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Like many "thought experiment paradoxes" I think this makes perfect sense if you get out of the habit of thinking too abstractly. The mathematical assumption i

Like many "thought experiment paradoxes" I think this makes perfect sense if you get out of the habit of thinking too abstractly. The mathematical assumption is that for the urn with the unknown number of red and green balls all distributions are equally likely and so P(red) = 0.5. But why should I assume they're all equally likely other than that it is part of the problem construction? The reason I prefer the known distribution is because this is a contrived game and my experience with contrived games has conditioned me to expect a "trick" such that I am somehow disadvantaged to lose in a way that is not clear to me. Perhaps whoever set up the game knows my favorite color is green. In probabilistic terms I have a Bayesian prior influencing my choice. In order to answer this paradox "correctly" I need to rely on my experience contrived problems like this to recognize I'm expected to abstract the problem and pretend that my prior does not exist. I find if you resist the urge to abstract in this way a stunningly large proportion of philosophical paradoxes cease to be paradoxes.

cubefox 3y ago
This explanation (suspected deception/trickery) is mentioned in the Wikipedia article, after the "info-gap" explanation: https://en.wikipedia.org/wiki/Ellsberg_paradox#Possible_explanations https://en.wikipedia.org/wiki/Ellsberg_paradox#Possible_expl... However, it does seem plausible to me that people would prefer the first urn even if they didn't suspect any trickery.
dash2 3y ago
You can't get away from the problem by assuming that my subjective P(red) ≠ 0.5. It only seems like this because the example was simplified. Look at the "one-urn paradox" version at https://en.wikipedia.org/wiki/Ellsberg_paradox https://en.wikipedia.org/wiki/Ellsberg_paradox. This clarifies that there is no subjective probability that can rationalise people's behaviour. If you prefer betting on red to betting on black, you should also prefer betting on red or yellow to betting on black or yellow.
- Hermitian909 3y ago
  IMO the one urn problem does nothing to get away from this. Recapping the one urn problem: 90 balls in one urn. 30 are red, 60 are black or yellow in unknown proportion. Users must pick a set of gambles called a scenario, then pick a particular gamble. All balls are equally likely to be pulled. I'll list out the possible probabilities based on possible ground state truth: Scenario 1: Gamble A: Receive $100 if you draw a red ball P(0.33~) Gamble B: Receive $100 if you draw a black ball P(0)-P(0.66~) Scenario 2: Gamble C: Receive $100 if you draw a red or yellow ball P(0.33~)-P(1) Gamble D: Receive $100 if you draw a black or yellow ball P(0.66~) Per my earlier argument, this scenario triggers my "someone is trying to trick and/or scam me" heuristic which creates an expectation that my choice of scenario will be predicted somehow, and that the color distribution will be chosen to maximally disadvantage me. I think most people would feel the same way and so I'd assume that most people would prefer gambles A and D respectively and that's exactly what happens.
  - dash2 3y ago
    Right, but note that you've made two arguments. The first was "you can explain this simply using 'P(red) ≠ 0.5'". The second is "you can explain this using a heuristic". The difference is that the first argument saves standard expected utility theory. The second does not. Someone who has this heuristic is violating expected utility. Maybe you didn't believe EUT to start with. But many people did, and so this paradox still resonates.
    - Hermitian909 3y ago
      > but note that you've made two arguments. Apologies if I was not clear, but I only intended to present the "heuristic" argument. > The second does not. Someone who has this heuristic is violating expected utility. Maybe you didn't believe EUT to start with. I did not think this was the case, but I think I disagree with any utilitarianism that did not allow me to weight probabilities via epistemic uncertainty. Another way to frame my argument here that people have a justified belief that the prima facie probabilities for these gambles are misleading in a way that disadvantages them. If you allow this to change the probabilities then EUT would seem to explain the behavior e.g. if I believe Gamble A has P(0.33~) and Gamble B is weighted down to P(0.3) due to epistemic uncertainty the experimental results match up with the math. If we allow the above calculation I think we're still in bounds for EUT.

cubefox 3y ago

This explanation (suspected deception/trickery) is mentioned in the Wikipedia article, after the "info-gap" explanation: https://en.wikipedia.org/wiki/Ellsberg_paradox#Possible_explanations https://en.wikipedia.org/wiki/Ellsberg_paradox#Possible_expl... However, it does seem plausible to me that people would prefer the first urn even if they didn't suspect any trickery.

dash2 3y ago

You can't get away from the problem by assuming that my subjective P(red) ≠ 0.5. It only seems like this because the example was simplified. Look at the "one-urn paradox" version at https://en.wikipedia.org/wiki/Ellsberg_paradox https://en.wikipedia.org/wiki/Ellsberg_paradox. This clarifies that there is no subjective probability that can rationalise people's behaviour. If you prefer betting on red to betting on black, you should also prefer betting on red or yellow to betting on black or yellow.

Hermitian909 3y ago
IMO the one urn problem does nothing to get away from this. Recapping the one urn problem: 90 balls in one urn. 30 are red, 60 are black or yellow in unknown proportion. Users must pick a set of gambles called a scenario, then pick a particular gamble. All balls are equally likely to be pulled. I'll list out the possible probabilities based on possible ground state truth: Scenario 1: Gamble A: Receive $100 if you draw a red ball P(0.33~) Gamble B: Receive $100 if you draw a black ball P(0)-P(0.66~) Scenario 2: Gamble C: Receive $100 if you draw a red or yellow ball P(0.33~)-P(1) Gamble D: Receive $100 if you draw a black or yellow ball P(0.66~) Per my earlier argument, this scenario triggers my "someone is trying to trick and/or scam me" heuristic which creates an expectation that my choice of scenario will be predicted somehow, and that the color distribution will be chosen to maximally disadvantage me. I think most people would feel the same way and so I'd assume that most people would prefer gambles A and D respectively and that's exactly what happens.
- dash2 3y ago
  Right, but note that you've made two arguments. The first was "you can explain this simply using 'P(red) ≠ 0.5'". The second is "you can explain this using a heuristic". The difference is that the first argument saves standard expected utility theory. The second does not. Someone who has this heuristic is violating expected utility. Maybe you didn't believe EUT to start with. But many people did, and so this paradox still resonates.
  - Hermitian909 3y ago
    > but note that you've made two arguments. Apologies if I was not clear, but I only intended to present the "heuristic" argument. > The second does not. Someone who has this heuristic is violating expected utility. Maybe you didn't believe EUT to start with. I did not think this was the case, but I think I disagree with any utilitarianism that did not allow me to weight probabilities via epistemic uncertainty. Another way to frame my argument here that people have a justified belief that the prima facie probabilities for these gambles are misleading in a way that disadvantages them. If you allow this to change the probabilities then EUT would seem to explain the behavior e.g. if I believe Gamble A has P(0.33~) and Gamble B is weighted down to P(0.3) due to epistemic uncertainty the experimental results match up with the math. If we allow the above calculation I think we're still in bounds for EUT.