A lot of game AI has just been set of arbitrary rules, but even in those cases, there are primitives that involve everything from good path finding algorithms, that account for flow and collision detection (think bridges, danger avoidance, primary vs secondary goal prioritization, etc.), to visibility and hearing driven reaction (very geometrically intensive) and player prediction. These are often very math and algorithm heavy, and thus far, far from solved.
I would argue that there is a lot of math involved in game AI today, much more so than there used to be, and that the simple dodge and chase mechanics that rely on scripted events are being phased out, or at least being used to incrementally augment more and more generic parameter based behavioral engines.
On what observations do you base this argument on? I'd be interested if you have links to specific papers/companies that are trying to integrate more complex AI into their games.
I've just graduated with an Artificial Intelligence degree and I, like the OP, had a strong interest in AI for games and tried quite a bit to find relevant research projects/work in this. The key point is that the purpose of AI in games is not to build the most realistic experience, but always to make the game fun. The reading I did revealed that most pathfinding was simple A* search, and the whole idea that you can 'cheat' in the game (ie the NPCs actually know where the character is even if they're not meant to) makes most of the 'AI' trivial. I read a nice piece on AI in Left4Dead2 where basically the whole dynamically generated levels/difficulty are simply hard coded based on map and player position.
I suggest you go do more research then, AI is very math heavy matrix algebra, bayesian statistics, calc etc. Look at the AI in crysis it has machine environmental learning in it.
HDS:
http://web.media.mit.edu/~jorkin/aibooks.html
I would say the most content heavy books on those links and most definitely not the ones that deal exclusively with game programming, but rather NLP, planning, and data mining (reading some of those textbooks now actually). Like you correctly say, AI is very math heavy and a lot of success stories come from Bayesian statistics. But Bayesian statistics is really only applicable when trying to reason about uncertainty with prior assumptions (see no free lunch theorem) - how and why would need to apply this in a game environment? The game engine knows where the player is at any time, what he is doing, and this is relayed to the engine to make the game fun. I don't think games strive to be realistic by trying to imitate human behaviour - realism is conveyed more through lovely textures, graphics, and story as a work of art.
I think these misconceptions come from the idea that AI always has to do with imitating human behaviour. In the academic world today, there is at least a clearer breakdown as to what AI encompasses, see:
http://en.wikipedia.org/wiki/Artificial_intelligence#Goals. The big fields of research today are very different from what you may expect, especially in the state of the art approaches.
What I was getting at is this:
Never heard of the phrase. Nor has Google apparently. Taking a wild guess I'd say you mean:
http://en.wikipedia.org/wiki/Reinforcement_learning
RL is a very active field of research with some great applications to robotics, trading agents, but I really haven't seen RL applied to computer games. It's a very tough field and I doubt extending NPCs with RL is going to make a game like Crysis fun.
The application of AI to Crysis I think you are referring to is the Bayesian skill rating system TrueSkill. This is a nice example of some 'hardcore mathematics/stats needed for AI'. In multiplayer games, like Crysis or Halo, teams of players play matches and either win or lose. Image that the 'skill' of a player is a hidden variable in all of this - you directly observe game outcomes and want to infer individual skill, to ultimately assign ratings to players on which the game system can carry out match making. It is clear that there is true uncertainty now that can be handled by a Bayesian framework.
Here is the TrueSkill paper:
http://research.microsoft.com/pubs/67956/NIPS2006_0688.pdf , here's a more readable FAQ:
http://research.microsoft.com/en-us/projects/trueskill/faq.aspx. To me this is a 'true' application of AI (more specifically machine learning) to games, and an active area of research in CS at Cambridge & Edinburgh. If I remember correctly, it is used by almost all Xbox Live games. To understand how TrueSkill works and build on it, you're going to need to understand probabilistic graphical models, bayesian statistics & machine learning, on top a basic foundation of LA. I haven't seen of work like this developed in industry - Microsoft is of course supported by Microsoft Research which does cool stuff like this.
Good luck HDS and if you are still interested in AI, consider it more generally, there are a lot of fun areas like Natural Language Processing(see:
http://www.inf.ed.ac.uk/teaching/courses/inf2a/schedule.html), Machine Learning (
http://www.inf.ed.ac.uk/teaching/courses/iaml/), Vision (
http://www.inf.ed.ac.uk/teaching/courses/av/), Agent based AI (
http://www.inf.ed.ac.uk/teaching/courses/inf2d/timetable/), Machine Translation (
http://www.inf.ed.ac.uk/teaching/courses/mt/lectures.html). Consider doing a masters in AI at a UK university like Edinburgh, there are scholarships, and a very strong place to do a PhD.
Edit: Here's a MDP used in Civ:
http://people.csail.mit.edu/camato/publications/LearningInCiv-final.pdf