Three Dollar Quill

So, I was saying last night that I had a bunch of candidate comparison operators for surcomplex numbers such that * - * = 0.

News! I came up with another. "Exactly like the existing * - * = * operators, except that both players can either make one or two moves per turn." This is... interesting, because it should have no effect on numbers at all, and only matter for nimbers. One thing it doesn't change from the existing operators is the equation > - > = *. For what it's worth, I don't think any set of operators guarantees x - x = 0 for all (quasi-)nimbers.

I'm writing this early, so I'm going to try to finally define some of these terms.

In the context of surreal numbers, I believe the full definition is that we have sets of numbers (construed broadly), and we combine those sets into ordered pairs. Those ordered pairs are then separated into equivalence classes, and those equivalence classes are numbers. Now, these equivalence classes are defined in terms of either a pair of comparison operators, or one operator applied in both directions. The comparison operators are defined recursively, with the pair of empty sets acting as the base case. It is a requirement that the comparison relations hold for identical pairs, and if both relations hold for two different pairs, they're in the same equivalence class. (Which means that the relations need to have a bunch more properties, in order to ensure that they actually define an equivalence relation.) Now, for the surreal numbers, all of this is established and proven.

Incidentally, the "nimbers" are what happens when you have a pair that fails to satisfy both relations at once for 0. In other words, they are neither greater-than-or-equal-to 0, nor less-than-or-equal-to 0.

Anyway, it wasn't immediately obvious how to apply these ideas to triples instead of pairs; I was leaning heavily on the "game" interpretation of the numbers, in which 0 is a loss for the first player, positive numbers are a win for Blue, the player represented by the left set, and negative numbers are a win for Red, the player represented by the right set. Having Blue, Amber, and Red in every game raised questions about how to handle scoring; I couldn't figure out any way for a player to decide between the other two players to take the losing move. This is why my various ideas have focused around reducing the number of players to 2. The main issue I have with my various options is that they tend not to line up intuitively with what we'd expect from a three-player game. We can say that the "goal equation" for such a reduction is > + > = -1, which I believe some of the more complex schemes I came up with do satisfy, so I guess I'll have to give them another shot.

For now, though, I really need to wrap things up for the night.

Good night.