Rock-Paper-Scissors

The game

RPS is a game that probably everybody played when he/she was a kid.

It's a two player game with 3 strategies, you can call "Rock," "Paper" or "Scissors." Rock beats scissors (the scissors loose their sharpness as they cut into the rock), scissors beats paper (it can cut it) and paper beats rock (as the paper can cover the rock and thus disable it).

There is no way of playing (against another intelligent player) such that you can, on the long run (i.e. play it infinitely long) play better than a tie.

Dynamics in populations playing the game

The dynamics of a population of individuals playing the game, is pretty interesting. You can imagine individuals that are programmed (i.e. genetically) to play one of the three strategies, against all other individuals in the population. Using Darwinian selection and reproduction, a new generation of individuals is created using on the individuals of the previous generation.

From this system, quite some interesting dynamics can be observed, if an infinitely large population of individuals is assumed.

• If you start with an initial populations where 1/3 of the individuals play rock, 1/3 play scissors and 1/3 play paper, then the population will always stay in that very same configuration.

• The same thing happens if you start with a population of individuals that all either play one of the three strategies, unless genetic mutation of the individual' chosen strategies is allowed.

• If you choose some other initial population (where all strategies are represented by the individuals, and not all for 1/3 of the population), then the dynamics of the population's configuration seems to continuously run around the central fixed point of 1/3, 1/3, 1/3, unless mutation is allowed where the population will start spiralling towards the central fixed point.

My personal interests in the game

Now we could wonder what would happen if we drop the assumption of infinitely large populations, and stochastically predict the limit behavior (the long run behavior) of that system? Some hypotheses:

• Even if no mutation is assumed, the central fixed point will be lost as the population will always have a (small) probability to move away from the center and end up in one of the other fixed points where it will never be able to escape from.

• If mutation is assumed, the system will no longer be spiralling towards the center fixed point, as it could easily (but less probably) move away from the central fixed point too. It can move to one of the three corners of the state space, but it wouldn't get stuck there because mutation will be able to move the population away from the fixed point. There's a delicate balance of mutation rate (the speed at which mutations occur in the population) and selection pressure within the population, based on the size of the population and payoff received when individuals win, lose or play a tie game.

I'm currently building and analyzing similar systems to study the influence of finitely sized populations when playing the RSP game.

Results

Th picture, shown below, shows some of the results I'm getting out of my experiments. However, as I haven't published this yet, I can't explain what the picture really depicts, so you'll only be able to look at how great this picture is :) I'm slowly starting to gather enough material to start writing a paper on this subject, so some more information might be available soon.