The goal of game theory in this case is as follows: to find a mixed strategy to follow for the game which optimizes overall wins.
An optimal strategy in a reciprocal game such as this is one which wins more often against any deviation from itself.
It is important to note that for mixed strategies, you have to choose as randomly as possible with the frequency of each pick being in line with the ratio found as being optimal. Predictablility is your enemy, as if an opponent knows what you are going to do, he has a heads up on countering it effectively.
Randomness insures that even if he knows your strategy, he will have a harder time countering a specific instance, because he can't accurately predict your exact course of action.
I can't recall exactly how one goes about determining the ratio for the mix of pure strategies ("Pick 1", "Pick 2", etc), but I do note that 1 has better overall odds then 2 or 3 in the scenario that the enemy is equally likely to pick 1-3 randomly.
If both limit themselves to 1 or 2, then 3 becomes a winning pick, oddly enough, while 1 or 2 tend to break even. (I guess 3 does have a use in a 3-person game..)
Both pick 1-3 randomly
1 gains 4/9 on average.
2&3 lose 2/9 on average.
4+ tends to break even..
Both pick 1 or 2 randomly
1&2 tends to break even.
3+ gains 1/2 on average.
Of course, your opponents might also behave based on this information, and could change their strategy AND the resulting effectiveness of the pure options. The whole thing loops back on itself, which is why mixed strategies with randomness come into being..