At the tactical level, the AI for A&A is a minmax on a zero-sum game tree with some fuzziness. We have a structure that represents game state (board and pieces), the relative score of each player for that state (value of troops and land income), and a set of transitions (i.e. moves) from that state to new states. Transitions can weighted by probability of outcome (this is what the simulators in other answers supply). The relative values of starting state and ending states can be compared to determine which moves seem good or bad. These results are scaled by the weight (likely outcome). More sophisticated implementations would go deeper into "promising" branches to check opponent payoff (i.e. whether this move give an opponent an even better counterattack opportunity).
This is a very, very big game tree. Pruning will be essential, and there are some obvious things to try ignoring (attacking with less than all your troops, for example). But it's relatively straightforward AI problem and there is a lot of info about solving it on the web. Maybe it's a bit harder because of the 5 players, 2 teams mechanic.
In addition to picking battles, there's the purchase, non-combat move, and deployment phases. The interesting problem with these choices is their impact will be felt 2-4 steps down the game tree. It might be possible to model this into the tactics tree, or its own separate tree, but the breadth intimidates me. 2-4 turns, 5 actors, 1-11 (on average) new pieces, 12 or so possible pieces, multiple possible deployments... If I had to try this, I'd tackle it in separate stages. Movement can be handled as first minimizing opponent payoff in one transition. If payoff is minimized without movement, stack to strongest front. If no strongest front, go to front of preference. I'd probably hardcode deployment, and placement of any additional factories, until everything else was working and I had lots of free time. I'd try purchasing as a kind of lookup table based on current troop IPC value, or if playing with infantry defends on 1, try purchasing with troop ratios (that is, try to have 6 infantry to 2 artillery to 2 tanks to 1 plane, etc.)
At the "grand strategy" level (i.e. if no strongest front), fronts of preference are defined per standard strategy--Axis prefer to attack Russia, then Britain, then US and Allies prefer to attack Germany, then Japan.
Of course, there are other ways to do it and mine is just one step removed from brainstormed. There's surely better.