- The two countries have been involved in a tug-of-war over (the) control of the island.
- In fierce rivalry for first place, finally Raman’s group won the tug-of-war game.
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- The two countries have been involved in a tug-of-war type of fight for the control of the island.
- In a fierce competition for the first place, Raman’s group finally won the game of tug-of-war.
- The two countries have been involved in a tug-of-war over (the) control of the island. <-- correct as written. It means they have been fighting between the two of them over who gets control of the island.
- In a fierce competition, Raman’s team finally won the tug-of-war. <-- there is no need to add ‘game’ at all. As there are only two teams, it is also not natural to add ‘for first place’.)
Possibly use ‘game’ to show it is such and not a metaphor and better 'the game of …