Let me commence this discourse with a confession: Whenever someone proposes to play a Risk-like game, I try to alter the proposal or I try to avoid that particular gaming session. I don't like Risk-like games, and in my eyes they are closer to a waste of time than to an enjoyable game.
Before I shall try to expound my harsh words, I want to deliver a precise definition of what I call Risk-like game. A Risk-like game has its players control territorial entities on the board; players conquer regions from other players; the more regions a player controls, the better; to obtain additional strength, players need to conquer additional territory; to win the game, players need to meet certain goals which are either linked directly to territorial possessions or are made easier obtainable by territorial possessions (e. g., "occupy 18 regions with 2 armies each" or "exterminate the black armies").
All basic risk-like games, no matter whether they are called Risk, Quest for the Dragon Lords, Mare Nostrum, Attack!, you-name-it, share some common vices:
To sum up: Risk-like games are problematic for intrinsic reasons. If you do know your fellow gamers and their probable reactions quite well, it's worse. Sessions get outright boring. Risk-like games differ in the kind of resources players are supposed to conquer, in the way combats are resolved and in the goal to reach. But this is not enough to be enjoyable, at least not in my opinion. And perhaps the worst thing: Risk-like games tend to fill up a whole gaming session.
I don't know of too many successful attempts to overcome the deficiencies of Risk-like games. Those I do know however belong to my all-time favorites.
There are first of the all the Britannia-like games. Britannia-like games have players to control peoples of different strength, different objectives, and different interests. The one player gains most of his victory points early in the game, the other one late. Every player shares common interests and common conflicts with any other player. If a player is especially successful with a certain people, this is of course helpful for winning the game. But there isn't yet a definite decision. His powerful people will whither and fall, and we'll see how he'll do with the next one. Britannia-like games need extremely thorough playtesting because players should obviously have fair chances of winning. After 50+ games of Britannia, I can say that except for purple, chances are equal (at least there's not much deviation for purple - this player can be very sure not to be last and will probably be second). Maharaja is a mere catastrophe. If I ever had bought this one, I would sue the company. I've played Hispania twice, and it looks quite well-balanced. I haven't played Rus yet.
Britannia-like games reshuffle the deck (or recast the dice, whichever metaphor you prefer) every once in a while with the arrival of new peoples. This eliminates issues 1 and 3. You cannot win the game with one successful people, and alliances don't last long enough for the sure backstab. Issue 2, ganging up on the leader, isn't an issue either because normally you don't quite know who's leading (well, in Britannia, I would know because I know how many victory points at any given time is more than a fair share, but this knowledge cost me dozens of gaming sessions). Even issue 4 is somewhat alleviated. Players don't stay neighbors all game long. There are several turns in which wounds have time to heal, and when a major aggression (aka Major Invasion) takes place, old enemies might find themselves fighting back against the new common threat.
There's another, entirely different way of making a game of territorial conquest enjoyable. This is Empires of the Ancient World, a masterpiece of game design by Martin Wallace.
Empires of the Ancient World at first looks much more similar to a basic Risk-like game. No different peoples, no objective cards, same victory point conditions for everyone. Players like me find themselves quickly conquering and conquering. Alas, you will quickly notice that the more you conquer, the more onerous it gets. Well, you get victory points for the countries you conquer. You even get a very small reward in playing force. But ultimately, you have just much longer frontiers to guard with upset barbarian tribes (err fellow players) beyond. The more you entangle yourself in reinforcing the army, the more you will fall behind economically. In fact, you not only score by conquest but also by placing trade cubes. The trading cubes are not entirely separate from the conquest mechanism, but quite so. Brave conquerors will soon realize that other players simply score more with some trade and some conquest than by simply rushing with their armies to the end of the world. But as soon as you start to improve your trading position, your military conquests are bound to crumble.
I have once played Empires of the Ancient World with four other players. In the middle of the game, I had reduced two of them to one region each. Both of them found their way into the game again. I think I finished third, and one of my victims ranked second.
Empires of the Ancient World elegantly solves issue 1. This solves at the same time more or less issue 2, because leading is difficult per se, no need for ganging up accordingly. Issue 3 isn't that much present either because peaceful characters can have their way and trade, thus avoiding conflict without toppling the game balance. In other words: If you have a non-aggressive player in Risk, this is bliss for his neighbors. S/he's no threat; s/he is as good as non-present. In Empires of the Ancient World, our shy friend will trade, and successful trade is one way to win the game. So our non-aggressive friend is indeed a danger, perhaps not military but certainly economically and therefore in terms of victory points. So you have to cope with him, in one way or another. You can't simply concentrate on the others.
Now, what about issue 4? There was this player who was reduced to one region at the middle of the game and ranked second at the end. This is unthinkable in a basic Risk-like game, and that's why unceasingly pursuing the player who robbed you of all chances of victory is comprehensible, yes indeed sensible (for reasons of deterrence). In Empires of the Ancient World however it is worthwhile to forget about past grudges since you keep your chance of winning; better concentrate on ranking first in the end. This solves issue 4 if the players are cool-blooded and not that much emotional.
Neither Britannia-like games nor Empires of the Ancient World solve issue 5. I therefore discourage playing them or any other Risk-related game with players whose playing behavior you cannot accept or who object to your playing behavior.