Good fundamental question.
I'll ask, if you were to select a LED today, which one might it be?
If it's a game, then there also should be LEDs for each player? How many players?
There are "addressable LEDs". See
https://www.espruino.com/Individually+Addressable+LEDs
which might be a way to go.
The most cost-effective may very well be a microprocessor.
Another concept that you need to learn is that mechanical switches bounce, which means they don't switch once.
there is both "software debouncing" and "hardware debouncing".
I sort of want to suggest looking at
www.picaxe.com and also the Arduino platform.
An idea might be to build something simple that's repeated. e.g. A PICaxe controls one string of lights (e.g. 16) and all the switches. Each PICAxe gets an enable signal to know when to respond. It might need a clear and win signal as well.
Have another PICAxe deal with which player is UP.
I could envision simple lines of a small number of LEDs. I could also envision an oval "track" with LED lanes as well. the number of lanes = number of players. Corners might be tough. Draw lines at angles through the LEDs on the corners number.