Ties

One unique thing about 3-moves, from the perspective of game mechanics, is that 3 outcomes allows better/worse/same outcomes—and ties are an important mechanic for games.

If you don’t have ties or no-change outcomes, then every episode requires you to either be perfect, or suffer constant attrition. Imagine trying to play Dark Souls⁠ where every single attack to you does damage or you damage them, with no alternatives like “dodge” or “block”—just hit or be hit. It’d be even harder to play, and if you stack encounters like an ordinary game, it’s basically demanding that you play a perfect move every time, or else you’ll be attrited long before you finish. So that’s why 2 is inadequate, and 3 is good.

But then at ≥4, you’re no longer adding any important mechanics: there is nothing like “tie” which is as generic and widespread in games. (And if you need more outcomes or more complexity, you can simply make sub-games which are themselves RPS: you might have ranged weapons vs melee vs magic, and within each kind of weapons, another RPS.)

Information Maximization

Can we formalize this observation about 3 seeming to be special?

I think it’s because 2 forces outcomes to be “better vs worse”, so can require a lot of games to ever approximate equality (imagine 2 people flipping fair coins: they both have the same expected value⁠ but they will have different percentages until they reach quite large numbers of coin flips, as the binomial convergence to 50% is slow). At 3, you get substantially more information from each comparison (rather than log₂(2) = 1 bit of information, you get log₂(3) = 1.58 bits or 58% more), enabling efficient short matches like ‘best of 3’. At 4, the outcomes start potentially overlapping and becoming redundant or uninformative (adding 25% options only increases information to log₂(4) = 2 or 21% more), without speeding up matches much.

From the perspective of a judge or observer deciding who wins the overall game, the details of which player played which is uninteresting, because they will be playing Nash equilibria; a series of RPS games is just a series of win/tie/loss, which could be written down as a base-3⁠ number like ‘2012210’, say, for a sequence of RPS games like tie/win/loss/win/win/… Each number is a ‘path’ through a ‘game tree’; and we can then do something like Pascal’s ⁠problem of points⁠ or sequential testing⁠ or ⁠tournaments/noisy sorting⁠: we can keep playing RPS games, accumulating the wins/ties/losses as a sufficient statistic⁠ and updating until a strong-enough posterior confidence is reached about which player is better, and stop playing there. (This would be a common enough Bayesian decision theory⁠ problem.)

Games can be counted and enumerated and can have odd connections to number systems (eg. ⁠Nim⁠ or Conway’s ⁠surreal numbers⁠ came out of the game of Go via ⁠combinatorial game theory⁠). And if we are writing down RPS games as numbers, then there is something interesting about the ‘RPS number system’ being base-3: that is the most ‘efficient’ integer base (because it is the closest to e⁠), which optimizes “radix economy”⁠, which is how many digits you have to write out multiplied by how many possible digits there are.

So if we are trying to ‘think through’ a game tree as a sequence, an RPS game is easiest to remember and think about: a binary game’s sequences like ‘1010010’ or a 4-move game like ‘013’ are both bulkier than an RPS like ‘0122’.⁠^⁠2⁠ We could further write it like ⁠balanced ternary⁠ as +1 (win) / 0 (tie) / −1 (loss), and then it is easy to keep a running tally, and find the Bayes-optimal threshold to stop & declare the winner (eg. Ortegaet al2019⁠); this is like how players will do ‘best-of-5’ and halt if one player wins 3 in a row.

Since only integer numbers of moves are allowed, the optimum is achieved at b = 3.

That is, compared to a binary game (b = 2) or a 4-move game (b = 4), a 3-move game requires fewer “units of decision-cost” (or nodes in the decision tree) on average to accumulate the necessary information for 99% confidence. The binary game gives less information per round (requiring more rounds/depth), and while the 4-move game gives more bits per round, it “pays” by having more nodes per round.

Thus, if we try to minimize the ‘psychic cost’ of playing a game repeatedly, we will find that binary games are somehow just a bit tedious to play enough times to be sure who won, while 4-move games are more decisive, but a bit of a headache. We will keep being subtly pulled towards the 3-move game, as 2-moves won’t be “enough” and 4-moves is “a bit too much”—while 3-moves feels “just right”. And as games copy earlier game mechanics, we would see continual evolution towards the 3-move game as the Platonic ideal of games.