# Is there a construction of surreal games that isn't related to actual games?

Is there a way to construct games that isn't based on games? For example, construction of the surreal numbers. I haven't seen anything about Dedekind cuts that allow for the lower set to be larger than the upper set, so that approach seems like it won't work. I'm curious about this primarily because I'm not actually that interested in games like domineering or kayles & just want to be able to construct games (mathematically) independent of positions in specific games. Additionally, is it impossible to multiply games? Or has this just not been worked out yet?

Related: Conway games and Induction Principle for games [Note: I want to be able to handle transfinite & loopy games]

• While the rules are derived from the properties of actual games, you can of course just take those rules as axiomatic without referring to any meaning. The mathematics of the axiomatic system doesn't care whether you try to model something with it. For example, the natural numbers work well even if you don't apply them to counting things in the real world. May 24 '20 at 8:13
• @celtschk I agree. I guess I am A) unclear of what the axioms for games are & B) lacking examples of this in practice (everything I have seen always introduces games & explains them in the context of actual games). May 24 '20 at 8:23
• @MatthewDaly I have looked through ONAG. Chapter 7 still takes the 'actual games' approach. I'm looking more for something along the lines of "defined differently, usually in terms of a version of graphs" (from the answer in the related link above). May 24 '20 at 8:35
• BTW, what do you mean with “loopy games”? With the most obvious definition, they are explicitly excluded in combinatorial game theory (one condition is that games always have to finish in finitely many rounds; mathematically this boils down to the axiom of foundation). May 24 '20 at 8:35
• After looking around in that book, it seems that Aaron Siegel defines games as graphs (actually, bigraphs) instead of as pairs of games (the way Conways). If that fits your need, definition 1.2 on page 281 seems to be exactly what you asked for. May 24 '20 at 8:56

I haven't seen anything about Dedekind cuts that allow for the lower set to be larger than the upper set, so that approach seems like it won't work.

The "Dekekind cut"-like construction of Surreal numbers is not actual Dedekind cuts, and if you want a construction of loopfree (non-loopy) games you can simply drop the inequality condition for the two sets.

Following the slightly informal treatment in Claus Tøndering's Surreal Numbers - An Introduction, a game is just a pair of sets of previously created games. This can be made more formal using ordinals, as in Definition VIII.1.1 of Siegel's Combinatorial Game Theory. Siegel basically defines the games with formal birthday (ordinal) $$\alpha$$ as $$\tilde{\mathbb{G}}_{\alpha}=\left\{(\mathscr{G}^L,\mathscr{G}^R):\mathscr{G}^L,\mathscr{G}^R\subset\displaystyle{\bigcup_{\beta<\alpha}}\tilde{\mathbb{G}}_{\beta}\right\}$$, and then a long game (a possibly transfinite loopfree game) is an element of any $$\tilde{\mathbb{G}}_{\alpha}$$.

Additionally, is it impossible to multiply games?

You can certainly apply the definition of multiplication of surreal numbers to arbitrary games in the above sense. The problem is that unlike with numbers (or "nimbers"/"impartial games"), the "product" of arbitrary games does not respect equality. If $$G_1=G_2$$ and $$H_1=H_2$$, then it is possible that $$G_1H_1\ne G_2H_2$$.

I want to be able to handle transfinite & loopy games

As celtschk mentioned in a comment, loopy games are defined elsewhere in Siegel, in Definition VI.1.2. Essentially, a loopy game is defined as $$((V,E^L,E^R),x)$$ where $$V$$ is a set (you could think of it as the set of positions of the game) $$x\in V$$ (the starting position), and $$E^L,E^R$$ are sets of ordered pairs of elements of $$V$$ (showing which position transitions Left and Right can do). For those familiar with graph theory, $$(V,E^L)$$ or $$(V,E^R)$$ are digraphs, and Siegel calls $$(V,E^L,E^R)$$ a "bigraph" and $$x$$ the "start vertex".

• Thanks for yet another concise answer! After seeing so many game trees I'm kind of suprised this approach wasn't more obvious to me.. May 24 '20 at 20:23