A simple rock, paper, scissors (RPS) system of direct counters is a perfectly solid and legitimate basis for a strategy game provided that the rock, paper, and scissors offer unequal risk/rewards. Better still is if those rewards are unclear, meaning that players cannot easily determine the exact values of the rewards. The following video is not an example of that, but it's pretty exciting looking.
An Example of $10/$3/$1
Consider a strictly equal game of RPS with clear payoffs. We'll play 10 rounds of the game, with a $1 bet on each round. Which move should you choose? It makes absolutely no difference whether you choose rock, paper, or scissors. You'll be playing a pure guess. Since your move will be a pure guess, I can't incorporate your expected move into my strategy, partly because I have no basis to expect you to play one move or another, and partly because I really can't have any strategy to begin with.
Now consider the same game of RPS with unequal (but clearly defined) payoffs. If you win with rock, you win $10. If you win with scissors, you win $3. If you win with paper, you win $1. Which move do you play? You clearly want to play rock, since it has the highest payoff. I know you want to play rock. You know I know you know, and so on. Playing rock is such an obvious thing to do, you must realize I'll counter it ever time. But I can't counter it (with paper) EVERY time, since then you could play scissors at will for a free $3. In fact, playing scissors is pretty darn sneaky. It counters paper--the weakest move. Why would you expect me to do the weakest move? Are you expecting me to play paper just to counter your powerful rock? Why wouldn't I just play rock myself and risk the tie? You're expecting me to be sneaky by playing paper, and you're being doubly sneaky by countering with scissors. What you don't realize is that I was triply sneaky and I played the original obvious move of rock to beat you.
That may have all sounded like double-talk, but it's Yomi Layer 3 in action. And it had quite a curious property: playing rock was both the naive, obvious choice AND the triply sneaky choice.
Math Says There is a Solution
You might say that even with unequal payoffs, there's still an optimal way to play. The optimal solution is called a mixed strategy, meaning that it involves randomly choosing your moves, but obeying certain percentages.
You should play rock 10/14ths of the time, scissors 3/14ths, and paper 1/14th. If you play against another player who is playing suboptimally (for example, he plays paper 100% of the time), you can change your strategy to exploit him (by playing scissors 100%). But the optimal mixed strategy above means that no one can exploit you to do better.
While that is the math answer, three related factors creep into the real-world application of that strategy:
1) People are very bad at actually playing randomly, especially at specific percentages such as 3/14ths.
2) When people fail to play randomly, they are probably falling back on tendencies they do not know they have, but that you can detect and exploit.
3) People cannot help but let their personalities spill over into decisions about how conservative (playing paper) or risky (playing rock) they are.
Fighting games rely heavily on RPS. They have both overall games of RPS going on as well as many rapid fire situations of RPS. Virtua Fighter games can even have 5 sets of RPS take place in a period of 2 seconds! Really!
Virtua Fighter's overall system of RPS is as follows: attacking beats throwing, throwing beats blocking or reversing, and blocking and reversing beats attacking. To be clear, let's define terms.
An attack is a move that deals damage. An attack has an initial startup phase where it can't yet do damage (a punch extending), a short phase where it actually can do damage (the sweet spot of the punch), and a recovery phase (the arm retracts). If the defender is blocking correctly, an attack will not damage him, but he can be thrown.
A throw is a special type of move that instantly grabs an opponent whether he's blocking or not and does damage. The catch is, a throw will not grab an opponent who attacking (specifically, a throw will fail if the opponent's move is in startup or hitting phase).
A reversal is a special type of move that grabs an incoming attack. Reversals usually look like throws, but they work at the exact opposite times. A reversal only works when the opponent's move is in startup or hitting phase, which are, incidentally, the only times a throw would fail.
Even these explanations are simplified, but the RPS system is basically there. Attack the opponent. If they tried to throw you, you'll hit them. If they block or reverse your attack, they nullified your attack. If you expect them to block, you can throw. If they expect you to throw, they can attack.
The fighting game Dead or Alive basically uses this same system, except that the risk/reward for doing a reversal is much different. Reversals are difficult and relatively rare in Virtua Fighter, but they're incredibly easy and do a ridiculous amount of damage in DOA. Reversals are so effective, in fact, that they can paralyze the enemy into not attacking for fear of being reversed. Of course, that's when you throw them....
While psychology makes it difficult for people to deal with unequal payoffs, it can be even more difficult to deal with unclear payoffs. Imagine that you are making an RPS decision in the fighting game above, and you must consider "how bad" it would be if you guessed wrong and got hit by an attack (as opposed to guessing wrong and getting thrown or reversed). How much damage will you take?
It depends on which character you are figthing, and which character you are. It depends on the distance between you and the opponent, and on the timings involved: maybe it's likely he'll do a launcher into a juggle combo or maybe that's not reasonable but you fear he might do a stagger into a ground combo. Is your character's back toward the wall, meaning that your opponent could get extra damage from a wall combo? Is your back near the edge of the ring, meaning you might lose the entire round to a ring out? How good is your opponent at doing combos? Is he likely to really maximize his damage, or just get one hit?
It's extremely unclear what the payoffs of this situation are. A half-second later when you are in another guessing game, the payoffs will be different (maybe the distance between you changed) and it will still be very unclear. Cutting through all that and making a reasonable guess requires knowledge of the game, of the opponent, and the presence of mind to put it all together. It is a real skill (I call it valuation) and a very valid skill to test. If you can make payoffs unequal AND unclear, then you've already gone a long way toward making a good strategy game.
RPS in RTS
Real-time strategy games such as StarCraft also use the RPS system. Like fighting games there's the concept of RPS on large scale and a small scale. On the small scale, particular units are designed to counter each other in a RPS way. A marine dies to a guardian. A guardian dies to a corsair. A corsair dies to a marine. Abstractly, there are 6 categories of unit. Ground units can either attack 1) other ground units, 2) air units, or 3) both. Air units can attack 4) other air units, 5) ground units, or 6) both. Pure ground-to-ground units usually beat both other types of ground units, yet lose to both types of air units that can attack ground. Similarly, pure air-to-air units usually beat both other types of air units, but loose to both types of ground units than can attack air.
RPS is not limited purely to units countering each other though. Real-time strategy games also have the concept of trading off powerful units now for a strong economy now, which leads to even more powerful units later. So on one extreme, a Zerg player in StarCraft might sacrifice his entire economy to get a quick attack force ("6 pool" is the term). This will likely beat a player who chose the other extreme of playing for pure economy and no immediate attack force (by building double oven triple hatcheries). A moderate build (pool on 9th peon, one sunken colony) will likely defend against the early attacker's rush, though. Surviving the rush, the moderate build will have a much superior economy and win in the end. However, this moderate build will produce an inferior economy to the player who built 2 or 3 hatcheries and went for pure economy.
In Starcraft, the early rush is a very, very risky strategy. It's all or nothing. You'll either win right away off it, or your rush will fail and you'll almost surely lose. Because of this, the early rush isn't all that common (depending on the map), but the very threat that the opponent might play the early rush is enough to stop you from playing for pure economy every time.
Finally, notice how hard it is to determine the actual payoffs in StarCraft. If your correct guess results in a battle between a few enemy Zealots and several of your Marines, what is the payoff? How many Marines will you lose? It depends on the micromanagement skill of both players, the terrain, and whether each player even focuses on the battle at all (maybe there's a more pressing battle somewhere else on the map). A lot of the goodness of StarCraft's design is that it's full of RPS with unequal and unclear payoffs.
Back to Basics
I'll leave you with this glimpse into the crazy world of people who don't seem to care about unequal or unclear payoffs.