It seems like you could create a dead man's switch using arbitrary participants. You distribute a secret to every participant and then to attempt to activate the dead man's switch they raise k to the power s mod p and pass it to the next participant. As long as you act as a participant each time and raise the passed value to some invalid s then the answer that is arrived at won't be the final secret.
As long as you participate every round the wrong answer will be arrived at, but as soon as you don't participate the right answer will be arrived at.
Any singular party refusing to cooperate would destroy the deadman's switch so malicious activation would be tough.
Designing it so it can tolerate failures would be the hard part.
EDIT: I am wrong, this isn't that great. It's really hard to hide information that can be recovered without a secret being revealed.