This is deeply related to the idea of utility functions; It’s utility, not any measured resource, that you’re optimizing for. The relationship between points/scores/money/friends/etc. and utility is non-linear.
Of course, games are a specific case where the utility (winning) is well-defined, and usually relative to other players, where life is much more complicated, ambiguous, and a mix of competitive and cooperative games.
games are a specific case where the utility (winning) is well-defined
Lots of board games have badly specified utility functions. The one that springs to mind is Diplomacy; if a stalemate is negotiated then the remaining players “share equally in a draw”. I’d take this to mean that each player gets utility 1/n (where there are n players, and 0 is a loss and 1 is a win). But it could also be argued that they each get 1/(2n), sharing a draw (1/2) between them (to get 1/n each wouldn’t they have to be “sharing equally in a win”?).
Another example is Castle Panic. It’s allegedly a cooperative game. The players all “win” or “lose” together. But in the case of a win one of the players is declared a “Master Slayer”. It’s never stated how much the players should value being the Master Slayer over a mere win.
Interesting situations occur in these games when the players have different opinions about the value of different outcomes. One player cares more about being the Master Slayer than everyone else, so everyone else lets them be the Master Slayer. They think that they’re doing much better that everyone else, but everyone else is happy so long as they all keep winning.
In Diplomacy I’ve never heard the 1/(2n) argument from that sentence. All it’s saying is that if you are part of the draw, the person who survived with 1 supply center gets the same result as the one with all 17 on the other side of the line. Whether players actually treat it that way is up to them, of course.
But of course, my natural instinct is that winning alone is a special thing, and that winning outright is more than twice as good as a 2-way draw. When thinking about whether a 2-way draw is more or less than twice as good as a 4-way draw, I’m not sure.
In Castle Panic I think part of the fun is deciding how much you care about the title versus winning the battle, where the right answer is not zero but not enough to *seriously* risk losing the battle over that...