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The rock-paper-scissors game between two people can substitute coin-tossing:
It's played until one of them wins, and that's equivalent to tossing a coin
to see who wins. (Let's forget about the slight non-randomness of human
behavior in this context. It's not the point here.)
This method of deciding between two things at random between to people
(or two groups of people, eg. teams) is often used when there's no proper
equipment available (in this case a coin).
Another way of putting it: Two people playing rock-paper-scissors once
(ie. until one of them wins) is equivalent to rolling a 1d2: If the first
person wins, it's like rolling a 1, and if the second person wins, it's
like rolling a 2.
This gave me an idea for a small thinking problem:
- How should two people play rock-paper-scissors in such way that it's
completely equivalent to rolling a 1d6? Or is it theoretically even
possible?
- How about other die sizes, such as rolling a 1d10 or a 1d20? How about
more difficult sizes, such as 1d3 or 1d5? Is it possible to achieve a
perfect equivalent with rock-paper-scissors?
- How should they play to get the equivalent of throwing multiple dice?
For example, the equivalent of rolling 2d6? How about 3d6 or 4d6? How
about even more complicated situations, such as 2d20 or 3d20?
- Would the problem become easier if there were more than two people
available?
- Would the problem become easier if a more complicated variant of the
rock-paper-scissors game would be used (eg. one with five options, each
option beating two others and being beaten by the remaining two)?
(If you don't understand dice notation, consult the
http://en.wikipedia.org/wiki/Dice_notation page.)
--
- Warp
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I was so enthralled in writing about my idea that I forgot to mention one
important detail:
Of course the idea in the problem is whether those things can be done
*more efficiently* (ie. with less rounds) using rock-paper-scissors than
by coin-tossing. (Since RPS is completely equivalent to coin-tossing, then
all the problems can be solved in the exact same way as with coin-tossing,
so the RPS element itself would become irrelevant in that way. However, the
whole idea is whether there's a more efficient/faster way of solving the
problems given that RPS is more complicated and has more options than
coin-tossing.)
--
- Warp
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Warp wrote:
> This gave me an idea for a small thinking problem:
>
> - How should two people play rock-paper-scissors in such way that it's
> completely equivalent to rolling a 1d6? Or is it theoretically even
> possible?
Both people show a number of fingers between 0 and N-1, if we're both
showing the same number I win, otherwise you win. Then the chance of me
winning assuming best play is 1/N. You can easily tweak this to get any
rational number.
> - How should they play to get the equivalent of throwing multiple dice?
> For example, the equivalent of rolling 2d6? How about 3d6 or 4d6? How
> about even more complicated situations, such as 2d20 or 3d20?
If you use toes as well as fingers, then sure.
> - Would the problem become easier if there were more than two people
> available?
I think it's highly unlikely.
> - Would the problem become easier if a more complicated variant of the
> rock-paper-scissors game would be used (eg. one with five options, each
> option beating two others and being beaten by the remaining two)?
Yup!
The next (and more difficult question) that follows is this: Given a
die/dice which we want to mimic, what's the smallest number of options
that the players can be given in a rock-paper-scissors variant which
gives the same odds of winning? I don't have an answer for this offhand.
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Kevin Wampler <wam### [at] u washingtonedu> wrote:
> Warp wrote:
> > This gave me an idea for a small thinking problem:
> >
> > - How should two people play rock-paper-scissors in such way that it's
> > completely equivalent to rolling a 1d6? Or is it theoretically even
> > possible?
> Both people show a number of fingers between 0 and N-1, if we're both
> showing the same number I win, otherwise you win. Then the chance of me
> winning assuming best play is 1/N. You can easily tweak this to get any
> rational number.
But the idea is to get a random number between 1 and 6.
> > - How should they play to get the equivalent of throwing multiple dice?
> > For example, the equivalent of rolling 2d6? How about 3d6 or 4d6? How
> > about even more complicated situations, such as 2d20 or 3d20?
> If you use toes as well as fingers, then sure.
Note that rolling a 2d6 is *not* the same as rolling a 1d12 (or, more
precisely, 1d11+1).
--
- Warp
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Still ignoring that RPS isn't completely random: This wouldn't seem very
efficient to me but it's definitely possible to simulate a die with any number
of sides playing Rock-Paper-Scissors. For each player generate a binary number
based on RPS wins or losses against any other person (opponent or not*).
Discard the entire set and re-do if the result is zero or exceeds the number of
die-sides simulated.
*An opponent's dice-rolling score must not be affected by your dice-rolling, nor
should your own second die be affected by your first, so a series of RPS rounds
must be played independently for each virtual "die".
For example, Player_A "rolls" a virtual six sided die by playing three rounds of
Rock-Paper-Scissors against Player_B or anybody else, and takes that score if
the result is in the range 1 - 6.
On a somewhat different note, and actually a little more inefficient, I've
always liked the beauty of selecting between three items (or more by
elimination) using coin flips. Flip a coin for each item, odd flip wins. If
all three are the same, start over.
Charles
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Warp wrote:
>> If you use toes as well as fingers, then sure.
>
> Note that rolling a 2d6 is *not* the same as rolling a 1d12 (or, more
> precisely, 1d11+1).
I should have read the dice notation page before I posted. In that
case, if you have N players you can all play rick-paper-scissors at
once, and the winner (if there's no tie) is the person who wins the most
of the pairwise games. This should be like a 1dN dice roll. I suppose
you could always play multiple rounds to mimic a MdN roll.
If you're limited to two players it's a bit trickier, but I think it's
still pretty easy to mimic a 1dN roll with the following game: Both
players pick a number between 0 and N-1, then you add the numbers mod N
and use the result as the value of the "roll".
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Warp <war### [at] tagpovrayorg> wrote:
> I was so enthralled in writing about my idea that I forgot to mention one
> important detail:
>
> Of course the idea in the problem is whether those things can be done
> *more efficiently* (ie. with less rounds) using rock-paper-scissors than
> by coin-tossing. (Since RPS is completely equivalent to coin-tossing, then
> all the problems can be solved in the exact same way as with coin-tossing,
> so the RPS element itself would become irrelevant in that way. However, the
> whole idea is whether there's a more efficient/faster way of solving the
> problems given that RPS is more complicated and has more options than
> coin-tossing.)
>
> --
> - Warp
I think coin tossing between two players has something inherently more efficient
than RPS. Namely with RPS, there's no way to avoid ties where you have to re-do
the round. With coin tossing, it doesn't really matter who calls it - it's
called before the toss and there is no tie short of the coin landing on its
edge.
Charles
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Charles C wrote:
> I think coin tossing between two players has something inherently more efficient
> than RPS. Namely with RPS, there's no way to avoid ties where you have to re-do
> the round. With coin tossing, it doesn't really matter who calls it - it's
> called before the toss and there is no tie short of the coin landing on its
> edge.
This is true only if you're talking about standard rock-paper-scissors,
which Warp isn't. He's talking about two-player games of the same
nature which are allowed to have different rules, and it's pretty easy
to construct such a game that doesn't allow for ties. For instance,
both players show zero or one fingers, if the result is even player 1
wins, but if the result is odd player 2 wins. Assuming best play,
either player wins with 50% probability and there are no ties.
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Warp wrote:
> But the idea is to get a random number between 1 and 6.
Not having thought too hard, I would say you could basically number the
hands, so rock=0, scissors=1, paper=2, and have people throw. The modulo sum
or difference could be a number between 0 and 2, so you get more bits per throw.
Too busy at the moment to try to deveop the idea further, but if it's
acceptable to just throw fingers (i.e., from 0 to 5 fingers), there's your
1d6 in one throw right there.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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Warp wrote:
> (Let's forget about the slight non-randomness of human
> behavior in this context. It's not the point here.)
Except in the Big Bang Theory tv show. :-)
(A TV series I highly recommend if you can watch it. Very funny.)
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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