rivalry
n. C / Un. a situation where people, teams, or companies compete against each other for a long time. It often happens when two sides want the same thing, like a trophy or more customers.
n. a state of competition or antagonism between two or more parties striving for the same objective or superiority in a specific field. Often implies a long-standing or historical relationship between the competitors.
The intense rivalry between the two football teams lasts for decades.
In the smartphone market, the rivalry between the two leading brands drives constant innovation and lower prices for consumers.
While healthy rivalry can stimulate economic growth and efficiency, extreme market dominance by two players may lead to aggressive tactics that eventually stifle smaller competitors.
From rival + -ry.
Often used with the prepositions 'between' (to name the parties) or 'for' (to name the goal).