Who ever wrote this board is not taking many factors into consideration, lets hypotheses:

First let's set the usual rules of the game: two players stand regularly about a meter (ish) in front of each other with their arms not fully extended waiving their fist from the elbow approximately six times until they present their chosen result. The winner is decided because in theory, the physical properties of on the results will either immobilize or destroy the other.

For argument sake, let's say both players have personal space/physical contact issues and leave 20 - 30cm between their hands and selected outcome. Force and acceleration must then be limited to this distance and the size of the objects (whether with physical or metaphorical objects) those of a clenched fist and an forward facing open palm.

We have now enough points of reference to begin applying physics (and a bit of anatomy) to defend papers potential to indeed beat/absorb the impact of rock.

Physical:

Both players are each holding a physical rock (either a smooth pebble like rock or one with sharp edges) and a piece of (A4 sheet) 20-30cm apart. If the player with the rock has only said distance (maintaining it's current anatomical posture) to give force/acceleration to his object and paper by being passive the same distance to absorb the impact (with an forward facing palm), it is rather unlikely (unless the rock has pronounced sharp edges) to rip through the piece of paper and therefor destroy it's original physical properties. Sure, paper may crumble but it would have done its job of immobilizing the rock and can always be straightened out again. It would really come down to where the applied pressure points and density (to and of) the paper are/is. In the given example, paper wins (likely).

Metaphorical:

If both players have same strengths and build (applying the same rules), unless the rock player has mastered Bruce Lees two finger punch building ridiculous acceleration rates over relatively short distances, rock is clearly the looser.

Wasn't this fun?