I have always been in the mindset that people are naturally competitive and need very little pressure to keep score and challenge each other. All that gamers need to compete is an environment where one player can do something better than another, either directly or indirectly. The best example of this is the Super Smash Brother Melee competitive community. When simply given a fun game where one player wins and another loses, many people around the world decided that they were the best at the game. After posting online, others who said that they were the best challenged the other players to a competition. These players were “the big fish in their small pond” that the book talked about, but they also slowly found and created larger ponds and some found they were big enough to get into bigger ponds. Then the larger fish decided they wanted to try to create an even larger pond. This grew and grew until tournaments that lasted entire weekends with thousands of entrants and prize pools nearing 6 digits became the norm. The best part, Nintendo actually actively discouraged this community until very recently. (If you want more watch the smash documentary Here: https://www.youtube.com/watch?v=NSf2mgkRm7Q&list=PLoUHkRwnRH-IXbZfwlgiEN8eXmoj6DtKM)
Why do gamers chase achievements? Dopamine is one hell of a drug. Progression and achievement in games has been proven to release dopamine in the brain.
So some gamers are chasing achievements to get their next hit of dopamine and feel that excitement of winning and moving forward. This one if fairly straightforward, giving players a goal to strive for means they will want to get to it.