I want to push on one thing, though. I’m sceptical of the claim that the ergodicity economics agent violates independence.
As I understand it, the EE agent has a fixed objective: maximize the time-average growth rate of wealth, which is equivalent to maximizing expected log terminal wealth. When the stochastic environment changes — say from multiplicative to additive dynamics — the optimal per-bet policy changes. In the multiplicative case, you Kelly-bet (which looks like log utility applied locally). In the additive case with many independent bets, you behave roughly linearly with each bet (because log is approximately linear for small additive increments relative to total wealth).
But is this actually a violation of independence? Independence says: if you prefer lottery A to lottery B, then mixing both with a common lottery C at the same probability shouldn’t reverse that preference. It’s a constraint on your ranking of probability distributions over outcomes.
What the EE agent is doing seems different. They have a fixed preference over (distributions over) outcomes (log terminal wealth, or equivalently, time-average growth rate). When the dynamics change, the mapping from available actions to outcome distributions changes, so the optimal action changes. But the preference ordering over final outcomes hasn’t changed — the agent still prefers higher log wealth to lower log wealth. It’s the decision problem that’s different, not the preferences.
To put it another way: an EU maximizer with log utility would make exactly the same choices as the EE agent in every case you describe. They’d Kelly-bet in multiplicative environments and behave more linearly in additive ones, because that’s what maximizing expected log wealth requires in each setting. But the EU maximizer with log utility satisfies independence by construction. So how can the EE agent be violating independence while making identical choices?
I think the thing that looks like a context-dependent utility function is really a context-dependent policy derived from a fixed utility function under different dynamics. These seem importantly different, and I’m not sure the independence axiom is violated by the latter.
Oh, you just apply different ergodic transformations to different lotteries, of course.
Also, beware that besides this wrong example, the linked paper contains other basic misconceptions about EE, like for example the claim that EE is equivalent to log utility.
You have to apply different transformations to different lotteries, because EE requires that all lotteries be transformed such that the result is ergodic. There is no single transformation function that can make a multiplicative lottery ergodic while also making an additive lottery ergodic.
the linked paper contains other basic misconceptions about EE, like for example the claim that EE is equivalent to log utility.
It does not make that claim. The claim was that there are multiple transformation functions that can make multiplicative bets ergodic, but in practice, EE proponents always use the logarithm function, which produces a decision theory that’s equivalent to log utility for the special case of multiplicative bets.
As I understand it, the EE agent has a fixed objective: maximize the time-average growth rate of wealth, which is equivalent to maximizing expected log terminal wealth.
EE has the objective to maximize time-average growth rate, but it is generally not equivalent to maximizing expected log terminal wealth. This is the single most common misunderstanding of ergodicity economics, probably(
To put it another way: an EU maximizer with log utility would make exactly the same choices as the EE agent in every case you describe. They’d Kelly-bet in multiplicative environments and behave more linearly in additive ones, because that’s what maximizing expected log wealth requires in each setting. But the EU maximizer with log utility satisfies independence by construction. So how can the EE agent be violating independence while making identical choices?
That is exactly my point that they are not making identical choices.
For multiplicative dynamics, yes, they coincide exactly. For additive dynamics, they diverge: the log-utility maximizer remains risk-averse (because log has curvature everywhere), while EE prescribes risk-neutrality (linear evaluation) for additive dynamics. The EE agent with an additive gamble would accept bets that the log-utility maximizer would reject.
You say:
In the additive case with many independent bets, you behave roughly linearly with each bet (because log is approximately linear for small additive increments relative to total wealth).
In the additive case, you behave within EE framework not roughly linearly, but just linearly, and it is not because log is approximately linear, but just because ergodic mapping is identity mapping in the case of linear dynamics.
And to address your core question of whether independence is actually violated: yes, it is, and here is one simple example similar to the one in the post itself.
Consider two compound lotteries sharing a common component C. Independence says your ranking of the non-common parts should not depend on what C is. But in EE, the ergodic mapping is applied to the dynamics of the entire wealth process, not branch by branch. When C is a multiplicative process, the overall dynamics are multiplicative, the ergodic mapping yields log, and you might prefer gamble A on the non-common branch. When C is an additive process, the overall dynamics are additive, the ergodic mapping yields linear (identity), and you might prefer gamble B on the non-common branch. Your preference between A and B has flipped depending on the common component, which is exactly what independence forbids.
This happens because the ergodic mapping is a global property of the stochastic process. You cannot decompose a compound lottery into branches, apply the ergodic mapping to each branch separately, and recombine.
I think the thing that looks like a context-dependent utility function is really a context-dependent policy derived from a fixed utility function under different dynamics. These seem importantly different, and I’m not sure the independence axiom is violated by the latter.
If this were true, I’d agree that independence isn’t violated.
“Maximize time-average growth rate” is a fixed preference, but it’s a preference over something richer than probability distributions over terminal wealth. It’s a preference over trajectory-classes-indexed-by-dynamics: you’re evaluating not just “what’s the distribution of my final wealth” but “what’s the distribution of my final wealth given the dynamic process I’m embedded in.” And vNM utility is defined over probability distributions over outcomes, not over dynamics-outcome pairs. The EE meta-preference cannot be expressed as a vNM utility function over outcomes because the dynamics are not outcomes, unless you trivially define the dynamics as “outcomes” and then it works “by definition”.
Thanks for the detailed reply! I want to push back on the claim that the EE agent and the log-utility maximizer make different choices
You say that in the additive case, the EE agent is exactly risk-neutral while the log-utility maximizer remains risk-averse. But consider what a log-utility maximizer actually does when facing a long sequence of independent additive bets.
For a single additive bet with payoff x, the contribution to log wealth is ln(W + x) - ln(W) ≈ x/W for small x relative to W. So the log-utility maximizer treats each bet approximately linearly. And as the number of independent additive bets grows, each individual bet becomes smaller relative to total wealth (because wealth is growing via the accumulated positive-EV bets), making the linear approximation increasingly exact.
In the limit of infinitely many independent additive bets, the log-utility maximizer’s per-bet behavior converges to exact risk-neutrality — which is exactly what EE prescribes. So in the regime where the EE prescription is most cleanly motivated (many repeated bets, which is the whole setup for time-average reasoning), the two agents converge.
Where they diverge is for large additive bets relative to current wealth. But here I think the log-utility maximizer is actually more faithful to the “maximize time-average growth rate” objective than the EE agent is. If you’re risk-neutral about an additive bet that could cost you half your wealth, you’re exposing yourself to ruin risk, which destroys your time-average growth rate. The identity mapping tells you the additive process is ergodic, but being ergodic doesn’t mean you should be risk-neutral about large bets within it — not if your goal is to maximize the growth rate of your own single trajectory.
So it seems to me that either (a) the bets are small relative to wealth, and the log-utility maximizer behaves (approximately, converging to exactly) like the EE agent, or (b) the bets are large relative to wealth, and the log-utility maximizer is arguably more correct about maximizing time-average growth. In neither case do the two agents clearly diverge in a way that supports the independence violation argument.
What am I missing?
If I’m right then again it doesn’t seem like the EE agent is really violating independence, given that its behaviour is replicated by a vnm agent that just values its final log wealth
I probably don’t have time to answer to this in detail. The way I see you could understand this is to imaging a completely different dynamic, neither multiplicative nor additive, something weirder, like for example square root or raising to power of 2. For each of that dynamic, EE will give different answers, because ergodic mapping is different for each and also—sometimes—dramatically different from log utility, or linear utility.
But I don’t think EE would give different answers! EE would give the answer that would maximise the time-averaged geometric growth rate. That would be the same answer that would maximise the expected final log wealth after a long series of identical decisions.
So yes, what you say is exactly the misconception about EE. EE will give different answers for each different dynamic, because time-average is calculated differently for each dynamic. Please see for example section 6.6. in the book on EE.
Which of dynamic consistency and consequentialism is not obeyed in EE situations like the one you suggested and how?
My intuition is that
Both are obeyed, and EE can in fact be expressed as a vnm utility function (prove me wrong by answering the above question! It’s an intuition because i haven’t yet been able to pin down how to express EE as a vnm utility function, more below)
Utility theory axioms imply that some utility function exists, not its form.
Said utility function depends on your preferences. If i have a buck and you offer me repeated 50-50 odds of winning 50% of everything or losing 40% of everything its a bad bet—unless this is the only way for me to make money and i want to buy something for two bucks. In this case, a chance is better than nothing.
More generally, EE is specifying a certain set of preferences where having a higher expected wealth is less important than ensuring the growth is measure 1. This is a useful set of preferences in the real world, but neither necessitated nor argued against by the axioms of utility theory.
I expect that if my preferences are about what will happen in the limit of the game, and i care about what happens most of the time, that the utility function will converge to the EE solution, regardless of the specific dynamics. (Is that defining dynamics as outcomes?)
Nice post!
I want to push on one thing, though. I’m sceptical of the claim that the ergodicity economics agent violates independence.
As I understand it, the EE agent has a fixed objective: maximize the time-average growth rate of wealth, which is equivalent to maximizing expected log terminal wealth. When the stochastic environment changes — say from multiplicative to additive dynamics — the optimal per-bet policy changes. In the multiplicative case, you Kelly-bet (which looks like log utility applied locally). In the additive case with many independent bets, you behave roughly linearly with each bet (because log is approximately linear for small additive increments relative to total wealth).
But is this actually a violation of independence? Independence says: if you prefer lottery A to lottery B, then mixing both with a common lottery C at the same probability shouldn’t reverse that preference. It’s a constraint on your ranking of probability distributions over outcomes.
What the EE agent is doing seems different. They have a fixed preference over (distributions over) outcomes (log terminal wealth, or equivalently, time-average growth rate). When the dynamics change, the mapping from available actions to outcome distributions changes, so the optimal action changes. But the preference ordering over final outcomes hasn’t changed — the agent still prefers higher log wealth to lower log wealth. It’s the decision problem that’s different, not the preferences.
To put it another way: an EU maximizer with log utility would make exactly the same choices as the EE agent in every case you describe. They’d Kelly-bet in multiplicative environments and behave more linearly in additive ones, because that’s what maximizing expected log wealth requires in each setting. But the EU maximizer with log utility satisfies independence by construction. So how can the EE agent be violating independence while making identical choices?
I think the thing that looks like a context-dependent utility function is really a context-dependent policy derived from a fixed utility function under different dynamics. These seem importantly different, and I’m not sure the independence axiom is violated by the latter.
EE isn’t VNM because it violates completeness—there are lotteries that can’t be compared to each other. An example (from here):
There is no single transformation function that makes both of these lotteries ergodic, so EE has no way of saying which is better.
I’m not sure whether EE violates independence; like you, I’m not convinced, but I’d have to think about it more to say with confidence.
Oh, you just apply different ergodic transformations to different lotteries, of course.
Also, beware that besides this wrong example, the linked paper contains other basic misconceptions about EE, like for example the claim that EE is equivalent to log utility.
You have to apply different transformations to different lotteries, because EE requires that all lotteries be transformed such that the result is ergodic. There is no single transformation function that can make a multiplicative lottery ergodic while also making an additive lottery ergodic.
It does not make that claim. The claim was that there are multiple transformation functions that can make multiplicative bets ergodic, but in practice, EE proponents always use the logarithm function, which produces a decision theory that’s equivalent to log utility for the special case of multiplicative bets.
EE has the objective to maximize time-average growth rate, but it is generally not equivalent to maximizing expected log terminal wealth. This is the single most common misunderstanding of ergodicity economics, probably(
That is exactly my point that they are not making identical choices.
For multiplicative dynamics, yes, they coincide exactly. For additive dynamics, they diverge: the log-utility maximizer remains risk-averse (because log has curvature everywhere), while EE prescribes risk-neutrality (linear evaluation) for additive dynamics. The EE agent with an additive gamble would accept bets that the log-utility maximizer would reject.
You say:
In the additive case, you behave within EE framework not roughly linearly, but just linearly, and it is not because log is approximately linear, but just because ergodic mapping is identity mapping in the case of linear dynamics.
And to address your core question of whether independence is actually violated: yes, it is, and here is one simple example similar to the one in the post itself.
Consider two compound lotteries sharing a common component C. Independence says your ranking of the non-common parts should not depend on what C is. But in EE, the ergodic mapping is applied to the dynamics of the entire wealth process, not branch by branch. When C is a multiplicative process, the overall dynamics are multiplicative, the ergodic mapping yields log, and you might prefer gamble A on the non-common branch. When C is an additive process, the overall dynamics are additive, the ergodic mapping yields linear (identity), and you might prefer gamble B on the non-common branch. Your preference between A and B has flipped depending on the common component, which is exactly what independence forbids.
This happens because the ergodic mapping is a global property of the stochastic process. You cannot decompose a compound lottery into branches, apply the ergodic mapping to each branch separately, and recombine.
If this were true, I’d agree that independence isn’t violated.
“Maximize time-average growth rate” is a fixed preference, but it’s a preference over something richer than probability distributions over terminal wealth. It’s a preference over trajectory-classes-indexed-by-dynamics: you’re evaluating not just “what’s the distribution of my final wealth” but “what’s the distribution of my final wealth given the dynamic process I’m embedded in.” And vNM utility is defined over probability distributions over outcomes, not over dynamics-outcome pairs. The EE meta-preference cannot be expressed as a vNM utility function over outcomes because the dynamics are not outcomes, unless you trivially define the dynamics as “outcomes” and then it works “by definition”.
Thanks for the detailed reply! I want to push back on the claim that the EE agent and the log-utility maximizer make different choices
You say that in the additive case, the EE agent is exactly risk-neutral while the log-utility maximizer remains risk-averse. But consider what a log-utility maximizer actually does when facing a long sequence of independent additive bets.
For a single additive bet with payoff x, the contribution to log wealth is ln(W + x) - ln(W) ≈ x/W for small x relative to W. So the log-utility maximizer treats each bet approximately linearly. And as the number of independent additive bets grows, each individual bet becomes smaller relative to total wealth (because wealth is growing via the accumulated positive-EV bets), making the linear approximation increasingly exact.
In the limit of infinitely many independent additive bets, the log-utility maximizer’s per-bet behavior converges to exact risk-neutrality — which is exactly what EE prescribes. So in the regime where the EE prescription is most cleanly motivated (many repeated bets, which is the whole setup for time-average reasoning), the two agents converge.
Where they diverge is for large additive bets relative to current wealth. But here I think the log-utility maximizer is actually more faithful to the “maximize time-average growth rate” objective than the EE agent is. If you’re risk-neutral about an additive bet that could cost you half your wealth, you’re exposing yourself to ruin risk, which destroys your time-average growth rate. The identity mapping tells you the additive process is ergodic, but being ergodic doesn’t mean you should be risk-neutral about large bets within it — not if your goal is to maximize the growth rate of your own single trajectory.
So it seems to me that either (a) the bets are small relative to wealth, and the log-utility maximizer behaves (approximately, converging to exactly) like the EE agent, or (b) the bets are large relative to wealth, and the log-utility maximizer is arguably more correct about maximizing time-average growth. In neither case do the two agents clearly diverge in a way that supports the independence violation argument.
What am I missing?
If I’m right then again it doesn’t seem like the EE agent is really violating independence, given that its behaviour is replicated by a vnm agent that just values its final log wealth
I probably don’t have time to answer to this in detail. The way I see you could understand this is to imaging a completely different dynamic, neither multiplicative nor additive, something weirder, like for example square root or raising to power of 2. For each of that dynamic, EE will give different answers, because ergodic mapping is different for each and also—sometimes—dramatically different from log utility, or linear utility.
But I don’t think EE would give different answers! EE would give the answer that would maximise the time-averaged geometric growth rate. That would be the same answer that would maximise the expected final log wealth after a long series of identical decisions.
So yes, what you say is exactly the misconception about EE. EE will give different answers for each different dynamic, because time-average is calculated differently for each dynamic. Please see for example section 6.6. in the book on EE.
Which of dynamic consistency and consequentialism is not obeyed in EE situations like the one you suggested and how?
My intuition is that
Both are obeyed, and EE can in fact be expressed as a vnm utility function (prove me wrong by answering the above question! It’s an intuition because i haven’t yet been able to pin down how to express EE as a vnm utility function, more below)
Utility theory axioms imply that some utility function exists, not its form.
Said utility function depends on your preferences. If i have a buck and you offer me repeated 50-50 odds of winning 50% of everything or losing 40% of everything its a bad bet—unless this is the only way for me to make money and i want to buy something for two bucks. In this case, a chance is better than nothing.
More generally, EE is specifying a certain set of preferences where having a higher expected wealth is less important than ensuring the growth is measure 1. This is a useful set of preferences in the real world, but neither necessitated nor argued against by the axioms of utility theory.
I expect that if my preferences are about what will happen in the limit of the game, and i care about what happens most of the time, that the utility function will converge to the EE solution, regardless of the specific dynamics. (Is that defining dynamics as outcomes?)