I have a question about this entirely divorced from practical considerations. Can we play silly ordinal games here?
If you assume that the other agent will take the infinite-order policy, but then naively maximize your expected value rather than unrolling the whole game-playing procedure, this is sort of like ω+1. So I guess my question is, if you take this kind of dumb agent (that still has to compute the infinite agent) as your baseline and then re-build an infinite tower of agents (playing other agents of the same level) on top of it, does it reconverge to A∞ or does it converge to some weird Aω2?
I think you’re saying Aω+1:=[ΔAω→ΔA0], right? In that case, since A0 embeds into Aω, we’d have Aω+1 embedding into Aω. So not really a step up.
If you want to play ordinal games, you could drop the requirement that agents are computable / Scott-continuous. Then you get the whole ordinal hierarchy. But then we aren’t guaranteed equilibria in games between agents of the same order.
I suppose you could have a hybrid approach: Order ω+1 is allowed to be discontinuous in its order-ω beliefs, but higher orders have to be continuous? Maybe that would get you to ω2.
I have a question about this entirely divorced from practical considerations. Can we play silly ordinal games here?
If you assume that the other agent will take the infinite-order policy, but then naively maximize your expected value rather than unrolling the whole game-playing procedure, this is sort of like ω+1. So I guess my question is, if you take this kind of dumb agent (that still has to compute the infinite agent) as your baseline and then re-build an infinite tower of agents (playing other agents of the same level) on top of it, does it reconverge to A∞ or does it converge to some weird Aω2?
I think you’re saying Aω+1:=[ΔAω→ΔA0], right? In that case, since A0 embeds into Aω, we’d have Aω+1 embedding into Aω. So not really a step up.
If you want to play ordinal games, you could drop the requirement that agents are computable / Scott-continuous. Then you get the whole ordinal hierarchy. But then we aren’t guaranteed equilibria in games between agents of the same order.
I suppose you could have a hybrid approach: Order ω+1 is allowed to be discontinuous in its order-ω beliefs, but higher orders have to be continuous? Maybe that would get you to ω2.