Simon McGregor on Wed, 28 Jul 2010 06:13:23 -0700 (MST)

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[game-lang] Another quick tangential comment about integers and reals

Some friends and I occasionally played a game of my design (for 3 or
more players) which used the integers. The rules are simple: each
player secretly chooses an integer and all choices are revealed
simultaneously. If the choices are all different, the player who chose
the second-highest number wins. If any choices are the same, and one
player chose a higher number than all the others, that player wins. If
two players are tied for the highest choice, nobody wins. (I didn't
try to analyse this game properly!)

In regard to rationals and reals, imagine the following: a game is
played by moving points on a board in R2, up to 1 unit's distance at a
time (with the Euclidean metric). We can model this using the
rationals, because rational arithmetic is adequate to determine
whether two points with rational coordinates are within a rational
distance of one another.
However, suppose that the provably optimal move in some situation is
to move your point as far as possible towards another point. Then (in
general), no move in the rational model is optimal, although there's
an optimal move in the continuous model. Uh-oh!

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