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"On Independence Axiom" by Ihor Kendiukhov
The Fifth Fourth Postulate of Decision Theory
In 1820, the Hungarian mathematician Farkas Bolyai wrote a desperate letter to his son János, who had become consumed by the same problem that had haunted his father for decades:
"You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone... Learn from my example."
The problem was Euclid's fifth postulate, the parallel postulate, which states (in one of its equivalent formulations) that through any point not on a given line, there is exactly one line parallel to the given one. For over two thousand years, mathematicians had felt that something was off about this postulate. The other four were short, crisp, self-evident: you can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any center and radius, all right angles are equal. The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its [...]
---
Outline:
(00:09) The Fifth Fourth Postulate of Decision Theory
(04:58) A Tale of Two Utilities
(09:49) Independence Is Sufficient but Not Necessary for Avoiding Exploitation
(09:55) The strongest case for independence
(12:31) Sufficiency, not necessity
(14:08) Resolute choice
(15:10) Sophisticated choice
(16:36) Ergodicity economics as a naturally resolute framework
(19:26) The broader landscape
(21:17) Allais and Ellsberg Behavior Is Rational
(21:21) Allais Paradox
(25:40) Ellsberg Paradox
(29:37) How LessWrong Has Engaged with This
(30:05) Armstrongs Expected Utility Without the Independence Axiom (2009)
(32:20) Scott Garrabrants comment (2022) -- Updatelessness and independence
(35:50) Academians VNM Expected Utility Theory: Uses, Abuses, and Interpretation (2010)
(38:37) Fallensteins Why You Must Maximize Expected Utility (2012)
(42:40) Just Give Up on EUT
---
First published:
March 8th, 2026
Source:
https://www.lesswrong.com/posts/MsjWPWjAerDtiQ3Do/on-independence-axiom
---
Narrated by TYPE III AUDIO.
In 1820, the Hungarian mathematician Farkas Bolyai wrote a desperate letter to his son János, who had become consumed by the same problem that had haunted his father for decades:
"You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone... Learn from my example."
The problem was Euclid's fifth postulate, the parallel postulate, which states (in one of its equivalent formulations) that through any point not on a given line, there is exactly one line parallel to the given one. For over two thousand years, mathematicians had felt that something was off about this postulate. The other four were short, crisp, self-evident: you can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any center and radius, all right angles are equal. The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its [...]
---
Outline:
(00:09) The Fifth Fourth Postulate of Decision Theory
(04:58) A Tale of Two Utilities
(09:49) Independence Is Sufficient but Not Necessary for Avoiding Exploitation
(09:55) The strongest case for independence
(12:31) Sufficiency, not necessity
(14:08) Resolute choice
(15:10) Sophisticated choice
(16:36) Ergodicity economics as a naturally resolute framework
(19:26) The broader landscape
(21:17) Allais and Ellsberg Behavior Is Rational
(21:21) Allais Paradox
(25:40) Ellsberg Paradox
(29:37) How LessWrong Has Engaged with This
(30:05) Armstrongs Expected Utility Without the Independence Axiom (2009)
(32:20) Scott Garrabrants comment (2022) -- Updatelessness and independence
(35:50) Academians VNM Expected Utility Theory: Uses, Abuses, and Interpretation (2010)
(38:37) Fallensteins Why You Must Maximize Expected Utility (2012)
(42:40) Just Give Up on EUT
---
First published:
March 8th, 2026
Source:
https://www.lesswrong.com/posts/MsjWPWjAerDtiQ3Do/on-independence-axiom
---
Narrated by TYPE III AUDIO.
View all episodes
By LessWrong
4.8
1212 ratings
The Fifth Fourth Postulate of Decision Theory
In 1820, the Hungarian mathematician Farkas Bolyai wrote a desperate letter to his son János, who had become consumed by the same problem that had haunted his father for decades:
"You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone... Learn from my example."
The problem was Euclid's fifth postulate, the parallel postulate, which states (in one of its equivalent formulations) that through any point not on a given line, there is exactly one line parallel to the given one. For over two thousand years, mathematicians had felt that something was off about this postulate. The other four were short, crisp, self-evident: you can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any center and radius, all right angles are equal. The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its [...]
---
Outline:
(00:09) The Fifth Fourth Postulate of Decision Theory
(04:58) A Tale of Two Utilities
(09:49) Independence Is Sufficient but Not Necessary for Avoiding Exploitation
(09:55) The strongest case for independence
(12:31) Sufficiency, not necessity
(14:08) Resolute choice
(15:10) Sophisticated choice
(16:36) Ergodicity economics as a naturally resolute framework
(19:26) The broader landscape
(21:17) Allais and Ellsberg Behavior Is Rational
(21:21) Allais Paradox
(25:40) Ellsberg Paradox
(29:37) How LessWrong Has Engaged with This
(30:05) Armstrongs Expected Utility Without the Independence Axiom (2009)
(32:20) Scott Garrabrants comment (2022) -- Updatelessness and independence
(35:50) Academians VNM Expected Utility Theory: Uses, Abuses, and Interpretation (2010)
(38:37) Fallensteins Why You Must Maximize Expected Utility (2012)
(42:40) Just Give Up on EUT
---
First published:
March 8th, 2026
Source:
https://www.lesswrong.com/posts/MsjWPWjAerDtiQ3Do/on-independence-axiom
---
Narrated by TYPE III AUDIO.
In 1820, the Hungarian mathematician Farkas Bolyai wrote a desperate letter to his son János, who had become consumed by the same problem that had haunted his father for decades:
"You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone... Learn from my example."
The problem was Euclid's fifth postulate, the parallel postulate, which states (in one of its equivalent formulations) that through any point not on a given line, there is exactly one line parallel to the given one. For over two thousand years, mathematicians had felt that something was off about this postulate. The other four were short, crisp, self-evident: you can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any center and radius, all right angles are equal. The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its [...]
---
Outline:
(00:09) The Fifth Fourth Postulate of Decision Theory
(04:58) A Tale of Two Utilities
(09:49) Independence Is Sufficient but Not Necessary for Avoiding Exploitation
(09:55) The strongest case for independence
(12:31) Sufficiency, not necessity
(14:08) Resolute choice
(15:10) Sophisticated choice
(16:36) Ergodicity economics as a naturally resolute framework
(19:26) The broader landscape
(21:17) Allais and Ellsberg Behavior Is Rational
(21:21) Allais Paradox
(25:40) Ellsberg Paradox
(29:37) How LessWrong Has Engaged with This
(30:05) Armstrongs Expected Utility Without the Independence Axiom (2009)
(32:20) Scott Garrabrants comment (2022) -- Updatelessness and independence
(35:50) Academians VNM Expected Utility Theory: Uses, Abuses, and Interpretation (2010)
(38:37) Fallensteins Why You Must Maximize Expected Utility (2012)
(42:40) Just Give Up on EUT
---
First published:
March 8th, 2026
Source:
https://www.lesswrong.com/posts/MsjWPWjAerDtiQ3Do/on-independence-axiom
---
Narrated by TYPE III AUDIO.
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