Some years ago, I worked abortively on a project whose goal was to generate games that would represent various viewpoints via their mechanics. That turned into something different, a system for auto-reskinning WarioWare games so the reskinned game at least loosely "makes sense" (while not necessarily representing any particular viewpoint).
In any case, I still became interested in tracking the ways in which games, especially newsgames (small editorial games on current events), make rhetorical points, both via what Ian Bogost calls "procedural rhetoric" (making a point via the game rules), and through other kinds of rhetoric, like the way the game is positioned. I started collecting such games, and thinking about how their rhetorics can be categorized. I've come up with six general types. I'm not sure this is exhaustive, especially as far as covering all possible rhetorics, but it seems to cover most of the rhetoric that existing newsgames actually employ.
You are thrust into an unwinnable game, which is unwinnable because you're trying to employ (or forced to employ) a strategy that simply will not work. The goal is to highlight in some way the futility of the strategy, partly by showing you that no matter what variants you try out within the space of possibilities the game allows, they all fail. The canonical example is Gonzalo Frasca's September 12, intended to highlight how bombing terrorists is a futile strategy, because it just produces more of them. A fictional precursor might be the game/simulation Global Thermonuclear War in the film WarGames, which serves the rhetorical purpose of showing that there is no winning strategy in a nuclear exchange, because you always die ("strange game... the only winning move is not to play").
You are thrust into an unwinnable game, which is unwinnable because you're stuck in an untenable/unfair/absurd situation. The goal here isn't to highlight the failure of a strategy, but the crappiness of the situation. Either the situation inherently sucks, or it sucks because you've been given unreasonable goals, and/or you've been given insufficient tools/resources. It's possible this should be split into additional rhetorically meaningful categories, depending on who/what exactly is being blamed for the unwinnability: is it the rules of the situation, the nefariousness of an enemy, the inadequacy of your preparations or supplies, or your implied choice of goals? Some examples include: Operation: Pedopriest, where you're a Vatican official trying to both protect children from priests and also protect priests from police/parents/media; Food Import Folly, where you try to stop tainted food imports at customs; and Al Quaidamon, where you try to treat war-on-terror prisoners well enough to satisfy human-rights advocates.
A main rhetorical use of simulation logics is basically to highlight sliders, and point out that they exist and have effects, and might interrelate. Chris Crawford's Balance of the Planet is an influential older example, pointing out the interrelationship of many ecosystem elements; and AntiWarGame is a more recent newsgame example.
Highlight some situation by having the player simply play through it (either just to remind them of it, or to highlight its absurdity). The ambiguous 1999 game Pico's School is a possible example, as is the more recent newsgame Points of Entry.
A normal game mechanic that's been skinned with a news event. It's not necessarily impossible to make an interesting political game this way, and indeed my former research project was based on the assumption that it would be possible (which I still think is the case). Nonetheless, this category is somewhat notorious for really superficial games, like John Kerry Tax Invaders and Kerryopoly.
The way a game is billed, or the fact that something is represented as a game, or something else outside the game, is (at least part of) the point. A good example is Harpooned, a whale-hunting game that is deadpan billed as a "Japanese Cetacean Research Simulator", satirizing the Japanese government's official explanation for their whaling fleet. Another possible example is Presidential Pong, which depicts presidential debates as a game of ping-pong.
Follow-up: For an in-depth case study of the category "general mechanics with a skin", Mike Treanor, Michael Mateas, and Noah Wardrip-Fruin published a nice paper at FDG 2010 on how to use the mechanics of the game Kaboom! in all sorts of meaningful ways.