As we all know, Rock-Paper-Scissors is a game where each option defeats one other option and loses to one other option. This has the important effect that regardless of what you choose, you don't have an advantage before the game starts (you will win if you chose -eg- rock and the other player chose scissors, and lose to paper if you have chosen rock).

That said, due to a linguistic ambiguity (how the game's terms translate to Greek), in some areas (including, luckily, my old neighborhood) the game got a fourth option. It was "glue", because "paper" got translated as "sheet of paper", and the greek term for "sheet" is homophonic to the greek term for glue: κόλα and κόλλα respectively.

So this variation of the game has four options, but as can easily be seen it leads to imbalance, because you cannot have equal number of win/lose if the remaining options (you choose 1 of 4, 3 remain) aren't perfectly divisible by two. So you end up with one of the options winning against more enemy options.

To be more illustrative:

Now, as far as I recall, since the children weren't aware of either having thus made up a new option, nor that the dynamic changed, the option Glue defeated the objects one would realistically expect it to. It defeated two (paper, for it folded it, and scissors, because it glued together the dangerous edges) and lost to the rock (it got crushed). But do notice that Rock also becomes more powerful than the other old options, cause it has two options to defeat. In fact Rock becomes the most powerful option, since it also defeats Glue, the other powerful one.

While one could choose to play as Rock (or Glue) and still lose, the possibility of a win would be higher. But then something important happens: people start to notice the difference in dynamic, which causes them to be more weary of choosing Rock, and even wearier of choosing Glue. So by the end Glue is almost never chosen, while Rock reverts to being a regular option which risks defeat by paper, that in time becomes more popular. Scissors, on the other end, in time became the least popular option to choose, for reasons we can gather from the above summation... (while it defeats the popular Paper, it will lose to everything else and starts as less popular than the other 'lose to everything else' option: Paper).

I think it is interesting, cause it goes to show that what nominally is the best option, may become unpopular if enough calculation takes place, which calculation takes into account not which option had the biggest probability of winning but which option defeats that option.

Perhaps of more practical interest is that such dynamic elements can be used (eg in a formal logic system) to push problematic (here symbolized by "overpowered") elements out of the way, while still making use of the dynamic of the default (pre-altered) system.

This sentence confuses me. It seems to me like you're implying that there is a time when the probability of choosing rock exceeds the probability of choosing glue when in fact the Nash equilibrium strategy is 13 rock, 13 paper, 13 glue and 03 scissors.

For others who want to check those numbers: note that glue dominates scissors (both beat paper, lose to rock, and glue beats scissors), so scissors should never be played. With that simplification, it's an ordinary game of rock-paper-scissors, except "scissors" is now called "glue".

Please read my edited reply to lsusr.

You mean Nash equilibrium strategy? Rock-Paper-Scissors is a zero-sum game, so Pareto optimal is a trivial notion here.

Fixed.

Edit (I rewrote this reply, cause it was too vague in the original :) )Very correct in regards to

every playeractually having identified this (indeed, if all players are aware of the new balance, they will pick up that glue is a better type of scissors so scissors should not be picked). But imagine a player comes in and hasn't picked up this identity, while (for different reasons) they have picked upan aversion to choose rock from previous players.Then scissors still has a chance to win (against paper), and effectively rock is largely out, so the triplet scissors-paper-glue has glue as the permanent winner. This in turn (after a couple of games) is picked up and stabilizes the game as having three options for all (scissors no longer chosen), until a new player who is unaware joins.Essentially the dynamic of the 4-choice game allows for periodic returns to a 3-choice, which is what can be used to trigger ongoing corrections to other systems.

Regardless of what the new player does, there is no reason to ever play scissors. I don't see any interesting "4-choice dynamic" here. Perhaps you should pick a different example with multiple Nash equilibria.

You are confusing "reason to choose" (which is obviously not there; optimal strategy is trivial to find) with "happens to be chosen". Ie you are looking at what is said from an angle which isn't crucial to the point.

Everyone is aware that scissors is not be chosen at any time if the player has correctly evaluated the dynamic. Try asking a non-sentence in a formal logic system to stop existing cause it evaluated the dynamic, and you'll get why your point is not sensible.

You may be interested in alpha-rank. It's an Elo-esque system for highly 'nontransitive' games - ie, games where there're lots of rock-paper-scissors-esque cycles.

At a high level, what it does is set up a graph like the one you've drawn, then places a token on a random node and repeatedly follows the 'defeated-by' edges. The amount of time spent on a node gives the strength of the strategy.

You might also be interested in rectified policy space response oracles, which is one approach to finding new, better strategies in nontransitive games.

Thank you, I will have a look!

My own interest in recollecting this variation (an actual thing, from my childhood years) is that intuitively it seems to me that this type of limited setting may be enough so that the inherent dynamic of 'new player will go for the less than optimal strategy', and the periodic ripple effect it creates, can (be made to) mimic some elements of a formal logic system, namely the interactions of non-sentences with sentences.

So I posted this as a possible trigger for more reflection, not for establishing the trivial (optimal strategy in this corrupted variation of the game) ^_^

To see what other rock-paper-scissor scholars have to say on this, you might want to investigate the controversial "dynamite" option.