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If it depends on physical ability, this opens up more things than I would consider sports to be one - including darts,bar billiards, shove ha'penny etc.
Indeed. Darts and billiards, weirdly, can be considered sports and are certainly treated similarly by aficionados.
@dancing james re: esports - if you haven’t already, watch a video of elite StarCraft or call of duty players. The people that get to that level need exceptional motor control, reaction speed, and spatial awareness, as well as creativity, and strategic and tactical thinking. I think a strong case can be made to consider them ‘real’ sports.
Chess is an example of a game that can also be a sport: a normal match is a game, whereas blitz chess (or whatever the name of the timed version of chess is, I forget) introduces an element of physicality that grants an advantage to the more physically adept player who is able to move their pieces and tap their clock more quickly. Whether or not that is enough for it to be a determining factor in the competition, I don’t know, but if it were very determinant then it could also be considered a sport under this working definition.
My conclusion thus far to distinguish between a sport and a game is that a sport hinges on the physical abilities of the participants, whereas the physical actor in a game is irrelevant as long as the decisions are made by the same person.