Especially older games had no real means of playing with or against other humans. Some game concepts also simply are not compatible with multi-player play of any kind. That does not mean, such games cannot be played competitively. Many of those games give a player some sort of score or other means for comparison with other players. Speed-runners for example try and play the single-player in as shorter time as possible. The players with the lower time are considered better, including keeping records of achieved times and proof videos showing how/that the player actually did it.
Classic games on older computers such as a C64 or an arcade machine simply gave points for certain events inside the game to the player. At the end the score was recorded, often along with the name of the player.
Other games are kind of open-end, meaning, you cannot actually win them but will be faced with a situation, that will defeat the player eventually. In such settings the higher time of surviving is considered the goal. This can even be combined with cooperational play, like the Invasion Mode of UT2004, where the goal is to survive waves of incoming hostile aliens and not fragging each other.
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