In your example, the utility function is fixed (linear with a jump) and the optimal strategy adapts to future opportunities via standard backward induction. In EE, the evaluation function itself is derived from the dynamics rather than assumed independently of them. Multiplicative dynamics yield log, additive dynamics yield linear, other dynamics yield whatever the ergodic mapping produces. This isn’t “fixed utility function, different strategy in different trees” but “different dynamics, different evaluation function.”
This matters because standard sequential decision theory requires you to bring a utility function to the table before you can do any backward induction. Where does it come from? In your example, you simply stipulate it (linear with a jump). In standard economic practice, it’s a free parameter fitted to behavior. EE’s claim is that you don’t need to assume or fit a utility function: the dynamics of the process determine it.
Imagine an agent with a simple indexical utility function.
If the agents wealth is increasing like then the utility is (1,a,b)
If the agents wealth is increasing like then the utility is (0,a,b)
(Note that this agent doesn’t care what happens at any finite time. They care about the infinite asymptotic limit of their wealth)
This agent gives priority to the leftmost term, only optimizing the terms to the right if it faces a tie on the left.
I think this agent behaves like the ergodic economics agent on these 2 test environments.
Ps. I don’t understand what ergodic economics would do in more complicated environments. Lets say it doesn’t yet know whether it’s future bets will be linear or exponential. So it doesn’t know if it should use wealth or log wealth to calculate the bet in front of it. What then?
Standard expected utility theory deals most easily with the case where the number of rounds is finite.
So. How do we deal with an infinite number of rounds?
First, to avoid the difficulties of infinite backwards induction, assume you take a single choice. You choose one policy out of the set of all policies.
A potential result of a policy is a wealth trajectory, a function from the naturals to the reals, representing your wealth after each timestep.
Pick a nonprinciple ultrafilter U on the naturals.
We look at the equivilance classes formed by A~B iff is U-large.
This gives us a non-standard model of the reals.
Finite sums and products are easy to define on this nonstandard model, but the naive infinite sum depends on the choice of representitive of the equivilance class.
The expectation calculation involves an infinite sum.
I think that, given an infinite sequence of events (random bools) then it makes sense to define
Ultrafilters aren’t in the usual sigma algebra by default, but I think this definition makes sense.
Now our nonstandard model of the reals is something it makes sense to define a supremum in.
So, given our wealth sequence we can define the median sequence as
Note that W and A are sequences, so the median is a sequence (up to equivilance class) . This comparison is actually an infinite sequence of comparisons, and we are using the probability that an infinite sequence of bools is in an ultrafilter construction (above)
Now we just define the expectation as where f(x) is the median (except with x=0.5) Eg f(x) represents the sequence that you have a chance x of exceeding.
I think that this setup yields your ergodic dynamics as just expected utility maximization. (Admittedly on a nonstandard model of the reals)
In your example, the utility function is fixed (linear with a jump) and the optimal strategy adapts to future opportunities via standard backward induction. In EE, the evaluation function itself is derived from the dynamics rather than assumed independently of them. Multiplicative dynamics yield log, additive dynamics yield linear, other dynamics yield whatever the ergodic mapping produces. This isn’t “fixed utility function, different strategy in different trees” but “different dynamics, different evaluation function.”
This matters because standard sequential decision theory requires you to bring a utility function to the table before you can do any backward induction. Where does it come from? In your example, you simply stipulate it (linear with a jump). In standard economic practice, it’s a free parameter fitted to behavior. EE’s claim is that you don’t need to assume or fit a utility function: the dynamics of the process determine it.
Previous comment was too complicated.
Imagine an agent with a simple indexical utility function.
If the agents wealth is increasing like
If the agents wealth is increasing like
(Note that this agent doesn’t care what happens at any finite time. They care about the infinite asymptotic limit of their wealth)
This agent gives priority to the leftmost term, only optimizing the terms to the right if it faces a tie on the left.
I think this agent behaves like the ergodic economics agent on these 2 test environments.
Ps. I don’t understand what ergodic economics would do in more complicated environments. Lets say it doesn’t yet know whether it’s future bets will be linear or exponential. So it doesn’t know if it should use wealth or log wealth to calculate the bet in front of it. What then?
Standard expected utility theory deals most easily with the case where the number of rounds is finite.
So. How do we deal with an infinite number of rounds?
First, to avoid the difficulties of infinite backwards induction, assume you take a single choice. You choose one policy out of the set of all policies.
A potential result of a policy is a wealth trajectory, a function from the naturals to the reals, representing your wealth after each timestep.
Pick a nonprinciple ultrafilter U on the naturals.
We look at the equivilance classes formed by A~B iff
This gives us a non-standard model of the reals.
Finite sums and products are easy to define on this nonstandard model, but the naive infinite sum depends on the choice of representitive of the equivilance class.
The expectation calculation involves an infinite sum.
I think that, given an infinite sequence of events (random bools)
Ultrafilters aren’t in the usual sigma algebra by default, but I think this definition makes sense.
Now our nonstandard model of the reals is something it makes sense to define a supremum in.
So, given our wealth sequence
Note that W and A are sequences, so the median is a sequence (up to equivilance class) . This comparison is actually an infinite sequence of comparisons, and we are using the probability that an infinite sequence of bools is in an ultrafilter construction (above)
Now we just define the expectation as
I think that this setup yields your ergodic dynamics as just expected utility maximization. (Admittedly on a nonstandard model of the reals)