I’m interested in what happens if individual agents A, B, C merely have a probability of cooperating given that their threshold is satisfied. So, consider the following assumptions.
⊢□a(□e(□wE)→(□wE))→□xA
⊢□b(□e(□wE)→(□wE))→□yB
⊢□c(□e(□wE)→(□wE))→□zC
⊢□xA∧□yB∧□zC→□wE
The last assumption being simply that w is low enough. Given these assumptions, we have ⊢□wE via the same proof as in the post.
So for example if x,y,z are all greater than two thirds, there can be some nonzero w such that the agents will cooperate with probability w. In a sense this is not a great outcome, since viable w might be quite small; but it’s surprising to get any cooperation in this circumstance.
The interesting thing about this—beyond showing that going probabilistic allows the handshake to work with somewhat unreliable bots—is that proving ⊢□wE rather than ⊢E is a lot different. With ⊢E, we’re like “And so Peano arithmetic (or whatever) proves they cooperate! We think Peano arithmetic is accurate about such matters, so, they actually cooperate.”
With the conclusion ⊢□wE we’re more like “So if the agent’s probability estimates are any good, we should also expect them to cooperate” or something like that. The connection to them actually cooperating is looser.
I’m interested in what happens if individual agents A, B, C merely have a probability of cooperating given that their threshold is satisfied. So, consider the following assumptions.
⊢□a(□e(□wE)→(□wE))→□xA
⊢□b(□e(□wE)→(□wE))→□yB
⊢□c(□e(□wE)→(□wE))→□zC
⊢□xA∧□yB∧□zC→□wE
The last assumption being simply that w is low enough. Given these assumptions, we have ⊢□wE via the same proof as in the post.
So for example if x,y,z are all greater than two thirds, there can be some nonzero w such that the agents will cooperate with probability w. In a sense this is not a great outcome, since viable w might be quite small; but it’s surprising to get any cooperation in this circumstance.
The interesting thing about this—beyond showing that going probabilistic allows the handshake to work with somewhat unreliable bots—is that proving ⊢□wE rather than ⊢E is a lot different. With ⊢E, we’re like “And so Peano arithmetic (or whatever) proves they cooperate! We think Peano arithmetic is accurate about such matters, so, they actually cooperate.”
With the conclusion ⊢□wE we’re more like “So if the agent’s probability estimates are any good, we should also expect them to cooperate” or something like that. The connection to them actually cooperating is looser.