![ole_b_peters's tweet photo. As part of the Ergodicity Economics 2025 conference, we spent the last few days meeting in person in (stormy) Cascais, Portugal. This year, the aim of the conference was to get the EE crowd to meet the ecological rationality crowd around Gerd Gigerenzer.
Over dinner, Gerd brought up the two-envelopes paradox and how ecological rationality might address it. I found this worth sharing because it illustrates how ergodicity economics and ecological rationality can meet, and how the two approaches, vastly different in some ways, can be joined to support each other.
The two-envelopes paradox
Here comes the paradox. I have two envelopes, and I give you one of them. I tell you that one envelope contains twice as much money as the other, and I give you the option to swap envelopes with me.
Should you do it? The paradox (which we'll resolve formally below), is this: if you compute the expected value of what's in my envelope, you'll find that it's (2+1/2)/2=1.25 times whatever is in the envelope you currently have. This suggests that you should swap. But once you've swapped, you're in the same situation as before, you didn't learn anything new, so the same reasoning applied again will again tell you to swap and so on ad infinitum.
Intuitively, it also feels that it cannot possibly make a difference to swap because swapping doesn't change your situation in any way.
Ecological rationality to the rescue
Ecological rationality suggests the following solution. Instead of tangling yourself in mathematical imponderables, get real, and develop a heuristic for making the choice. Here is the heuristic Gerd mentioned: think about the context, your surroundings, and come up with a plausible range of what might be in the envelopes. For example, you may be at an academic conference. It's unlikely that you'll find $10,000 in the envelope. Something symbolic like $10 or $20 would make sense, given the context.
So you might choose $12 as a threshold, then open the envelope you've currently got, and keep it if it contains more than $12, otherwise swap.
The heuristic (unlike the mathematical analysis) allows you to make a definite decision, it requires no complicated computations and gets the job done quickly. Also, it's designed to be fairly good -- you can rely on your intuition about the situation, implicitly taking a lot of information into account.
I found this approach utterly refreshing -- if we're really talking about a real situation, and the story about the envelopes isn't just some cryptic encoding of an exercise in probability theory, then forget probability theory and focus on the real-world problem at hand!
Ergodicity economics insights
Expected value is meaningless here
Nonetheless, let's see how ergodicity thinking can be informative here. Let's go back to the formal problem. The first thing we realize is that the expected value has absolutely nothing to do with the problem. The expected value is the ensemble-average over all imagined possible worlds of what might be in the envelope I'm holding. But you don't have access to the ensemble of all imagined possible worlds, you're stuck in this one universe.
Tragically, the name "expected value" with its entirely inappropriate connotations, has become the standard word for referring to the average over the infinite statistical ensemble. Of course it's not what you should "expect": whatever is in my envelope, it's certainly not 1.25 times what's in yours -- it's either 0.5 or 2 times what's in yours. The "expected value" is not even in the set of possible values.
Once we try to attach physical meaning to the expected value, and once this really sinks in, we realize that making this choice (and many others) based on expected value is as sensible as measuring your left shoe lace and swapping envelopes if it's longer than Julius Cesar's right arm (whose length you don't know).
Long term growth
But how would an ergodicity economist address the problem? I think maybe like this: try embedding the problem in time; make it part of a dynamic, and think of the choice you're making as the result of a behavioral protocol. Then judge the different options by how they affect you in the long run (not in expectation). How?
You're currently holding an envelop. Let's call the (unknown) amount of money in that envelop x(0). You have the option to swap, or to keep the envelop. What's the growth rate of your wealth in the long run under either decision protocol?
Let's check option A) keep your envelope. In this case, the amount of money you have will stay what it is, x(t)=x(0). The long-time additive growth rate of your wealth will be
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
What about option B), swapping? If you choose to swap envelopes now, you will learn nothing new, so you will repeat the same choice at the next time step. In other words, you will keep swapping envelopes. If you've currently got the good envelope, your wealth will evolve according to
x(t)= x(0)/2 if t is odd
and
x(t)=x(0) if t is even
If you've currently got the bad envelope, it will be 2 x(0) and x(0). In either case, the long-term additive growth rate of your wealth will again be zero
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
That's because in the large-t limit the contribution of a doubling or halving of the amount of money contributes nothing to the time-average growth rate.
Paradox lost
So: ergodicity economics resolves the foundational puzzle and arrives at the intuitive result: whether you swap or not is irrelevant according to the simple criterion of maximizing time-average growth rates. This exposes the invalidity of the formal analysis that established the paradox in the first place, and we quite like that.
Back to reality: EE and ER synthesis
If you now ask an ergodicity economist what should be done in the real situation, I'd suspect he or she would say the first-order decision criterion of ergodicity economics gives no answer, and we have to look for extra information to make the choice.
Enter Ecological Rationality: it's exactly right -- look for more information than what we had immediately translated into a probabilistic mathematical model. Is this happening at an academic conference, or perhaps on live TV? Do you know me? Am I winking at the envelope in my hand and smiling at you? Am I trustworthy? Are we friends? Might I be trying to help you out?
The EE2025 conference was a wonderful opportunity to start a serious dialogue between the two communities around EE concepts and Ecological Rationality concepts. I look forward to continuing it.](https://pbs.twimg.com/media/GnDaKWAXMAAi533.jpg)
![ole_b_peters's tweet photo. As part of the Ergodicity Economics 2025 conference, we spent the last few days meeting in person in (stormy) Cascais, Portugal. This year, the aim of the conference was to get the EE crowd to meet the ecological rationality crowd around Gerd Gigerenzer.
Over dinner, Gerd brought up the two-envelopes paradox and how ecological rationality might address it. I found this worth sharing because it illustrates how ergodicity economics and ecological rationality can meet, and how the two approaches, vastly different in some ways, can be joined to support each other.
The two-envelopes paradox
Here comes the paradox. I have two envelopes, and I give you one of them. I tell you that one envelope contains twice as much money as the other, and I give you the option to swap envelopes with me.
Should you do it? The paradox (which we'll resolve formally below), is this: if you compute the expected value of what's in my envelope, you'll find that it's (2+1/2)/2=1.25 times whatever is in the envelope you currently have. This suggests that you should swap. But once you've swapped, you're in the same situation as before, you didn't learn anything new, so the same reasoning applied again will again tell you to swap and so on ad infinitum.
Intuitively, it also feels that it cannot possibly make a difference to swap because swapping doesn't change your situation in any way.
Ecological rationality to the rescue
Ecological rationality suggests the following solution. Instead of tangling yourself in mathematical imponderables, get real, and develop a heuristic for making the choice. Here is the heuristic Gerd mentioned: think about the context, your surroundings, and come up with a plausible range of what might be in the envelopes. For example, you may be at an academic conference. It's unlikely that you'll find $10,000 in the envelope. Something symbolic like $10 or $20 would make sense, given the context.
So you might choose $12 as a threshold, then open the envelope you've currently got, and keep it if it contains more than $12, otherwise swap.
The heuristic (unlike the mathematical analysis) allows you to make a definite decision, it requires no complicated computations and gets the job done quickly. Also, it's designed to be fairly good -- you can rely on your intuition about the situation, implicitly taking a lot of information into account.
I found this approach utterly refreshing -- if we're really talking about a real situation, and the story about the envelopes isn't just some cryptic encoding of an exercise in probability theory, then forget probability theory and focus on the real-world problem at hand!
Ergodicity economics insights
Expected value is meaningless here
Nonetheless, let's see how ergodicity thinking can be informative here. Let's go back to the formal problem. The first thing we realize is that the expected value has absolutely nothing to do with the problem. The expected value is the ensemble-average over all imagined possible worlds of what might be in the envelope I'm holding. But you don't have access to the ensemble of all imagined possible worlds, you're stuck in this one universe.
Tragically, the name "expected value" with its entirely inappropriate connotations, has become the standard word for referring to the average over the infinite statistical ensemble. Of course it's not what you should "expect": whatever is in my envelope, it's certainly not 1.25 times what's in yours -- it's either 0.5 or 2 times what's in yours. The "expected value" is not even in the set of possible values.
Once we try to attach physical meaning to the expected value, and once this really sinks in, we realize that making this choice (and many others) based on expected value is as sensible as measuring your left shoe lace and swapping envelopes if it's longer than Julius Cesar's right arm (whose length you don't know).
Long term growth
But how would an ergodicity economist address the problem? I think maybe like this: try embedding the problem in time; make it part of a dynamic, and think of the choice you're making as the result of a behavioral protocol. Then judge the different options by how they affect you in the long run (not in expectation). How?
You're currently holding an envelop. Let's call the (unknown) amount of money in that envelop x(0). You have the option to swap, or to keep the envelop. What's the growth rate of your wealth in the long run under either decision protocol?
Let's check option A) keep your envelope. In this case, the amount of money you have will stay what it is, x(t)=x(0). The long-time additive growth rate of your wealth will be
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
What about option B), swapping? If you choose to swap envelopes now, you will learn nothing new, so you will repeat the same choice at the next time step. In other words, you will keep swapping envelopes. If you've currently got the good envelope, your wealth will evolve according to
x(t)= x(0)/2 if t is odd
and
x(t)=x(0) if t is even
If you've currently got the bad envelope, it will be 2 x(0) and x(0). In either case, the long-term additive growth rate of your wealth will again be zero
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
That's because in the large-t limit the contribution of a doubling or halving of the amount of money contributes nothing to the time-average growth rate.
Paradox lost
So: ergodicity economics resolves the foundational puzzle and arrives at the intuitive result: whether you swap or not is irrelevant according to the simple criterion of maximizing time-average growth rates. This exposes the invalidity of the formal analysis that established the paradox in the first place, and we quite like that.
Back to reality: EE and ER synthesis
If you now ask an ergodicity economist what should be done in the real situation, I'd suspect he or she would say the first-order decision criterion of ergodicity economics gives no answer, and we have to look for extra information to make the choice.
Enter Ecological Rationality: it's exactly right -- look for more information than what we had immediately translated into a probabilistic mathematical model. Is this happening at an academic conference, or perhaps on live TV? Do you know me? Am I winking at the envelope in my hand and smiling at you? Am I trustworthy? Are we friends? Might I be trying to help you out?
The EE2025 conference was a wonderful opportunity to start a serious dialogue between the two communities around EE concepts and Ecological Rationality concepts. I look forward to continuing it.](https://pbs.twimg.com/media/GnDZ3r7W4AAg7x9.jpg)
Olivia
Online
Ergodicity Economics 2026
@EE_Conf_LML
Registration for EE2026 now open at
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Two comments on yesterday's talks at #ErgodicityEconomics2026.
Still time to register and join today's sessions.
@youreyes_stell Free seats have been filled, yes. You can still sign up, though.
To make sure you don't miss any future events and freebees, join our mailing list at
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Who to follow
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#EE2026 videos are now up and can be watched by participants.
Check your email if you've registered, or register now if you haven't.
The conference itself starts next week Thursday, 26-03-2026, with a tutorial day about the EE textbook.
@michaelschiltz Recordings can now be viewed by participants, pre-conference.
...and the program is ready!
Day one this time will start with a textbook-based introduction to EE.
The aim of the meeting is to practice the `ergodicity squint' as we re-consider big questions of social organization, business and finance.
EE2026 now open for abstracts and registration at https://t.co/pEWeYFppyY
This time, we will focus on messages from EE for finance and society. Speakers include Jutta Steiner and Mike Green @profplum99.
Day 1 will use the EE textbook to set the stage.
New book: An Introduction to Ergodicity Economics
A new textbook by SFI External Professor Ole Peters and Alexander Adamou draws on physics to re-examine core assumptions in economic theory. It presents a formal challenge to expected-value thinking and shows how non-ergodicity can arise naturally in economic models.
The book introduces ergodicity economics as a “third way” alternative to expected-utility theory and behavioral critiques of rationality. It is the first textbook in this emerging field, and it offers a new way to think about decision-making under uncertainty.
https://t.co/YAsd5sonsk
The Ergodicity Economics textbook is out!
Here's a thread about how it came into being, and where you can get your copy.
1/thread🧵
Almost 20 years ago, I started thinking about the ergodicity problem in the context of economics. That turned out to be surprisingly fruitful, and now there's a book about it.
As part of the Ergodicity Economics 2025 conference, we spent the last few days meeting in person in (stormy) Cascais, Portugal. This year, the aim of the conference was to get the EE crowd to meet the ecological rationality crowd around Gerd Gigerenzer.
Over dinner, Gerd brought up the two-envelopes paradox and how ecological rationality might address it. I found this worth sharing because it illustrates how ergodicity economics and ecological rationality can meet, and how the two approaches, vastly different in some ways, can be joined to support each other.
The two-envelopes paradox
Here comes the paradox. I have two envelopes, and I give you one of them. I tell you that one envelope contains twice as much money as the other, and I give you the option to swap envelopes with me.
Should you do it? The paradox (which we'll resolve formally below), is this: if you compute the expected value of what's in my envelope, you'll find that it's (2+1/2)/2=1.25 times whatever is in the envelope you currently have. This suggests that you should swap. But once you've swapped, you're in the same situation as before, you didn't learn anything new, so the same reasoning applied again will again tell you to swap and so on ad infinitum.
Intuitively, it also feels that it cannot possibly make a difference to swap because swapping doesn't change your situation in any way.
Ecological rationality to the rescue
Ecological rationality suggests the following solution. Instead of tangling yourself in mathematical imponderables, get real, and develop a heuristic for making the choice. Here is the heuristic Gerd mentioned: think about the context, your surroundings, and come up with a plausible range of what might be in the envelopes. For example, you may be at an academic conference. It's unlikely that you'll find $10,000 in the envelope. Something symbolic like $10 or $20 would make sense, given the context.
So you might choose $12 as a threshold, then open the envelope you've currently got, and keep it if it contains more than $12, otherwise swap.
The heuristic (unlike the mathematical analysis) allows you to make a definite decision, it requires no complicated computations and gets the job done quickly. Also, it's designed to be fairly good -- you can rely on your intuition about the situation, implicitly taking a lot of information into account.
I found this approach utterly refreshing -- if we're really talking about a real situation, and the story about the envelopes isn't just some cryptic encoding of an exercise in probability theory, then forget probability theory and focus on the real-world problem at hand!
Ergodicity economics insights
Expected value is meaningless here
Nonetheless, let's see how ergodicity thinking can be informative here. Let's go back to the formal problem. The first thing we realize is that the expected value has absolutely nothing to do with the problem. The expected value is the ensemble-average over all imagined possible worlds of what might be in the envelope I'm holding. But you don't have access to the ensemble of all imagined possible worlds, you're stuck in this one universe.
Tragically, the name "expected value" with its entirely inappropriate connotations, has become the standard word for referring to the average over the infinite statistical ensemble. Of course it's not what you should "expect": whatever is in my envelope, it's certainly not 1.25 times what's in yours -- it's either 0.5 or 2 times what's in yours. The "expected value" is not even in the set of possible values.
Once we try to attach physical meaning to the expected value, and once this really sinks in, we realize that making this choice (and many others) based on expected value is as sensible as measuring your left shoe lace and swapping envelopes if it's longer than Julius Cesar's right arm (whose length you don't know).
Long term growth
But how would an ergodicity economist address the problem? I think maybe like this: try embedding the problem in time; make it part of a dynamic, and think of the choice you're making as the result of a behavioral protocol. Then judge the different options by how they affect you in the long run (not in expectation). How?
You're currently holding an envelop. Let's call the (unknown) amount of money in that envelop x(0). You have the option to swap, or to keep the envelop. What's the growth rate of your wealth in the long run under either decision protocol?
Let's check option A) keep your envelope. In this case, the amount of money you have will stay what it is, x(t)=x(0). The long-time additive growth rate of your wealth will be
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
What about option B), swapping? If you choose to swap envelopes now, you will learn nothing new, so you will repeat the same choice at the next time step. In other words, you will keep swapping envelopes. If you've currently got the good envelope, your wealth will evolve according to
x(t)= x(0)/2 if t is odd
and
x(t)=x(0) if t is even
If you've currently got the bad envelope, it will be 2 x(0) and x(0). In either case, the long-term additive growth rate of your wealth will again be zero
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
That's because in the large-t limit the contribution of a doubling or halving of the amount of money contributes nothing to the time-average growth rate.
Paradox lost
So: ergodicity economics resolves the foundational puzzle and arrives at the intuitive result: whether you swap or not is irrelevant according to the simple criterion of maximizing time-average growth rates. This exposes the invalidity of the formal analysis that established the paradox in the first place, and we quite like that.
Back to reality: EE and ER synthesis
If you now ask an ergodicity economist what should be done in the real situation, I'd suspect he or she would say the first-order decision criterion of ergodicity economics gives no answer, and we have to look for extra information to make the choice.
Enter Ecological Rationality: it's exactly right -- look for more information than what we had immediately translated into a probabilistic mathematical model. Is this happening at an academic conference, or perhaps on live TV? Do you know me? Am I winking at the envelope in my hand and smiling at you? Am I trustworthy? Are we friends? Might I be trying to help you out?
The EE2025 conference was a wonderful opportunity to start a serious dialogue between the two communities around EE concepts and Ecological Rationality concepts. I look forward to continuing it.
![ole_b_peters's tweet photo. As part of the Ergodicity Economics 2025 conference, we spent the last few days meeting in person in (stormy) Cascais, Portugal. This year, the aim of the conference was to get the EE crowd to meet the ecological rationality crowd around Gerd Gigerenzer.
Over dinner, Gerd brought up the two-envelopes paradox and how ecological rationality might address it. I found this worth sharing because it illustrates how ergodicity economics and ecological rationality can meet, and how the two approaches, vastly different in some ways, can be joined to support each other.
The two-envelopes paradox
Here comes the paradox. I have two envelopes, and I give you one of them. I tell you that one envelope contains twice as much money as the other, and I give you the option to swap envelopes with me.
Should you do it? The paradox (which we'll resolve formally below), is this: if you compute the expected value of what's in my envelope, you'll find that it's (2+1/2)/2=1.25 times whatever is in the envelope you currently have. This suggests that you should swap. But once you've swapped, you're in the same situation as before, you didn't learn anything new, so the same reasoning applied again will again tell you to swap and so on ad infinitum.
Intuitively, it also feels that it cannot possibly make a difference to swap because swapping doesn't change your situation in any way.
Ecological rationality to the rescue
Ecological rationality suggests the following solution. Instead of tangling yourself in mathematical imponderables, get real, and develop a heuristic for making the choice. Here is the heuristic Gerd mentioned: think about the context, your surroundings, and come up with a plausible range of what might be in the envelopes. For example, you may be at an academic conference. It's unlikely that you'll find $10,000 in the envelope. Something symbolic like $10 or $20 would make sense, given the context.
So you might choose $12 as a threshold, then open the envelope you've currently got, and keep it if it contains more than $12, otherwise swap.
The heuristic (unlike the mathematical analysis) allows you to make a definite decision, it requires no complicated computations and gets the job done quickly. Also, it's designed to be fairly good -- you can rely on your intuition about the situation, implicitly taking a lot of information into account.
I found this approach utterly refreshing -- if we're really talking about a real situation, and the story about the envelopes isn't just some cryptic encoding of an exercise in probability theory, then forget probability theory and focus on the real-world problem at hand!
Ergodicity economics insights
Expected value is meaningless here
Nonetheless, let's see how ergodicity thinking can be informative here. Let's go back to the formal problem. The first thing we realize is that the expected value has absolutely nothing to do with the problem. The expected value is the ensemble-average over all imagined possible worlds of what might be in the envelope I'm holding. But you don't have access to the ensemble of all imagined possible worlds, you're stuck in this one universe.
Tragically, the name "expected value" with its entirely inappropriate connotations, has become the standard word for referring to the average over the infinite statistical ensemble. Of course it's not what you should "expect": whatever is in my envelope, it's certainly not 1.25 times what's in yours -- it's either 0.5 or 2 times what's in yours. The "expected value" is not even in the set of possible values.
Once we try to attach physical meaning to the expected value, and once this really sinks in, we realize that making this choice (and many others) based on expected value is as sensible as measuring your left shoe lace and swapping envelopes if it's longer than Julius Cesar's right arm (whose length you don't know).
Long term growth
But how would an ergodicity economist address the problem? I think maybe like this: try embedding the problem in time; make it part of a dynamic, and think of the choice you're making as the result of a behavioral protocol. Then judge the different options by how they affect you in the long run (not in expectation). How?
You're currently holding an envelop. Let's call the (unknown) amount of money in that envelop x(0). You have the option to swap, or to keep the envelop. What's the growth rate of your wealth in the long run under either decision protocol?
Let's check option A) keep your envelope. In this case, the amount of money you have will stay what it is, x(t)=x(0). The long-time additive growth rate of your wealth will be
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
What about option B), swapping? If you choose to swap envelopes now, you will learn nothing new, so you will repeat the same choice at the next time step. In other words, you will keep swapping envelopes. If you've currently got the good envelope, your wealth will evolve according to
x(t)= x(0)/2 if t is odd
and
x(t)=x(0) if t is even
If you've currently got the bad envelope, it will be 2 x(0) and x(0). In either case, the long-term additive growth rate of your wealth will again be zero
lim₍ₜ → ∞₎ 1/t [x(t)-x(0)]=0.
That's because in the large-t limit the contribution of a doubling or halving of the amount of money contributes nothing to the time-average growth rate.
Paradox lost
So: ergodicity economics resolves the foundational puzzle and arrives at the intuitive result: whether you swap or not is irrelevant according to the simple criterion of maximizing time-average growth rates. This exposes the invalidity of the formal analysis that established the paradox in the first place, and we quite like that.
Back to reality: EE and ER synthesis
If you now ask an ergodicity economist what should be done in the real situation, I'd suspect he or she would say the first-order decision criterion of ergodicity economics gives no answer, and we have to look for extra information to make the choice.
Enter Ecological Rationality: it's exactly right -- look for more information than what we had immediately translated into a probabilistic mathematical model. Is this happening at an academic conference, or perhaps on live TV? Do you know me? Am I winking at the envelope in my hand and smiling at you? Am I trustworthy? Are we friends? Might I be trying to help you out?
The EE2025 conference was a wonderful opportunity to start a serious dialogue between the two communities around EE concepts and Ecological Rationality concepts. I look forward to continuing it.](https://pbs.twimg.com/media/GnDaNpnXEAAsVnz.jpg)
EE2025 is over now, sadly. Let us know your thoughts, and let's start thinking about EE2026!
!! The EE2025 conference has officially started !!
As a registered participant, you can now watch the pre-recorded videos online, and leave comments, in preparation for the live online conference on 13-14 March.
Not registered yet? What are you waiting for?
..which can be found here https://t.co/uzQL64riRE
Reminder: if you'd like to attend the EE2025 in-person meeting, the deadline for letting us know is coming up.
Drop us a line via the online form.
Fabulous!
This is really fantastic. Hope it turns into a trend. If anyone out there feels inspired, get in touch with me right here.
@krpdeadlifts https://t.co/KtcbCFTUDw
Early-bird registration ends this Friday, 20 December.
And the program is now complete. This will be absolutely fantastic!
@krpdeadlifts Have a look at the conference website. It's a very clever format: all videos are pre-recorded, made available to participants a week in advance, and are streamed and discussed live at the conference.
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