I gave this a very quick review while still on my first cup of coffee.  The answer is not very detailed, but then there were a lot of things I caught even with a quick scan.

The public fields at the top of the `RockPaperScissor` class should either be `private fields` or `public properties`.  The one exception to public fields are constants. Note a static readonly value cannot be a constant, so if you have any of those they should be properties (be it public or private).

If you use public properties, the naming should be Pascal-cased, so `isPlaying` should be `IsPlaying` (if you think it should be exposed publicly).  Private fields or properties should begin with lowercase; public fields or properties should begin with uppercase.

`isPlaying` is initialized to `true`.  It should be `false` and you should set it to `true` at the top of `PlayGame`.  You may want to put a check at the top to make sure a game is not already in progress.

I'd like to see a enum for { Rock, Paper, Scissors } instead of literals "ROCK", "PAPER", and "SCISSORS".  This requires better validation for the user input to map that input to the correct enum.  I would also recommend using a `switch` instead of cascading `if-else-if's`.

`ShowResult(int flagNum)` could be improved.  I don't like flagNum as an integer and would prefer to see an enum of GameResult = { InProgress, ComputerWins, PlayerWins, Draw }.  There should be a class level public readonly property to hold the GameResult.  And `ShowResult` should not take any parameters but instead read that class level public readonly property.