(6) Continued. So extensive and intensive influence on their own will vary inversely, but obviously this can't be the whole story, and would make for a very boring game -- philosophical movements would simply isolate themselves into little bubbles as their legacy becomes more esoteric and they become less and less in touch with the spirit of the times. In reality, much more than time is relevant here, which brings us to Institutions.
Every player starts with an Institution -- the player's Academy or Lyceum -- and over time develops the ability to found new ones. Institutions affect both extensive and intensive influence, although in what degree will depend on the particular Institution; in general, the more Institutions you have, the more rapidly your intensive influence develops, while at the same time an Institution serves as a platform for persuading others by improving the aesthetic and logical presentation of your arguments (even a very passive Institution can slightly increase the chances of persuading passers-by). They also contribute research points to the development of Doctrines. Over time Institutions can also be made more powerful: for instance, Schools can develop into Colleges, which can be joined together into Universities.
Very strong! However, there is an obvious drawback to Institutions: they require resources, and the more you have, the more they require, and the more powerful your Institution, the more it ties up resources. For the purposes of our toy model, we can consider two kinds of resources, land and money. At the City level of play, land is a limited resource -- due to external factors (i.e., population growth and chance) it can occasionally increase, but this can't be counted on. While some temporary Institutions (like Correspondence Networks or the Internet) do not require a physical location (at least any of any serious significance for our purposes), most major Institutions do, and there is only so much to go around. (It's an interesting question whether we should have a location effect. One can imagine designing the cities as centered around an Agora or Forum, the locus of city government, and Institutions in the pricier land near it giving a somewhat greater advantage in influencing the government, or Institutions closer to residential areas having greater weight with the populace. But we will set aside such things here.) Obviously land will be a greater issue at the City level of play than at the Empire level of play.
Far more important in general, however, is money, which represents (relatively) unlimited kinds of resources that have to be collected and then used. Institutions are a constant drain on such resources, and when those resources dry up, Institutions fail. (For practical purposes a game should probably make an unrealistic exception for the starting Institution.) In terms of the game, each player could receive a very small income, which can be augmented as more people join the philosophical movement and are thus able to support it. Bad relations with the city government can increase tax burdens on Institutions, and good relations can both decrease tax burdens and, if they become excellent, make possible occasional direct support from the government, just as the Hellenistic schools survived as long as they did in part because of support from cities and, eventually, the Roman Empire. Likewise good relations with a religion can increase the chances of occasional windfall, in which religious donations are used to support your Institutions. But stewarding this small stream is quite important: if you overextend yourself in Institutions, you may lose them all.
(7) Institutions are powerful, but they are relatively passive pieces on the board. More active are Texts. Texts come in two kinds: stand-alone and critical. Stand-alone Texts increase your extensive influence, both passively (people are more likely to come to your side on your own) and actively (they give you advantage in Debate); they also contribute to your intensive influence. However, in real life, texts form networks and chains as one text criticizes and refutes another. Lady Mary Shepherd's book on causation, for instance, cannot be understood at all without reference to Hume's work. Critical Texts allow you to nullify the influence of an opposing movement's Texts (each Text being able to be linked to a specific opposing Text), reducing the ability of those texts to have an effect (how much they reduce it will be randomly determined when the Text is created); Critical Texts can nullify other Critical Texts, allowing for some complicated interaction.
Texts require resources in order to be created, but unlike Institutions, Texts are not in themselves constant drains on resources, nor do they require any land. They do, however, depend on a kind of resource which we might call iterability. One of the more important material constraints on philosophical arguments in real life is their ability to be carried through time -- some arguments ceased to be used and some positions cease to be held not because they were ever refuted but just because they stopped being repeated. This is most clearly the case with philosophical texts. Whether you think Lady Mary Shepherd was a complete refutation of David Hume as a matter of rational argument, in some sense it doesn't really matter: even if she was, Hume's works were continually reprinted, and Shepherd's works were not, so that eventually there were generations who had read Hume but had never even heard of Shepherd. The Stoic Chrysippus was a famously prolific author in his day; but we have not a single one of his works, so the effect of his having written them all has long since ceased to be significant. In other words: Texts do not last forever on their own. As time goes on, the chances of losing the Text increase. With certain Institutions -- Libraries, Scriptoria, Publishing Houses -- you can increase the iterability of your Texts, and possibly even keep your Texts going indefinitely, but Institutions, again, require land and money. (I think there's a good case for arguing that Stand-alone Texts should have their interability increased slightly by Critical Texts attaching to them, which would add an interesting quirk to game play, since under certain circumstances using a Text to criticize a Text may increase the length of time it has an effect.)
(8) While more active than Institutions, Texts are still fairly passive, and both Institutions and Texts are relatively slow in how they work. If you really want to dominate the city, you need to get out and convince people! And this brings us to Debates, one-time events by which you can increase the number of people who are persuaded by you and perhaps steal followers from other philosophical movements -- or even, if you're very lucky, capture certain kinds of Institution from them! As the number of people in your philosophical movement grows, you will be able to assign limited portions of them to engage with other growing philosophical movements in a Debate. Through Institutions, Texts, and Doctrines, and by developing extensive influence generally, you develop points that allow you to outmaneuver others in Debate.
Debates basically have two targets: the general populace and the opposing movement. The general populace, of course, follows the Atmosphere of the city, whereas the philosophical movement has its own traits. This is important for their effects: if you do very well in an argument, you are more likely to convince people who share more traits with you. A second factor relevant to how effective a Debate will be (if you win) is intensive influence: opposing philosophers are much harder to convince than the general population, due to their greater intensive influence; and it is easier to pull away people from a movement that has relatively little intensive influence than to do so from one that has considerable intensive influence.
(9) Every so often in history a philosophical movement benefits from extraordinary talent; it's not a predictable thing, although perhaps education and the like can increase the chances. So it makes sense to have a Great Mind feature. Great Minds show up quasi-randomly; they can found Institutions and create Texts at cheaper prices, help you develop Doctrines, and participate in Debates. All other things being equal they are more powerful Debaters than you'd get simply by selecting part of your population, but I think for our purposes we can divide Great Minds according to approach: some are Traditional (call them Sages), others Encyclopedic (call them Researchers), some are Genealogical (call them Critics). Which you get is affected partly by chance, partly by the Institutions and Doctrines you have, partly by the Atmosphere of the city, and partly by the traits of your own movement. In Debates, Sages will do best against other Traditional movements, Researchers against other Encyclopedic movements, and Critics against other Genealogical movements. In terms of the rest, each kind of Great Mind has the most effect for the society that shares its primary trait.
(10) All these give the basic elements of play, but there are more complicated dynamics to consider, and three in particular: interaction with the city government, interaction with religion, and general interaction (and especially alliance and bad blood) with other philosophical movements. More on this in a later post.