Personnally, I don't think the concept of Senate, Consul or Dictator will work in a succession game of 7 persons. It will take a lot of time and most decisions (such as building wonders) are very straightforward. I say, let's try, at least for our first SG, a normal game where every 10 turns we switch player (and maybe 20 turns in the latter periods). Only usefulness of a senate comes when we need to declare war. At the same time, the mandate for war should include the power for the current player to make all necessary alliances. Sure, sometimes we are going to sign unwanted alliances, but that can reflet bad consuls or bad emperors. Plus in times of war, you don't question autority
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Suggestion for the first 10 turns :
1 to 3 : Conquer Epirus with as little losses as possible and hurry Legio/Miles Socius in conquered cities. On the second turn, when we have possibly captured at least 1, if not the 2 cities of Epirus, sign ROP with all nations except Cartago and Magna Graecia.
Micromanage : this is about the only part that I micromanage. Build in Neapolis the most costly thing (60 shield). Make sure that either a) it will be done in 12 turns in order to change it to Mare Nostrum at the exact time we discover First Punic war or b) build a little something and after that be sure to have the 60 shields accumulated for the 12 turn. We can change production when we discover a tech by clicking on What's the big picture and then Domestic Advisor and then change Neapolis' production which will be completed as soon as you quit the Advisors panel.
4 to 12 : Well, Magna Graecia can be a usefull ally in a war against Cartago as they will lose Syracusa to Cartago (at least in my games they always do). What I normally do, is put most of my army in Messana and wait for Cartago's declaration of war (we need to make it very angry by demanding a city in tribute for 10-20 times, they declare war soon after). After a few turns, I take Syracuse and Panormus. Meanwhile, the other Consular army with a couple of Miles Socius go after Corsica and Sardeigne. If we play this right, by the end of the war, 250BC-230BC, we can have all western Mediterranean islands, Sicile and Valentia Edetanorum. If we are lucky, the greeeks are going to lose everything excep bovianum and we can take their coastal cities and declare war on them right after. Anyways, the first 20 turns are going to be critical for the rest of the game. So, in case of continuous bad luck, we can esaily restart.