Most children are taught to play Rock, Paper, Scissors at an early age. It's a fun game to teach kids, and one where the strategy of winning (which in theory should be totally random), is based on psychology. It's a game about predicting the next move of your opponent. If you can accurately predict the symbol your opponent will ‘throw’ you are always able to select a different symbol and get a win. |

This is possible because of the cyclic nature of the rules for the game:

Rock |

Scissors |

Paper |

The cyclic property of this scoring system is given a fancy name: Intransitivity

What does that mean? A *Transitive* function or property is something in which all comparisons are performed according to the same dependent criteria. For example, if **Albert** is older than **Bob**, and **Bob** is older than **Charlie**, then we know that **Albert** must be older than **Charlie** too.

This is because However, if I told you that You can read a little more about intransitive games on my posting about How to win free drinks from your friends using dice. |

There are numerous articles on the web that talk about strategy from the psychological perspective. I’m not going to repeat this work. If you want to read more on that angle, Google is your friend. There are also some very interesting articles about strategies for writing computer 'bots to play against human players.

What I am going to look at in this article is sensitivity analysis based on the different concentrations and populations of players and moves (I’ll explain all in just a moment).

First, however, let’s take a detour and talk a little biology. Imagine there are three organisms living on a fictitious island in the Pacific Ocean. These are the Rock Wolf, the Scissor Rat, and the Paper Snake (I think you can see where I am going with this). The Rock Wolf feeds entirely on the Scissors Rats. Scissor Rats’ exclusive diets are Paper Snakes, and Paper Snakes attack, with extreme prejudice any Rock Wolves they see. None of these animal species is cannibalistic to their own species. Each animal has a unique prey, and a unique nemesis. |

On this island, there is a natural state of dynamic equilibrium with the three species balancing each other out. Now imagine what would happen if, suddenly, half the snakes disappeared? How would this effect the ecosystem? How would it change the balance and the ecosystem? |

Imagine there are a lot of people in a room, and they are going to play a giant game of *rock, paper, scissors*. However instead of autonomy in selection, their choice of Rock, Paper, Scissors is pre-ordained. They are automatons and always play the same move.

To play each round, two random people are selected from the pool (the players still in the game) and challenge one another. Each plays his/her allocated move. The winner goes back into the pool to fight another day. The loser leaves the game. (In the event of a tie, both players return back to the pool).

Over time, the number of people in the pool gets fewer. Eventually we’ll be left with just one class of participants. Because, obviously, when we are down to just one remaining class there is nobody who they can attack, or they can get attacked by.

(In fact, once any one of the populations gets to zero it's a sure conclusion for the winner of the game).

The class that survives is determined to be the winner (this could be as few as one person). I hope you can see the analogy between the |

The probability of being the winning tribe depends of the concentrations (relative ratios) of the starting players. The starting conditions effect the chances of surviving to the end. Obviously, if the ratios of Rock, Paper and Scissors at the start of the game are the same, the probabilities of being the last tribe standing are the same (This should be obvious from symmetry, and from the fact the intransitive property that each class is the same with exactly one strength and one weakness each). |

To work out what happens with non-equal starting conditions, we need to model the system.

To model this system, I wrote code to perform this contrived Rock, Paper, Scissors game using a random number generator to select two opponents for each round. Each game was played to completion. To get smooth results, I repeated this exercise 10,000 times for all combitions of starting players from 1-50 of each symbol. I counted the number of times the Rock team won, the number of times the Paper team won, and the number of times the Scissor team won. I also recorded the average number of rounds that were required to produce a winner.

Below are some sample results. The three columns on the left show the number of starting players for each symbol. The three columns to the right of these are the number of times (out of 10,000 games), that each symbol was the winning item. The final column shows the average number of battles until a winner is found. |

First a few obvious results:

# Rocks | # Paper | # Scissors | Rock Wins | Paper Wins | Scissor Wins | AVG moves | |
---|---|---|---|---|---|---|---|

1 | 1 | 1 | 3,317 (33.17%) | 3,292 (32.92%) | 3,391 (33.91%) | 2.000 | |

2 | 2 | 2 | 3,333 (33.33%) | 3,356 (33.56%) | 3,311 (33.11%) | 5.705 | |

3 | 3 | 3 | 3,355 (33.55%) | 3,338 (33.38%) | 3,307 (33.07%) | 9.889 | |

4 | 4 | 4 | 3,352 (33.52%) | 3,360 (33.60%) | 3,288 (32.88%) | 14.289 |

It's no great surprise, as mentioned earlier, that if the distribution of starting entities is symmetrical, so are the results. As you can see, when the number of starting *rocks*, *paper* and *scissors* are equal, the percentage time each wins is approx the same at 33.3% (when averaged out over 10,000 games - the minor differences down to the quality of the random number generator and the sample size).

The average number of moves required to win increases with the number of people playing (another obvious result!) The number of moves required to win with just one of each symbol is obviously two. It does not matter which combination of the three are put together for the first round; one will obviously win, and one will lose. There are then just two entities left (by definition these will be different), so it is game over after this round. As soon as we have more than one of any symbol there can be a tie. A tie increases the number of rounds the game is played, even though nobody is eliminated. |

Things get *really* interesting when we start to adjust the relative concentrations.

Below, I've kept the number of starting *Rocks* and starting *Papers* at ten, and adjusted the number of starting *Scissors*. Can you predict what will happen?

# Rocks | # Paper | # Scissors | Rock Wins | Paper Wins | Scissor Wins | AVG moves | |
---|---|---|---|---|---|---|---|

10 | 10 | 10 | 3,351 (33.51%) | 3,309 (33.09%) | 3,340 (33.40%) | 41.606 | |

10 | 10 | 9 | 2,802 (28.02%) | 3,304 (33.04%) | 3,894 (38.94%) | 40.172 | |

10 | 10 | 8 | 2,307 (23.07%) | 3,304 (33.04%) | 4,389 (43.89%) | 38.870 | |

10 | 10 | 7 | 1,819 (18.19%) | 3,273 (32.73%) | 4,908 (49.08%) | 37.852 | |

10 | 10 | 6 | 1,299 (12.99%) | 3,310 (33.10%) | 5,391 (53.91%) | 36.989 | |

10 | 10 | 5 | 900 (9.00%) | 3,190 (31.90%) | 5,910 (59.10%) | 36.646 | |

10 | 10 | 4 | 521 (5.21%) | 3,401 (34.01%) | 6,078 (60.78%) | 36.462 | |

10 | 10 | 3 | 222 (2.22%) | 3,866 (38.66%) | 5,912 (59.12%) | 36.244 | |

10 | 10 | 2 | 78 (0.78%) | 4,804 (48.04%) | 5,118 (51.18%) | 35.659 | |

10 | 10 | 1 | 6 (0.06%) | 6,611 (66.11%) | 3,383 (33.83%) | 33.336 |

This is fascinating stuff. As we **decrease** the number of *scissors*, at first, the percentage of times that the scissors will win *increases!!!*

How can this be? Well, the nemises for the scissors is, of course, the rock. With a lower number of scissors, it's more likely that a battle will take place between the rock and the paper (which the paper will win), reducing the number of rocks.

This pattern increases as the number of starting When the number of The chances of When there are no |

The keys to winning in this survival race are not just to be strongest yourself, but to make sure that enemy of your emeny is also strong. It's a cliché, but *"the enemey of your enemy"* really *is* your friend (until you have to finally eat him too to win!)

*"The enemy of my enemy is my friend"*

Here is the same data plotted as individual percentages. The |

As we increase the population sizes, however, the *paper* does become more sensitive to changes in *scissors*. Here is a similar graph showing the percentages chance of winning starting with 50 of each symbol instead. You can find the tabular data here.

Interestingly, for these data sizes, the peak chance for *scissors* to survive is when we start with 13 of them. This results in a percentage chance of victory at over 98% when battling 50 *Rock* and 50 *paper*! Who would have predicted that?

Another interesting phenomenon is that, as the population sizes increases, the stability of *Rocks* remains constant longer (and higher) as the *scissors* decrease *cf.* this being *paper* when the population is small.

The chance of a *rock* winning falls monotically (it never increases again), unlike *paper*, which does. When the number of starting *scissors* is 17, the chances *paper* winning is at a minimum, and this percentage chance increases with the number of *scissors*. It passes the chance for *rock* winning at 13, and goes on to overtake *scissors* at 1.

The average number rounds to find a winner has an interesting shape curve too, with a noticeable peak, and a shallow minimum. It's expected that as the number of *scissors* increases the number of moves should increase (there are, after all, more people playing the game!), but increasing also lessens the chance of a tie, which uses a round and causes no elimination.

Below is a table of keeping the number of *scissors* constant, and this time reducing the starting numbers of both *Rocks* and *Paper*.

# Rocks | # Paper | # Scissors | Rock Wins | Paper Wins | Scissor Wins | AVG moves | |
---|---|---|---|---|---|---|---|

10 | 10 | 10 | 3,351 (33.51%) | 3,309 (33.09%) | 3,340 (33.40%) | 41.606 | |

9 | 10 | 9 | 3,324 (33.24%) | 2,839 (28.39%) | 3,837 (38.37%) | 38.557 | |

8 | 10 | 8 | 3,148 (31.48%) | 2,334 (23.34%) | 4,518 (45.18%) | 35.645 | |

7 | 10 | 7 | 2,856 (28.56%) | 1,880 (18.80%) | 5,264 (52.64%) | 33.034 | |

6 | 10 | 6 | 2,352 (23.52%) | 1,513 (15.13%) | 6,135 (61.35%) | 30.531 | |

5 | 10 | 5 | 1,664 (16.64%) | 1,131 (11.31%) | 7,205 (72.05%) | 28.346 | |

4 | 10 | 4 | 1,026 (10.26%) | 916 (9.16%) | 8,058 (80.58%) | 26.631 | |

3 | 10 | 3 | 445 (4.45%) | 746 (7.46%) | 8,809 (88.09%) | 25.704 | |

2 | 10 | 2 | 94 (0.94%) | 637 (6.37%) | 9,269 (92.69%) | 26.586 | |

1 | 10 | 1 | 4 (0.04%) | 1,014 (10.14%) | 8,982 (89.82%) | 31.224 |

Here there are multiple things going on. In the last row, where *paper* dominates, it's chances of victory are still slim because, to win outright, it needs to defeat the *scissors* at some point. There is only one piece in game that can take out the *scissor*, and that is the one *rock* in the game. Sadly for *paper* if he meets the rock in game before the *scissors*, he takes away his only chance of being able to get a win.

Changing two of the three variables at the same time is hard to visualize, and also changing the denominator of the sample each time does not create a strictly level playing field (each time an additional A better way to plot this data is in the form of a Ternary Mixture Plot, and the use of concentrations. This is the kind of thing Scientists (especially Chemists) do when looking at mixtures of three substances. |

Imagine we have a *50 ml* flask. In this we wish to place various combinations of three liquids: **R**, **P**, **S** to test their properties.

We are constrained, because the flask can only hold *50 ml*, and we also want to see the effect of different concentrations of mixtures. We can do this by making sure the flask is always full, then selecting one of the liquids, say **R**, and looping through each volume of this first liquid from *0-50 ml*. Then, we will fill the remaining space with the various combinations of **P** and **S** to top up the flask.

Since the volume in the flask is constant, this allows us to appreciate how the change in concentration effects things.

This data can be plotted inside an |

Below is the Ternay Plot for the all the mixture of starting conditions for *Rock, Paper, Scissors*

The data used for this plot is all combitions of starting conditions up to 50 people.

RED |
Red is used show the percentage of times |

GREEN |
Green is used show the percentage of times |

BLUE |
Blue is used show the percentage of times |

The brighter the color, the higher the percentage chance of that class being victorous. Each of the three primary colors are mixed for each pixel. As we expect, there is rotational symmetery in the plot.

The top left corner shows pure *Rock* and as we move away from this, in a direction perpendicular to the opposite wall, we decrease the concentration of *Rocks* in the starting conditions.

Similarly for |

These three plots merge together to form the final chart. Pretty!

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