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Nothing Beats Rock


Earlier today, Aubrey Hesselgren linked to a Numberphile video about Rock, Paper, Scissors, with the tongue in cheek assertion that all competitive games “reduce” to RPS:


He went on to make it clear that he was being ironic, but admitted that the thought did haunt him a little:

…although, admittedly, it has been a thought that has haunted me. genuine nightmares. “oh my god! all games are same! This is what i am doing with my life? this is all there is? ahh!?”

I know how he feels. The fact that we are able to map the abstract structure of some games onto others is kind of spooky. For me, the nightmare lurking beneath the surface of every strategy game is Nim, the take-away game played by the tragic, existentially glamorous characters in Alain Resnais’ Last Year at Marienbad. (I may have gotten this nightmare from trying to read books that I’m not smart enough to understand.)

But, actually, watching that video had the opposite effect on me. And here’s why:

First of all, Numberphile is awesome. I like everything about it, but mostly I like the way they always write equations on big sheets of brown butcher paper using sharpies.

This particular video was about a paper from some Chinese game theorists who studied a large number of iterated R/P/S games. They observed that, on average, players don’t use the “perfect” strategy of randomly selecting a move. Instead, there are noticeable patterns based on the outcome of the previous round – there is a slight tendency to keep the same move after winning, and to pick a new move after losing (win-stay/lose-shift). This general tendency allows a savvy player to do better than average by assuming her opponent is playing win-stay/lose-shift and playing accordingly.

Sound familiar? Yes, this is basically Donkeyspace! Although neither the scientists nor Numberphile seems to recognize the larger implications.

What are the larger implications? Essentially, that the “fancy” strategy that takes advantage of the typical tendencies of average players is actually just level 1 of a virtually endless process of adaptive strategic responses. Yes, I can do better than average by assuming my opponent is playing win-stay/lose-shift. But I can do even better than that. Sometimes my opponent will be playing the strategy that beats w-s/l-s (let’s call it win-stay/lose-shift prime, or w-s/l-s’), I can change gears to take advantage of her (by playing w-s/l-s”) and the sooner I do, the better. Sometimes my opponent will notice that I’m playing w-s/l-s” and change gears back to perfect random (rnd). That’s ok, w-s/l-s” does fine against rnd (as good as possible in fact). But occasionally I’ll be up against an opponent who will notice what I’m doing and change gears to w-s/l-s”’. I need to be on the lookout for that.

My real goal is to efficiently discover what I’m up against – a complete fish who always plays rock, a standard donkey playing w-s/l-s, a medium sized shark playing w-s/l-s’ or a killer with the capacity of identifying my own gear-changing strategy and adapting to me.

This changes Rock Paper Scissors from a “solved” game with a known perfect strategy, to a very subtle game of reading my opponent. The first step of this reading is knowing the overall tendencies of the player population I am in (the local meta) as outlined in the paper. But this is just step one, because any particular opponent in this population might be a fish, a donkey, a shark, or a killer (or a killer camouflaged as a donkey.) The real game is how quickly and accurately I extrapolate from my opponent’s moves to understand her true strategy, which will sometimes include a gear-changing strategy based on her interpretation of me. (Hm – maybe we can develop an Optimal Meta Adaptation Response that will even give us an edge on the killers…)

You see where this is going, right? Even if all competitive games did ultimately reduce to R/P/S under the hood, we would still be at the bottom rung of a very tall ladder. (As long as we have the right kind of iterated, pair-wise tournament structure.)

But it’s even better than that. Because imagine this chaotic dance of mutual signal-processing and gear-shifting adaptation taking place around a core game that isn’t even solved – where, unlike R/P/S, you don’t have the base level of perfect-random as a starting point. That’s where competitive games reside, a turbulent swarm of moves and counter-moves, some random, some overloaded with meaning, some listening to hear the shape of the whole, the whole flock turning and turning, searching through problem-space, and searching through search-space, and beyond.


  1. Let’s assume that all competitive games reduce to RPS. We would then just have to forbid (or some other way eliminate) the optimal strategy of playing randomly in order to get deep and rather interesting competition. See for example the Computer RoShamBo Programming Competition, where pretty much the only rule is that you cannot submit a bot the plays randomly.
    What the empirical results of the competition seem to show is that as long as there are no (or few) completely random players in the lineup, the prediction and counter-prediction game has lots of layers.

    Which sort of serves to reinforce your point.

    Monday, May 11, 2015 at 3:32 pm | Permalink
  2. >This changes Rock Paper Scissors from a “solved” game with a known perfect strategy, to a very subtle game of reading my opponent.

    My problem with “reading the opponent” is that it’s very difficult to know whether you ever actually did so, or if you just got lucky.

    One of my favorite moments in game design history is when a bunch of the top Yomi players started playing using RNGs to choose their cards, and did just as well (or better) than they had without it. (The admins said that it was “cheating” to use RNGs to choose your combat card, by the way – good luck enforcing that when you’re drawing random cards every turn…)

    Humans are well-known to err on the side of “seeing patterns where there are none”. So to me, when someone says “hah, I made a good read, I *knew* you were going to do that”, to me it’s like seeing Jesus on toast.

    I view “what the opponent will do” as basically a source of randomness in games, and since I see it that way, I think it’s important to give players at least a few “turns” (or “moments”) to see the input of the other player before it affects him. This way, player choices become input randomness for the other player (rather than output randomness). By doing this, we create competitive games that really aren’t RPS-reliant at all.

    Monday, May 11, 2015 at 3:49 pm | Permalink
  3. Frank Lantz wrote:


    Short sessions + lots of iteration = sufficient sample size to reveal the edge.

    I fully admit that this edge might attenuate dramatically the further up the ladder you go. But edge is edge.

    Monday, May 11, 2015 at 5:35 pm | Permalink
  4. humans are bad at randomness & randomness is an advantage, so a human+RNG is going to outperform just a human. (same way humans aided by computer search can outperform both human and computer players at Chess)

    RPS to pick what affects your opponent in a few turns is still RPS. (unless it’s Nim ha ha)

    and on Nim – Grundy-Sprague gives a precisely defined set of games equivalent to Nim; break one of those conditions and it’s probably not. so you can tell /exactly/ when you need to worry. (but even then – even though Nim is completely solved it can be hard to compute that solution in realistic play situations where you don’t have unlimited time, and even though a game may be Nim-equivalent that equivalence can also be hard to compute. so in practice you still have to decide on heuristics to use and that’s interesting-ish.)

    yeah “edge is edge”

    Monday, May 11, 2015 at 5:59 pm | Permalink
  5. @Frank Lantz

    The issue becomes that the level of iterations you need is well beyond the rate of play that most players play at. So the noise vastly outweighs the edge. Especially once you consider how hard it is for someone to actually even determine WHAT a players range even is. Also in game valuation tends to trump everything anyway.

    Tuesday, May 12, 2015 at 11:38 am | Permalink
  6. Frank Lantz wrote:


    “noise vastly outweighs the edge”

    Yes, but it’s *this* problem, which is a problem about information, evidence, statistics, knowledge, certainty, etc. that makes this topic so interesting for me, the idea that this whole problem space exists beyond the perfect Nash Equilibrium solution that we might otherwise think closes the book on this game.

    Tuesday, May 12, 2015 at 10:17 pm | Permalink
  7. Devin Becker wrote:

    Interesting article. It begs two questions for me though:

    1) Where do you draw the line in competitive play between in-game strategy/tactics and metagame dynamics that emerge from the system or tournament structure but that aren’t explicitly part of the game system itself?

    2) Doesn’t the “all competitive play reduces to RPS” idea only work if the choices are essentially equal in utility?

    Jason Rohrer’s new game Cordial Minuet has a lot of good Donkey Space to it as well if you haven’t checked it out.

    Thursday, May 14, 2015 at 3:13 am | Permalink

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