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My question is: Is it possible to counter Dash Hopes by paying 5 life if you are already on 4 life or less?

It sounds like a weird question so I will put it in context.
My friends and I play large (5/6 player) multiplayer games and I like to use Exsanguinate late game. The problem is that it often gets countered. Therefore I decided to add Dash Hopes to my deck.
I plan to cast my Exsanguinate for 6 mana with a Dash Hopes in hand for safety. This means that each opponent will lose 4 life to Exsanguinate but if one of them chooses to counter my Dash Hopes they will lose 5 life but save the others from losing life.
Obviously my logic is that no one will sacrifice themselves to save their opponents from losing life. The only reason they would is if they are already on 4 or less life and the Exsanguinate was going to kill them anyway. In which case they would counter the spell just to spite me since they would die either way.
I know spells like Exsanguinate can put you on negative life but is it possible for a player to put themselves on negative life intentionally? It says in the rules text of Dash Hopes that "A player can't choose the "extortion" (i.e. pay 5 life) option unless he or she can actually do it." However, the argument could be made that you are still paying 5 life even though it kills you.

Sorry if this question is a bit long, first time poster.

You can't pay life you don't have.  If a player is at 5 life exactly, they can pay 5 life to counter Dash Hopes. However, being at 4 life or less makes it impossible to pay 5 life.