hope is all about giving others the courage to press onwards with the hope that things will work out for the best, and does that better than all other Hope classes, an interesting dynamic could be in place here. But, with the Lord of Doom changing how things happen and the Muse believing that having the hope that the desired outcome will come to pass will MAKE it come to pass, the two are liable to oppose each other.
The Lord could be in an antagonistic position, messing with the players of a session by constantly changing the rules on the fly, dooming them to failure and performing a motley of other shenanigans. However, the Muse of Hope might be able to DENY the Lord of Doom’s rules through inspiring their teammates and creating a surge of hope-power that lets them shrug off the Lord’s power over them!
However, an interesting consideration is that the MUSE could be the antagonist of the situation, becoming a crusader-type that is after a destructive goal and is armed with the power of their belief! The Lord and his allies must then scramble to place structural roadblocks in the Muse’s path, attempting to deny them the ability to see their goals through!
…WOW, that was way longer than I thought it was gonna be. Time for the araneamode tag.