How is this game where you write a kanji on your hand supposed to work?
It was indeed explained more fully in the manga in volume 3, chapter 18.

Origin of the "game"



Yes, the most people writing the same character win, and yes, you can theoretically write any kanji character there is, as long as you explain how you could make the race more exciting with the idea represented by that kanji.

Hijiri and Raimu both won, but Hijiri was especially excited because in the manga,


   it took the girls 15 minutes to play the game to decide on what to do at the school festival, and Hijiri was the only one who wrote a different kanji character in that first round of vote so she felt kind of a loner.


And in the Battle of the Red Cliff, only Zhou Yu was in charge of carrying out the fire attacks because Wu had more military power and was more hard-pressed for survival, just like Hijiri is compared to Raimu.

Lastly, I believe it to be an error on the part of the sub group to translate what Rin wrote as "silver" (銀) when in fact she wrote "bells" (鈴).