Player elimination is something many people hate in games. Knowing you are losing and having to stay in the game is also something many people hate in games. The main thing it leads to is dissatisfaction. As a designer you can't have both, so you need to select a path. You can't make everyone happy.
Some games have catch up mechanisms as a way to settle this issue and never let a player feel like they are out of the game and just filling a chair. Many people like that, but it is problematic. Think of a game that allows you to make mistakes and always know that "it will be ok". This is because some mechanism will bump you forward or push the leader backwards. This is what catch up mechanisms do. So wait, what importance do my decisions have if making mistakes doesn't make you lose?
NONE!
Without naming names, think of games that always end at 1-2 point difference no matter what players do. I have a few in mind instantly, but I said no naming names, so I won't. I love knowing that good decisions on my behalf will make me the winner while bad ones make me lose the game. That said, I enjoy when catch up mechanisms keep you in the game, but not when they dictate a tight final scoring every single time.
My "grail game" is the new version of Through the Ages. It is a harsh game. You can be nothing short of destroyed half way through the game and it is a long game. This game has a nice resigning mechanism that works really well. A 4 player game can easily shift into a 3 player game and a 3 player into a 2 player. We already don't mind winning a game half through and just starting up a new one. This kind of solution fits mostly heavy gamers.
Different types of gamers enjoy different solutions for this issue. Gamers are found on a spectrum. Not that spectrum. They are always found somewhere between the enjoying the game itself (actually playing it) and enjoying winning games. Also there's the intensity of enjoyment that differs between players, since not everyone like gaming on the same level and gives it the same priority.
Look at this beautiful plot I made:
Most gamers I know move slightly below the dashed line, meaning they enjoy playing but also love winning. They will continue playing even after a streak of loses and have fun. I call them "fun" players because they will enjoy almost every gaming session and will most likely be open to more types of gaming experiences. Competitive gamers however, will enjoy being on the winning side much more and will thrive for winning. They enjoy the competition in and of itself, they want to improve the way they play and out smart their opponent after they understand why they lost.
For heavy gamers don't create extreme catch up mechanisms, it's not a good idea. While a harsh game that allows for no mistake is not a smart choice for the light gamer (Unless it is a very short game and no one really cares...).
Fit the solution to the gamer that you think will enjoy you game.