We've talked recently about combinatorializing connection games, but what about point-based games. Here's a general definition of a point-based game:
Players make moves and earn points during the game. This continues until there are no possible moves. When all moves have been made, the player with the most points wins.
What makes this not-quite-combinatorial is that it is not necessarily the case that the last player to move is the one who wins the game.
Flume is an example of such a game. Players score "a point" for each piece they play, and at the end the player with the most of their pieces on the board wins. Since there are an odd number of spaces, there will be no tie.
What happens, then, if you add two games of Flume together? What if you add a game of Flume to a game of Hex?
The way I've always envisioned these games working is as follows. Let's say the game ends with the left and right players each with their point totals (called left-points and right-points, respectively). Then a new game immediately starts with value: left-points - right-points, the winner of which wins the whole thing.
Thus, if you play a game of Flume and at the end the left player (Blue) played 9 disks while the right player (Red) played 16, then the result is game of value: -7, which the right player should win.
Is this how the "combinatorialization" is usually conceived? Is there another good way to handle this?
EDIT: Fixed a typo in the title. (Feb. 10, 2012)
The Best Writing on Mathematics 2016
3 days ago