expression
fighting to be first; scrambling to take the lead
Describes people competing eagerly to be the first to do something, often in a rush or with a sense of urgency. Used adverbially with verbs like する, 出る, 走る.
子どもたちは先を争ってバスに乗り込んだ。
The children scrambled to get on the bus first.
新製品の発売日には、客が先を争って店に押し寄せた。
On the release day of the new product, customers rushed to the store, fighting to be first.
Both mean trying to be first, but 我先に emphasizes selfishness or disregard for others, while 先を争って focuses on the competitive scramble itself.
Fixed expression from 先 (さき, 'ahead; first') + を + the te-form of 争う (あらそう, 'to compete; to fight for'), literally 'competing for the lead'.