This blog is a sequel to this blog post, so if you haven't read that
We would like to design a game like rock paper scissors with the following axioms
- Symmetry: The isomorphisms form a transitive permutation group.
- Unambiguous: unless all players select the same action, there is never a draw.
- Single winner: for all combinations, there should be a single winner.
- Order invariance: The winner is selected indifferently from the order the players go in.
- Backtracking: Each losing player should have an action that if they would have chosen that action, and all other players kept the same action, they would have won.
We have already proven that this does not work if the number of players is larger than 3. For two players, rock-paper-scissors fits all axioms, so the one game we still have to find is a three-player game.
First, let us assume that there is a cycle that is also an isomorphism. This is sufficient for axiom 2, but not necessary.
Axiom 5 can be interpreted differently, saying that for every pair(in case of 3 players) there exists play where this pair loses. And as each triple has exactly one pair that loses, this also mains that there have to exist more triples than pairs, which is only true if there are at least five actions. For five actions, it is easy to see that the following winning strategies have a transitive permutation group (the one spanned by (01234)), and that each pair has a single winning action.
{012} -> 2
{013} -> 1
{014} -> 1
{023} -> 3
{024} -> 0
{034} -> 0
{123} -> 3
{124} -> 2
{134} -> 4
{234} -> 4
This can also be seen as drawing the five numbers on a wheel. We always pick the number wherefore there exists one number lower, but not one higher. (where 0 is one higher than 4). For example:
If we pick 034, then 3 for 3, there is a number one higher, but not one lower, so it loses. for the number 4, there is both one higher (0) and one lower (3), so it also loses. For 0, there is one lower (4), but not one higher, so zero wins.
If we pick 134, for 1, there is no higher nor lower, so it loses. For 3, there is one higher number, but and no lower one, so it loses, but for 4, there is a number one lower, but not one higher, so it loses.
Instead of using boring numbers, we can use the 5 elements according to Taoism.
We only need a single cycle, so we are using the generating one. So the game goes as follows:
- Each player selects an element, and all players reveal it at the same time.
- If all players have selected the same element, it is a draw.
- If two players have selected the same element, and a third one has a different element, the third player wins.
- Otherwise, the player whos element is generated by another element, but does not generate another element wins. So for example, Fire wins if Wood is there, and Earth is not.
So in practice, there are 125 choices, 5 are draws, 60 are won by other players playing duplicates, and 60 by 3 different elements.
If you want to play this game in real life, I suggest using scissors for fire, paper for metal(like a sheet of metal), Rock for earth (obviously), closed scissors for wood, and pinky for water, altough the choice is completely free, as long as it is consistent.