Which leads me to needing: There's some kind of object; some of which are desirable for some players (some of the time?) and to others are actively harmful.
Got a couple of ideas for it.
One is a tile chain. There are four colours, each tile outputs one; and consumes either one, or a chain, for more points. If you can't use it for either, then it's actively harmful to your score. So each player is after different inputs and outputs to make their different chains.
Another is something more like simple yahtzee matching. Try to make a combination, and cash it in. Or hold it and try to go for a more difficult combination.