William M. Briggs

Statistician to the Stars!

The Philosophy Of Rock Paper Scissors

Way my dad and I used to play is that when somebody won they got to grab the hand of the loser and then, with the fore- and middle-fingers only, slap the loser on the wrist. My dad would lick his fingers (“to cut down on air resistance”) before whacking me.

Say, these days that would be child abuse. Well, the government knows best, right?

Briefly, you and simultaneously your opponent select rock, paper, or scissors. Rock breaks scissors, paper covers rock, and scissors cut paper. There are 9 outcomes: 3 of which are rock for you, and rock, paper, or scissors for your enemy; et cetera. Three of the 9 outcomes are ties, in which nobody gets slapped; there are 3 ways for you to lose, and 3 ways for you to win.

Now, given just those premises, and none other, what is the probability you don’t get slapped? Well, “don’t get slapped” means tying or winning; and since there are 6 ways for these things to happen, the chance is 2/3. Similarly, there is a 1/3 chance for you to lose, and 1/3 probability you get to slap.

These deduced probabilities are correct assuming only the premises describing the rules of the game. Which implies, and it is true, that the probabilities are not likely to be correct assuming other premises. What other premises? There are an infinite number of premises we might choose, but what we’re after are premises that help us win the game.

The thing to emphasize is that we know with certainty the non-human premises, and so know with certainty the non-human probabilities. Rock paper scissors is thus similar to poker where we have a good handle on the probabilities; but where in poker they are harder to memorize, yet in poker we know there are consistent and good players.

There are also consistently winning Rock Paper Scissors players. Like 2008 champion Sean “Wicked Fingers” Sears. That means, like poker, human premises must exist that change the odds.

In any number of places you’ll read that the way not to get beat is to make your pick “randomly.” This is impossible. No matter what, you must cause your pick, and no cause in the universe is ontologically “random.” Suppose you decide to divide up a minute into thirds and pick based on the secondhand of your watch. If your opponent does not know this, he has no way of guessing what you’ll do (except that you must choose, of course), and so to him your guess is “random”—which means only unknown. But to you, the choice is determined, caused by your decision and the state of the time.

If your opponent catches you sneaking peeks before each round, then he’ll too know what you’re going to do—creating a new and probative and deductive premise for him—and you’ll consistently lose.

That’s the way to win, too. By searching for patterns, i.e. premises, which your opponent is using, knowingly or not. Bin Xu and pals think they’ve discovered patterns many people use. In their arxiv paper “Cycle frequency in standard Rock-Paper-Scissors games: Evidence from experimental economics” they posit that winners often repeat their winning pick, and losers select the next object in the cycle (lose with scissors and move to rock). Armed (handed?) with these premises allows you to change the odds.

Until your opponent figures out his mistake; and when he does, he can use that knowledge against you. Figuring you are figuring on a rock (since your opponent just lost on a scissors), and thus you’d pick paper, your opponent upends the algorithm and sticks with scissors.

Noted RPS expert Sharisa Bufford, a member of the prestigious USA Rock Paper Scissors League, is sure that “Girls always throw scissors first. Guys always throw rock.” If that’s so, these are winning premises. Unless your opponent knows these rules, too. And you know they know. And you know they know you know…you know?

Now it’s true you cannot “pick randomly” but must always cause a choice. But that does not mean it’s impossible to discover a strategy which your opponent cannot guess—where your opponent may be some “sophisticated computer” (computers are only dumb unthinking distillations of fractions of human brains). All it takes to create impossible-to-discover picks are to create picks which your opponent cannot guess beyond the standard premises (a tautology, really).

Maybe that’s memorizing a stream of numbers generated by some process which remains unknown to your opponent. Or maybe that means changing your picking algorithm based on circumstance. All that matters is that your opponent cannot guess.

Incidentally and curiously, computer programs which test “randomness” (i.e. unpredictability) always turn a blind eye to the algorithm which generated the sequence they’re testing.

Lastly, to prove (as I’ve done before) that I can pick a number you cannot guess with certainty, no matter what your resources, I’m thinking of a number between 0 and 4. What is it? (It’s hidden in the code to this page.)

31 Comments

  1. Yeah, agreed, perhaps the best one can do is choose sequences with typical properties (averages, moments, etc). These typical properties can be mathematically defined and shown to be secure against almost all strategies (like girls-scissor-boys-rock).
    What is a good “random generator” or a good poker player? One whose sequential play is not just unknown but hard to infer by any method.

  2. I lived in a small dorm/hall in college. At the beginning of every new year, we would have a big first meeting to welcome all of the new members to the hall. It was a tradition to have a tournament of rock-paper-scissors, of which the winner was promised a reward of some kind.

    Being informal, the older members of the hall would pair off (both would “lose”) as the new, excited, and eager to participate freshman played seriously, trying to win the hidden reward. The elders yelled, hollered, and generally increased the energy and excitement. Eventually, only the new members would be playing, and a poor freshman would win the “reward” (a dip in a nearby fountain).

    Sometimes there is a strategy at hand that you have no chance of detecting.

  3. James: An excellent example of why one must always ask what the reward is. Without that information, one does not play!

    It’s is mildly disturbing that Rock Paper Scissors has tournaments…….Anyway, if anyone wants to consistently win at the game, play me. I am hopelessly bad at the game.

    Briggs: “Hidden” in the code is not exactly correct. It’s in the code and it’s easy to find. Maybe just say it’s in the code? The reason the number cannot be guessed is it has basically only one limitation, that being it is between 0 and 4. It does not have to be a whole number, it is not limited in decimal places, giving very close to an infinite number of choices. Allowing the number to go to 5000 decimal places or more creates so many possibilities I’m not sure there’s any way to up the odds. If we could do this thousands of times, then run the data through a super computer and search for a pattern, maybe we could marginally up the odds. As long as you don’t know what we are doing, of course!

  4. One’s own pattern may be difficult or impossible to infer, but if your opponent’s is not, you’d be leaving precious hand slaps on the table by sticking with your esoteric system.

  5. Yes, Gary …
    … and since the premise for the enhanced game of Rock-Paper-Scissors-Lizard-Spock was to DECREASE the number of ties … that didn’t work out very well, did it?

  6. Hidden? Hmmm. Please define your use of “hidden.” I think the term you menat was embedded, Nonetheless …

  7. *meant

  8. The number is 2.976.

  9. Sheri:

    It does not have to be a whole number, it is not limited in decimal places …

    That WOULD be infinite, but given the “code” to produce this page IS finite, we are looking at “finite”, “impossibly finite”, let’s call it “imfinite” 😉

  10. john b:

    Here is the claim: Lastly, to prove (as I’ve done before) that I can pick a number you cannot guess with certainty, no matter what your resources, I’m thinking of a number between 0 and 4. What is it? (It’s hidden in the code to this page.)

    Key on, “no matter what your resources.”

    My resources were the page we are on, right mouse click, view page source, and cntrl-F searching for “number.”

    Using those resources, I guessed the number with 100% certainty.

  11. john b,
    Nerds of a feather…

    Like on this blog.

  12. Jim

    Way cool … Yes … 2.976 … but then it was no longer a guess

  13. john b: You make a good point. The number being “hidden” in the code is another limiting factor. How limiting would depend on Brigg’s desire to put a number with a huge decimal string into the code. How far would he go to make the point?!

  14. Possibly, consistent winners use tells much as is done in poker. If a person is very fast, as in Wyatt Earp fast, he would detect tells in how the opponent is throwing soon enough to alter his own throw and fast enough not to be detected. I wonder if there are procedures in place to prevent this. They might use screens and an appropriately placed moderator to prevent this, sort as is done in high level duplicate bridge tournaments.

    Go Sears!

    A terrible picture of rock, paper, scissors, by the way. 😉

  15. Gary

    Thanks for the compliment …

    But I can only aspire to nerddom …

    I’m at sea with many of Brigg’s posts …

    I usually wait for other comments before I can dig in

  16. Sheri:

    Jim has the answer … it’s correct .. but no longer guessed

    He “guessed” at “how” to arrive at the solution, but thereafter it was no longer a guess

  17. Scotian

    Given the “Chinese” characters, it must be an oriental variant … I wonder if it reduces “tells”. (Is oriental PC? I grew up in MN where our major airline was “Northwest Orient”.

    An aside, I only recently realized, by watching the film recently, that North by Northwest was not actually a position on the compass rose but that Cary Grant actually flew North on Northwest Airlines.

  18. john b: I know what the answer is. I did not “guess” at how to find it, nor did Jim, probably. I know how to find the source code and how to search for what I want. I learned that from past experience (though the original learning may have been based on a guess or by deduction, I don’t remember). I just don’t post the number since others may want to look for it.

    If Briggs were to “hide” the number, he could do fun things like inserting the number, top to bottom (or bottom to top), at various points throughout the code. Then you would have to search for each number and be sure you have them in the right order. This would qualify more as “hidden” than just inserting the statement that “X” is the number. To really complicate things, he can insert numbers over 9, so there are numbers too large to use included. Or, use the double-digit numbers and make a number with many decimal places using the paired numbers. One then would have to decide if the numbers were decoys or part of the number. The possibilities are multiple!

  19. It’s Japanese. The frames are “slug” “frog” “snake”. Frog defeats the slug, slug defeats the snake, and snake defeats the frog.

  20. Even after looking in the code – is it still a guess at the number Briggs was thinking, with a probability of 1, conditioned on this evidence?

    That evidence being that Briggs told us he’s hide the number in the code, that it was the number he was thinking of, that he’s telling the truth, that the computer didn’t change the number around, that he didn’t accidentally type it incorrectly, etc etc.

    Pedantic, but is it the right way to think about our knowledge of this number?

  21. That Random guy is always being picked at or having stuff thrown at him. He should lead a more ordered life, but then, he wouldn’t be Random.

  22. The “Unless your opponent knows these rules, too. And you know they know. And you know they know you know…you know?” section would benefit from a reference to Vizzini’s sequence with the Man in Black in Princess Bride…

    Vizzini: But it’s so simple. All I have to do is divine from what I know of you: are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool, you would have counted on it, so I can clearly not choose the wine in front of me.
    Man in Black: You’ve made your decision then?
    Vizzini: Not remotely. Because iocane comes from Australia, as everyone knows, and Australia is entirely peopled with criminals, and criminals are used to having people not trust them, as you are not trusted by me, so I can clearly not choose the wine in front of you.
    Man in Black: Truly, you have a dizzying intellect.
    Vizzini: Wait till I get going! Now, where was I?
    Man in Black: Australia.
    Vizzini: Yes, Australia. And you must have suspected I would have known the powder’s origin, so I can clearly not choose the wine in front of me.
    Man in Black: You’re just stalling now.
    Vizzini: You’d like to think that, wouldn’t you? You’ve beaten my giant, which means you’re exceptionally strong, so you could’ve put the poison in your own goblet, trusting on your strength to save you, so I can clearly not choose the wine in front of you. But, you’ve also bested my Spaniard, which means you must have studied, and in studying you must have learned that man is mortal, so you would have put the poison as far from yourself as possible, so I can clearly not choose the wine in front of me.
    Man in Black: You’re trying to trick me into giving away something. It won’t work.
    Vizzini: IT HAS WORKED! YOU’VE GIVEN EVERYTHING AWAY! I KNOW WHERE THE POISON IS! “

  23. Rock is strong.

  24. e

  25. “Ruk (rock), protect!”

    To quote an old Star Trek (TOS) line (kind of the antithesis of Asimov’s Three Laws of Robotics – see earlier posts):

    RUK: I was programmed by Korby. I cannot harm him.
    KIRK: The old ones programmed you, too, but it became possible to destroy them.
    RUK: That was the equation! (seizes Kirk) Existence! Survival must cancel out programming.
    KIRK: That’s it, Ruk! Logic! You can’t protect someone who’s trying to destroy you!

  26. Two handed gesture. One hand open in a vertical position with the pinky edge facing out, the other hand in a fist on top of the fingers of the other hand. (mushroom cloud).

    Nuclear explosion incinerates everything. 🙂

  27. I gather that this game–rock/paper/scissors–is an important example in game theory. Here are some links:
    http://gametheory101.com/Rock_Paper_Scissors.html
    and
    http://en.wikipedia.org/wiki/Rock-paper-scissors
    I quote from the Wikipedia article:
    “It is impossible to gain an advantage over a truly random opponent. However, by exploiting the weaknesses of nonrandom opponents, it is possible to gain a significant advantage.[19][20] Indeed, human players tend to be nonrandom.[20][21]”
    The article also discusses algorithms.
    My final point–playing checkers with my 5 year old grandson is much more fun than rock/scissors/paper.

  28. You fooled me Briggs, I was expecting an integer, and all I got was this lousy rational number.
    I have never played rock-paper-scissors, nor paper-scissors-rock, nor…, am I missing out on anything?

  29. Philosophy of rock-paper-scissors! Does the philosophy lie in being slapped or the psychology behind it?

    When my children were young, rock-paper-scissors was how they decided who go first when they played checkers. Fair and square, supposedly. No parental supervision required.

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