The Secret Garden of Rock-Paper-Scissors

Summary: People have tried extending rock-paper-scissors to more than 3 options, with some success. If you allow more pairings to be ties, this uncovers a rich garden of different game dynamics and strategies.

As far as a battle of wits go, its hard to find a more balanced game than rock-paper-scissors. Its simplicity means that all strategy is scraped away, and all that’s left is to cold read the other person’s soul. Come, stare into my eyes, and I’ll see the weapon you’ll choose. Your heart says to crush, your mind says to cut, and your hands yearn to smother. I know the future already, and I’ve lined up the perfect parry. Are you ready? Best 2 out of 3, and we’ll go on 3-2-1-shoot. Oh by the way – I always go rock.

Choose Your Weapon

What if we wanted more than 3 options?

This is explored in depth in this wonderful video released yesterday by Fractal Philosophy, which inspired this post. We are going to use that as a jumping off point, and then go further.

If we want to have a game where every weapon is a viable option, then we can’t bump RPS’s 3 to 4. We have to jump all the way to 5 choices with the game Rock – Paper – Scissors – Lizard – Spock. Each weapon beats two others and loses to two. The big downside of this game is complexity – I’m not a Trekkie and have no idea how Spock and Lizard are supposed to relate to the others. The upside is that ties are less common. For RPS 1/3 of all games will be a tie, while for RPSLS this drops to 1/5 – good if you want more decisive outcomes.

Both these games are formally called “paradoxical tournaments”. Tournament means that it’s a game without ties, and paradoxical means that every move can be countered. The video above ends up at the smallest “two-paradoxical tournament”, which means that every pair of moves has a common counter. This doesn’t happen until you get to n=7 moves, but ensures that a game built on these RPS dynamics will have a wide range of viable strategies.

Balancing Act

But something’s bugging me. The definition above says there aren’t any ties, but two people can still both throw paper. Ties are already part of the game – why are we throwing them out? Let’s allow them, relaxing the tournament constraint to “weak tournaments.” To keep things interesting we’ll require the game to be connected, so that the chain of wins and losses reaches every option and you don’t have a split game of RPS floating separately from the rest of the game. We’ll draw some graphs, and each node will be labeled with the number of matchups it wins, ties, and loses: W-T-L.

The first graph we already know about, and the last “ring” structure is boring. Both are regular games, where each choice has the same number of wins and loss connections going in and out. But the middle ones are interesting! By allowing ties, we have allowed materially different strategies. Either you can go for a safe “tanky” 1-2-1 strategy that only loses to one but also only beats one, or you can go for a “glass cannon” 2-0-2 strategy that packs a punch against two others but also has more weaknesses. Since each node has the same number of winning and losing matchups, each still has the same odds of winning and we call this a balanced game.

Let’s give that second game an elemental theme. I came up with Fire – Water – Grass – Clay – Sand. Water and fire are the two powerful but vulnerable 2-0-2 choices. Water puts out fire and covers sand. Fire bakes clay and burns grass. The three 1-2-1 earthy choices all tie against each other, and beat / lose to one of the powerful ones. Grass absorbs water, clay contains water, and sand smothers fire. I’m not sure how you would translate these into hand signals, but it’s easy to remember and has some interesting dynamics!

At n=5, the four above are the only balanced games available. But at n=6 things get more interesting. There are 16 balanced games, 5 of which are regular. Here’s a look at some of them:

Choose Your Character

Each node in a balanced game has the same odds of winning or losing each turn.

What if they didn’t? What if some options had stronger offense than defense, or stronger defense than offense? Somewhat surprisingly, this doesn’t break the game!

So let’s allow for imbalanced strategies. Our new rule to keep things interesting is that all options must be viable strategies under optimal play. Formally, this means that if you play the game over and over again, every option has positive probability in a resulting Nash Equilibrium. This excludes Rock – Paper – Scissors – Match in the video above (0:55 – 3:20), because Match is a strictly worse choice than the others. We’ll call games that fit our criteria an inclusive game.

Here are all the inclusive games for n=4. We have games with meaningfully different strategies now: strong 2-0-1’s, weak 1-0-2’s, and balanced 1-1-1’s. But due to the relationships between the nodes, the strong and weak strategies are equally likely to win you the game! Optimally playing these games means that you mix up your strategies and randomly make your choice according to these probabilities.

Let’s give that first graph a cops theme. I call this one Witness – Cop – K-9 – Perp. The Witness withholds the evidence the Cop needs, and doesn’t bother the K-9. The Cop controls the K-9, and both of them catch the Perp. And the Perp warns the Witness that snitches get stitches. Now we really get an interesting game. Is it better to be the 2-0-1 cop, which is the strongest option? The balanced 1-1-1 Witness which beats the Cop? The weak 1-0-2 Perp that can counter the Witness but loses to the Cop and K-9? Or the balanced 1-1-1 K-9 which doesn’t interact with the Witness at all? The beauty of these games is that the math to find out how everything balances has already been figured out, so you don’t need to chase these chains of recursion. If we solve for a Nash Equilibrium, we see that choosing Witness 40% of the time is a valid strategy to hit a balanced win/loss ratio under optimal play. But the other 60% of the time you still need to split the rest of your choices evenly between the three others.

Now, in a head-to-head game of WCKP, everyone shows up already knowing these odds and you could argue that this doesn’t have any more game depth than baseline RPS. But it’s a much richer narrative canvas. Does the person across from you hate cops? Do they like making barking noises when they play a K-9? These are the psychological hooks you can use to win the mind games that lie at the heart of RPS.

Lumpers and Splitters

The middle n=4 graph above seems like it might have similar dynamics, but if we look closely it’s not actually that interesting. This is just Rock – Paper – Scissors with rock split into two identical choices. Boulder – Brick – Paper – Scissors might technically have more play options, but is really the same game as RPS with another tie.

In my opinion, these games don’t add anything interesting. We can check for them by seeing if they can be reduced to a size n-1 game by combining two nodes that win & lose against the same options, and tie each other. We’ll call games which can’t be simplified further twin-free.

If we do this, we find out that the elemental game is actually the Cops game in disguise! The clay and grass options both beat water and lose to fire, and so can be combined into one. This turns them into the equivalent of the Witness in the Cops game.

Elemental GameCops GameWater 🌊 (2-0-2)Cop 👮♂ (2-0-1)Fire 🔥 (2-0-2)Perp 👤 (1-0-2)Sand ⛱ (1-2-1)K9 🐕🦺 (1-1-1)Clay 🏺 + Grass 🌱 (1-2-1)Witness 🤐 (1-1-1)

Decision Paralysis

If memorizing the optimal strategy for the Cops game is too complicated, I have good news. That strategy isn’t actually the only one – games can have multiple equilibrium points. For example, take the strategy where you choose the K-9 and Witness each 50% of the time. They cancel out each other’s weaknesses, so there’s no way for your opponent to gain an advantage over you.

Or you could also choose Cop, Perp, and Witness each 1/3 of the time and ignore K-9 altogether. This also balances.

In fact, there is a whole continuum of optimal strategies, and they connect the two above. If A is the K-9/Witness 50% strategy and B is the Cop/Perp/Witness 33% strategy, then all the strategies that lie on the line Ax + B(1-x) are also optimal. The 20% / 40% / 20% / 20% strategy we found above is what happens when you happen to set x=0.4. Another way to think of this is that A and B set endpoints that bound this family of optimal strategies – Witness is played frequently between 33-50% of the time, Cop and Perp are played sparingly at 0-33%, and K-9 has a wide range of viable play from 0-50%.

We need a way to somehow choose a representative solution from the continuum of possibilities. For this exploration we will choose the solution that maximizes the minimum probability, to show what optimal play looks like when choices are relatively balanced between all options. Finding the space of optimal play quickly gets into deep math with interesting results especially at higher n – for example, in our construction odd n games might have a unique solution, but no even n game does. But let’s not dawdle, we need to keep moving.

Diamond in the Rough

We have our hard criteria, but how do we figure out which games are more interesting than others? This is subjective, but here are a few metrics I’ve found to rank them. These are listed for each game on the plots at the end, so you can compare how these metrics change games.

Orbits. This measures how many unique strategies you can choose from. Formally, two nodes share an orbit if you can swap them without changing the game, or cycle through them for groups of nodes. Ring games only have a single orbit. Clay and Grass in the elemental game have the same orbit, meaning that choosing between them is entirely up to the player’s aesthetics. I think more choices between meaningfully different strategies makes for a more interesting game and narrative choices.