Plants vs. Zombies amazon is a very popular game.
The game, according to Wikipedia, is a type of game called Tower defense. As you can see in the video, each type of plant has particular properties, such as shield level, energy level, fire power, and position etc. Same for zombies. If you watched the video, you might wonder what's the optimal strategy, given the number of incoming zombies, their type, and position.
This aspect is similar to many games, for example Second Life's Tech Warfare, and lots other video games. In a sense, hundreds or thousands of these video games arbitrarily creates a math optimization problem.
If you are the designer of the game, you also might wonder what's the optimal strategy. In fact, if you are math oriented, you might start with a formalization of the math core of the game. To understand this, let's use the game chess as a example. Given a chess game, for example, you know all the positions of the pieces, you want to have a algorithm that can tell you who's winning and by how much. In other words, a score system or evaluation system. This is almost the first thing you need to do.
In the “plants vs zombies” game, the cute plants, the monstrous zombies, their animations, are not relevant to the core mathematical aspect of the game. The core math aspect is just several plants, their types, their position, available plants, and each plant type's attack/defense properties. Same for zombies. Once you have the game play abstracted this way, you could easily create different games that feels vastly different, yet mathematically the same. The zombies could be aliens, monster, or missiles. The plants can be policemen, warriors. It could be naval warfare, or like Sim City's building vs nature. Or, the whole game can be abstract like chess, where the zombies are just black abstract pieces, and the plants are just white pieces, and their types represented by different shapes of the piece.
I don't think many games started with a formal system in their gameplay. But am pretty sure some games are started as some abstract gameplay system, then theme and animation are added on. For example, i think TRON's Light Cycle game might be such. 〔☛ TRON Light Cycle Optimal Strategy〕
This optimal strategy and formulation of a game reminds me of Cellular Automata as discussed by Stephen Wolfram's A New Kind of Science. You can easily generate almost infinite varieties of simple systems, yet it's basically impossible to derive some theorems or optimal strategy for it.
Also, in programing games, you have to create a AI system of the characters controlled by computer. (⁖ those ghosts chasing pacman; the paths they take.) If you have some formalization of the gameplay, i suppose it might help in designing the AI.
Video games is truely fantastically fascinating. I think they are the forefront of computing and research in many aspects.