The strategy of Politeia share some similiarities with its sister game Demokratia, as the objective in both games is to exhaust your supply of citizens. In race games like this, the long-term strategy needs to take into account the short-term tactical opportunities so let's start with them.
Getting citizens to the board is accomplished in three steps. First, you acquire citizens as talents. Second, you turn those talents into citizens. Third, you use those citizens to increase your production of talents/citizens while also decreasing those of your opponents. To accomplish this, you have nine actions at your disposal:
The talent actions
The first group of actions can be characterized as talent actions - actions that give you talents.
The Tax action gives you talents directly. This is good for short term needs, where you are short of talents and need them as soon as possible.
The Export action gives you talents in exchange for resources. This action potentially gives you more talents than the tax action if you can collect valuable sets of resources but also costs you two actions to actually get the talents (first import to get resources, then export to get talents). This is good for long term needs, where you had the talents you needed and could afford waiting for new talents.
The Produce action is a middle way where you get both talents and resources. This is good if you are short of both talents and resources.
The Trade action is a double action where you get to take or sell resources but only one. Hence, this is weaker than the import/export actions, that lets you take or sell multiple resources, and should only be used if you have an odd resource and either need to sell it now (to get talents) or to complete with another resource and sell later.
The talent actions should be used to ensure a stable income both in the short term and in the long term. Too many talents on the hand typically means that you have wasted turns where the talents could have been used for other actions. Too few talents on the hand means that you run a risk not affording actions. To avoid this, strive to maintain a hand of 4-6 talents and 2-4 resources.
The investment actions
The second group of actions can be characterized as investment actions - actions that lets you invest talents for future cost savings or revenue increasings.
The Build action lets you acquire building cards that save future action costs.
The Import actions gives you resources that you can collect in valuable sets and sell for talents with the export action.
The Produce action is both a talent action and an investment action. It gives you both one talent and one resource as discussed above.
The Trade action is both a talent action and an investment action. It gives you only one resource as discussed above but at least you get to choose which resource.
The investment actions should be used early in the game to continuously improve the talent flow. The build action is better the earlier you take it, since it saves you talents in every action afterwards, but be sure to use it for actions that fit your strategy. Typically, one build card in each action is a good spread. The import action can be very powerful, since a complete set of resources will give you enough talents for many round, but be sure not to import more resources than you will have time to export.
The citizen actions
The third group of actions can be characterized as citizen actions - actions that let you turn talents into citizens.
The Mobilize action lets you turn talents into citizens in the leader circles. This is good for defending against a hostile takeover or preparing for an upcoming conflict.
The Intrigue action lets you turn talents into citizens in the people circles. This is good for preparing a hostile takeover and improving your talent actions.
The Colonize action lets you turn talents into citizens in the colonist circles. This is good for improving your talent actions.
The citizen actions are necessary to get citizens on the board that can earn talents. They are also the "victory point" actions that determine the winner. Unlike many other games, Politeia has no big end game scoring actions and hence no turning point where you need to switch from talents to citizens. Instead you must maintain a steady flow of citizens by having talents at hand and taking at least one citizen action every turn. Consequently, opportunities to disrupt other players' citizen flow rather than generating more citizens is the the best way to catch up/keep runners up behind.
The opponent actions
The fourth group of actions can be characterized as opponent actions - actions that lets you interfere with opponent citizens.
The Influence action lets you improve your strength in conflicts. It does not earn you any talents directly so you need to combine it with a conflict to get a return on your investment (either by being the attacker or by defending against an attacker).
The Maneuver action lets you reallocate citizens. It doesn't increase your number of citizens but in combination with other opponent actions, it can help turning a conflict disadvantage to an advantage.
The Attack action lets you attack and hopefully take over an opponent's board position. This not only improves your talent flow but also hurts that of an opponent. In addition, it can be used to trigger a well-timed Greco-Persian War but more about that below.
The opponent actions are the main actions to bash leaders (if you're behind) or hold back runner ups (if you're ahead). Use them together with diplomacy by agreeing on cooperative actions or threatening with retaliation. Remember that a citizen lost due to conflict is removed from the board rather than returned to the stock so it doesn't hurt your current victory points but potentially your future ones. Thus, you should spend few citizens in the beginning, to prevent opponents outside the conflict from taking over vacant areas, and more citizens towards the end, when the there is no time to take advantage of vacant areas.
The action board
The action board determines two things: the cost of actions and the availability of actions.
The cost of actions is dependent on which actions the other players take. Popular actions will get more markers and become more expensive. Less popular actions will get less markers and become less expensive. By playing tactically and pick the cheapest available action from an action group, you can often accomplish the same action objective for a lower cost. If you have no option but to take expensive actions, try to drop your citizen on the most expensive as this will give you more actions on your turn.
The availability of actions is dependent on their positions on the randomly set up action board. Of particular interest is how the different action groups are spread. If all groups are spread around the board, it should be easy to take turns with actions from all groups for a balanced strategy. However, if any groups are concentrated, it may be necessary to sometimes take turns without actions from specific groups and there may also be opportunities to block opponents.
On the other hand, concentrated groups may open up for powerful combinations. Some examples include:
Import-Trade/Produce-Export: This combination lets you build a valuable set and sell it in one turn.
Influence-Maneuver-Attack: This combination lets you build up for a powerful attack and launch it in one turn.
Influence-Mobilize-Intrigue: This combination lets you build up a strong presence to trigger a war with a Greek victory, earning you talents.
Finally, you should think carefully where to drop action markers and where to drop your citizen. Dropping action markers where there are opponent citizen will leave them an extra action marker. Dropping your citizen there is even worse as it will leave you with no action markers when it's your turn again. On the other hand, you can use this to your advantage by dropping your citizen on attractive actions to either discourage your opponents from dropping their citizens there or force to leave you an extra action marker.
The Greco-Persian War
The Greco-Persian War serves a balancing mechanism that will keep a citizen count on the board that is not too big, nor too small. However, don't let this hide the strategic opportunities of a well-timed war. If you are weak, consider removing citizens from the board and trigger a war that will be lost and hurt the strongest players the most. If you are strong, consider adding citizens to the board and trigger a war that will be won and benefit the strongest players (you!) the most.
To summarize, there are three important elements of successful strategy in Politeia:
Balance your actions between talent actions, investment actions, citizen actions and opponent actions.
Be observant on tactical actions opportunities, such as taking cheap actions, leaving expensive
actions to your opponents and blocking their actions with your citizen.
Monitor the military and political level so that you can predict when a Greco-Persian War will be
triggered, which side will win and how to allocate
your citizens to be on the right side.