Rock-Paper-Scissors-Lizard-Spock-Starfish-Unicorn-Shotglass-Segway

[major_clanger]

In a friends-locked post of mine purplecthulhu commented on a report about a US court ordering the parties to resolve a trivial dispute via Rock-Paper-Scissors by suggesting that the variant with Lizard and Spock be used. I responded by suggesting that the higher the court the more complex the game should be, and proposed Rock-Paper-Scissors-Lizard-Spock-Starfish-Unicorn-Shotglass-Segway for the Supreme Court.

Now cuboid_ursinoid wants that as a T-Shirt. So I suppose we need some rules.

It turns out that N-option generalised Rock-Paper-Scissors works so long as:

- There is an odd number of options.
- Each options beats exactly half the other options and is beaten by the other half of them.

So, trivially, in classic RPS, Rock is beaten (wrapped) by paper and beats (breaks) scissors, and so on. RPS-5 adds two new options:

Spock vaporises Rock and breaks Scissors but is poisoned by Lizard and disproven by Paper.
Lizard eats Paper and poisons Spock but is crushed by Rock and decapitated by Scissors.

Thus RPS-5 has another 7 interactions (we've just listed Spock/Lizard twice) to add to the three in classic RPS for a total of 10.

We thus need more interactions. Indeed, since for RPS-N, there are N(N-1)/2 interactions, for RPS-9 we need 36 interactions of which we already know 10, so we need 26 more! Remember, each new option must defeat half the others and be defeated by the other half of them.

I can think of a few:

Spock rides Segway
Spock dissects Starfish
Shotglass inebriates Spock
Unicorn impales Spock
Rock trips Segway
Starfish grips Rock
Rock smashes Shotglass
Unicorn kicks Rock

...but we need more. So, fire away!

Comments

[cobrabay.livejournal.com]

I'm with cuboid_ursinoid, I want that as a T-Shirt.

I see the lizard interactions as a progression,
Lizard eats starfish
Lizard washes down meal with drink from shotglass
Rock fired by crude cannon into lizard
Segway runs over stunned lizard

[ciphergoth.livejournal.com]

http://www.umop.com/rps25.htm

[drplokta]

And also http://www.umop.com/rps101.htm

[cdave]

https://docs.google.com/spreadsheet/ccc?key=0Apir0tcIKsxrdHZjTC1MZ3B6cmt1dDJIVWtHbXktSkE

[cdave]

In a tangentially related question, it seems clear that in the US game mechanisms are not strongly covered by copyright. Do you know off the top of your head what's the situation in the UK is?

[major-clanger.livejournal.com]

In the UK, you can't patent the general method of play of a game, because it's akin to a business process, mathematical theory or computer programme. If you write down the rules then copyright subsists in that document, but an independent re-write doesn't infringe that copyright (non-fiction works don't have a 'plot' per se, as per the Da Vinci Code case.) The name of the game and a logo might be registered as trade marks, and there might be graphical copyright in a game board. Design right might also exist in aspects of a game, e.g. the shape of playing pieces.

RPS apparently goes back centuries, and there is no clear single 'original' statement of the rules that I'm aware of, so I very much doubt that copyright would apply.

[davidwake.livejournal.com]

But you can patent genes, which have been around for millions of years.

[a-cubed.livejournal.com]

The names are all trademarked, often including major designer names (Reiner Knizia is a friend of my brother, so I know some of this). THe artwork is all copyrighted as are the board designs, non-generic piece designs, card designs (not sure about the short contents of cards since they probably fail the de miminis in most cases, though Magic the Saddening may exceed the very low theshold the courts allow these days) etc. Precise game-play mechanics are a really important part of any game, of course, but commercially they're a tiny part of selling a game.
There exist board game "fan-fics" actually, generally new boards for existing games which are played by the standard rules, just on the different board with existing pieces. Like written fan-fic different manufacturers and designers have different attitudes and some have been endorsed and become official releases, whereas others have been "cease-and-desisted" or sued.
My brother (or failing that Reiner) would know more details if people are burning to know.

[makyo.livejournal.com]

N-option generalised Rock-Paper-Scissors
One of my students brought this up in a tutorial a year or so back, and we came to the same conclusion as you with the odd-N case: each option beats exactly ½(N-1) other options, is beaten by exactly ½(N-1) options, and draws with itself. For the even-N case, we concluded it worked if each option beats exactly ½N-1 other options, is beaten by exactly ½N-1 other options, and draws with itself and one other option.

So, for example, for a 4-option game where the options are A, B, C and D, we have: A beats B, which beats C, which beats D, which beats A; A draws with C and B draws with D.

Less symmetric is drdoug's variant "B52 Bomber and Small, Unsuspecting Yorkshire Hamlet", the rules for which are left as an exercise to the reader.

[ciphergoth.livejournal.com]

I like that last variant. But what are the hand gestures?

[steer.livejournal.com]

IIRC (and this is from about 20 years ago) the B52 bomber looks like two spread palms joined at the thumbs to form a butterfly sort of shape. The small unsuspecting Yorkshire Hamlet is indistinguishable from rock.

[ciphergoth.livejournal.com]

Do you have to hum Ride of the Valkyries if you're the bomber?

[steer.livejournal.com]

It would very much help add drama. ISTR drdoug would usually play SUYH on the grounds that "it was bound to work eventually" but memory fades.

[ciphergoth.livejournal.com]

Also, when you say "indistinguishable from rock", is that before or after the encounter with the B52 bomber?

[steer.livejournal.com]

Both -- it was bombed back into the stone age. :-)

[steer.livejournal.com]

Incidentally, drdoug's children convinced me that non-symmetric RPS can be interesting. That is, it is not necessary for an RPS variant to have the same number of wins and losses to be interesting. Consider, plays A,B,C,D, E.

A beats B, C, D
B beats C, E
C beats D, E
D beats B, E
E beats A

The mixed strategy Nash Equilbrium here is to play A 1/3 E 1/3 and B, C and D 1/9th each.
[Intuitive proof, consider B,C,D together, this forms an "outer" RPS with A, {B,C,D} and E as moves -- if you choose B,C and D then that is an "inner" RPS].

I think this is the only 5 move version where all plays are useful without draws (no strategy is strictly dominated).

Naively, you might think to avoid playing B,C and D as they have a lesser chance of winning. If you do this your opponent will play E and you will only ever lose or draw so you need to throw in B C or D sometimes.

I cannot find a 4 move asymmetric version where one strategy does not strictly dominate another except for a "degenerate" uninteresting solution.
A > B,C
B > D
C > D
D > A

(B and C draw). B and C are identical plays and it is RPS with two variants for one move -- uninteresting -- rock paper, scissors A, scissors B.

I can't find a 5 move asymmetric version with equal mixed strategy nash equilibrium probabilities for all plays (even allowing draws).

[steer.livejournal.com]

Also I probably should have done work instead of spending time on that.

[major-clanger.livejournal.com]

I view such diversions as the mental equivalent of going out for a quick jog, so you should probably feel slightly virtuous for exercising the grey matter.

[steer.livejournal.com]

Well, except what I was *meant* to be doing was refactoring some Java code which emulates a number of routers in a distributed testbed -- which, believe me, stimulates the grey matter.

[major-clanger.livejournal.com]

You can patent the application of a gene, which is a slightly different thing from patenting the gene itself. So if you discover that gene 4A324 codes for a protein that acts as a potent hair restorer, you can get a patent for that use. Now, it may well be that about the only thing one can actually do with gene 4A324 is use it to make said protein, in which case you've got a patent over the sole practical use of that gene. But strictly speaking it's still the use that is patentable, not the gene itself.

As to why the applications of mathematical discoveries can't be patented but the applications of scientific discoveries can... well, there's no hard-and-fast good explanation so far as I understand it.

[drdoug.livejournal.com]

A possible interesting asymmmetric four-way:

A > B, C
B > C
C > D
D > A

[steer.livejournal.com]

Ah.... this is the sort of thing I started from. However, A weakly dominates B. If you played B then A would have always been a better or as good strategy and B can be removed from the game as you should never play it. There are no circumstances where B is a better move than A and there are circumstances where A is better than B. At this point is becomes regular RPS.

I guess you could argue "headology" that B would somehow psychologically unsettle your opponent but, to be honest, on me at least it would reassure me not unsettle me.

The mixed strategy nash equilbrium in that situation is to play A, C and D 1/3 of the time and never B.

[I think when I wrote the first entry I accidentally said strictly dominates rather than weakly dominates.]

[drdoug.livejournal.com]

Doh. You are quite right on all points here. Including accidentally saying strictly dominates rather than weakly dominates. :-) Which is less of a mistake than mine.

I suspect I've (now) duplicated your work here and am moderately convinced. Except I have two further points:

First, my new favourite four way, which is so good it needs instantiating with names: Rock/Paper/Scissors/Bodhisattva. Rock blunts Scissors, Scissors cuts Paper, Paper wraps Rock, Bodhisattva has escaped the cycle of suffering by transcending such attachments to attain nirvana, and so neither beats nor is beaten by the others.

Second, trying to understand why it's possible for five but not four, I had a brief glimpse of how to visualise this in N dimensions, but it went before I could do it properly. But it did convince me that the way to prove this would be to map it on to algebraic topology and knot theory. (Actual proof left as an exercise for the reader.)

you could argue "headology" that B would somehow psychologically unsettle your opponent but, to be honest, on me at least it would reassure me not unsettle me

You are talking to the person who regularly played Small Unsuspecting Yorkshire Hamlet here.

[drdoug.livejournal.com]

This is pretty much as I would say, except that the Small Unsuspecting Yorkshire Hamlet is correctly played with the fist pointing down, i.e. with the knuckles up, representing the chimneys on the roofs of the dwellings.

[drdoug.livejournal.com]

Oh, and for the reference of other readers, the non-symmetric version my kids (and their mates) developed is: Rock, paper, scissors, laser gun, super laser gun, water bottle.

RPS work as usual. Laser gun beats everything except super laser gun, and super laser gun beats everything except rock, which smashes it. Water bottle makes paper disintegrate, and contains rock. Scissors cut water bottle.

(Presentation and some analysis on my journal.)

Game theorists leave out paper, on the grounds that it is weakly dominated by water bottle. But game theorists are beaten by laser gun, super laser gun, rock and probably scissors too (they're very sharp scissors). Game theorists drink the water from the bottle, though, so they are not quite as underpowered as paper.

[steer.livejournal.com]

That just shows that even when you think you know a game, there are subtleties. A minute to learn, a lifetime to master.

[steer.livejournal.com]

Love the Bodhisattva move. I was trying to think of a 4 move variant even including draws but failed because when I arrived at the answer that "move D cannot possibly beat anything" it did not occur to me this was a satisfactory solution.

[surliminal.livejournal.com]

Gad I started this didn;t I!

[cartesiandaemon.livejournal.com]

Would it be too tedious to suggest numbering the options and representing them in binary with fingers? That would lessen the flavour, but make it a lot easier to play without arguments like "of course [water/fire] beats [fire/water], duh!"