Word for mental exertion

One definition of sport is:

an activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment.

Is there a word for:

an activity involving mental exertion and skill in which an individual or team competes against another or others for entertainment.

AKA (e-sport), but not all mental exertion/skill based activities are done on a computer or game platform. Would competition be the best word use (I feel like this word can apply to either physical or mental events though)?

I couldn't find a single word , but I found this idiom to be very close to what you want (from Farlex Dictionary of Idioms, via TheFreeDictionary.com):

a battle of wits

a situation in which two people or groups try to defeat each other by using their intelligence (often + between )

Example

"The two young geniuses entered into a battle of wits as they competed for first place in the national trivia championship. As each candidate tried to defend his stance on the hot-button issue, a battle of wits began as they each produced seemingly endless lists of statistics to support their views."

I feel this idiom best suits OP's needs since this is all encompassing when it comes to contextual usage.

You might be able to use game. Relevant definitions, from Merriam-Webster:

1 a (1) : activity engaged in for diversion or amusement : play

3 a (1) : a physical or mental competition conducted according to rules with the participants in direct opposition to each other

Although not all games would fit your requirement of mental exertion and skill, I think any activities that do fit your definition could be called games—and most games, even of the very physical type, also require at least some strategy/mental exertion, at least to remember the rules¹. This term also seems a bit more specific to me than competition (which isn't always for entertainment).

¹ There is a fair amount of overlap between game and sport, with the key difference largely hinging on

• How organized the competition is: games require organization and rules, but not all definitions of sport have this requirement; and

• Physical activity: most definitions of sport, including the one you use, require this—although there is a fair amount of dispute about the level of prowess/exertion required—while definitions of game don't generally require physical exertion, although they do allow for it.

There's a good discussion of the distinction in this answer to the question "Difference between 'game' and 'sport'".