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I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|}

I see that pseudo-surreal numbers seem to have an interesting branch of game theory.

Still having a form of {x|y}, where 'x' and 'y' are sets of surreal numbers, this includes integer and real numbers, as they have equivalent surreal numbers.

But pseudo-surreal numbers don't have to follow the other rules, that result in ordinary valid surreal numbers, such as the left term being less than the right term.

How can pseudo-surreal numbers be described?


Is "The value that is both less than 2 and more than 4." a reasonable description of {4|2}?

Is {2|2} analogous to "Between 2 and 2." or "The value that is both less than two and more than 2.", or 0 as that is the oldest value in that range? With the latter interpretation, does {0|0} behave differently?

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  • $\begingroup$ These are still "games" in the sense of On Numbers and Games, right? Just special games so that, after one move by either player, it is a number. $\endgroup$ – GEdgar Mar 21 '15 at 11:53
  • $\begingroup$ I think Conway uses the notation $\large{*}$ for $\{0|0\}$. $\endgroup$ – Akiva Weinberger Mar 30 '15 at 2:11
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As GEdgar said, these "pseudo-surreal numbers" are "games" in the technical sense of Combinatorial Game Theory. Basically, instead of thinking of this as a number system any more, we can think of these as game positions of a game with two players (conventionally called Left and Right, but they might correspond to, say, Black and White in a game of Chess). The "normal play convention" is that the first player to be stuck without a move loses the game, and the two sets tell you the moves.

For example, in $1=\{0|\}$, Left can move to $0=\{|\}$, and right has no moves available. Therefore, whether Left goes first or Right goes first, Left will win for sure. That makes this game "positive": it's better for Left than $0=\{|\}$.

If you want to know how these new values fit with the surreal number line, it's a little complicated: $\{0|0\}$, conventionally and hereafter referred to as $*$ (as columbus8myhw pointed out) is less than every positive surreal (positive surreals are clear benefits for Left compared to this game), and greater than every negative surreal (negatives are benefits for Right) but if you look at the definition of $\le$, this incomparable to $0$. We might say that it's "confused with $0$".

$\{4|2\}$ is a little weirder. It's greater than any surreal less than $2$, and less than any surreal greater than $4$, but confused with everything in the "closed interval" between. As an aside, this is more conveniently thought of as $\{1|-1\}+3$, which might be written as $3\pm1$.

If you would like to learn more about combinatorial game theory, I would recommend getting Lessons in Play (perhaps at a university library), but there are many resources online. One introduction aimed at advanced high school students is at https://simomaths.wordpress.com/2012/08/02/combinatorial-game-theory-i/. I have a slightly drier/more advanced treatment at https://combinatorialgames.wordpress.com/welcome-to-the-blog/. There are also plenty of PDFs and slides floating around the internet with a variety of different takes on the basics.

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games (aka fuzzy numbers aka nimbers)!! the most common example being $*=\{0|0\}$ which can also be stated as $*=\{0\}$ or "star is a $fuzzy$ zero".

the interesting thing about games! is that different types of games can be analyzed. the normal game form of $g=\{g_L|g_R\}$ is a 2 player form. thus $g=\{g_x\}$ can be seen as a 1 player form & $g=\{g_a|g_b|g_c\}$ as a 3 player form.

alternatively, games can analyzed as sub-games such that in the above representation the word player can be substituted with the word game.

these ideas can be combined such that an analysis over some number of subgames each with it's own number of players.

for instance, a game! broken down into 3 subgames (a 1 player game, a 2 player game & a 3 player game) :

$$g_0=\{g_1=\{g_x\}|g_2=\{g_l|g_r\}|g_3=\{g_a|g_b|g_c\}$$

also, games of the form {1|-1} are called hot games. based on my understanding at this point, it appears {1}={1|-1} or "fuzzy 1 equals plus/minus 1" such that {x} can be seen to be equivalent to |x|.

it is also intereting to note that 0={*}. based on the equivalent idea of absolute value we might say that * has an absolute value of 0 & inversely that 0 has an absolute value of * or that the absolute value of 0 is fuzzy.

tiny & miny are fun as well! $$tiny(x)=\{0|\{ 0 | -x \} \}$$ $$-tiny(x) = miny(x) = \{\{x|0\}|0\}$$ of additional interest is the fact that $$tiny(0) = \{0|*\} = UP$$ $$miny(0) = \{*|0\} = DOWN$$ $$tiny(*)=\{0|tiny(0)\}$$ $$miny(*)=\{miny(0)|0\}$$

an interesting pattern emerges when repeating the process of tiny or miny eg. $tiny(tiny(x))$ or $miny(miny(x))$ w/ 0 & *. try it out!

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    $\begingroup$ Unfortunately, your answer contains some inaccuracies/unclear statements. I don't believe fuzzy numbers (in the technical sense) are related to game theory at all (though I would believe that someone said "star is a fuzzy zero" in relation to games). Most of the games described by the OP are not nimbers. I am not certain what you are getting at with "1 player form"; impartial 2-player games are often written that way. And I'm not sure how you are proposing to break down a game into a 3 player game and a 2 player game..."Games of the form $\{-1|1\}$" is a little unclear. At the very least,... $\endgroup$ – Mark S. Jun 2 '18 at 22:50
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    $\begingroup$ I'll say that not every hot game is a switch. If anything should be called "fuzzy 1", I might nominate $1+*$, not $\pm 1=\{-1|1\}$, but I believe there is no definition of "fuzzy" in the CGT literature. I don't see any connection between $\{x\}$ and $|x|$; it seems like you were making some sort of analogy between the usual meaning of $\pm 1$ and the game meaning, but it's not clear to me. And since the meaning of $\pm$ is very different in the two cases, I worry you risk confusing people. $|x|$ has no standard definition outside of the surreals in combinatorial game theory. $\endgroup$ – Mark S. Jun 2 '18 at 22:50

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