Saucy, Ruby
=

    def saucy(total_degrees, degrees_left, total_people, people_left)
    
      current_winner_path = './data/saucy_current_winner'
      previous_cake_path = './data/saucy_previous_cake'
    
      first_move = (total_people == people_left)
    
      current_winner = first_move ? 0 : File.read(current_winner_path).to_i
      previous_cake = first_move ? total_degrees : File.read(previous_cake_path).to_i
    
      last_slice = previous_cake - degrees_left
      current_winner = [current_winner, last_slice].max
    
      File.open(previous_cake_path, 'w') { |f| f.puts(degrees_left)   }
      File.open(current_winner_path, 'w'){ |f| f.puts(current_winner) }
    
      if first_move
        degrees_left
      else
        average_left = degrees_left.fdiv(people_left).ceil
        beats_past_players = current_winner + 1
        beats_future_players = degrees_left/2 - (people_left-2)*13
        [average_left, beats_past_players, beats_future_players].max
      end
    
    end
    
    p saucy(*ARGV.map(&:to_i))

Saucy is willing to accept slightly less than half of the remaining cake, so long as it's more than anyone else has gotten or is likely to get (based on secret sauce).