Going to the green state means you can’t get to the purple state as quickly.
On a deep level, why is the world structured such that this happens? Could you imagine a world without opportunity cost of any kind?
In a complete graph, all nodes are directly connected.
Equivalently, we assumed the agent isn’t infinitely farsighted (γ<1); if it were, it would be possible to be in “more than one place at the same time”, in a sense (thanks to Rohin Shah for this interpretation).
The opposite of this, is that if it were possible for an agent to be in more than one place at the same time, they could be infinitely farsighted. (Possibly as a consequence of FTL.)
In a complete graph, all nodes are directly connected.
Surprisingly, unless you’re talking about K1 (complete 1-graph), opportunity cost still exists in Kn (n>1). Each round, you choose where to go next (and you can go to any state immediately). Going to one state next round means you can’t go to a different state next round, so for any given action there exists a reward function which incurs opportunity cost.
Definition. We say opportunity cost exists at a state s if there exist child states s1,s2 of state s such that V∗R(s1)≠V∗R(s2) for some reward function R. That is, s has successor states with different (optimal) AUs for some reward function.
The opposite of this, is that if it were possible for an agent to be in more than one place at the same time, they could be infinitely farsighted. (Possibly as a consequence of FTL.)
Things get weird here, depending on your theory of identity and how that factors into the planning / reward process? Can you spell this out some more?
There is a star, many light years away. If you exist in two locations at once simultaneously, from which the star is visible, and those two locations are not the same distance from the star, then intuitively, by seeing the star first from the closer position, you can know what it will look like from the second before it happens.
Less trivially, by altering the relative speeds of the two versions (with FTL telepathy), and setting up suitable devices for signaling, I think in theory, this would enable turning FTL into time travel. (Person A performs a calculation, and sends the results to Person B. Since Person A is the future version of person B, and they’re the same person in two places at once simultaneously, then by ‘de-synchronizing them right’ a message can be sent into the past.)
In a complete graph, all nodes are directly connected.
The opposite of this, is that if it were possible for an agent to be in more than one place at the same time, they could be infinitely farsighted. (Possibly as a consequence of FTL.)
Surprisingly, unless you’re talking about K1 (complete 1-graph), opportunity cost still exists in Kn (n>1). Each round, you choose where to go next (and you can go to any state immediately). Going to one state next round means you can’t go to a different state next round, so for any given action there exists a reward function which incurs opportunity cost.
Definition. We say opportunity cost exists at a state s if there exist child states s1,s2 of state s such that V∗R(s1)≠V∗R(s2) for some reward function R. That is, s has successor states with different (optimal) AUs for some reward function.
Things get weird here, depending on your theory of identity and how that factors into the planning / reward process? Can you spell this out some more?
There is a star, many light years away. If you exist in two locations at once simultaneously, from which the star is visible, and those two locations are not the same distance from the star, then intuitively, by seeing the star first from the closer position, you can know what it will look like from the second before it happens.
Less trivially, by altering the relative speeds of the two versions (with FTL telepathy), and setting up suitable devices for signaling, I think in theory, this would enable turning FTL into time travel. (Person A performs a calculation, and sends the results to Person B. Since Person A is the future version of person B, and they’re the same person in two places at once simultaneously, then by ‘de-synchronizing them right’ a message can be sent into the past.)