#mi kama tawa anpa tawa supa tawa e lanpan e sona jo e toki pona tawa e sitelen toki ni ken. pilin pona o tawa sina.

[mayajadeart]

I love that jan Misali first learned about the academic field of game theory when people kept mentioning it in the comments of their video about hangman, and now they're on tumblr posting shit like this. Truly a jan who speaks for our generation. sina pali e ijo pi namako nasa mute la mi sona ala o awen. pona! anu... pona ala... anu pona! pona ala li sama e pona. anu... ala? mi sona ala! ;-;

the additive group of day-2 normal partizan games

recently I've been trying to gain a deeper understanding of one very small subset of the class of normal partizan games: the games that are born on or before day 2. I've explained the general concept on main before, so I'll assume you're already familiar with the idea.

to recap, there are 22 games under the normal play convention that end after at most two moves. three moves extends that to 1474, and beyond that many moves it's an open question.

normal partizan games can be added together to form new normal partizan games. analyzing when a complicated game can be broken down into the sum of multiple other simpler games is one of the most important concepts in combinatorial game theory.

the sum of games has the properties that it's associative ((F + G) + H = F + (G + H)) and commutative (G + H = H + G), just like regular addition. so what I'm interested in is what happens if you ignore the game-ness of these games and just explore the behavior of the sums of these particular small games as an abelian group. that is, given two games G and H expressed in the form of some number of day-2 games all added together (allowing duplicates), when does G equal H? and I uh, almost understand it? but not everything.

the generative set

while these 22 games are all very simple, we don't really need all 22 of them. I'm not 100% sure how small of a generative set you can reduce them to, but I've been able to cut it down to just ten:

*2 = {0,*|0,*}

½ = {0|1}

-½ = {-1|0}

↑ = {0|*}

↓ = {*|0}

±1 = {1|-1}

±½ = {½|-½}

{½|-½*}

{1|0,*}

{0,*|-1}

the ones I've put in red are not actually day-2 games, but they can be expressed as the sums of other day-2 games, so they're fine to use. they could be replaced with actual day-2 games, but I think these particular forms make the math easier. and the last two are uh, weird. I don't know what their deal is.

anyway, all day-2 games (and by extension all sums of day-2 games) can be represented as the sum of some combination of members of this generative set. I won't explain how for all of them because it's pretty trivial a lot of the time, but here are some of the highlights:

the zero game can be reached by adding any game to its negative, or alternatively it just doesn't need to be part of the generative set because the "empty sum" comes for free

the switches {1|0} and {0|-1} can be represented as ½±½ and -½±½, respectively

the hot game {1|*} has the property that {1|*} + {1|*} = 1 + * = 1*. this suggests that it could be broken down into a "one half" component and a "star halves" component. */2 is not a unique value (there are in fact infinitely many games which when added to themselves equal star), but {½|-½*}, one half less than {1|*}, does indeed have the property that when added to itself you get * as we'd expect, and since * is its own negative, three copies of {½|-½*} can be added together to get its negative.

adding these games together

a lot about the behavior of the sums of games is already very well-documented. as I said, this is one of the key concepts in combinatorial game theory.

within this generative set, three games are their own negatives (*2, ±1, and ±½), which simplifies things a lot. they essentially don't interact with anything else in the group. "star halves" works in a similar way, as described earlier.

numbers and arrows also don't interact with each other. ↑ is infinitesimal, smaller than every positive surreal number but still positive.

if we ignore the hot fuzzy game {1|0,*} and its negative, everything else then is pretty straightforward. games in this group can be broken down into a number part, an arrow part, a "star halves" part, a *2 part, and two switch parts. the sum of two games in this group can then be handled in all of those parts individually without needing to worry about any of these parts bleeding into each other.

but what of the hot fuzzy games {1|0,*} and {0,*|-1}? do they interact with anything else?

and I don't have the answer to that. they're really weird! {1|0,*} is hot (players have an incentive to move) and fuzzy (whoever makes the first move wins), but {1|0,*} added to itself is positive! I don't think these games will ever combine with something in this group to form something that could have been expressed without using them, but I also don't know how I would go about proving that.

anyway that's just what I've been thinking about recently

[janokenmun]

"o moli e ona kepeken pali pona" LON ALA. IKE NASA NIMI

a akesi ala alasa ale anpa ante anu awen e en epiku esun ijo ike ilo insa jaki jan jasima jelo jo kala kalama kama kasi ken kepeken kijetesantakalu kili kin kipisi kiwen ko kokosila kon ku kule kulupu kute la lanpan lape laso lawa leko len lete li lili linja lipu loje lon luka lukin lupa ma mama mani meli meso mi mije misikeke moku moli monsi monsuta mu mun musi mute n namako nanpa nasa nasin nena ni nimi noka o oko olin ona open pakala pali palisa pan pana pi pilin pimeja pini pipi poka poki pona pu sama seli selo seme sewi sijelo sike sin sina sinpin sitelen soko sona soweli suli suno supa suwi tan taso tawa telo tenpo toki tomo tonsi tu unpa uta utala walo wan waso wawa weka wile

“kill them with kindness” Wrong. CURSE OF RA 𓀀 𓀁 𓀂 𓀃 𓀄 𓀅 𓀆 𓀇 𓀈 𓀉 𓀊 𓀋 𓀌 𓀍 𓀎 𓀏 𓀐 𓀑 𓀒 𓀓 𓀔 𓀕 𓀖 𓀗 𓀘 𓀙 𓀚 𓀛 𓀜 𓀝 𓀞 𓀟 𓀠 𓀡 𓀢 𓀣 𓀤 𓀥 𓀦 𓀧 𓀨 𓀩 𓀪 𓀫 𓀬 𓀭 𓀮 𓀯 𓀰 𓀱 𓀲 𓀳 𓀴 𓀵 𓀶 𓀷 𓀸 𓀹 𓀺 𓀻 𓀼 𓀽 𓀾 𓀿 𓁀 𓁁 𓁂 𓁃 𓁄 𓁅 𓁆 𓁇 𓁈 𓁉 𓁊 𓁋 𓁌 𓁍 𓁎 𓁏 𓁐 𓁑 𓀄 𓀅 𓀆