TITLE: Is perfect play possible in continuous rock-paper-scissors? game "step size" vs. "acceleration"
QUESTION [6 upvotes]: The first part of my question is simple: Is every game continuous in time and strategy-space also a game of perfect information with a good equilibrium? For example, consider rock-paper-scissors. The discrete version has no nash equilibrium - a perfectly uniform random mixed strategy is the best option.
Continuous rock-paper-scissors, by contrast, allows players to move at some limited velocity (consider 2 cases, acceleration is limited and acceleration is infinite) through a "strategy space" s.t. R+P+S = 1, and (0.5, 0.5, 0) vs. (0,1,0) returns 0.5 to player one and -0.5 to player 2, while (1,0,0) returns 1 to player one, -1 to player 2. To avoid the "go directly to the middle" strategy, it's fine to remove (1/3,1/3,1/3) or some disk around it from the strategy plane.
So, is continuous RPS effectively a game of perfect information?
For a more dramatic example, consider the stock market as a game. If it were continuous, would randomness essentially be removed? Would a player also need to explicitly know the strategies of all other players as individuals, or only the end result of those strategies (i.e. value of stocks at a given point in time) in order to play perfectly?
For a more realistic example, consider a hunt between a dog and hare. Strategies for them are the direction they choose to run in the pursuit. The dog has reflexes r, the time it takes him to notice the hare's change in direction. The rabbit has acceleration a. Ignore the dog's acceleration for now. As r*a becomes extremely small (i.e. the dog's reflexes are swift relative to the hare's acceleration), does this effectively converge to a continuous, perfect-information game (specifically the game of the homicidal chaffeur), or is the difference still important? Specifically, suppose that the dog can only make decisions on pursuit directions at increments equal to r - I don't want it to be a continuous game with a lagging signal.

REPLY [4 votes]: If I interpret your question correctly, continuous RPS isn't a game of perfect information.  I'm assuming what I know at any instant is A) my strategy at any past instant, and B) the outcome of our two strategies matched up against each other at any past instant.  
But I can't deduce your strategy from knowing mine and the outcome; consider my strategy of (1,0,0) and an outcome of 0 for both players.  I can't tell if you played (1,0,0) or (0,.5,.5) at that moment.  In the continuous game, you could be moving anywhere you please on the line connecting those two strategies while I stayed at (1,0,0), and I would be none-the-wiser.
It is, however, true that in any continuous game where your opponent's strategy can be completely deduced from the outcome, you have perfect information.  This follows by reverse-engineering his strategy at all previous moments, and taking the limit.  Unfortunately, if the outcome is represented by a single real number, then the strategy space for such a game has to be essentially 1-dimensional, which means they are rather boring.