In game design, you can think of your game as about getting to a winner, or discarding all the losers.
Simplest example, Snakes and Ladders. Snakes and Ladders, one player is trying to be separated from the group by being the winner. Their entire plan is advancing themselves. That winner has nothing to do with the losers – they’re just trying to take more steps ahead on their own.
You can look at a game like Jungle Speed, which is a sorting engine to find the loser – the winner finishes the game (and can step out) and then the next player to win, and so on, the game intact until the players decide to stop playing or until a loser is chosen by the process of the game.
Red Dragon Inn is a game about knocking out as many players as possible, by as many means possible – you can impoverish them, you can inebriate them, you can beat them up, you can even do some stuff outside the rules. In this way, players are trying to make everyone else lose, and resist losing as long as possible.
In Cockroach Poker, a rare example, all the players together pick one of them to lose. It starts out as a test of wits, then slowly becomes a game of trying to find a hole in each others’ defenses, trying to find the player who’s most convenient to get rid of… and making sure, as sure as you can that it’s not yours.
When you’re dealing with a cooperative game it’s the same basic thing: You want to advance the plan of the winners, which is everyone. And deathmatch games, where everyone is trying to survive the longest, are games where you want to lose the least.
Think about this when you design your game. Think about if your game is trying to push people to advance their own plan, primarily, or if they want to inhibit the plans of others.