If you really want dynamic variable names, may be you can use a Hash, than your can set the key dynamic
file = File.new("games.log", "r")
lines = {}
i = 0
file.lines do |l|
l.split do |p|
if p[1] == "InitGame"
lines[:"Game_#{i}"] = Hash.new
i = i + 1
end
end
end