The mechanics of admin government is fun and interesting and added more dimension to government besides feudalism. But it seems that while it might be fun for the player to interact with it, it doesn't seem like the AI is actually being aggressive in gaining influence and using the influence to invest in candidacy besides the imperial throne.
AI noble families barely invest any influence on candidacy of governorships in the themes. Outside of their children, they barely invest any family members into the theme governorship candidacy. AI also tend to castrate their family members and making it harder to have enough eligible male candidates for governorships after a few generations.
Neither would AI see or notice "cheap to invest" candidacy for governorships. AI seems to mostly invest their influence on the governorships their family owns, and never look at other governorships or even admin counties if there's some 'spare' positions for their family members.
Nor does the AI adopt people into their families at all. A small family might be more interested in adopting courtiers with decent stats into the family and then promote them as candidates into various admin governments.
I think there can be some suggestions on how to improve admin government gameplay, not just for the players but how AI approach things.
1. Value/prestige of provinces should have a ranking system. In the Byzantine empire, there is a clear ranking of themes in terms of prestige and hierarchy.
en.wikipedia.org
The Anatolic Theme has long been seen as the most valuable theme, and it should be a theme that many noble families would seek to have appointments in rather than some smaller and less developed themes.
So an AI should be happy if they get promoted to a more valuable theme, or be more unhappy if they get demoted to a less prestigious theme.
2. System of promotion/demotion in governorships
We can maybe have a 'cursus honorum' system whereby an character have to go through a series of appointments first before gaining a bigger office/governorship. So an AI Noble family might have to first invest influence to help their child become a admin-count first, and only after gaining and holding that office, they will be eligible for higher governorship offices. For non-noble families, admin counts can potentially invest influence so they can rise in status and get promoted into duchy-admin governor. So this would help solve the issue of lack of eligible candidate for admin noble families. A noble family with only female heirs might not get much sway to be nominated as a successor to a theme governor over that of a admin count in that province. So a possible solution for such noble family might be to marry their female heirs to the new (non-noble) admin count and then have that admin count basically found a new house within that dynasty.
3. Experience as governorship
There should be some sort of experience traits for governors. Those who have experience of being a governors should be able to gain more skills in governorship, and also increase eligibility for promotion to higher offices.
4. To actually invest influence to become emperor, candidates needs to gain support of themes/governors.
Right now the mechanics of just allowing everyone to invest influence for the imperial throne is throwing AI off in how to spend and invest their influence. Either restrict the ability to invest influence to only powerful or dominant noble families, or you have to ensure only those who have support of a certain percentage of themes will be eligible.
Also this can also limit or restrict investment of influence to imperial throne to some of the more senior/prestigious themes. So a governor of the Anatolic Theme should be able to have more chance of becoming emperor than a governor of a lesser province.
AI noble families barely invest any influence on candidacy of governorships in the themes. Outside of their children, they barely invest any family members into the theme governorship candidacy. AI also tend to castrate their family members and making it harder to have enough eligible male candidates for governorships after a few generations.
Neither would AI see or notice "cheap to invest" candidacy for governorships. AI seems to mostly invest their influence on the governorships their family owns, and never look at other governorships or even admin counties if there's some 'spare' positions for their family members.
Nor does the AI adopt people into their families at all. A small family might be more interested in adopting courtiers with decent stats into the family and then promote them as candidates into various admin governments.
I think there can be some suggestions on how to improve admin government gameplay, not just for the players but how AI approach things.
1. Value/prestige of provinces should have a ranking system. In the Byzantine empire, there is a clear ranking of themes in terms of prestige and hierarchy.

Anatolic Theme - Wikipedia
The Anatolic Theme has long been seen as the most valuable theme, and it should be a theme that many noble families would seek to have appointments in rather than some smaller and less developed themes.
So an AI should be happy if they get promoted to a more valuable theme, or be more unhappy if they get demoted to a less prestigious theme.
2. System of promotion/demotion in governorships
We can maybe have a 'cursus honorum' system whereby an character have to go through a series of appointments first before gaining a bigger office/governorship. So an AI Noble family might have to first invest influence to help their child become a admin-count first, and only after gaining and holding that office, they will be eligible for higher governorship offices. For non-noble families, admin counts can potentially invest influence so they can rise in status and get promoted into duchy-admin governor. So this would help solve the issue of lack of eligible candidate for admin noble families. A noble family with only female heirs might not get much sway to be nominated as a successor to a theme governor over that of a admin count in that province. So a possible solution for such noble family might be to marry their female heirs to the new (non-noble) admin count and then have that admin count basically found a new house within that dynasty.
3. Experience as governorship
There should be some sort of experience traits for governors. Those who have experience of being a governors should be able to gain more skills in governorship, and also increase eligibility for promotion to higher offices.
4. To actually invest influence to become emperor, candidates needs to gain support of themes/governors.
Right now the mechanics of just allowing everyone to invest influence for the imperial throne is throwing AI off in how to spend and invest their influence. Either restrict the ability to invest influence to only powerful or dominant noble families, or you have to ensure only those who have support of a certain percentage of themes will be eligible.
Also this can also limit or restrict investment of influence to imperial throne to some of the more senior/prestigious themes. So a governor of the Anatolic Theme should be able to have more chance of becoming emperor than a governor of a lesser province.
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