Some thoughts from designing two-player games:
1) Damage and score are equivalent. This isn't quite true: score can be unbounded if there's another condition for the game ending, but a target score is equivalent to a target amount of damage; competing to be the first to deal 100 damage is identical to competing to be the first to score 100 points. There are some perceptual differences: numbers going up feel different to numbers going down, and being able to recover damage done to yourself feels quite different to being able to reduce your opponent's score. The abstractness of score is generally more appropriate for more abstract games.
In multiplayer (in the sense of "more than two players") this symmetry is broken, and score/damage lead to quite different dynamics.
2) Everything is zero-sum. In multiplayer games it's common for different objects to have different values for each player, and this can add a lot of depth. In a two-player game, everything has the same value for both players; anything that helps one player is just as much hurting the other. If there's something only one of us can get, it's value to me is "what good it does for me to have it" + "what good it does for me for you not to have it". The value to me of you not having it is exactly how good it would be from your perspective for you to have it, so this equation is symmetric in the players - for example, something worth 3 points to me and 5 to you makes a difference of 3+5=8 points between our scores, no matter which of us gets it.
So lots of mechanics common in multiplayer games just don't work very well in two-player. If something is more effective for one player because of synergies with other powers, it's equally desirable for their opponent to prevent them getting. If an auction determines who gets something, each player should be willing to pay the exact same amount. If an action benefits an opponent as well as yourself, that is a strict disadvantage.
As a side-effect, when balancing a game sometimes the value of an object turns out to be double what you might expect - for example, in Exuberant Struggle sometimes a "new life" powerup appears; collecting this is actually a swing of *two* lives, because it gives you one and prevents your opponent getting one. (If you're in a situation where you can't possibly get the powerup, there is a third option - destroy it so that your opponent can't get it; a swing of one life, equivalent to killing them!) In ADAM ATOMIC's Terratri, capturing an opponent's territory is worth twice as much as capturing unoccupied territory but has the same cost - this gives rise to a degenerate "snaking" tactic of continually exchanging territory (Vertex Dispenser would suffer from the same problem if capturing enemy vertices were not sufficiently more slow and risky than capturing neutral ones).
3) In order to break this symmetry and get each player doing something different (without which the game is not very interesting) it's necessary to obscure the choices somehow so that it's hard to select the best option*. Some level of obscurity also helps to soften the essential viciousness of a pure zero-sum game - e.g. in games like Magic, Race for the Galaxy, Dominion (2-player), both players have a constructive feeling of "building up", even though with each action one player's raw chance of victory decreases.
Obscurity can be achieved through:
- Computational complexity (as in Chess, etc.). There may be an optimal choice, but it's very hard to work out what it is. If there's a tight time limit on making decisions then the difficulty of finding the best move doesn't have to be so high, and if there is unlimited time available and the complexity is too high this can lead to "analysis paralysis" as a player tries to compute their ideal move.
- Randomness or hidden information, so that it's impossible to determine the best choice using the information currently available. This isn't a completely satisfactory method because there will still be a statistically optimal choice based on what you do know (often this optimal choice is to randomly select between several alternatives with specified weights on each).
- Difficulty of execution, so that even if you know the best move it is still challenging to perform it.
- Secrecy of rules: simply not giving players access to what they need to know to make the best choice. This option is interesting to me, because it's only really possible with videogames, but it's not entirely satisfactory - it's more elegant for a game to offer difficult choices even when the rules are fully understood. It's also not very reliable; a sufficiently motivated player can always reverse engineer the rules.
* While obscurity is of special importance to two-player games, it is a key element of games in general, and in this they are distinct to puzzles - you want a puzzle to be eventually solved, even if the solution may be very difficult, but it's better for a game never to be completely solved.